Solutions to Chapter 4 Overfitting.ipynb

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      "D:\\git\\Learning-From-Data-A-Short-Course\n",
      "['D:\\\\git\\\\Learning-From-Data-A-Short-Course', 'C:\\\\ProgramData\\\\Anaconda3\\\\python37.zip', 'C:\\\\ProgramData\\\\Anaconda3\\\\DLLs', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib', 'C:\\\\ProgramData\\\\Anaconda3', '', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib\\\\site-packages', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib\\\\site-packages\\\\win32', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib\\\\site-packages\\\\win32\\\\lib', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib\\\\site-packages\\\\Pythonwin', 'C:\\\\ProgramData\\\\Anaconda3\\\\lib\\\\site-packages\\\\IPython\\\\extensions', 'C:\\\\Users\\\\zhang\\\\.ipython', 'D:\\\\git\\\\Learning-From-Data-A-Short-Course\\\\libs']\n"
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   "source": [
    "# Add lib input sys.path\n",
    "import os\n",
    "import sys\n",
    "nb_dir = os.getcwd() #os.path.split(os.getcwd())\n",
    "print(nb_dir)\n",
    "if nb_dir not in sys.path:\n",
    "    sys.path.append(nb_dir)\n",
    "\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import libs.linear_models as lm\n",
    "import libs.data_util as data\n",
    "\n",
    "import sklearn\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import minimize\n",
    "import math\n",
    "from sklearn.preprocessing import normalize\n",
    "from functools import partial\n",
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "import cvxopt\n",
    "\n",
    "%matplotlib inline\n",
    "print(sys.path)"
   ]
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    "#### Exercise 4.1\n",
    "\n",
    "* For $h(x) \\in \\mathcal{H}_2$, we have $h(x) = a_0 + a_1x + a_2x^2$, the parameters are $a_0, a_1, a_2$.\n",
    "* For $h(x) \\in \\mathcal{H}_{10}$, we have $h(x) = \\sum^{10}_{k=0} a_k x^k$.\n",
    "\n",
    "For any given $h(x) \\in \\mathcal{H}_2$, we have $h(x) = a_0 + a_1x + a_2x^2 = \\sum^{10}_{k=0} a_k x^k$ where $a_k = 0$ when $k\\gt 2$. So $h(x) \\in \\mathcal{H}_{10}$ as well. We thus conclude $\\mathcal{H}_2 \\subset \\mathcal{H}_{10}$."
   ]
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    "#### Exercise 4.2\n",
    "Reading exercise.\n",
    "\n",
    "#### Exercise 4.3\n",
    "\n",
    "* (a) Consider a given $\\mathcal{H}$\n",
    "  * If the best approximation from $\\mathcal{H}$ is less complex than the initial target function, then when we increase the complexity of $f$, the deterministic noise in general should increase, since it'll be harder for functions in $\\mathcal{H}$ to fit the target function. There'll be a higher tendency to overfit. \n",
    "  * If the best approximation from $\\mathcal{H}$ is more complex than the initial target function, then when we increase the complexity of $f$, the deterministic noise in general may decrease first, reducing the deterministic noise and there'll be a lower tendency to overfit. But once the complexity of $f$ exceeds the best function approximation from $\\mathcal{H}$, and if we continue increase the complexity of $f$, we will increase the deterministic noise and thus increase the tendency to overfit.\n",
    "  \n",
    "* (b) Given a fixed $f$ \n",
    "  * If the best approximation from $\\mathcal{H}$ is less complex than the target function, then when we decrease the complexity of $\\mathcal{H}$, we increase the deterministic noise thus increasing the tendency of overfit. \n",
    "  * If the best approximation from $\\mathcal{H}$ is more complex than the target function, then when we decrease the complexity of $\\mathcal{H}$, we will decrease the deterministic noise thus decreasing the tendency of overfit. Well, if we continue to decrease the complexity of $\\mathcal{H}$, passing the point where its complexity is equal to $f$, we start to increase the deministic noise again and thus increasing overfit.\n",
    "  \n",
    "#### Exercise 4.4\n",
    "\n",
    "Let's compute $E_{in}(w)$: \n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= \\frac{1}{N}\\sum^N_{n=1}\\left(w^Tz_n - y_n\\right)^2\\\\\n",
    "&= \\frac{1}{N} \\|Zw - y\\|^2\\\\\n",
    "&= \\frac{1}{N} (Zw - y)^T(Zw - y)\\\\\n",
    "&= \\frac{1}{N} (w^TZ^T - y^T)(Zw - y)\\\\\n",
    "&= \\frac{1}{N} w^TZ^TZw - w^TZ^Ty - y^TZw + y^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "On the other hand, the equation (4.3) is\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= \\frac{(w-w_{lin})^TZ^TZ(w-w_{lin})+y^T(I-H)y}{N}\\\\\n",
    "&= \\frac{(w^T-w^T_{lin})Z^TZ(w-w_{lin})+y^Ty-y^THy}{N}\\\\\n",
    "&= \\frac{w^TZ^TZw-w^TZ^TZw_{lin}-w^T_{lin}Z^TZw+w^T_{lin}Z^TZw_{lin}+y^Ty-y^THy}{N}\\\\\n",
    "&= \\frac{w^TZ^TZw-w^TZ^TZ(Z^TZ)^{-1}Z^Ty-\\left((Z^TZ)^{-1}Z^Ty\\right)^TZ^TZw}{N}\\\\\n",
    "&+\\frac{\\left((Z^TZ)^{-1}Z^Ty\\right)^TZ^TZ(Z^TZ)^{-1}Z^Ty+y^Ty-y^TZ(Z^TZ)^{-1}Z^Ty}{N}\\\\\n",
    "&= \\frac{w^TZ^TZw-w^TZ^Ty-y^TZ(Z^TZ)^{-T}Z^TZw}{N}\\\\\n",
    "&+\\frac{y^TZ(Z^TZ)^{-T}Z^TZ(Z^TZ)^{-1}Z^Ty+y^Ty-y^TZ(Z^TZ)^{-1}Z^Ty}{N}\\\\\n",
    "&= \\frac{w^TZ^TZw-w^TZ^Ty-y^TZ(Z^TZ)^{-1}Z^TZw}{N}\\\\\n",
    "&+\\frac{y^TZ(Z^TZ)^{-1}Z^Ty+y^Ty-y^TZ(Z^TZ)^{-1}Z^Ty}{N}\\\\\n",
    "&= \\frac{w^TZ^TZw-w^TZ^Ty-y^TZw+y^Ty}{N}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This agrees with the result derived above. So we proved equation (4.3).\n",
    "\n",
    "* (a) The value of $w$ that minimizes $E_{in}$ is $w_{lin}$, since it makes the first term $0$, which is a quadratic term on $w$ and is greater or equal to $0$ all the time. The second term doesn't depend on $w$. \n",
    "\n",
    "* (b) The minimum in-sample error when $w=w_{lin}$ is simply the second term, i.e. $\\min E_{in}(w) = \\frac{y^T(I-H)y}{N}$.\n",
    "\n",
    "#### Exercise 4.5 [Tikhonov regularizer]\n",
    "\n",
    "* (a) $\\sum^Q_{q=0}w^2_q = [w_1, w_2, \\dots, w_Q]\\begin{bmatrix}w_1\\\\ w_2\\\\ \\dots\\\\ w_Q\\end{bmatrix} = w^Tw$, so $\\Gamma = I$.\n",
    "\n",
    "* (b) $\\left(\\sum^Q_{q=0}w_q\\right)^2 = [w_1, w_2, \\dots, w_Q]\\begin{bmatrix}1\\\\1\\\\\\dots\\\\1\\end{bmatrix}[1,1,\\dots,1]\\begin{bmatrix}w_1\\\\ w_2\\\\ \\dots\\\\ w_Q\\end{bmatrix}$, so $\\Gamma = [1,1,\\dots,1]$.\n",
    "\n",
    "#### Exercise 4.6\n",
    "\n",
    "If we use the hard-order constraint, with less parameters, the perceptron's VC dimension decreases, and it's less likely to classify the same amount of points with more parameters. If we use the soft-order constraint, it won't change the signs of $(w^Tx)$ even when $w$ is small. So we'll still be able to classify the points.\n",
    "\n",
    "#### Exercise 4.7\n",
    "* (a) Note that the expectation w.r.t. $\\mathcal{D}_{val}$ is equivalent to $x$ because the $y$ are assumed to be generated by a true $f(x)$.\n",
    "\n",
    "\\begin{align*}\n",
    "\\sigma^2_{val} &= Var_{\\mathcal{D}_{val}}\\left[E_{val}(g^{-})\\right]\\\\\n",
    "&= Var_{\\mathcal{D}_{val}}\\left[\\frac{1}{K}\\sum_{x_n \\in \\mathcal{D}_{val}} e\\left(g^{-}\n",
    "(x_n),y_n\\right)\\right]\\\\\n",
    "&= \\frac{1}{K^2}\\left[Var_{\\mathcal{D}_{val}}\\sum_{x_n \\in \\mathcal{D}_{val}} e\\left(g^{-}\n",
    "(x_n),y_n\\right)\\right]\\\\\n",
    "&= \\frac{1}{K^2}\\left[\\sum_{x_n \\in \\mathcal{D}_{val}} Var_{\\mathcal{D}_{val}} \\left[e\\left(g^{-}\n",
    "(x_n),y_n\\right)\\right]\\right]\\\\\n",
    "&= \\frac{1}{K^2}\\left[\\sum_{x_n \\in \\mathcal{D}_{val}} Var_{x} \\left[e\\left(g^{-}\n",
    "(x_n),y_n\\right)\\right]\\right]\\\\\n",
    "&= \\frac{1}{K^2}\\left[\\sum_{x_n \\in \\mathcal{D}_{val}}\\sigma^2(g^{-})\\right]\\\\\n",
    "&= \\frac{1}{K}\\sigma^2(g^{-})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "* (b) In classification problem, $e\\left(g^{-}(x), y\\right) = 1(g^{-}(x) \\ne y)$. We have\n",
    "\n",
    "\\begin{align*}\n",
    "E_x\\left[e\\left(g^{-}(x), y\\right)\\right] & = P(g^{-}(x) \\ne y)\\times 1 + P(g^{-}(x) = y)\\times 0\\\\\n",
    "& = P(g^{-}(x) \\ne y)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So the variance is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\sigma^2(g^{-}) &= Var_x\\left[e\\left(g^{-}(x), y\\right)\\right] \\\\\n",
    "&= E_x\\left[\\left(e - E_x[e]\\right)^2\\right]\\\\\n",
    "&= P(g^{-}(x) \\ne y)\\left[(1-E_x[e])^2\\right] + \\left(1-P(g^{-}(x) \\ne y)\\right)\\left[(0-E_x[e])^2\\right]\\\\\n",
    "&= P(1-P)^2 + (1-P)P^2\\\\\n",
    "&= P(1-P)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) In the end\n",
    "\n",
    "\\begin{align*}\n",
    "\\sigma^2_{val} &= \\frac{1}{K}\\sigma^2(g^{-})\\\\\n",
    "&= \\frac{P(1-P)}{K}\\\\\n",
    "&= \\frac{-(P-0.5)^2+0.25}{K}\\\\\n",
    "&\\le \\frac{1}{4K}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) The squared error $e\\left(g^{-}(x),y\\right)$ is unbounded. The variance of it is also unbounded. So there's no uniform upper bound for $Var_{\\mathcal{D}_{val}}\\left[E_{val}(g^{-})\\right] = \\frac{1}{K}\\sigma^2(g^{-})$. \n",
    "\n",
    "* (e) For regression with squared error, if we train using fewer points (smaller $N-K$) to get $g^{-}$, then the resulting $g^{-}$ will be worse, the expectation of the squared error $E\\left[e\\left(g^{-}(x),y\\right)\\right]$ becomes larger. For continuous, non-negative random variables, higher mean often implies higher variance, so $\\sigma^2(g^{-})$ will be higher.\n",
    "\n",
    "* (f) When we increasing the size of validation set $K$, the error between $E_{val}(g^{-})$ and $E_{out}(g^{-})$ is $\\frac{\\sigma(g^{-})}{\\sqrt{K}}$. It can drop in the case of classification. But for regression, it depends on which of $\\sigma(g^{-})$ or $K$ increases faster, so the  $E_{val}(g^{-})$ as an estimate of $E_{out}$ can become worse or better.\n",
    "\n",
    "Does it mean for classification the estimate will always become better when we increase the K? \n",
    "\n",
    "But note, the $E_{out}(g^{-})$ is only for the hypothesis $g^{-}$, which can be pretty bad when $K$ is large. So for classification problem, even the error between $E_{out}(g^{-})$ and $E_{val}(g^{-})$ goes to zero, but the  $E_{out}(g^{-})$ can be quite large.\n",
    "\n",
    "\n",
    "#### Exercise 4.8\n",
    "\n",
    "For each model $\\mathcal{H}_m$, $g^-_m$ is independently learned of from the validation set. When we take the expectation w.r.t. validation data set, the $g^-_m$ doesn't change, the expectation depends only on $x_n$, as in the derivation of equation (4.8). \n",
    "So I think $E_m$ is an unbiased estimate for the out-of-sample error $E_{out}(g^-_m)$.\n",
    "\n",
    "#### Exercise 4.9\n",
    "\n",
    "When $K$ increases, there are less points available to train the models, so the learned models become worse, and $E_{val}(g^-_m)$ becomes larger for each model $m$. $g^-_{m^*}$ is the model with the lowest validation error among all models, so  $E_{val}(g^-_{m^*})$ will also increases with $k$. The same logic applies to $E_{out}(g^-_{m^*})$ as well since $E_{val}(g^-_{m^*})$ is an estimate of $E_{out}(g^-_{m^*})$.\n",
    "\n",
    "When $K$ increases to certain large value, there are much less points $N-K$ available to train the models, complex models converge to simple models(TODO), so the optimistic validation error $E_{val}(g^-_{m^*})$  is closer to all validation errors $E_{val}(g^-_{m})$ for all models. Since each single validation error $E_{val}(g^-_{m})$  is an unbiased estimate of the $E_{out}(g^-_{m})$, the optimistic validation error $E_{val}(g^-_{m^*})$ converges to $E_{out}(g^-_{m^*})$ as well.\n",
    "\n",
    "#### Exercise 4.10 TODO"
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    "#### Problem 4.1\n",
    "\n",
    "As we can see from below graphs, when we increase the order, the graph of monomials $\\phi_i(x)$ don't seem to become more complex. This does not correspond to the intuitive notion of increasing complexity."
   ]
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\n",
      "text/plain": [
       "<Figure size 288x288 with 16 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### Problem 4.1 \n",
    "\n",
    "def monomials(x, i):\n",
    "    return np.exp(i*np.log(np.abs(x)))\n",
    "    #return np.power(x, i) #This causes overflow\n",
    "\n",
    "xs = np.arange(-1, 1, 0.02) #(1,50,1)\n",
    "figsize = plt.figaspect(1)\n",
    "f, axs = plt.subplots(4, 4, figsize=figsize)\n",
    "for i in np.arange(0,16):\n",
    "    ys = monomials(xs, i)\n",
    "    k = int(i / 4)\n",
    "    j = int(i - 4*k)\n",
    "    axs[k][j].plot(xs, ys)\n",
    "    #axs[k][j].set_yscale('log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.2\n",
    "\n",
    "$z = \\left[L_0(x), L_1(x), L_2(x) \\right]^T = \\left[1, x, \\frac{3}{2}x^2 - \\frac{1}{2}\\right]^T$. For the hypothesis with $w=[1,-1,1]^T$, we have\n",
    "\n",
    "$h(x) = w^Tz = [1, -1, 1]\\begin{bmatrix}1\\\\ x \\\\\\frac{3}{2}x^2 - \\frac{1}{2}\\end{bmatrix} = 1-x+\\frac{3}{2}x^2 - \\frac{1}{2} = \\frac{3}{2}x^2 - x + \\frac{1}{2}$. \n",
    "\n",
    "This is degree-2 polynomial."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.3\n",
    "\n",
    "* (a) \n",
    "\n",
    "\\begin{align*}\n",
    "L_0(x) &= 1\\\\\n",
    "L_1(x) &= x\\\\\n",
    "L_2(x) &= \\frac{3}{2}x^2 - \\frac{1}{2}\\\\\n",
    "L_3(x) &= \\frac{5}{2}x^3 - \\frac{3}{2}x\\\\\n",
    "L_4(x) &= \\frac{35}{8}x^4 - \\frac{15}{4}x^2+\\frac{3}{8}\\\\\n",
    "L_5(x) &= \\frac{63}{8}x^5 - \\frac{35}{4}x^3+\\frac{75}{40}x\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Legendre(x, K):\n",
    "    \"\"\"Compute Legendre at x up to the Kth Legendre Polynomials\n",
    "    \"\"\"\n",
    "    \n",
    "    L0, L1 = 1, x\n",
    "    if K < 0:\n",
    "        raise ValueError(\"The order of Legendre polynomial can't be negative\")\n",
    "    \n",
    "    if K == 0:\n",
    "        return [L0]\n",
    "    elif K == 1:\n",
    "        return [L0, L1]\n",
    "\n",
    "    Ls = [L0, L1]\n",
    "    Lkm1, Lkm2 = L1, L0\n",
    "    for k in range(2, K+1):\n",
    "        Lk = (2*k-1)*x*Lkm1/k - (k-1)*Lkm2/k\n",
    "        Ls.append(Lk)\n",
    "        Lkm2 = Lkm1\n",
    "        Lkm1 = Lk\n",
    "    return Ls\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "xs = np.arange(-1,1,0.05) #Legendre polynomials are defined within [-1, 1]\n",
    "figsize = plt.figaspect(1)\n",
    "f, axs = plt.subplots(2, 3, figsize=figsize)\n",
    "np_arrays = []\n",
    "K=6\n",
    "\n",
    "for x in xs:\n",
    "    y = np.array(Legendre(x, K))\n",
    "    np_arrays.append(y)\n",
    "ys = np.vstack(np_arrays)\n",
    "\n",
    "for i in np.arange(0,K):\n",
    "    k = int(i / 3)\n",
    "    j = int(i - 3*k)\n",
    "    axs[k][j].plot(xs, ys[:,i])    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.3 (b) \n",
    "\n",
    "We use induction. It's easy to see that when $k=0$ and $k=1$, $L_k(x)$ is a linear combination of monomials $x^k, x^{k-2},\\dots$.\n",
    "\n",
    "Suppose this is also true for $L_m(x)$ where $m\\le k$, then when $m=k+1$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "L_{k+1}(x) &= \\frac{2k-1}{k}xL_{k-1}(x) - \\frac{k-1}{k}L_{k-2}(x)\\\\\n",
    "&= \\frac{2k-1}{k}x\\left(a_0x^k+a_1x^{k-2}+\\dots\\right) - \\frac{k-1}{k}\\left(b_0x^{k-1}+b_1x^{k-3}+\\dots\\right)\\\\\n",
    "&= \\frac{2k-1}{k}\\left(a_0x^{k+1}+a_1x^{k-1}+\\dots\\right) - \\frac{k-1}{k}\\left(b_0x^{k-1}+b_1x^{k-3}+\\dots\\right)\\\\\n",
    "&= c_0x^{k+1}+c_1x^{k-1}+\\dots\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $L_{k+1}(x)$ is a linear combination of monomials $x^{k+1}, x^{k-1},\\dots$. Thus\n",
    "\n",
    "\\begin{align*}\n",
    "L_k(-x) = (-1)^kL_k(x)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "#### Problem 4.3 (c)\n",
    "\n",
    "* $k=1$, $LHS=(x^2-1)\\frac{dL_1(x)}{dx} = (x^2-1)$, and $RHS=xL_1(x) - L_0(x) = x^2 -1$. So the equation is correct when $k=1$.\n",
    "\n",
    "* Suppose for all $m\\le k$, we have $\\frac{x^2-1}{m}\\frac{dL_m(x)}{dx}=xL_m(x)-L_{m-1}(x)$.\n",
    "* When $m=k+1$, we have (To make the display simple, we use $L_k$ to represent $L_k(x)$.\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{dL_{k+1}(x)}{dx} &= \\frac{2(k+1)-1}{k+1}\\left(L_k + x\\frac{dL_k}{dx}\\right) - \\frac{k}{k+1}\\frac{dL_{k-1}}{dx}\\\\\n",
    "&= \\frac{2k+1}{k+1}\\left(L_k + x\\frac{k}{x^2-1}\\left(xL_k-L_{k-1}\\right)\\right) \\\\\n",
    "&- \\frac{k}{k+1}\\frac{k-1}{x^2-1}\\left(xL_{k-1} - L_{k-2}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take $L_{k-2} = \\frac{(2k-1)xL_{k-1} - kL_k}{k-1}$ into above equation we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{dL_{k+1}}{dx} &= \\frac{2k+1}{k+1}\\left(L_k + x\\frac{k}{x^2-1}\\left(xL_k-L_{k-1}\\right)\\right) \\\\\n",
    "&- \\frac{k}{k+1}\\frac{k-1}{x^2-1}\\left(xL_{k-1} - \\frac{(2k-1)xL_{k-1} - kL_k}{k-1}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Multiply both sides with $x^2-1$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "(x^2-1)\\frac{dL_{k+1}}{dx} &= \\frac{2k+1}{k+1}\\left((x^2-1)L_k + kx\\left(xL_k-L_{k-1}\\right)\\right) \\\\\n",
    "&- \\frac{k(k-1)}{k+1}\\left(xL_{k-1} - \\frac{(2k-1)xL_{k-1} - kL_k}{k-1}\\right)\\\\\n",
    "&= \\frac{2k+1}{k+1}\\left(\\left((k+1)x^2-1\\right)L_k - kxL_{k-1}\\right) \\\\\n",
    "&- \\frac{k}{k+1}\\left(\\left(k-1\\right)xL_{k-1} - \\left(2k-1\\right)xL_{k-1} +kL_k\\right)\\\\\n",
    "&= \\frac{2k+1}{k+1}\\left(\\left((k+1)x^2-1\\right)L_k - kxL_{k-1}\\right)- \\frac{k^2}{k+1}\\left(L_k-xL_{k-1}\\right)\\\\\n",
    "&= \\left((2k+1)x^2-(k+1)\\right)L_k-kxL_{k-1}\\\\\n",
    "&= x\\left((2k+1)xL_k-kL_{k-1}\\right)-(k+1)L_k\\\\\n",
    "&= x(k+1)L_{k+1} - (k+1)L_k\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So we have\n",
    "\\begin{align*}\n",
    "\\frac{x^2-1}{k+1}\\frac{dL_{k+1}}{dx} &=xL_{k+1} - L_k\\\\\n",
    "\\end{align*}\n",
    "\n",
    "#### Problem 4.3 (d) \n",
    "\n",
    "From problem 3.4(c), we have \n",
    "\\begin{align*}\n",
    "(x^2-1)\\frac{dL_{k}}{dx} &= k(xL_{k} - L_{k-1})\\\\\n",
    "(x^2-1)\\frac{dL_{k-1}}{dx} &= (k-1)(xL_{k-1} - L_{k-2})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take derivative w.r.t $x$ on $\\frac{d}{dx}(x^2-1)\\frac{dL_k}{dx}$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{d}{dx}(x^2-1)\\frac{dL_k}{dx}&= \\frac{d}{dx}\\left(kxL_k - kL_{k-1}\\right)  \\\\\n",
    "&= k\\left(L_k + x\\frac{dL_k}{dx}\\right) - k\\frac{dL_{k-1}}{dx}\\\\\n",
    "&= kL_k + k\\left(x\\frac{dL_k}{dx} - \\frac{dL_{k-1}}{dx}\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Replace the above computed $\\frac{dL_{k}}{dx}$ and $\\frac{dL_{k-1}}{dx}$, we obtain\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{d}{dx}(x^2-1)\\frac{dL_k}{dx}&= kL_k + k\\left(x\\frac{dL_k}{dx} - \\frac{dL_{k-1}}{dx}\\right)\\\\\n",
    "&= kL_k + \\frac{k}{x^2-1}\\left(xk(xL_{k} - L_{k-1}) - (k-1)(xL_{k-1} - L_{k-2})\\right)\\\\\n",
    "&= kL_k + \\frac{k}{x^2-1}\\left(kx^2L_{k} - \\left((2k-1)xL_{k-1} - (k-1)L_{k-2}\\right)\\right)\\\\\n",
    "&= kL_k + \\frac{k}{x^2-1}\\left(kx^2L_{k} - kL_k\\right)\\\\\n",
    "&= kL_k + k^2L_k\\\\\n",
    "&= k(k+1)L_k\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (e) It doesn't hurt if we can prove that the equation is correct for $l\\le k$ for any given $k$. We use induction to prove this.\n",
    "  * When $k=0$, $l$ can only be $0$, so we have $\\int^1_{-1}dxL_0L_0 = \\int^1_{-1}dx = 2 = \\frac{2}{2k+1}$. The equation is thus correct when $k=0$.\n",
    "  * Assume that the equation is correct when $k\\le K$, let's look at the case when $k = K+1$.\n",
    "  \n",
    "\\begin{align*}\n",
    "L_{K+1}L_l &= \\frac{2K+1}{K+1}xL_{K}L_l - \\frac{K}{K+1}L_{K-1}L_l\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We need work out the integration of $\\int^1_{-1}xL_{K}L_ldx$. Let's do it.\n",
    "\n",
    "\\begin{align*}\n",
    "\\int^1_{-1}xL_{K}L_ldx &= \\frac{1}{2}(x^2-1)L_KL_l|^1_{-1} - \\int^1_{-1}\\frac{1}{2}(x^2-1)\\left(L_KdL_l + L_ldL_K\\right)dx\\\\\n",
    "&= - \\int^1_{-1}\\frac{1}{2}(x^2-1)\\left(L_KdL_l + L_ldL_K\\right)dx\n",
    "\\end{align*}\n",
    "\n",
    "Plug the result of problem (c), i.e. $\\frac{x^2-1}{k}\\frac{dL_{k}}{dx} =xL_{k} - L_{k-1}$ for $k=K$ and $k=l$ into above equation, we have\n",
    "\n",
    "\\begin{align*}\n",
    "-2\\int^1_{-1}xL_{K}L_ldx &= \\int^1_{-1}(x^2-1)\\left(L_KdL_l + L_ldL_K\\right)dx\\\\\n",
    "&= \\int^1_{-1}\\left(L_K(x^2-1)dL_l + L_l(x^2-1)dL_K\\right)dx\\\\\n",
    "&= \\int^1_{-1}\\left(L_K\\left(lxL_l-lL_{l-1}\\right) + L_l\\left(KxL_K-KL_{K-1}\\right)\\right)dx\\\\\n",
    "&= \\int^1_{-1}\\left(\\left(l + K\\right)xL_KL_l -\\left(lL_KL_{l-1} + KL_lL_{K-1}\\right)\\right)dx\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Move the term with $xL_KL_l$ to the same side, we have\n",
    "\\begin{align*}\n",
    "(2+K+l)\\int^1_{-1}xL_{K}L_ldx &= \\int^1_{-1}\\left(lL_KL_{l-1} + KL_lL_{K-1}\\right)dx\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Let's consider different cases of different $l$ values to compute the $\\int^1_{-1}xL_{K}L_ldx$.\n",
    "\n",
    "  * If $l \\lt K-1$, then clearly by our assumptions using induction, we have $\\int^1_{-1}xL_{K}L_ldx = 0$. Plug into the formula for $\\int^1_{-1}L_{K+1}L_ldx$ we have \n",
    "  \n",
    "\\begin{align*}\n",
    "\\int^1_{-1}L_{K+1}L_ldx &= \\frac{2K+1}{K+1}\\int^1_{-1}xL_{K}L_l - \\frac{K}{K+1}\\int^1_{-1}L_{K-1}L_l\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * If $l = K-1$\n",
    "\n",
    "\\begin{align*}\n",
    "\\int^1_{-1}xL_{K}L_{K-1}dx &= \\frac{1}{2K+1}\\int^1_{-1}\\left((K-1)L_KL_{K-2} + KL^2_{K-1}\\right)dx\\\\\n",
    "&= \\frac{1}{2K+1}\\int^1_{-1}KL^2_{K-1}dx\\\\\n",
    "&= \\frac{2K}{(2K-1)(2K+1)}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\int^1_{-1}L_{K+1}L_{K-1}dx &= \\frac{2K+1}{K+1}\\int^1_{-1}xL_{K}L_{K-1} - \\frac{K}{K+1}\\int^1_{-1}L_{K-1}L_{K-1}\\\\\n",
    "&= \\frac{2K+1}{K+1}\\frac{2K}{(2K-1)(2K+1)} - \\frac{K}{K+1}\\frac{2}{2K-1}\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * If $l = K$\n",
    "  \n",
    "\\begin{align*}\n",
    "\\int^1_{-1}xL_{K}L_Kdx &= \\frac{1}{2(K+1)} \\int^1_{-1}\\left(KL_KL_{K-1} + KL_KL_{K-1}\\right)dx\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\int^1_{-1}L_{K+1}L_Kdx &= \\frac{2K+1}{K+1}\\int^1_{-1}xL_{K}L_K - \\frac{K}{K+1}\\int^1_{-1}L_{K-1}L_K\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * If $l = K+1$\n",
    "  \n",
    "\\begin{align*}\n",
    "\\int^1_{-1}xL_{K}L_{K+1}dx &= \\frac{1}{2K+3} \\int^1_{-1}\\left((K+1)L_KL_K + KL_{K+1}L_{K-1}\\right)dx\\\\\n",
    "&= \\frac{K+1}{2K+3}\\frac{2}{2K+1}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\int^1_{-1}L_{K+1}L_{K+1}dx &= \\frac{2K+1}{K+1}\\int^1_{-1}xL_{K}L_{K+1} - \\frac{K}{K+1}\\int^1_{-1}L_{K-1}L_{K+1}\\\\\n",
    "&= \\frac{2K+1}{K+1}\\frac{K+1}{2K+3}\\frac{2}{2K+1}\\\\\n",
    "&= \\frac{2}{2K+3}\\\\\n",
    "&= \\frac{2}{2(K+1)+1}\\\\\n",
    "\\end{align*}  \n",
    "\n",
    "Where the seond term on the right hand side is $0$ because we just proved when $l =K-1$.\n",
    "\n",
    "We now conclude that the equation is correct for all $k$ and $l$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.4\n",
    "\n",
    "To scale the coefficients $a_q$, we need to compute the expectation $E_{a,x}[f^2]$ first. \n",
    "\n",
    "\\begin{align*}\n",
    "E_{a,x}[f^2(x)] &= E_{a,x}\\left(\\sum^{Q_f}_{q=0} a_qL_q(x)\\right)^2\\\\\n",
    "&= E_{a,x}\\left(\\sum^{Q_f}_{q=0} a^2_qL^2_q(x) + \\sum^{Q_f}_{p\\ne q} a_pa_qL_p(x)L_q(x)\\right)\\\\\n",
    "&= E_{a,x}\\left(\\sum^{Q_f}_{q=0} a^2_qL^2_q(x)\\right) + E_{a,x}\\left(\\sum^{Q_f}_{p\\ne q} a_pa_qL_p(x)L_q(x)\\right)\\\\\n",
    "&= \\sum^{Q_f}_{q=0} E_a a^2_qE_xL^2_q(x) + \\sum^{Q_f}_{p\\ne q} E_a[a_pa_q]E_x[L_p(x)L_q(x)]\\\\\n",
    "&= \\sum^{Q_f}_{q=0} E_xL^2_q(x)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have used the independence of random variables $a$ and $x$, and also $a$ is standard normal distributed, so $E_a[a_q] = 0$ and $E_a[a^2_q] = 1$.\n",
    "\n",
    "From problem 4.3(e) we have $\\int^1_{-1}L^2_k(x) = \\frac{2}{2k+1}$, also $x$ has a probability density $P(x) = \\frac{1}{2}$, so we have the expectation \n",
    "\n",
    "\\begin{align*}\n",
    "E_{a,x}[f^2(x)] &=\\sum^{Q_f}_{q=0} E_xL^2_q(x)\\\\\n",
    "&= \\sum^{Q_f}_{q=0}\\int^1_{-1}L^2_q(x)P(x)dx\\\\\n",
    "&= \\frac{1}{2}\\sum^{Q_f}_{q=0}\\int^1_{-1}L^2_q(x)dx\\\\\n",
    "&= \\frac{1}{2}\\sum^{Q_f}_{q=0}\\frac{2}{2q+1}\\\\\n",
    "&= \\sum^{Q_f}_{q=0}\\frac{1}{2q+1}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "To scale $E_{a,x}[f^2(x)]$ to $1$, we need to re-scale each $a_q$ by $\\sqrt{\\frac{1}{\\sum^{Q_f}_{q=0}\\frac{1}{2q+1}}}$.\n",
    "\n",
    "* (a) Notice that $E_{a,x}[f] = 0$, so in fact we have $E_{a,x}[y_n] = 0$. Once we normalize $E_{a,x}[f^2(x)]=1$, we make the variance of $f$ to be 1, which can be compared with the variance on the noise term $\\epsilon$, which is $\\sigma^2$. This makes it easier to compare between different order of polynomials. \n",
    "\n",
    "* (b) Suppose we have data set $x_1,\\dots, x_N$ and $y_1,\\dots,y_N$, we apply polynomial $\\Phi_Q(x)$ transformation to each $x_n$ and obtain $z_n = \\Phi_Q(x_n)$, so we have an input matrix of $Z$ with dimension $N\\times (Q+1)$. The best fit hypotheses to the data is then from the least square algorithm: \n",
    "\n",
    "\\begin{align*}\n",
    "w &= (Z^TZ)^{-1}Z^Ty\\\\\n",
    "g_Q(x) &= \\Phi_Q(x)(Z^TZ)^{-1}Z^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) Suppose we obtained a best fit hypothesis $g_K(z) = z^Tw = \\Phi^T(x)w$ to the data from $\\mathcal{H}_K$, here $K=2$ or $K=10$. Let $M=\\begin{bmatrix}1\\\\x\\\\x^2\\\\ \\vdots \\\\ x^K\\end{bmatrix}_{(K+1)\\times 1}$, then we have $g_K(x) = M^Tw$. If we can express $g_K(x)$ in terms of Legendre polynomials, i.e. $g_K(x) = \\sum^K_{i=0}c_iL_i(x)$ then we can compute $E_{out}$ easily as:\n",
    "\n",
    "\\begin{align*}\n",
    "E_{out} &= E_{x}[(g_K(x) - f(x))^2]\\\\\n",
    "&= E_{x}\\left[\\left(\\sum^K_{i=0}c_iL_i(x) - \\sum^{Q_f}_{q=0}c_qL_q(x)\\right)^2\\right]\\\\\n",
    "&= E_{x}\\left[\\left(\\sum^{\\max{K,Q_f}}_{i=0}b_iL_i(x)\\right)^2\\right]\\\\\n",
    "&= E_{x}\\left(\\sum^{\\max{K,Q_f}}_{i=0}b^2_iL^2_i(x) + \\sum^{\\max{K,Q_f}}_{i\\ne j}b_ib_jL_i(x)L_j(x)\\right)\\\\\n",
    "&= \\sum^{\\max{K,Q_f}}_{i=0}b^2_iE_{x}L^2_i(x)\\\\\n",
    "&= \\sum^{\\max{K,Q_f}}_{i=0}b^2_i \\frac{2}{2i+1}\\\\\n",
    "&= \\sum^{\\max{K,Q_f}}_{i=0}\\frac{2b^2_i}{2i+1}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Now let's figure out how to solve for $c_i$ that can make $g_K(x) = \\sum^K_{i=0}c_iL_i(x)$. \n",
    "First, let's define the matrix of Legendre coefficients of a polynomial of degree $K$, i.e. the coefficient vector for $L_0 = 1$ is $P_0 = \\begin{bmatrix}1\\\\0\\\\0\\\\ \\vdots \\\\0\\end{bmatrix}_{(K+1)\\times 1}$. Similarly for $L_1 = x$, we have $P_1 = \\begin{bmatrix}0\\\\1\\\\0\\\\ \\vdots \\\\0\\end{bmatrix}_{(K+1)\\times 1}$ and for $L_2 = \\frac{3}{2}x^2 - \\frac{1}{2}$, we have $P_2 = \\begin{bmatrix}- \\frac{1}{2}\\\\0\\\\\\frac{3}{2}\\\\ \\vdots \\\\0\\end{bmatrix}_{(K+1)\\times 1}$. Thus $L_K = M^TP_K$. we have\n",
    "\n",
    "\\begin{align*}\n",
    "g_K(x) &= \\sum^K_{i=0}c_iL_i(x)\\\\\n",
    "&= \\begin{bmatrix}1&x&x^2& \\dots & x^K\\end{bmatrix}\n",
    "\\begin{bmatrix}1&0&-\\frac{1}{2}&\\dots\\\\ 0&1&0&\\dots \\\\ 0 & 0 & \\frac{3}{2} \\dots \\\\ \\dots & \\dots & \\dots & \\dots \\\\\\end{bmatrix}\n",
    "\\begin{bmatrix}c_0\\\\c_1\\\\c_2\\\\ \\vdots \\\\c_K\\end{bmatrix}\\\\\n",
    "&= M^T\\begin{bmatrix}P_0 & P_1 & P_2 & \\dots & P_K\\end{bmatrix}C\\\\\n",
    "&= M^TPC\\\\\n",
    "\\end{align*}\n",
    "\n",
    "On the other hand, from least square solution, we have $g_K(x) = M^Tw$, compare the two equations we see that \n",
    "$PC = w$, i.e. $C = P^{-1}w$. This is the formula we are going to use to solve for coefficients $c_0, c_1, \\dots, c_K$. \n",
    "* (d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import libs.data_util as data\n",
    "\n",
    "def run_one_validation_experiment(Qf, N, sigma_square, normalized_aqs, test_xs, test_ys):\n",
    "    \"\"\"Generate one set of training data for a given model and compute the test error\n",
    "    hypo_q: The degree of hypothesis set\n",
    "    \n",
    "    \n",
    "    \"\"\"\n",
    "        \n",
    "    xs, ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    \n",
    "    # Hypothesis with degree 2\n",
    "    Z2 = data.polynomial_transform(2, xs)\n",
    "    #print('xs: ', xs.shape, 'Z2: ', Z2.shape, 'ys: ', ys.shape)\n",
    "    w2 = np.matmul(np.linalg.inv(np.matmul(Z2.transpose(), Z2)), np.matmul(Z2.transpose(), ys))\n",
    "\n",
    "    # Hypothesis with degree 10\n",
    "    Z10 = data.polynomial_transform(10, xs)\n",
    "    w10 = np.matmul(np.linalg.inv(np.matmul(Z10.transpose(), Z10)), np.matmul(Z10.transpose(), ys))\n",
    "    #print('w2: ', w2.shape, 'test_xs: ', test_xs.shape, test_ys.shape)\n",
    "    Eout2 = data.calc_Eout(w2, test_xs, test_ys)\n",
    "    Eout10 = data.calc_Eout(w10, test_xs, test_ys)\n",
    "\n",
    "    return Eout2, Eout10, xs, ys, test_xs, test_ys, w2, w10, normalized_aqs\n",
    "\n",
    "\n",
    "def calc_avg_Eouts(num_exps, Qf, N, sigma_square, normalized_aqs,\n",
    "                   test_xs, test_ys, calc_avg_Ein = False, calc_std = False):\n",
    "    \n",
    "    avg_Eout2, avg_Eout10 = 0, 0\n",
    "    avg_Ein2, avg_Ein10 = 0, 0\n",
    "    Eout2s, Eout10s = [], []\n",
    "    for ei in np.arange(num_exps):\n",
    "        res = run_one_validation_experiment(Qf, N, sigma_square, \n",
    "                                            normalized_aqs, test_xs, test_ys)\n",
    "        Eout2, Eout10, xs, ys, test_xs, test_ys, w2, w10, normalized_aqs = res\n",
    "        Eout2s.append(Eout2)\n",
    "        Eout10s.append(Eout10)\n",
    "        avg_Eout2 += Eout2\n",
    "        avg_Eout10 += Eout10\n",
    "        if calc_avg_Ein:\n",
    "            Ein2 = data.calc_Eout(w2, xs, ys)\n",
    "            Ein10 = data.calc_Eout(w10, xs, ys)\n",
    "            avg_Ein2 += Ein2\n",
    "            avg_Ein10 += Ein10\n",
    "\n",
    "    avg_Eout2 = avg_Eout2/num_exps\n",
    "    avg_Eout10 = avg_Eout10/num_exps\n",
    "    avg_Ein2 = avg_Ein2/num_exps\n",
    "    avg_Ein10 = avg_Ein10/num_exps\n",
    "    std_Eout2 = np.std(Eout2s)\n",
    "    std_Eout10 = np.std(Eout10s)\n",
    "    return avg_Eout2, avg_Eout10, avg_Ein2, avg_Ein10, std_Eout2, std_Eout10\n",
    "\n",
    "def calc_overfit(num_exps, Qf, N, sigma_square, normalized_aqs, test_xs, test_ys):\n",
    "    \n",
    "    Eout2s, Eout10s = [], []\n",
    "    for ei in np.arange(num_exps):\n",
    "        res = run_one_validation_experiment(Qf, N, sigma_square, \n",
    "                                            normalized_aqs, test_xs, test_ys)\n",
    "        Eout2, Eout10, xs, ys, test_xs, test_ys, w2, w10, normalized_aqs = res\n",
    "        Eout2s.append(Eout2)\n",
    "        Eout10s.append(Eout10)\n",
    "\n",
    "    diffs = np.array(Eout10s) - np.array(Eout2s)\n",
    "    mean_diff = np.mean(diffs)\n",
    "    std_diff = np.std(diffs)\n",
    "    return mean_diff, std_diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Out-of-Sample error: E_out2 = [0.6043702] ; E_out10 =  [28.41765633]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Out-of-Sample error: E_out2 = [0.66495586] ; E_out10 =  [268.63322742]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Duplicate Figure 4.1\n",
    "# 10th order target function\n",
    "Qf = 10\n",
    "N = 15\n",
    "sigma_square = 0.25\n",
    "test_N = 100\n",
    "tol = 1.0e-6\n",
    "\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "Eout2, Eout10, xs, ys, test_xs, test_ys, w2, w10, normalized_aqs = run_one_validation_experiment(Qf, N, sigma_square, \n",
    "                                                                                                 normalized_aqs, test_xs, test_ys)\n",
    "print('Out-of-Sample error: E_out2 =', Eout2, '; E_out10 = ', Eout10)\n",
    "xvals = np.arange(np.min(xs),np.max(xs)+tol,0.02)\n",
    "yvals = data.calc_legendre_array(normalized_aqs, xvals.reshape(-1, 1))\n",
    "ypred2 = data.calc_pred(w2, xvals)\n",
    "ypred10 = data.calc_pred(w10, xvals)\n",
    "plt.plot(xvals, yvals, label='True Target Function')\n",
    "plt.plot(xs, ys, '.', markersize=15, label='Training Data')\n",
    "plt.plot(xvals, ypred2, label='H2 Prediction')\n",
    "plt.plot(xvals, ypred10, label='H10 Prediction')\n",
    "plt.legend()\n",
    "plt.title(\"Noisy low-order target: 10th degree polynomial + noise\")\n",
    "#plt.ylim(-2, 1)\n",
    "plt.show()\n",
    "\n",
    "# 50th order target function\n",
    "#print('Start to generate non-noisy data...')\n",
    "Qf = 50\n",
    "N = 15\n",
    "sigma_square = 0.0\n",
    "test_N = 100\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "Eout2, Eout10, xs, ys, test_xs, test_ys, w2, w10, normalized_aqs = run_one_validation_experiment(Qf, N, sigma_square, \n",
    "                                                                                                 normalized_aqs, \n",
    "                                                                                                 test_xs, test_ys)\n",
    "print('Out-of-Sample error: E_out2 =', Eout2, '; E_out10 = ', Eout10)\n",
    "xvals = np.arange(np.min(xs),np.max(xs)+tol,0.02)\n",
    "yvals = data.calc_legendre_array(normalized_aqs, xvals.reshape(-1, 1))\n",
    "ypred2 = data.calc_pred(w2, xvals)\n",
    "ypred10 = data.calc_pred(w10, xvals)\n",
    "plt.plot(xvals, yvals, label='True Target Function')\n",
    "plt.plot(xs, ys, '.', markersize=15, label='Training Data')\n",
    "plt.plot(xvals, ypred2, label='H2 Prediction')\n",
    "plt.plot(xvals, ypred10, label='H10 Prediction')\n",
    "plt.legend()\n",
    "plt.title(\"Noiseless high-order target: 50th degree polynomial without noise\")\n",
    "#plt.ylim(-2, 1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Duplicate Figure 4.2 on Learning Curve\n",
    "num_exps = 5000\n",
    "Qf = 10\n",
    "sigma_square = 0.25\n",
    "test_N = 100\n",
    "max_N = 120\n",
    "Ns = np.arange(20, max_N+1, 5)\n",
    "avg_Ein2s, avg_Ein10s = [], []\n",
    "avg_Eout2s, avg_Eout10s = [], []\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "for N in Ns:    \n",
    "    avg_Eout2, avg_Eout10, avg_Ein2, avg_Ein10, _, _ = calc_avg_Eouts(num_exps, Qf, N, sigma_square, \n",
    "                                                                normalized_aqs, test_xs, test_ys, True)\n",
    "    avg_Ein2s.append(avg_Ein2)\n",
    "    avg_Ein10s.append(avg_Ein10)\n",
    "    avg_Eout2s.append(avg_Eout2)\n",
    "    avg_Eout10s.append(avg_Eout10)\n",
    "\n",
    "plt.plot(Ns, avg_Ein2s, label='Average E_in')\n",
    "plt.plot(Ns, avg_Eout2s, label='Average E_out')\n",
    "plt.legend()\n",
    "plt.title(\"Learning Curves for H_2\")\n",
    "#plt.ylim(-2, 1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(Ns, avg_Ein10s, label='Average E_in')\n",
    "plt.plot(Ns, avg_Eout10s, label='Average E_out')\n",
    "plt.legend()\n",
    "plt.title(\"Learning Curves for H_10\")\n",
    "plt.yscale('log')\n",
    "plt.ylim(0.1, 1000)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The stablized expected error from $\\mathcal{H}_{10}$ is much smaller than the error from $\\mathcal{H}_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2ba607bf198>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.arange(10), avg_Eout2s[-10:], label='Average Eout from H2')\n",
    "plt.plot(np.arange(10), avg_Eout10s[-10:], label = 'Average Eout from H10')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------ Working on Figure 4.3(a)\n",
      "N_coords:  (21, 60) sigma_coords:  (21, 60)\n",
      "Working on the  0 th row...with N =  20\n",
      "Working on the  1 th row...with N =  25\n",
      "Working on the  2 th row...with N =  30\n",
      "Working on the  3 th row...with N =  35\n",
      "Working on the  4 th row...with N =  40\n",
      "Working on the  5 th row...with N =  45\n",
      "Working on the  6 th row...with N =  50\n",
      "Working on the  7 th row...with N =  55\n",
      "Working on the  8 th row...with N =  60\n",
      "Working on the  9 th row...with N =  65\n",
      "Working on the  10 th row...with N =  70\n",
      "Working on the  11 th row...with N =  75\n",
      "Working on the  12 th row...with N =  80\n",
      "Working on the  13 th row...with N =  85\n",
      "Working on the  14 th row...with N =  90\n",
      "Working on the  15 th row...with N =  95\n",
      "Working on the  16 th row...with N =  100\n",
      "Working on the  17 th row...with N =  105\n",
      "Working on the  18 th row...with N =  110\n",
      "Working on the  19 th row...with N =  115\n",
      "Working on the  20 th row...with N =  120\n",
      "done!\n"
     ]
    }
   ],
   "source": [
    "%timeit\n",
    "max_qf = 100\n",
    "max_N = 120\n",
    "max_sigma_square = 2\n",
    "Qfs = np.arange(1, max_qf+1)\n",
    "Ns = np.arange(20, max_N+1, 5)\n",
    "sigma_squares = np.arange(0, max_sigma_square+1, 0.05)\n",
    "test_N = 100\n",
    "\n",
    "num_exps = 10000\n",
    "\n",
    "# Calculate the figure 4.3(a)\n",
    "print('------ Working on Figure 4.3(a)')\n",
    "Qf = 20\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "sigma_coords, N_coords = np.meshgrid(sigma_squares, Ns) #N is fixed on each row\n",
    "diff_sigmas = np.zeros(sigma_coords.shape)\n",
    "print('N_coords: ', N_coords.shape, 'sigma_coords: ', sigma_coords.shape)\n",
    "i = 0\n",
    "for N in Ns:\n",
    "    j = 0\n",
    "    print('Working on the ', i, 'th row...with N = ', N)\n",
    "    for sigma_square in sigma_squares:\n",
    "        test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "        avg_Eout2, avg_Eout10, _, _, _, _ = calc_avg_Eouts(num_exps, Qf, N, sigma_square, normalized_aqs, test_xs, test_ys)\n",
    "        diff = avg_Eout10 - avg_Eout2\n",
    "        diff_sigmas[i, j] = diff\n",
    "        j += 1\n",
    "    i += 1\n",
    "\n",
    "np.savetxt(\"diff_sigmas.csv\", diff_sigmas, delimiter=\",\", fmt='%12.8f')    \n",
    "print('done!')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------ Working on Figure 4.3(b)\n",
      "N_coords:  (21, 100) Qf_coords:  (21, 100)\n",
      "Working on the  0 th row...with N =  20\n",
      "Working on the  1 th row...with N =  25\n",
      "Working on the  2 th row...with N =  30\n",
      "Working on the  3 th row...with N =  35\n",
      "Working on the  4 th row...with N =  40\n",
      "Working on the  5 th row...with N =  45\n",
      "Working on the  6 th row...with N =  50\n",
      "Working on the  7 th row...with N =  55\n",
      "Working on the  8 th row...with N =  60\n",
      "Working on the  9 th row...with N =  65\n",
      "Working on the  10 th row...with N =  70\n",
      "Working on the  11 th row...with N =  75\n",
      "Working on the  12 th row...with N =  80\n",
      "Working on the  13 th row...with N =  85\n",
      "Working on the  14 th row...with N =  90\n",
      "Working on the  15 th row...with N =  95\n",
      "Working on the  16 th row...with N =  100\n",
      "Working on the  17 th row...with N =  105\n",
      "Working on the  18 th row...with N =  110\n",
      "Working on the  19 th row...with N =  115\n",
      "Working on the  20 th row...with N =  120\n",
      "done!\n"
     ]
    }
   ],
   "source": [
    "# Calculate the figure 4.3(b) \n",
    "print('------ Working on Figure 4.3(b)')\n",
    "max_qf = 100\n",
    "max_N = 120\n",
    "Qfs = np.arange(1, max_qf+1)\n",
    "Ns = np.arange(20, max_N+1, 5)\n",
    "test_N = 100\n",
    "num_exps = 100\n",
    "sigma_square = 0.1\n",
    "\n",
    "N_coords, Qf_coords = np.meshgrid(Qfs, Ns)\n",
    "diff_Qfs = np.zeros(Qf_coords.shape)\n",
    "print('N_coords: ', N_coords.shape, 'Qf_coords: ', Qf_coords.shape)\n",
    "\n",
    "\n",
    "i = 0\n",
    "for N in Ns:\n",
    "    j = 0\n",
    "    print('Working on the ', i, 'th row...with N = ', N)\n",
    "    for Qf in Qfs:\n",
    "        normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "        test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "        avg_Eout2, avg_Eout10, _, _, _, _ = calc_avg_Eouts(num_exps, Qf, N, sigma_square, normalized_aqs, test_xs, test_ys)\n",
    "        diff = avg_Eout10 - avg_Eout2\n",
    "        diff_Qfs[i, j] = diff\n",
    "        j += 1\n",
    "    i += 1\n",
    "    \n",
    "np.savetxt(\"diff_Qfs_raw.csv\", diff_Qfs, delimiter=\",\", fmt='%12.8f')    \n",
    "print('done!')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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guftvAypoRXRl2XXYFy7PRRkkeie6wA0i7qJ0nBFkSsSGRMzySN6hltAFWGt5JFZtNioUgLmZ43mZ4xNg/MR6z3P6gwuc4zgDTUt/t6zYBVhqQekJqUjQZHwyXzrVYc3cxVFeX0x+wIkzKLjAOY4zNLSJXSq1IGd3LKJO+kGqC/icM+pbgs5g4QLnOM5Qkt1LXJg/LPh+SIlmTqRiUdWWaE/O0wQGERc4x3FGgiaCZy5aWy5yCS3ilhdZ6QwELnCO47RQFtQxLBGYTZF1U2ZzC1k9J5UL5wwaLnCO47SQ3eeabaJWxNiG8emIyjiScnz5GDAHPIpyINlkphfgOM7g4uKWH4WZRFIunrcGlk/2eUVOKC5wjuOMDAsX9Lb0VzploEXknIHEXZSO4wwtRWJWJXJFlmnLfQUpA2MbxqciKhcx4blwA4wLnOM4Q0WpeC0lKO8tO0cieFNRlBVkRa6D5kBOH3CBcxxnaCgUn1SJrpag/QZ7zHni9/DhAuc4zlAQUqZrzdzFjC1LdRFPj+9E7Ery4dJWnDOYFAqcpGcBnwPmAxcB7zKze+JrV5jZAf1ZouM4s52Qnm4Q5alNzF00FQCSK3YZocvux5W5Kd2KGy7KLLjPAicR1SNfDnxP0mFm9ktg8z6szXEcp53YqkrC9xMRSxdCTkc75jV1TaibBlG3+7czs5QJ3NZmdnH888ckXQ1cLOlvgeFqIuc4M0yeReA5ZuG0WVU51f9brLVs7lqXwpYlazXiYSYDSZnASdK2ZnYfgJldLmkx0Se5Q19W5zhDSp5LDeIgiPjLd2Em4s8Fr5xc12GO0LVdy5knmIJ9vgRPERhsygTu34C/INUy0cyuk/Ri4F96vTDHGTbyumjnfSlyXPTX2IZxKBA8F7t8CvfHssEgDUZPQvvn2FaTcsS6CUh6BXAqsCmw2sxOyVyfA5wJPBf4A3Ckmd0maXNgNbAfkb6caWYf7uvi0+s0Gy5v416SfWqmF+E4FZQFRUDK2nALrjaVeXB5lAhebkPVgkCWPIttfP0YzN/yajPbv2RljbD/brKr3tX9PFpB4XolbQr8HHgpsA64Enitmd2YGvNmYB8zO1bSUcAiMztS0uuAw8zsKElzgRuBg8zstu5XXZ/KNAFJ+5jZdf1YjOOMCi1iFVsdcxreB5qtlCZjF4X1lxhYC3OuhQhbwuJ5a0ZtB+4A4GYzuwVA0rnA4URilXA4URAiwBrgNEkiis/YStJmwGOAPwP392ndbZQKnKSXEL3EX/dlNY4zoriYNUtHIldCXnTkiO+v7SjpqtTxKjNbFf88H7g9dW0dcGDm/qkxZrZR0n3A44jE7nDgDqLeC283s7t7sP4gyvLglgDvAF7ev+U4juP0lmwe2zAFjdiODeXhreCuEpeq8h4dOOYA4BGihkLbA9+V9I3EGuw3ZRbc54G9zez3nUwsaVeiTcidgUeJ/oVwamaMiDYyDwE2AEeb2TWdPM9xnNGko+4AHVhxMN0hoC4j5qJcB+yaOn4isL5gzLrYHbktcDfwOuBiM3sYuFPS94H9gYETuJOBz0s6xMz+1MHcG4F3mNk1kh4LXC3p0vRGJbAQ2Cv+cyBRcnnWFHYcZ5bQy1Y3CdlqJElSuCduT3ElsJekPYDfAEcRCVeaC4DXA/8LjAHfNDOT9GvgYElnEbkoFwCf7NvKMxQKnJl9KF7sV+nATWlmdxD5YTGzByTdROS3zW5UnmlRKOdaSdtJ2iW+13GcEacs2jQ0r62QEisumTtP6DphlJIE4j21FcAlRGkCp5vZDZJOBq4yswuIPHz/JelmIsvtqPj2TwNnAD8hcmOeMZNBiqVBJmZ2lqSuxUbS7sC+wA8zl/I2M+cTC2Pq/mOAYwB26nYxjuPMGGUJ8AlTltRxGdFZ1oDoZfDakvmY2YXAhZlzJ6R+fgg4Iue+B/POzxSVaQJmdlk3D5C0NZGL+m1mlg0XDdnMJI7uWQVRHlw363Ecpz9U5QJWuQTzLKqsGM3JDkhVNqnChW30CWqXI+mZwN7Alsk5Mzsz4L7NicTtbDM7P2dIyGam4zhDwsIFtIhZJ/tadVyFk1mrrkTY8gQtLyUgr2CzM5yEJHqfCBxEJHAXEgWGfI8oQrLsPhH5aW8ys48XDLsAWBEnEh4I3Of7b44zPGQF7ewSwaiiU0Ep2rMLEbTpUlvA6jmwfJJxxlg8bw0Tc6MxLnjDS4gFNwY8G/iRmS2T9ASiWmNV/BXwt8D1kq6Nz70H2A3AzFYSCeYhwM1EaQLuNHCcASZdympyGUwSiUaLYKyOHYcrgWNhnCWwfHJqjsXz1gD5wtG2x1ZCVsCKXI55eW5TwpZea7SC6Hp6zfPiS3NpW6/TW2JDaQmwp5mdLGk3YGczuyLo/qpalElz07hdzouAB4CfmNkzulx7R3gtSseZGUKKSU+wqFU8YpHIihpMC0VZv7bs80IJ2V8rqliSt8Y0eSJcVtuxSZ67n+wH3+t+ni236s96u0XSZ4nyqA82s7+QtD3wdTN7Xsj9IRbcVZK2I+rufTXwIBCkno7jjA5T5bFS4fdJFOKUEMyFRfMmIqFbPhZureUUnZ4S1KzoBQheiCW4JE7PTveRC73X6RsHmtl+kn4EYGb3SNoi9OaQKMo3xz+ulHQxsI0XX3acWU5G5CBf6KDCBVnSTSF73I3gFT0T2otgOwPFw3F3AwOQ9Hgiiy6I0CjKfYDdk/GSnlIQFek4zmwhLQxLC4SOeqJWRptll50voB9cE0Wv+1FtxZniU8AEsJOkDxLFhLwv9OaQKMrTgX2AG5hWTgNc4BxnFlH6xZ4SlDklzUe7FZjCNWStuOQ4I3TJ/Z2sw4Wt/5jZ2XH8x4uJ8qZfZWY3hd4fYsEtMLO9O12g4zjDTcfFjmOaahWUu46CnLupAJaCNYUIXWhj1cllwFYlY52OkLQJcJ2ZPRP4aSdzhAjc/0raO1Mk2XGcEaYTUetlz7uyqihtaQrAxLw4h+246X3APLELfs+MhdhekcVTB5rGzB6V9GNJu5nZrzuZI0Tgvkgkcr8lSntR9Gzbp5MHOo4zeJS6/iqCMPopbJVpCZncuyRpm7m01LacEruyd6uqlUk6xcAFrkfsAtwg6Qrgj8lJMzss5OYQgTudOGGbGtErjuMMPkF7WnlRijnuPuiBO7Ki7NciJqaiNSdOKE5NmAp08YjJYeNfu7k5JNH7m2Z2cDcPaRJP9HacZijb00poK25clSNWV0ACakfWrWfZkkAesJ60KBd1O0ivJ7umCRYxrqWe6D2AhFhwP5X0JeC/iVyUAHiagOOMGAUuuTRlFULmnEEjVfxDBS0kaTyPMitzKpk9mafgfcY2jE+tcxET7qDsEZIeYLrDzBbA5sAfzWybkPtDBO4xRML2stQ5TxNwnCGn5cu8AeqUx0qTLZWVJq9Qc66w1bTUgilIaE/W4V3Ae4uZPTZ9LOlVwAGh91e6KAcNd1E6Tm8IaUZah358+efWs4RgV2lhtZSEwPY7/XL5uYsSJK01s6B/moXsweXpyX1Ercu/1sH6usIFznH6Q1XD0jKa6gPX6f4bFOwXhu4R1ij/NbnMBa5XSHp16nATYH/ghWb2/JD7Q1yUWwJPB74SHy8mqmryRkkvMrO31Viv4zhDQkvwRfx3ukpJWb81iFyPZb3gsoKWJ0jpQshVYjfBotYebse1P6etA3hCjU7gWfpZnPmeTbZjzdwXNzDT0OwavjL180bgNuDw0JuDoiiBl5nZxvh4M+DrwEuB6/td5cQtOMeZGaosuqL2M9C+lxYUIFLSlidNbvNSplv0pJ/fae+5KvrVLmfP/be391/VvcAt1fhQWHDdEmLBzScqRHNffLwVMM/MHpE0WXyb4zijRF5QStIuJ80iJtpErk305qZ+TiVgs6w9mKPKckvEK8mHSwgSs4b6zzm9QdJHgA8AfwIuJmq+/TYzC3I2hwjcR4BrJX2LqIrJC4APSdoK+EYni3YcZ/SoE1XYJoA5rsVOng9d7r3ljXXBm0leZmb/LGkRsA44AricwE80pB/c5yVdSBSaKeA9ZrY+vvxPna3ZcZyhpiRHDPKtuBCWrj8byO8AnkebqDXUJqfFUi1wmzp9YfP470OAc8zsbknBNxcKnKSnm9lPJe0Xn7o9/ntnSTub2TUdLddxnJEi7abMJkAXiVx2v4zVreEf4yyJ/q7oCp48PxGhpkqFVfaeAxe7/vDfkn5K5KJ8c9zw9KHQmwuDTCStMrNjJF2ec9lmqnyXB5k4zsxRlSdWVM4qjzzxSwtfWtjyUggS8qy3pgtAV7XO8SCT3iFpe+D+OO5jLrCNmf026F5P9HYcJ5RO6lem6SavrYwiF2WTQlcmclrrAtcrJD0T2JsoZQ0AMzsz5N4yF+XzgNsTpZT0d0Q5cL8CTjKzu7tZtOM4w0duea/Mflw2CjJNqGDVJXlWtudbk50OCt2WTs+QdCJwEJHAXUiUkvk9IEjgNim59p/An+OHvAA4JZ70PmBVxyt2HMcpYM4Z0386YXJZSliXpv4QCVPypxt62f/OaWMMeDHwWzNbRpQmUJivn6UsinLTlJV2JLDKzMaBcUnXdrpax3GcLHnJ3ulvsbr1MNvGJxZe/JyFicXZaX+4bICJi16v+FPc2XujpG2AO4E9Q28uFThJm8UVTF4MHBN4n+M4A0LWWumJ9VFjDy5NZYHkVPmsMrdnlrJ9vnTpL2BK+LpmRUPzOFmukrQd8DngauBB4IrQm8uE6hzg25LuIgrR/C6ApKcwXdXEcZwBo7BpZwN7UnWiKIsoKs1VtJ6FqZ+rrLqqIJai69mIzqr8u3aGprbjUGFmb45/XCnpYqIIyutC7y8UODP7oKTLgF2Ar9t0uOUm+L9XHGegCGrz0oDYtXW/zhQpzivdlZxvo8uw/vScZT3asuLVloMHrXl4yycZJ2cMrfUtW3GB6wWKsrqXAHua2cmSdpN0gJkFWXGlrkYza/nPTtKWRP7PQyR9bljCTB1n1Gnra1ZzfLfPLLKySkntgyViW9qfrSSxOi12iRsyz1qrSjDPPbd8uuRucn+x0DkN8xngUeBg4GTgAaJ/TTwv5ObKvTRJTyUqk/JyYHuiOmDnA//Y2Xodx+k1WUur15F/LfPnpRKUVf1IXWsT5jrVQpLglDhtYQnjkWUXF3aeYNGUME0J3fLJfJFLWN5eTz6vS4Hbbz3jQDPbT9KPAMzsHklbhN5clgf3GWAB8AvgEuCNqRqUjuMMCTMR1l5pURYJV6igVSR0T/WvY1roxojcl4nQtVlzK4Fjix+Ztdrq79M5HfCwpE0BA4hLdT0aenOZBXc2sMLMgidzHMfJoy1JuqSm42QmpL+FwEolUwnpOUWh0+11JljE+PJE6Oa0WWzuipxxPgVMADtJ+iBRXtz7Qm8uCzL5fjerknQ6cChwp5k9M+f6QcDXgFvjU+eb2cndPNNxnCEkpxJKS8J2mmWtwle615hqyloUPbmIiSmh44Sa63Z6jpmdLelqolQ1Aa8ys5tC7+9lPtsXgNMoL6nyXTM7tIdrcJxZyZSllHX5NVx1vzFqlPtqOZf6OS+qMrT2ZdrdGNrmJxnnrsrmkbRD6vBOorS1qWuhpSJ7JnBm9h1Ju/dqfsdxpskKWtEXfBJhuDBTnHggBC+TcgDFaQd5lFlqReTVxhyjOAoTcpq1sohRCzOR9ArgVGBTYLWZnZK5PofIeHku8AfgSDO7Lb52PPBG4BHgLWZ2SQdLuIuowenG5JGpa0ZgNZPaAifpzUQvNB5XOemG50v6MbAeeKeZ3VDwzGOIK6ns1OUDHWeUqBK2CRYxvn66p1oSUZgIXRI/mLj6BkLoMuRZZglpEUoLT55VVVXoOSumyfis0CVzd9LQdRiIgzo+DbyUSGSulHSBmd2YGvZG4B4ze4qko4B/A46UtDdwFPAMYB7wDUlPNbNHai7jP4iKLH+fyHr7XioXO/xd6t4j6Tjg6cCTzOywirG7A/9TsAe3DfComT0o6RDgVDPbq+r53i7HcYpJC15a7Io6X/dK0ELz17qlSPCa6lqQZ8Xl9rHT0pFplyPp+UQdY14eHx8PYGYfTo25JB7zv5I2A34LPB54d3pselzdNcZJ3gcBrwUOAL4OfNYFvPfRAAAgAElEQVTMbi27L01tC87MPl33noJ57k/9fKGkz0ja0czuamJ+x5mNTAnWWlgYi9iSpeN9cUXmiSt03hkghOzcbbUmOyDUNTnk7CjpqtTxKjNLusTMB25PXVsHHJi5f2qMmW2UdB/wuPj82sy98ztZYGyxXR7nwB0FvJ8obe1zoXOU5cGVJnKb2cdDH1Iw/87A78zMJB1AVALsD93M6TjONGmx6wVl7tEWd+FxkUsvbVX1SvSK5u2mSHOVsEX5dD00U1Pcw/YNCe34XSUWp3LOZV19RWNC7q1E0lbA4USdbB5PVFxkPzO7vfTGDGUW3GPrLiqNpHOIzMsdJa0DTgQ2BzCzlUT5DG+StJGomPNRnfhYHcfpH0UuUJgWgvH1Yy3VQZI8s4l5DYpeVZubjN6UBauEuiGz5Na0HA3WAbumjp9IFCeRN2Zd7KLcFrg78N4Q7iSy1s4BbiYSyefFjbgxs/NDJqm9BzfT+B6c48wMtcUtXRkklUCdV+qqV9ZdaARmXZFrE7f5W/ZlD277/fe0F1/1/q7nKdszjAXr50S5Z78BrgRelw4CjGMxnmVmx8ZBJq82s9dIegbwJaI9s3nAZcBedYNMJH2BYsvPzOwNIfOE1qL8LPAEM3umpH2Aw8zsA6GLdRxn9Cm1aFZPVwlJR3VOsIhFTLQEwtRtbtoEeRGTi5gYtX23IOI9tRVEJRo3BU43sxsknQxcZWYXAJ8H/kvSzUSW21HxvTdIOg+4kSjE/7gOIigxs6ObeJdNAsZ8DjgeeDh+8HXEL+M4jtMUdXPY8u5P/nRKNvpytiZxm9mFZvZUM3uymX0wPndCLG6Y2UNmdoSZPcXMDjCzW1L3fjC+72lmdtFMvQOERVHONbMroojNKbrNf3McZ1g5a7pi/1TU4tzq2o55rsmEkLD+OsKVNzY0dSBrzeXlveUWa3YGjhCBu0vSk5mu5jwG3NHTVTmO0yjdtsxpa38D00JHlIqQVOxfNC8WhBPaS1nliUzIntvksmqBSotaE3lw6TnWzF3cbs3Nm/5xtOqYDA6S5pjZZNW5wvurgkwk7QmsAv4SuIeoOPISM/tVZ0vuDg8ycZxqqmpRFpEngrniVsbS9n20QhGrioZMzVlEP/fsiqzIssTpJulHkMkgIekaM9uv6lwRIRbcr8zsJXFewiZm9kAnC3UcpzfkVQ1Jf+m3WDbLMhVNoEVkaotZHinLrow6FmVh14Cl9aMuuxHErFXn9IY4T3o+8BhJ+zKdX7cNUwXnqgkRuFslXQx8Gfhm3YU6jtMbyoQtr0ZjEq0ImYofef3XQi2rmnTqJm3rJ5dQ0leuiG7TEJLfcVrs+pPmPat4OXA0UR5duqjIA8B7QicJcVE+BnglUeTkfsD/AOea2ffqrbcZ3EXpOPlUWXJ1aCQXrUErrhHLsi41VEsrcBdlD5C02Mw63uKstODM7E/AecB5krYnaqHwbaL8CMdxBoQWsUgsnpwWNCGiN7msAZHLE4iM6GWFK/0OMyJqaXpkxTrVSFpqZmcBu+eVjQwtFRlUbFnSC4lqgi0kymp/TY21Oo4zQ+TVo1xY4tZLi19TwRstQlkR9NKkqA1i6x8nmGSfbetuJgmpZHIrcC2RFfdPZvbHbh7oOM7MkrX00qKS5LclNCFyeXNMiV6AlReKC9pI8eT47xvN7CudThJiwT073drGcZzRIisM6YjFOamf6xQrLqKsHFep6AWQG2nZy+iPtBC7uDbNIZLeR1RFq6cCt7OkCbwWpePMClr2wVLni8RubMN4sMgVjRvbMN69tVjj/kZC/I9L/SxP9W6Yi4G7gK0kpQ0sERVb3iZkkhCB+xzwT8B/Es18naQvAS5wjjPiVIldOmQ+Kxp5hYqLajv2MqesqGByW6mt1XNyx7WxPK+Ihgtck5jZPwH/JOlrZnZ4p/N4LUrHcYLI27tL79ktYbwlD6+pQsWdVvTPrRWZFbGVmet3BNQGPrGj5Tg1kbQpsFU3c3gtSsdxOiIdoZkEqiSClxa7bhkj3wVaJXzp4s4J04WgY5aTEb2F7aKXR4gQOl1hZo9I2iBpWzO7r5M5vBal4ziN01i4f0FEZ5lLs67FF2TpQbnw3SFP9O4BcW+5BcClwFQEv5m9Jej+0I7e6VqU3WaXd4MLnONUUyUwwxRSn1ehBdojMevu41UJYS0XpwtcT5D0+rzzZvbFoPtDBS7z0F+b2W61b2wAFzjHaadQ0AK7CQy64IWKHDQrdIU937JCd6ILXK+Iy0XuZmY/q3tvUCWTvGd2eJ/jOA2TK24ZEWjLMcupIDIoIlfU6id0T69O2kIRlc1Mk0jK0MhLpyMkvRL4GLAFsIek5wAnm9lhQfe7Bec4w0ubuOVYN1MdBDaUt8qBmRO5hQvoukB0HnX26rrq0D1/y75YRHr2c42LftD9RH1ab7dIuho4GPiWme0bn7vezJ4Vcn+hBSfpeuLIyewl4AkdrNVxnAapctslX+5TX+Rxdb+WnnAZi65fllzaSptcBpN562U6by6kQ3dauMvy87LPgC7FzeklG83svkyaWrBVVuaiPLTjJTmO01Oybry8CMPkS3zqy3tePKBC6HohcnnrLVxnyu2XhPVPzKsnenPOKBa5tLi1CFvW3Zib0O30mZ9Ieh2wqaS9gLcAwSZsRy7KmcRdlM5sJW9vqshaaxOL+Ms6yQ3LisSU9RNbck0JXJEQ51lrRWsuIp3nln6fIDFrGndR9gRJc4H3Ai+LT10CfMDMHgq5v9MgE8dx+kBHogat1shKgDmwfJLx9WMsnreGCRZNdfhO6kA20uQ0u+547aE5bECw5ZR+33GmLT13N44UTzOz9xKJXG1c4BxngCgTNKghatCamFwici0spaN2NWWRnEXi1mkJrjbSLk2WTJ93F+Mo8HFJuxB1FDjXzG6oc3Now9OO8xAcxykmL1CkTNCgxNW2fLJW2HpurcgmxK3JFjWh75NbZSR1by/FzlMFeoaZvUjSzkRNtldJ2gb4cmg3m5BSXVN5CGZWOw+haXwPzhkV6opbx6631XNa9uCq9t+g8z24IrELseJKrdFQ+m21JevsU6L3bNuDSyPpWcA/A0ea2RYh94RYcCcBBwDfAjCzayXt3tEKHccJIs9y65TFJ5wN5Ecf5u27dRNgUtQ8Nd1ep4lEbKBFzPIKK4OH/w87kv4COBI4gqg/3LnAO0LvDxG4vDwEx3GaoiDCME3IF3Xel3zaDVkqbB24JkO4KOk0EHcZSML3E5FbxES7iFeE6+dFT0J72sCauYtZNK/VDZv3D4auLGOn15wBnAO81MzW1705ROC6ykNwHCefbLuZ3FYzcc7aVA5biqJ+a0X5YWUVTHrJRWu76C7Qobgl57L/WMgT1GROt/YGkhcBTwa2l3R3aHpAQsgeXDoPQUR5CO+vepCk04mSxe80s2fmXBdwKnAIsAE42syuqVqw78E5s4GyRO48gkL8a4paJ67KkGjKkARvVgLH5sxV4pbME/zaFlunVpnvwTWKpM2ADwHLgF8DmwBPJLLo3mtmDwfNUyfRO+mwamb3B4x9AfAgcGaBwB0C/AORwB0InGpmB1bN6wLnzDbaUgf6aH0FURE1mVe1BAqEDcr7rmVFr0DwgrsAZAlpdppHn9rlzCKB+wTwWODtZvZAfG4booDHP5nZW0PmqXRRSvoS0X9WjwBXA9tK+riZfbTsPjP7TkUwyuFE4mfAWknbSdrFzLxbuOOkSLsymyLIZRgY7l9mXVZGgrYlpFfQNiaTA1eVKlFHwLxr90xyKPBUS1lgZna/pDcBPwWaEThg73jiJcCFwLuIhK5U4AKYD9yeOl4Xn2sTOEnHAMcA7NTlQx3HKdkXyxG1IgErioQMcgs2FaCRFayVFfP2QrR2WZjzreV0iVmOe9HMHpHUSLHlhM0lbQ68CjjNzB6u84AS8sIyc+c1s1XAKohclA0823FmHZ0mZKcLFyd0JW6Qb2kdS+cuwiJ6ZYXtsnD652OBE3vzmFnMjZL+zszOTJ+UtJTIggsiROD+E7gN+DHwHUlPAir34AJYB+yaOn4iUDsM1HFGnY4jENN0WV0kK3KN5bJlSfbYuhG6XllpCel9wGQP0AWuaY4Dzpf0BiKPoQHPAx4D4cmhlQJnZp8C0nEdv5L0onprzeUCYIWkc4mCTO7z/TfHabb0VSJKTRZSTggRudKIxbL9srpC17SoVQkamaCWZp8+6zGz3wAHSjoYeAaRx+8iM7uszjxlDU+XmtlZkv6xYMjHyyaWdA5wELCjpHVE/8bZPF78SqL9vEOAm4nSBBrs4+s4w0PIXlgnXa5bxOe46v5pVeS5KrumqrTW8vjvIiFMBDAtSE0QIGrptAQXuN5gZt8Evtnp/WUW3Fbx34/tZGIze23FdSMyQx3HSdOAuMG0oCVCV2ptHdd6Tx7Z+5vsip3tU9f2jBMKnlMlgJ1QM9dubMN4o/WlnebwhqeOM8OUuSSbtJi62TNrUsyguFkpTFuTVYEtjZbdqlhbmrx/BGy5FbMmD07SDsCXgd2J4jNeY2b35Ix7PfC++PADZvbF+PxrgfcQ7autB5aa2V2drKWKkDy4JwL/AfxVvKDvAW81s3W9WJDjzGoGwBToSsxW57eoCRE0oCWJfU46oT0uYZYwuQzGaLVQgaDak1WEilov9jWHhHcDl5nZKZLeHR+/Kz0gFsETgf2JdONqSRcADxBVsNrbzO6S9BFgBVFR/8YJiaI8A/gSUTVniP4XPAN4aS8W5DizmrgeZUKT+17pL+na7sZOXICpNj0JRfUzgVoVWtJWXtl7Fbo8nW44nCi+AuCLRJ1m3pUZ83LgUjO7G0DSpcArgDVEASNbSfoDsA1RHEZPCKlFea2ZPafqXL9wF6UzavQqyCShyOWXppbg1RG7HCsu1IJrIaAcWJoyd2xdkQspaj2ELspfEbWfSVgV5xtXr0G618y2Sx3fY2bbZ8a8E9gyaUwq6V+ISmx9TNIYcDrwR+AXwIvM7JHuXiifEAvurji57pz4+LXAH3qxGMeZjeRWFSmw5Lpxi2Xvzea1tTC39TDdzWB8eQ2xy7HiEtbMXczYhvGu3wvaRTwbYJMmt0VPCYVjW35HfYqjvEtNBdTcVSbIkr4B7Jxz6b2B8+cW8oiLhrwJ2Be4hWj763ggqEN3XUIE7g3AacAniHypP4jPOY7TS3JErknS82UtoFLByxO7vC/dpCNALHLj68dYPG8NEywqdlUupati0kVCB5m9upoil0fr/aOVKGBmLym6Jul3Sd1gSbsAd+YMW8e0GxOiQh7fAp4Tz//LeK7ziPbwekJIovevgcN6tQDHcXI6YScWXSbQorFOAhmXX554FlUuyRWHbNJ2RYJ2InJBVtwABN44LVwAvB44Jf77azljLgE+JClxXb6MyFLbEthb0uPN7PdEsRw39WqhhXtwkk7IvRBhZvb+3iypHN+Dc2YjjZTrqqJESPK6jed2CEhELhG4GgnTuXtyJWsJJc9NWbTHmNcVvYxk7Us13p89uPn7G39/VfcTddG/TtLjgPOA3Yh6tR1hZndL2h841syWx+PeQJQOAPBBMzsjPn8sUTeAh4FfEfUC7cm2V5nAvSPn9FbAG4HHmdnWvVhQFS5wjtMMdVvmFDUshZKmpRDcwy0rdN1SR9jSVIlcniBrRZ+CTAZA4IaJoERvSY8lUtw3Ein3v5tZnt+157jAOU7vKBS9iq7ckCNyRQIHbYEnVcnVVZSlCmTX2bLWhGTN8bqKKpi0WZmxy1hrXeAGkVKBi5P1/hFYQpTvcGpexno/cYFznN5RatV1InJZKmpPhrgI6whgJ0nreWtwgRtOylyUHwVeTdSH7dNm9mA/F1aEC5wzG+nLHlwOSfDL1PMrRK4wX66qqHKKuvtgeXRSsquuuLqLcvApE7hHgUlgI62NSEUUZLJN75fXjgucM1soEpW6BCVS55CO7GwR2DoiNwR0Kqhpq65vid4ucLXwYsuOMyBkBQ3yoxc7pci9lkc2bSF0fX0VuawbtCJyM03onl9orty4lrrADSAhid6O48wAZaH5dUlyzgDGlsVClx0Uat1lEtDzWDxvTXMiVyRkWZKmpyvTveEy99ZwlUJ5Qnjr+3my3iDiAuc4A8KU1RSX7kqq6S9ZGglSuoJ+HmVWXjrKsDShOn7mwgUFLsoetfLJJVTYErJNT/OiN+M5x1kyLXbzcsYxfK5Wpx0XOMcZQFpchBnBKzIWknYyecITXOYrR+SqxK0J92kLRWW/QskTtpLntIidM1K4wDnOEJAVvDTZCMtsH7XaZEQuO0/jlltZ8eAyYQsVshD2n5M/pwvfUOMC5zhDTmG0I0zvq9UVunTdy5J7O7beqiri90rYqizB9PWkUHQRLn4Djwuc44wQhWLXidAVpCeUBb2U7lt1I2pNUSSOZVVXishUP3EGDxc4xxlRcsUuIAISivfYalcGaVrUqgQoT2xC+qd1YxU205/N6QEucI4zC2gJFikQuRBR61rQoDeWWpkV1YmF5aI1ErjAOc5sJNCSCxa3JoXtjovaQ/4HlX64VZ2OcYFznJqE1IXMqwQycBR0DJ9clsqbK+jkDRnBS6ykMqE7lnoil2WXhfn3VwWDNIEL2VDiAuc4FZTVhCzKLxsS+yM3+CT9TknVE6BF7CZY1FLuakrsqoSujshlqSN6/SJvTc7A4ALnOBlCK+cvYgKOi65lm3T2bQenRvHkynlyXJYdi12Z0JWJXF3BKBvfhJvTBWyocYFznB7QizJWudZi0T5aU8JXQIugF4hdi9BBq9glUYtZoUtEqUpY+rVHF/qcO3q7DKczvJuA41TQVNuaTggusVVEmdB1UOWk7N2D0wm6LcVVRJOVTUJIi/f8LfvTTWCL/Y0dG+gmcMfs6CbgAuc4Nel1+ara/dv6XMg+5F07FrsqoSsSsbqpAB1WKCnq9r1U4y5wA4gLnON0SV4zUMgXgjVzF7ft10GxqIVEYxZGdfZI+OqIebaUV2naQUgkZEB/tyqqcvmyc+X1i8t+hn1reOoCV4ue7sFJegVwKrApsNrMTslcPxr4KPCb+NRpZra6l2tynKbJq/wPcdHjjOC1NR2F2oIWhLcnc5zeWXCSNgV+DrwUWAdcCbzWzG5MjTka2N/MVoTO6xacM2y0WXgNClqb9ZYRtiY7godQ1pC1Y9dkU27JDihySaYZ2zDuFtyA0ksL7gDgZjO7BUDSucDhwI2ldznOiNFSE5LmrLQicUuLWiI4oc07q1x9HTcBTQta3YCSbIX/vDlDxS40ITyeL/2+ye8mLeLTndKLG9E6M0cvBW4+cHvqeB1wYM64xZJeQGTtvd3Mbs8Z4zgjQU/ELSVs6Vy9qS/n1XOKRSLDOEuaWWCaPEGrk1+WDtUPEbsmyBHPrLgvnrcmJXYucINILwVOOeey/tD/Bs4xs0lJxwJfBA5um0g6BjgGYKemV+k4Q0ZIInqhuMFwlZ0qy0MLFO3c8VUEWIqtguebnoNIL/fgng+cZGYvj4+PBzCzDxeM3xS428y2LZvX9+AcJ7zayvj6se7cg840ZXuBngc3kPTSgrsS2EvSHkRRkkcBr0sPkLSLmSU1AA4DburhehxnZJlzxnSU5pq5i6cDIeYBJ6RrRdK7ROthIsRFmrUes7+jRPC8tc7A0tM8OEmHAJ8kShM43cw+KOlk4Cozu0DSh4mEbSNwN/AmM/tp2ZxuwTlOde5d7SaloyZ6vawhmec27ZNF5BZcPTzR23GGlJAE8zKhSwgWvG7opVgOREHkQ1zgBhAXOMdpgJAecQm9ThOAYpFLU0vwuqHpPcCmBK2qkHKt57jADSIucI4TSKWI1QmkayDZOzTJG8ISvcuStLuhMMG7m+andSgTsqLAkdqpDS5wg4gLnOMUkCtoJSJWt+ByE+W66ohcmiYFL69WY94cpbUns4LSjagVCVpoRZSQKitt63OBG0S8H5zjFHDR2mqrraigch4tjVIBjksV7V02LXjpr+fGalPWpI411/ZenZDtD9evfm8QCVrdsl+hfeucGcUtOMcJJHTPK6Fu/cd0hfop665GP7dO1hBcOzJFncr9ufOEBLD0I4KzLDm87vM9inIgcYFznA4JbZPTSaHjXLHLUOc5tYWsSIS6KXDcaWTmMKQruMANJC5wjtMldcP165LXP66nYtZ009FeJkIPivi5wA0kvgfnOANO3TB/qBnqv3yyVYSOpVw4Cq/NQEWPPLGtI3pVe2j93At0GscFznF6QJ71VtS6pu6eVt4cXZO1vvJKelUJR1Epq36Tfm7VmndZWC5y2WsueEjaAfgysDtwG/AaM7snZ9zFwALge2Z2aOq8gA8ARwCPAJ81641jzgXOcZqibuHjmLYWNX1o5BlE2zpq5rAVjemV8KXXm/yeQ8SuSuTS5I2bfaL3buAyMztF0rvj43fljPsoMBf4+8z5o4Fdgaeb2aOSetYkxvfgHKcLyqr65wpbpy1bimhADEMtyNoduRMxqCMAdcSvmyan3faoa2P25MFJ+hlwkJndIWkX4Ftm9rSCsQcB78xYcFcArzOzmzt5fh3cgnOcDkmLW6WwwfSXaugXf5kYhlayDxCB8fVjHblJp6hK0q5j9aykWQuv15X+p/LhevuYHrCjpLRSrjKzVYH3PiHpAhOLXF0L7MnAkZIWAb8H3mJmv6g5RxAucI5Tg4ULKO+gfS/FX6pJ8EY3Lq26X/4FzTqz5O3pJaLXeAWSTkpn5ZH3e84GzKQp+gdDXcstu/5jgRPrTdExD9/XVHL5XWUWnKRvADvnXHpvA8+eAzxkZvtLejVwOvA3Dczbhguc45SQtdImyRG1OlbCTAVeQO3ctsoE7U5C9OuW0apLnTXWEYo8UUtYPtk/gesTZvaSomuSfpf08oxdlHfWnH4dkOS+TAAFmZ7d4wLnOCny9tRqWWnDSJElVDamkz2sXotbQpXwhgpbmajB4AQD9Z8LgNcDp8R/f63m/V8FDiay3F4I/LzR1aXwIBPHyRAkcn0kuz/Wl/5tZXSTXJ0nZmVCUfVevexIUEfg5m/ZnyAT7WXQxDdg50Exkh4HnAfsBvwaOMLM7pa0P3CsmS2Px30XeDqwNfAH4I1mdomk7YCz4/sfjO/5cdevlLdWFzjHKacomAS6T7YuC+5IFy9OVzPJe3aQ6Pa7VFZdMSuj7r4aNLNXlWd15gndLBK4YcIFznFqkg40gfodtLMUCVluO52c5xY9s1D0QoWurrDVdOGFRG6WpiZAD8L9cwgJijmxT6W6XOBq4QLnOF1SpxblBIumBK1SzChul5P3zLLqKRAgFlAcFRliyUChqHWVhkCNpqm9al9T2f3bBW4Q8SATx2mKTPuaRLQml6XEbG5+8eTJVD84ljIlcgsXVPSEy2nAmle7MnjfMCtaKxc2EgSSPL/rsmRVHcHrVCWpQydJ686M4xac43RAkdVWRt2O3wl57XLK3JONBqEkFlnd+5uKMKwStDL62oy0T5VM3IKrhVtwjhNAWbPTUNIWXR1qiVm30ZSxMLVZWifUfFYTUZ3dtsJJW1veeXtW4gLnOBnaxAw6ErQiQoSuoz20EEEo2TdLi1o68GXq3LyJ1sCZuqKXphd93MoKK7vYzUpc4ByHAlFLcxZtIldliRV14s7eWxV52bWw3XFR0N5R3ZqUXVU66UZkuqlj6eI2q/A9OGfWUyluReREL4bQSSoB1NxbyxOXmt0JShPMuy3b1aTQVEY49kPUfA9uEHELzpnV1BG3bDRj8rU6hzCRS3ca6IRcwSkqqdVp5GNKuMaXNyhovcStMqcAFzhnVpIVttJQ/AIuWhvPc9Z0K9Cs0JV18w51B+bth02wqFrwug306KSHXQguSE6fcIFzRp4QK60y36yKeI8uNFKym8Tnyn26TuiXRebi5vQRFzhnaOl47yxgvq7EjmmhG1s2neANwLz6c6Wrn/RE3CAnwbv7KadwUXNmCA8ycYaKbKX/Fs7KOdcAQeWyEnLWVeS2TNPp3lypuDXVYaATsRtGUeuqCooHmQwibsE5A01IzcWp0lfx+aK6jk2soRPLLuu2zJbqWjN3ceE+Wxl9a9uTWHdN9VkbJLIRmJ4vN1L0VOAkvQI4FdgUWG1mp2SuzwHOBJ5L1C/oSDO7rZdrcoaLqUAOyN3nKixYnIxvmFK3aEkyeN6eXJ4lB51bcz2l121pBo1RfKdZSM8ETtKmwKeBlxK1KL9S0gVmdmNq2BuBe8zsKZKOAv4NOLJXa3KGkxarKSV4c9LJ1zXELD1fR/t4BUIWImJ9E69u3JO9bCI6aIzCOziF9NKCOwC42cxuAZB0LnA4kBa4w4GT4p/XAKdJkg3bxqDTV7KC1xcK2uAkVO2r5bkTu20hs3jems7clN0EkLggOENELwVuPnB76ngdcGDRGDPbKOk+4HHAXT1cl+NM0eICzRLQ1BTarbKg9i7HwjhLop9Lqod0RKd1KstwYXOGkJ5FUUo6Ani5mS2Pj/8WOMDM/iE15oZ4zLr4+JfxmD9k5joGOCY+fCbwk54sevDYkdkj9rPlXWfLe8Lsetenmdlje/0QSRcT/V675S4ze0UD8ww0vbTg1gG7po6fCKwvGLNO0mbAtsDd2YnMbBWwCkDSVbMhvBX8XUeR2fKeMPvetR/PmQ2i1CSb9HDuK4G9JO0haQvgKOCCzJgLgNfHP48B3/T9N8dxHKcJembBxXtqK4BLiNIETjezGySdDFxlZhcAnwf+S9LNRJbbUb1aj+M4jjO76GkenJldCFyYOXdC6ueHgCNqTruqgaUNC/6uo8dseU/wd3VmmKEr1eU4juM4IfRyD85xHMdxZoyBFjhJu0q6XNJNkm6Q9Nb4/A6SLpX0i/jv7Wd6rU0gaVNJP5L0P/HxHpJ+GL/nl+NgnaFH0naS1kj6afzZPn+EP9O3x//t/kTSOZK2HJXPVdLpku6U9JPUuVlXmaoAAAgwSURBVNzPURGfknSzpOsk7TdzK69Pwbt+NP5v+DpJE5K2S107Pn7Xn0l6+cys2hlogQM2Au8ws78AFgDHSdobeDdwmZntBVwWH48CbwVuSh3/G/CJ+D3vISptNgqcClxsZk8Hnk30ziP3mUqaD7wF2N/MnkkUbJWUpBuFz/ULQDZsvehzXAjsFf85Bvhsn9bYFF+g/V0vBZ5pZvsAPweOB4i/o44CnhHf85m4dKHTZwZa4MzsDjO7Jv75AaIvwvlEJb6+GA/7IvCqmVlhc0h6IvB/gNXxsYCDiUqYwei85zbAC4giaDGzP5vZvYzgZxqzGfCYOM9zLnAHI/K5mtl3aM9bLfocDwfOtIi1wHaSdunPSrsn713N7OtmtjE+XEuU6wvRu55rZpNmditwM1HpQqfPDLTApZG0O7Av8EPgCWZ2B0QiCOw0cytrjE8C/ww8Gh8/Drg39T/QOiJxH3b2BH4PnBG7Y1dL2ooR/EzN7DfAx4BfEwnbfcDVjObnmlD0OeaV7hul934DkNQzG/V3HRqGQuAkbQ2MA28zs/tnej1NI+lQ4E4zuzp9OmfoKIS8bgbsB3zWzPYF/sgIuCPziPefDgf2IOrlvRWRqy7LKHyuVYzqf89Iei/RdsrZyamcYSPxrsPGwAucpM2JxO1sMzs/Pv27xL0R/33nTK2vIf4KOEzSbcC5RC6sTxK5cZJcxbxSZ8PIOmCdmf0wPl5DJHij9pkCvAS41cx+b2YPA+cDf8lofq4JRZ9jSOm+oUPS64FDgSWpKkwj+a7DyEALXLwP9XngJjP7eOpSusTX64Gv9XttTWJmx5vZE81sd6LN6W+a2RLgcqISZjAC7wlgZr8Fbpf0tPjUi4laKI3UZxrza2CBpLnxf8vJu47c55qi6HO8APi7OJpyAXBf4socVhQ1dH4XcJiZbUhdugA4StIcSXsQBdZcMRNrnO0MdKK3pL8Gvgtcz/Te1HuI9uHOA3Yj+hI5wszaijQPI5IOAt5pZodK2pPIotsB+BGw1Mwmy+4fBiQ9hyiYZgvgFmAZ0T+2Ru4zlfSvRE18NxJ9hsuJ9mOG/nOVdA5wEFF1+98BJwJfJedzjAX+NKKowg3AMjPrS4HiJih41+OBOUDS/WStmR0bj38v0b7cRqKtFe83NAMMtMA5juM4TqcMtIvScRzHcTrFBc5xHMcZSVzgHMdxnJHEBc5xHMcZSVzgHMdxnJHEBc5Bkkn699TxOyWd1NDcX5A0Vj2y6+ccEXcmuDxzfndJf4rLgt0k6Yo4ObdqvudIOqTmGpJnXSvpRkkrJRX+PyZpnqQ1RddT495TZx2Ze0+StEHSTqlzD3Y6n+MMEy5wDsAk8GpJO870QtLUrMD+RuDNZvainGu/NLN9464URwFvl7SsYr7nALUELvWs5wD7AHtTUkjZzNabWYj4dyxwMXcB7+hyDscZOlzgHIiSUVcBb89eyFpgyb/+JR0k6duSzpP0c0mnSFoSW0jXS3pyapqXSPpuPO7Q+P5N435aV8b9tP4+Ne/lkr5ElOCfXc9r4/l/Iunf4nMnAH8NrJT00bIXNbNbgH8kamODpAMk/SC28H4g6WmK+rOdDBwZW2NH5o2reM5G4AfAU+LqHR+N13y9pCPjZ++uuL+YpKMlnS/pYkW91D4Snz+FqBvBtZLOlrSVpP8n6cfxfEeWrSPm9PhddggY6zgjw2bVQ5xZwqeB65Iv1kCeDfwFURuRW4DVZnaAosa0/wC8LR63O/BC4MnA5ZKeAvwdUbmm50maA3xf0tfj8QcQ9dm6Nf0wSfOIeqk9l6iP2tclvcrMTpZ0MFEFmJDqGNcAT49//inwAjPbKOklwIfMbHEsmvub2Yr42dtkxwGLix4gaS5Raa4TgFcTWYTPJqqEcaWk7+Tc9hyijhmTwM8k/YeZvVvSitgqRNJiYL2Z/Z/4eNuA932QSOTeSlSBw3FmBS5wDgBmdr+kM4ksmz8F3nZlUk9Q0i+BRKCuB9KuwvPM7FHgF5JuIRKXlwH7pKzDbYlq9v0ZuCIrbjHPA75lZr+Pn3k2UW+5rwauNyFd7X1b4IuS9iKq+L55wT2h454s6dp4zNfM7CJJnwDOMbNHiIoRfzt+l+sy915mZvfF73Yj8CRa265A9Lv9WGy9/o+ZfTfgfQE+BVyr1F6r44w6LnBOmk8SWTdnpM5tJHZlSxJR/ciEdP3ER1PHj9L631a2HpwRicw/mNkl6QuKanH+sWB9eW1IOmFfpjunvx+43MwWKeo5+K2Ce0LHJXtwaULXnf59PkLO/59m9nNJzyXaH/ywpK+b2clVE5vZvbHb982Ba3Gcocf34Jwp4uLG5xEFbCTcRuQShKi3WZHlUsYRkjaJ9+X2BH4GXAK8SVE7JCQ9VVHj0zJ+CLxQ0o5xAMprgW/XWUgsTh8D/iM+tS3wm/jno1NDHwAemzouGhfCd4j2wDaV9Hgiq7NOdfmHU7+necAGMzuL6D32i89/WNKiink+Dvw9/g9bZ5bgAudk+XeifaKEzxGJyhXAgRRbV2X8jEiILgKONbOHiLoJ3AhcEwda/CcVX7yxO/R4onYzPwauMbOQVjNPjoNDbiIS8P8ws8RK/QiRJfR9IB21eTmwdxJkUjIuhAkid+SPgW8C/xy3DQplFdH+6NnAs4ArYjfoe4EPxGOeBZTOaWZ3xWuZU2/5jjOceDcBxxkBJF1iZi+f6XU4ziDhAuc4juOMJO6idBzHcUYSFzjHcRxnJHGBcxzHcUYSFzjHcRxnJHGBcxzHcUYSFzjHcRxnJHGBcxzHcUaS/w8DoBl91Jx2TgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.cm as cm\n",
    "diff_sigmas = np.loadtxt(open(\"diff_sigmas.csv\", \"rb\"), delimiter=\",\", skiprows=0)\n",
    "origin = 'lower'\n",
    "fig1, ax2 = plt.subplots(constrained_layout=True)\n",
    "levels = np.arange(-0.2, 0.2+0.01, 0.04) #[-0.2, -0.1, 0, 0.1, 0.2]\n",
    "#diff_sigmas = np.clip(diff_sigmas, -0.2, 0.2)\n",
    "c = ('#ff0000', '#ffff00', '#0000FF', '0.6', 'c', 'm')\n",
    "cmap = cm.get_cmap('jet') #rainbow\n",
    "CS = ax2.contourf(N_coords, sigma_coords, diff_sigmas, levels=levels, origin=origin, cmap=cmap)\n",
    "#CS = ax2.pcolor(diff_sigmas)\n",
    "#ax2.imshow(diff_sigmas, origin=origin, cmap=cmap)\n",
    "#CS2 = ax2.contourf(CS, levels= [-0.2, -0.1, 0, 0.1, 0.2], origin=origin, cmap=cmap)\n",
    "ax2.set_title('Impact of Sigma^2')\n",
    "ax2.set_xlabel('Number of Data Points, N')\n",
    "ax2.set_ylabel('Noise Level, \\Sigma^2')\n",
    "\n",
    "# Make a colorbar for the ContourSet returned by the contourf call.\n",
    "cbar = fig1.colorbar(CS)\n",
    "cbar.ax.set_ylabel('Overfit Measure')\n",
    "CS.cmap.set_under('blue')\n",
    "CS.cmap.set_over('red')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "diff_Qfs_raw = np.loadtxt(open(\"diff_Qfs_raw.csv\", \"rb\"), delimiter=\",\", skiprows=0)\n",
    "diff_Qfs = np.clip(diff_Qfs_raw, -0.2, 0.2)\n",
    "\n",
    "Qf_coords, N_coords = np.meshgrid(Qfs, Ns) #N is fixed on each row\n",
    "origin = 'lower'\n",
    "fig1, ax2 = plt.subplots(constrained_layout=True)\n",
    "levels = np.arange(-0.2, 0.2+0.01, 0.045)\n",
    "cmap = cm.get_cmap('jet')\n",
    "CS = ax2.contourf(N_coords, Qf_coords, diff_Qfs, levels=levels, origin=origin, cmap=cmap)\n",
    "ax2.set_title('Impact of Qf')\n",
    "ax2.set_xlabel('Number of Data Points, N')\n",
    "ax2.set_ylabel('Target Complexity, Qf')\n",
    "\n",
    "# Make a colorbar for the ContourSet returned by the contourf call.\n",
    "cbar = fig1.colorbar(CS)\n",
    "cbar.ax.set_ylabel('Overfit Measure')\n",
    "CS.cmap.set_under('blue')\n",
    "CS.cmap.set_over('red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.4 (d)\n",
    "\n",
    "The course slides of lecture 11 have better picture quality, so we look at them instead.\n",
    "\n",
    "* Impact of $\\sigma^2$, target function $Q_f = 20$\n",
    "  * When $N \\lt 80$, the overfit of $\\mathcal{H}_{10}$ is much larger than $\\mathcal{H}_2$, all over the maximum drawing limit $0.2$, regardless of the noise level. That's probably because the target function is more complex than both hypothesis sets, so the deterministic noises are large for both of them. However hypothesises in $\\mathcal{H}_{10}$ are more susceptible to noise than the simpler ones because they have more ways to go astray. \n",
    "  * For a given noise level, the overfit decreases with increasing $N$. The more data we have, the smaller the expected out-of-sample errors (by learning curve, through either through VC analysis or bias-variance analysis, the generalization error and variance both decrease with increasing $N$). However, as $N$ increases, more complex model,  $\\mathcal{H}_{10}$, is able to achieve a lower expected out-of-sample error than a simpler model, because the latter lack the degrees of freedom to fit more data. So the overfit decreases with increasing $N$. \n",
    "  * The larger the noise level $\\sigma$, the larger the overfit. This is similar with deterministic noise, more complex model has more degrees to mess up with noise, thus the larger overfit. \n",
    "  * We see that large $N$ decrease overfit, while large $\\sigma^2$ increases overfit. From the graph, it looks like the decrease and increase of overfit are kind of linear from these two factors. And the slopes are larger with larger $\\sigma^2$ and smaller with larger $N$. This can be seen from problem 3.11, where for a noisy target, the expected out-of-sample error is $E_{out} = \\sigma^2+\\frac{\\sigma^2}{N}trace\\left(\\Sigma(\\frac{1}{N}Z^TZ)^{-1}\\right)$, note $Z$ is the polynomial transformation of original data $X$. So for a given $\\sigma$ and $N$, we have the $overfit = \\frac{\\sigma^2}{N}\\left[trace\\left(\\Sigma_{10}(\\frac{1}{N}Z_{10}^TZ_{10})^{-1}\\right) - trace\\left(\\Sigma_2(\\frac{1}{N}Z_2^TZ_2)^{-1}\\right)\\right]$, where for very large $N$, the subtraction terms approaches 8, which is $d_{10} - d_{2}$.\n",
    "  \n",
    "* Impact of $Q_f$, $\\sigma^2 = 0.1$\n",
    "  * When $N \\le 80$ and $Q_f \\ge 10$, the overfit is large. This indicates that the deterministic noise overwhelms complex model than in simple model. \n",
    "  * When $Q_f \\le 10$, the deterministic noise is small. Also the noise $\\sigma^2 = 0.1$ is small as well. We are getting noiseless data from simple target function. As we increase $N$, the overfit decreases. When $Q_f$ is 1 or 2, both models can fit the data well without much overfit, and achieve similar level of expected out-of-sample errors, so the overfit goes to zero (the green area at the bottom). \n",
    "  * When $Q_f \\gt 10$, the deterministic noise is large, as $N$ increases the overfit decreases. \n",
    "  * For a relatively large $N$, e.g. $N=100$, it takes a lot of increase of $Q_f$ to increase the overfit. This is because the deterministic noise increases slowly once $Q_f$ passed some level, such as 20. This might because the polynomials are not much different from 20 to 100 in the domain of $[-1,1]$.\n",
    "\n",
    "#### Problem 4.4 (e)\n",
    "\n",
    "We take the average over many experiments to average over many data sets of size $N$, so we get better estimation of the expectation of out-of-sample errors, because the hypothesis we use to compute the out-of-sample errors depend on data used in training. \n",
    "\n",
    "If we use the variance (one-sigma rule) to select an acceptable number of experiments to average over, we obtain 20 as an appropriate number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Working with num_exps =  20\n",
      "Working with num_exps =  40\n",
      "Working with num_exps =  60\n",
      "Working with num_exps =  80\n",
      "Working with num_exps =  100\n",
      "Working with num_exps =  120\n",
      "Working with num_exps =  140\n",
      "Working with num_exps =  160\n",
      "Working with num_exps =  180\n",
      "Working with num_exps =  200\n",
      "Working with num_exps =  220\n",
      "Working with num_exps =  240\n",
      "Working with num_exps =  260\n",
      "Working with num_exps =  280\n",
      "Working with num_exps =  300\n",
      "Working with num_exps =  320\n",
      "Working with num_exps =  340\n",
      "Working with num_exps =  360\n",
      "Working with num_exps =  380\n",
      "Working with num_exps =  400\n",
      "Working with num_exps =  420\n",
      "Working with num_exps =  440\n",
      "Working with num_exps =  460\n",
      "Working with num_exps =  480\n",
      "Working with num_exps =  500\n",
      "Working with num_exps =  520\n",
      "Working with num_exps =  540\n",
      "Working with num_exps =  560\n",
      "Working with num_exps =  580\n",
      "Working with num_exps =  600\n",
      "Working with num_exps =  620\n",
      "Working with num_exps =  640\n",
      "Working with num_exps =  660\n",
      "Working with num_exps =  680\n",
      "Working with num_exps =  700\n",
      "Working with num_exps =  720\n",
      "Working with num_exps =  740\n",
      "Working with num_exps =  760\n",
      "Working with num_exps =  780\n",
      "Working with num_exps =  800\n",
      "Working with num_exps =  820\n",
      "Working with num_exps =  840\n",
      "Working with num_exps =  860\n",
      "Working with num_exps =  880\n",
      "Working with num_exps =  900\n",
      "Working with num_exps =  920\n",
      "Working with num_exps =  940\n",
      "Working with num_exps =  960\n",
      "Working with num_exps =  980\n",
      "Working with num_exps =  1000\n",
      "Working with num_exps =  1500\n",
      "Working with num_exps =  2000\n",
      "Working with num_exps =  2500\n",
      "Working with num_exps =  3000\n",
      "Working with num_exps =  3500\n",
      "Working with num_exps =  4000\n",
      "Working with num_exps =  4500\n",
      "Working with num_exps =  5000\n",
      "Working with num_exps =  5500\n",
      "Working with num_exps =  6000\n",
      "Working with num_exps =  6500\n",
      "Working with num_exps =  7000\n",
      "Working with num_exps =  7500\n",
      "Working with num_exps =  8000\n",
      "Working with num_exps =  8500\n",
      "Working with num_exps =  9000\n",
      "Working with num_exps =  9500\n",
      "done!\n"
     ]
    }
   ],
   "source": [
    "# Use variance to select the number of experiements\n",
    "Qf = 10\n",
    "sigma_square = 0.25\n",
    "test_N = 100\n",
    "N = 100\n",
    "\n",
    "avg_Eout2s, avg_Eout10s = [], []\n",
    "std_Eout2s, std_Eout10s = [], []\n",
    "\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "num_expp1 = np.arange(20,1000, 20)\n",
    "num_expp2 = np.arange(1000,10000, 500)\n",
    "num_expp = np.concatenate([num_expp1, num_expp2])\n",
    "for num_exps in num_expp:\n",
    "    if num_exps == 0:\n",
    "        continue\n",
    "    print('Working with num_exps = ', num_exps)\n",
    "    avg_Eout2, avg_Eout10, _, _, std_Eout2, std_Eout10 = calc_avg_Eouts(num_exps, Qf, N, sigma_square, normalized_aqs, test_xs, test_ys, False, True)\n",
    "    avg_Eout2s.append(avg_Eout2)\n",
    "    avg_Eout10s.append(avg_Eout10)\n",
    "    std_Eout2s.append(std_Eout2)\n",
    "    std_Eout10s.append(std_Eout10)\n",
    "    \n",
    "#np.savetxt(\"overfit_mean_diff_raw.csv\", mean_diffs, delimiter=\",\", fmt='%12.8f')    \n",
    "#np.savetxt(\"overfit_std_diff_raw.csv\", std_diffs, delimiter=\",\", fmt='%12.8f')    \n",
    "print('done!')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The optimal number of experiments:  20\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "min_id = np.argmin(avg_Eout10s)\n",
    "min_Eout = avg_Eout10s[min_id]\n",
    "lb = min_Eout - std_Eout10s[min_id]\n",
    "ub = min_Eout + std_Eout10s[min_id]\n",
    "best_id = None\n",
    "for idx in np.arange(min_id+1):\n",
    "    if avg_Eout10s[idx] >= lb and avg_Eout10s[idx] <= ub:\n",
    "        best_id = idx\n",
    "        print('The optimal number of experiments: ', num_expp[idx])\n",
    "        break\n",
    "        \n",
    "plt.errorbar(num_expp, avg_Eout10s, yerr=np.array(std_Eout10s), label='Expected Out-of-Sample Error H10')\n",
    "plt.legend()\n",
    "#plt.ylim(1.2,1.6)\n",
    "plt.title(\"Mean overfit vs. number of experiments\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.4 (f) TODO"
   ]
  },
  {
   "cell_type": "markdown",
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    "#### Problem 4.5\n",
    "\n",
    "If $\\lambda \\lt 0$, since $w^Tw$ is positive, to minimize the $E_{aug}(w)$, the learning algorithm will try to increase $|w|$ as much as possible. Then this corresponds to a soft order constraint of $w^Tw =\\infty$.\n",
    "\n",
    "#### Problem 4.6\n",
    "\n",
    "For $\\Gamma = I$ and $\\lambda \\gt 0$. \n",
    "* (a) Assume we have $\\|w_{reg}\\| \\gt \\|w_{lin}\\|$, then $w^T_{reg}w_{reg}=\\|w_{reg}\\|^2 \\gt \\|w_{lin}\\|^2 =w^T_{lin}w_{lin} $\n",
    "\n",
    "\\begin{align*}\n",
    "E_{aug}(w_{reg}) &= E_{in}(w_{reg}) + \\lambda w^T_{reg}w_{reg}\\\\\n",
    "&\\gt E_{in}(w_{reg}) + \\lambda w^T_{lin}w_{lin}\\\\\n",
    "&\\ge E_{in}(w_{lin}) + \\lambda w^T_{lin}w_{lin}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The last step is valid because $w_{lin}$ minimizes $E_{in}(w)$, so we have $E_{in}(w_{lin}) \\le E_{in}(w_{reg})$.\n",
    "\n",
    "so we have $E_{aug}(w_{reg}) \\gt E_{aug}(w_{lin})$. This is contradictary though because $w_{reg}$ is assumed to minimize $E_{aug}(w)$.\n",
    "\n",
    "So we must have $\\|w_{reg}\\| \\le \\|w_{lin}\\|$\n",
    "\n",
    "* (b) For linear models, we have $w_{reg} = \\left(Z^TZ+\\lambda I\\right)^{-1}Z^Ty$, then \n",
    "\n",
    "\\begin{align*}\n",
    "w^T_{reg}w_{reg} &= y^TZ\\left(Z^TZ+\\lambda I\\right)^{-T} \\left(Z^TZ+\\lambda I\\right)^{-1}Z^Ty\\\\\n",
    "&= y^TZ\\left(Z^TZ+\\lambda I\\right)^{-1} \\left(Z^TZ+\\lambda I\\right)^{-1}Z^Ty\\\\\n",
    "&= y^TZ\\left(Z^TZ+\\lambda I\\right)^{-2} Z^Ty\\\\\n",
    "&= u^T\\left(Z^TZ+\\lambda I\\right)^{-2} u\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $u=Z^Ty$.\n",
    "\n",
    "Now we show that $Z^TZ+\\lambda I$ has the same eigenvectors with correspondingly larger eigenvalues as $Z^TZ$. \n",
    "\n",
    "Assume $v$ is an eigenvector of $Z^TZ$ and $\\gamma$ is the corresponding eigenvalue, then we have \n",
    "$Z^TZv=\\gamma v$.\n",
    "\n",
    "\\begin{align*}\n",
    "(Z^TZ+\\lambda I)v &= Z^TZv + lambda I v\\\\\n",
    "&= \\gamma v + \\lambda v\\\\\n",
    "&= (\\gamma + \\lambda)v\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So $v$ is also an eigenvector of $Z^Z+\\lambda I$ and the corresponding eigenvalue is $\\gamma + \\lambda \\gt \\gamma$ since $\\lambda \\gt 0$. \n",
    "\n",
    "Also note, for any matrix $A$, the eigenvectors of $A^{-1}$ are the same as the eigenvectors of $A$, as $A^{-1}v = \\lambda^{-1}v$. The eigenvalues of $A^{-1}$ are inverse of eigenvalues of $A$, i.e. $\\lambda^{-1}$. Similarly, the eigenvectors of $A^{-2}$ are the same as the eigenvectors of $A$, but the eigenvalues of $A^{-2}$ are to the $-2$ power of eigenvalues of $A$, i.e. $\\lambda^{-2}$.\n",
    "\n",
    "Suppose the $v_1,v_2,\\dots,v_{d+1}$ are the eigenvectors of $Z^TZ$ with corresponding eigenvalues of $\\lambda_1, \\lambda_2, \\dots, \\lambda_{d+1}$, then write $u$ in the eigenbasis of these vectors, we have $u = \\sum^{d+1}_{k=1}c_kv_k$.\n",
    "\n",
    "Let $A=Z^TZ+\\lambda I$, then we have\n",
    "\n",
    "\\begin{align*}\n",
    "w^T_{reg}w_{reg} &= u^TA^{-2}u \\\\\n",
    "&= u^TA^{-2}(\\sum^{d+1}_{k=1}c_kv_k)\\\\\n",
    "&= u^T(\\sum^{d+1}_{k=1}c_kA^{-2}v_k)\\\\\n",
    "&= u^T(\\sum^{d+1}_{k=1}c_k\\lambda^{-2}_kv_k)\\\\\n",
    "&= (\\sum^{d+1}_{k=1}c_kv_k)(\\sum^{d+1}_{k=1}c_k\\lambda^{-2}_kv_k)\\\\\n",
    "&=\\sum^{d+1}_{k=1}\\lambda^{-2}_kc^2_k\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The second to last equation is because the eigenvectors are orthogonal to each other. \n",
    "\n",
    "The $\\lambda_k$s in the equation are eigenvalues of matrix $Z^TZ+\\lambda I$, the corresponding matrix for $w_{lin}$ without weight decay is $Z^TZ$, and $w_{lin}^Tw_{lin} = u^T(Z^TZ)^{-2}u = \\sum^{d+1}_{k=1}(\\lambda^s_k)^{-2}c^2_k$. Assume the eigenvalues of matrix  $Z^TZ$ are $\\lambda^s_1,\\lambda^s_2,\\dots,\\lambda^s_{d+1}$, where $\\lambda^s_k \\le \\lambda_k$ as proved above. So \n",
    "we have $w^T_{reg}w_{reg} \\le w_{lin}^Tw_{lin}$ and achieve the equality when $\\lambda = 0$.\n",
    "\n",
    "#### Problem 4.7\n",
    "\n",
    "For a matrix $Z$ that is $n\\times d$, let the SVD of $Z=U\\Gamma V^T$, where $\\Gamma$ is $d\\times d$ square diagnoal matrix, and $r$ of its diagonal elements are positive with the rest zeroes. $U$ is $n \\times d$ and $V$ is $d \\times d$. Also we have $U^TU=I_{d\\times d}$ and $V^TV = VV^T = I_{d\\times d}$. Note that $UU^T$ is not identity matrix. Let $Z^TZ$ have eigenvalues $\\sigma^2_1, \\sigma^2_2, \\dots, \\sigma^2_d$, define $a=U^Ty$, so we have\n",
    "\n",
    "\\begin{align*}\n",
    "Z^TZ &= V\\Gamma U^T U\\Gamma V^T\\\\\n",
    "&= V\\Gamma^2V^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So the eigenvalues in $\\sigma^2_1, \\sigma^2_2, \\dots, \\sigma^2_d$ (there might be zeroes) are the elements on the diagonal of matrix $\\Gamma^2$. \n",
    "\n",
    "\\begin{align*}\n",
    "H(\\lambda) &= Z(Z^TZ+\\lambda I)^{-1}Z^T \\\\\n",
    "&= U\\Gamma V^T \\left(V\\Gamma^2V^T + \\lambda VV^T\\right)^{-1} V\\Gamma U^T\\\\\n",
    "&= U\\Gamma V^T \\left(V\\left(\\Gamma^2 + \\lambda I \\right)V^T\\right)^{-1} V\\Gamma U^T\\\\\n",
    "&= U\\Gamma V^T V^{-T} \\left(\\Gamma^2 + \\lambda I \\right)^{-1} V^{-1} V\\Gamma U^T\\\\\n",
    "&= U\\Gamma^2 \\left(\\Gamma^2 + \\lambda I \\right)^{-1}U^T\\\\\n",
    "&= U\\Lambda U^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where the diagonal matrix $\\Lambda$ has elements of $\\frac{\\sigma^2_i}{\\sigma^2_i + \\lambda}$ on its $i$th diagnoal. This is because the matrix $\\Gamma^2 + \\lambda I$ is always invertible when $\\lambda \\gt 0$.\n",
    "\n",
    "Let $\\lambda = 0$, we have $H= U\\Gamma V^T \\left(V\\Gamma^2V^T \\right)^{-1} V\\Gamma U^T$, since $\\Gamma$ may not have inverse, since some of its diagnoal entries might be zero. For its inverse, let the non-zero entries to be inverse, and let the zero entries to keep zeroes. We have $H= UU^T$.\n",
    "\n",
    "Now it's time to compute $E_{in}(w_{reg})$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w_{reg}) - E_{in}(w_{lin}) &= \\frac{1}{N}y^T(I-H(\\lambda))^2y - \\frac{1}{N}y^T(I-H)^2y\\\\\n",
    "&= \\frac{1}{N}y^T\\left[(I-H(\\lambda))^2 - (I-H)^2\\right]y\\\\\n",
    "&= \\frac{1}{N}y^T\\left[(I-H(\\lambda))^T(I-H(\\lambda)) - (I-H)\\right]y\\\\\n",
    "&= \\frac{1}{N}y^T\\left[I-2H(\\lambda)+H(\\lambda)^2 - (I-H)\\right]y\\\\\n",
    "&= \\frac{1}{N}y^T\\left[-2H(\\lambda)+H(\\lambda)^2 + H\\right]y\\\\\n",
    "&= \\frac{1}{N}y^T\\left[-2U\\Lambda U^T+ U\\Lambda^2 U^T + UU^T\\right]y\\\\\n",
    "&= \\frac{1}{N}y^TU\\left[-2\\Lambda + \\Lambda^2  + I\\right]U^Ty\\\\\n",
    "&= \\frac{1}{N}a^T(I-\\Lambda)^2a\\\\\n",
    "&=\\frac{1}{N}\\sum^d_{i=1}a^2_i\\left(1-\\frac{\\sigma^2_i}{\\sigma^2_i + \\lambda}\\right)^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We thus finished the proof.\n",
    "\n",
    "#### Problem 4.8\n",
    "\n",
    "In the augmented error minimization with $\\Gamma = I$ and $\\lambda \\gt 0$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{aug}(w) &= E_{in}(w) + \\lambda w^T\\Gamma^T\\Gamma w\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take derivative w.r.t. $w$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla E_{aug}(w) &= \\nabla E_{in}(w) + \\lambda \\left(2\\Gamma^T\\Gamma\\right)w\\\\\n",
    "&= \\nabla E_{in}(w) + 2\\lambda w\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then the update rule is \n",
    "\n",
    "\\begin{align*}\n",
    "w(t+1) &= w(t) - \\eta \\nabla E_{aug}(w)\\\\\n",
    "&= w(t) - \\eta \\left(\\nabla E_{in}(w) + 2\\lambda w\\right)\\\\\n",
    "&= (1-2\\eta\\lambda)w(t) -  \\eta \\nabla E_{in}(w)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This is the weight decay rule: $w(t)$ decays before being updated by the gradient of $E_{in}(w)$.\n",
    "\n",
    "#### Problem 4.9\n",
    "\n",
    "* (a) $\\sqrt{\\lambda}\\Gamma$ has a dimension of $k \\times (d+1)$, while $Z$ has a dimension of $n\\times (d+1)$. $Z_{aug} = \\begin{bmatrix}Z\\\\ \\sqrt{\\lambda}\\Gamma\\end{bmatrix}$ and $y=\\begin{bmatrix}y\\\\ 0\\end{bmatrix}$.\n",
    "\n",
    "We also have $Z^T_{aug} = \\begin{bmatrix}Z^T&\\sqrt{\\lambda}\\Gamma^T\\end{bmatrix}$\n",
    "\n",
    "* (b) The least square solution with augmented data is\n",
    "\n",
    "\\begin{align*}\n",
    "\\left(Z^T_{aug}Z_{aug}\\right)^{-1}Z^T_{aug}y_{aug} &= \\left(\\begin{bmatrix}Z^T&\\sqrt{\\lambda}\\Gamma^T\\end{bmatrix} \\begin{bmatrix}Z\\\\ \\sqrt{\\lambda}\\Gamma\\end{bmatrix}\\right)^{-1} \\begin{bmatrix}Z^T&\\sqrt{\\lambda}\\Gamma^T\\end{bmatrix}\\begin{bmatrix}y\\\\ 0\\end{bmatrix}\\\\\n",
    "&= \\left(Z^TZ + \\lambda\\Gamma^T\\Gamma\\right)^{-1} Z^Ty\\\\\n",
    "&= w_{reg}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "#### Problem 4.10\n",
    "\n",
    "* (a) If $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\le C$, then $w_{reg} = w_{lin}$\n",
    "* (b) $E_{in}(w) = \\frac{1}{N}\\sum^N_{i=1}\\left(y_i - w^Tx_i\\right)^2$. $E_{in}(w)$ is thus a quadratic function of $w$, so the contours of constant $E_{in}$ are ellipsoids.  In this case, $w^T_{reg}\\Gamma^T\\Gamma w_{reg} = C$. First notice that the region $w^T\\Gamma^T\\Gamma w \\le C$ is convex. When $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$, $w_{lin}$ is outside of this region, as we move away from $w_{lin}$, the $E_{in}(w)$ increases (because $w_{lin}$ minimizes $E_{in}(w)$) but $w^T\\Gamma^T\\Gamma w$ decreases. The $w$ that satisfies $w^T\\Gamma^T\\Gamma w =C$ has the minimum possible $E_{in}(w)$ under constraint. So this is the solution $w_{reg}$.\n",
    "\n",
    "* (c) From part (b), consider that case when $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$, we have $w_{reg}^T\\Gamma^T\\Gamma w_{reg} = C$, let's replace the inequality constraint to equality constraint. \n",
    "\n",
    "Use Lagrange multiplier, we can instead minimize \n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) + \\lambda \\left(w^T\\Gamma^T\\Gamma w - C\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take derivative w.r.r. $w$, and let it equal to $0$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla E_{in}(w_{reg}) + 2\\lambda_C \\Gamma^T\\Gamma w_{reg} &= 0\\\\\n",
    "\\Gamma^T\\Gamma w_{reg} &= -\\frac{1}{2\\lambda_C}\\nabla E_{in}(w_{reg}) \\\\\n",
    "\\end{align*}\n",
    "\n",
    "Since we know that $w_{reg}^T\\Gamma^T\\Gamma w_{reg} = C$, (or take derivative w.r.t. $\\lambda$), we have \n",
    "\n",
    "\\begin{align*}\n",
    "C &= -\\frac{1}{2\\lambda_C}w^T_{reg}\\nabla E_{in}(w_{reg})\\\\\n",
    "\\lambda_C &= -\\frac{1}{2C}w^T_{reg}\\nabla E_{in}(w_{reg})\n",
    "\\end{align*}\n",
    "\n",
    "Also note that when $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\le C$, $w_{reg} = w_{lin}$, we have $\\nabla E_{in}(w_{reg})= 0$, thus $\\lambda_C = 0$.\n",
    "\n",
    "Combine above results, we see that $w_{reg}$ minimizes $E_{in}(w) + \\lambda_Cw^T\\Gamma^T\\Gamma w$. \n",
    "\n",
    "* (d) \n",
    "  * If $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\le C$, we have $w_{reg} = w_{lin}$, and $\\lambda_C = 0$.\n",
    "  * If $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$, Consider the norm to the surface of $w^T\\Gamma^T\\Gamma w = C$, it's $2\\Gamma^T\\Gamma w$.\n",
    "\n",
    "We claim that the norm and the graident of $E_{in}$ have to be in the opposite directions at the point $w_{reg}$, which minimizes the augmented error. \n",
    "\n",
    "To prove this is correct, Let's look at the projection of $\\nabla E_{in}(w)$ onto the norm. If this projection is along the same direction as norm, then we can move along the opposite of this projection, which will decrease $E_{in}(w)$ (because we move along some component of $-\\nabla E_{in}(w))$ and decrease $w^T\\Gamma^T\\Gamma w$ (it's also the opposite of norm), which will move to region of $w^T\\Gamma^T\\Gamma w \\lt C$. This contradicts our conclusion in part(b) that  the optimal $w_{reg}$ satisfies $w_{reg}^T\\Gamma^T\\Gamma w_{reg} = C$. So the projection of $\\nabla E_{in}(w)$ must be at the opposite of the norm. \n",
    "\n",
    "Now suppose that in addition to the projection component, there's a non-zero component of $\\nabla E_{in}(w)$ at $w_{reg}$ that is perpendicular to the norm. If we move along the opposite of this component, we stay on the surface, but we'll decrease the $E_{in}$, this contradicts with the assumption that $w_{reg}$ minimizes the agumented error.\n",
    "\n",
    "Now we proved that the norm at $w_{reg}$, i.e. $\\Gamma^T\\Gamma w_{reg}$ is in opposite direction of the gradient of $E_{in}$ at $w_{reg}$, i.e. $\\nabla E_{in}(w_{reg})$. From the derivation of part (c) we have following equation\n",
    "  \n",
    "\\begin{align*}\n",
    "\\Gamma^T\\Gamma w_{reg} &= -\\frac{1}{2\\lambda_C}\\nabla E_{in}(w_{reg}) \\\\\n",
    "\\end{align*} \n",
    "\n",
    "This shows that we must have $\\lambda_C \\gt 0$.\n",
    "\n",
    "  * If $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$. \n",
    "\n",
    "Note that $w_{reg}$ depends on $C$. We first work out $\\frac{d\\nabla E_{in}(w_{reg})}{dC}$. We know that $E_{in}(w_{reg}) = \\frac{1}{N}\\|Xw-Y\\|^2 = \\frac{1}{N}(Xw-Y)^T(Xw-Y) = \\frac{1}{N}(w^TX^TXw - w^TX^Ty - y^TX^Tw + y^Ty)$\n",
    "\n",
    "\\begin{align*}\n",
    "\\nabla E_{in}(w_{reg}) &=  \\frac{2}{N}\\left(X^TXw -X^Ty\\right)\\\\\n",
    "w^T_{reg}\\nabla E_{in}(w_{reg}) &=  \\frac{2}{N}\\left(w^T_{reg}X^TXw_{reg} - w^T_{reg}X^Ty\\right)\\\\\n",
    "\\frac{d\\left(w^T_{reg}\\nabla E_{in}(w_{reg})\\right)}{dC} &= \\frac{2}{N}\\frac{dw}{dC}|_{w_{reg}}\\left(2X^TXw_{reg} -X^Ty\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Also take derivative w.r.t. $C$ on surface $w^T\\Gamma^T\\Gamma w = C$, we have $2\\Gamma^T\\Gamma w \\frac{dw}{dC} = 1$, at $w = w_{reg}$, we have $w_{reg}^T\\Gamma^T\\Gamma w_{reg} = C$ when $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$, that is $2w_{reg}^T\\Gamma^T\\Gamma w_{reg}\\frac{dw}{dC}|_{w_{reg}}=1$, we obtain\n",
    "$\\frac{dw}{dC}|_{w_{reg}} = \\frac{1}{2C}w^T_{reg}$.\n",
    "\n",
    "\n",
    "Let's now take derivative of $\\lambda_C$ w.r.t. $C$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{d\\lambda_C}{dC} &= \\frac{1}{2C^2}w^T_{reg}\\nabla E_{in}(w_{reg}) -\\frac{1}{2C}\\frac{d\\left(w^T_{reg}\\nabla E_{in}(w_{reg})\\right)}{dC} \\\\\n",
    "&= \\frac{1}{2C^2}\\frac{2}{N}\\left(w^T_{reg}X^TXw_{reg} - w^T_{reg}X^Ty\\right) -\\frac{1}{2C}\\frac{2}{N}\\frac{dw}{dC}|_{w_{reg}}\\left(2X^TXw_{reg} -X^Ty\\right) \\\\\n",
    "&= \\frac{1}{NC^2}\\left(w^T_{reg}X^TXw_{reg} - w^T_{reg}X^Ty\\right) -\\frac{1}{2C}\\frac{2}{N}\\frac{1}{2C}w^T_{reg}\\left(2X^TXw_{reg} -X^Ty\\right) \\\\\n",
    "&= \\frac{1}{NC^2}\\left(w^T_{reg}X^TXw_{reg} - w^T_{reg}X^Ty\\right) -\\frac{1}{NC^2}\\left(w^T_{reg}X^TXw_{reg} - \\frac{1}{2}w^T_{reg}X^Ty\\right) \\\\\n",
    "&= -\\frac{1}{2NC^2}w^T_{reg}X^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "From previous derivation, $\\lambda_C = -\\frac{1}{2C}w^T_{reg}\\nabla E_{in}(w_{reg})$, we know that $\\lambda_C \\gt 0$, so we have  $w^T_{reg}\\nabla E_{in}(w_{reg}) \\lt 0$ because $C \\gt 0$. \n",
    "\n",
    "\\begin{align*}\n",
    "w^T_{reg}\\nabla E_{in}(w_{reg}) &= \\frac{2}{N}\\left(w^T_{reg}X^TXw_{reg} - w^T_{reg}X^Ty\\right)\\\\\n",
    "w^T_{reg}X^TXw_{reg} &\\lt w^T_{reg}X^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Then\n",
    "\\begin{align*}\n",
    "\\frac{d\\lambda_C}{dC} &= -\\frac{1}{2NC^2}w^T_{reg}X^Ty\\\\\n",
    "&\\lt -\\frac{1}{2NC^2}w^T_{reg}X^TXw_{reg}\\\\\n",
    "&\\le 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where the last inequation comes from the fact that $X^TX$ is positive semi-definite matrix. \n",
    "\n",
    "  * Proof: For any non-zero vector $v$, $v^TX^TXv = (Xv)^T(Xv) \\ge 0$ since $Xv$ is a vector.\n",
    "  \n",
    "So we have proved that $\\frac{d\\lambda_C}{dC} \\lt 0$, $\\lambda_C$ is a strictly decreasing function of $C$ when $w^T_{lin}\\Gamma^T\\Gamma w_{lin} \\gt C$.\n",
    "\n",
    "#### Problem 4.11\n",
    "\n",
    "* (a) \n",
    "\n",
    "\\begin{align*}\n",
    "w_{lin} &= (Z^TZ)^{-1}Z^Ty\\\\\n",
    "&= (Z^TZ)^{-1}Z^T(Zw_f+\\epsilon)\\\\\n",
    "&= (Z^TZ)^{-1}Z^TZw_f+ (Z^TZ)^{-1}Z^T\\epsilon\\\\\n",
    "&= w_f+ (Z^TZ)^{-1}Z^T\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The average function $\\bar{g}$ is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\bar{g}(x) &= E_{\\mathcal{D}}[g(x)]\\\\\n",
    "&= E_{\\mathcal{D}}[w^T_{lin}x]\\\\\n",
    "&= E_{\\mathcal{D}}[\\left(w_f+ (Z^TZ)^{-1}Z^T\\epsilon\\right)^Tx]\\\\\n",
    "&= E_{Z,\\epsilon}[w^T_fx + \\epsilon^T Z(Z^TZ)^{-T} x]\\\\\n",
    "&= E_{Z,\\epsilon}[w^T_fx] + E_{Z,\\epsilon}[\\epsilon^T Z(Z^TZ)^{-1} x]\\\\\n",
    "&= w^T_fx + E_{Z,\\epsilon}[\\epsilon^T]E_{Z,\\epsilon}[Z(Z^TZ)^{-1}]x\\\\\n",
    "&= w^T_fx\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So the average function $\\bar{g}(x)$ is exactly the target function $f(x)$, i.e. $\\bar{g}(x) = f(x)$. \n",
    "We can compute the bias as: \n",
    "\n",
    "\\begin{align*}\n",
    "bias &= E_{\\mathcal{D},x}[\\left(\\bar{g}(x) - f(x)\\right)^2]\\\\\n",
    "&= E_{\\mathcal{D},x}[\\left(w^T_fx - w^T_fx\\right)^2]\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (b) Now we can compute variance\n",
    "\n",
    "\\begin{align*}\n",
    "var &= E_{\\mathcal{D},x}\\left[\\left(g^{\\mathcal{D}}(x)-\\bar{g}(x)\\right)^2\\right]\\\\\n",
    "&= E_{Z, \\epsilon,x}\\left[\\left(w^T_{lin}\\Phi(x)-w^T_f\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= E_{Z, \\epsilon,x}\\left[\\left(((Z^TZ)^{-1}Z^T\\epsilon)^T\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= E_{Z, \\epsilon,x}\\left[\\left(\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= E_{Z, \\epsilon,x}\\left[\\left(\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right)^T\\left(\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right)\\right]\\\\\n",
    "&= E_{Z, \\epsilon,x}\\left[\\Phi^T(x)(Z^TZ)^{-1}Z^T\\epsilon\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right]\\\\\n",
    "&= E_{Z, x}E_{\\epsilon}\\left[\\Phi^T(x)(Z^TZ)^{-1}Z^T\\epsilon\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right]\\\\\n",
    "&= E_{Z, x}E_{\\epsilon}\\left[trace\\left(\\Phi^T(x)(Z^TZ)^{-1}Z^T\\epsilon\\epsilon^T Z(Z^TZ)^{-1}\\Phi(x)\\right)\\right]\\\\\n",
    "&= E_{Z, x}E_{\\epsilon}\\left[trace\\left(Z(Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ)^{-1}Z^T\\epsilon\\epsilon^T \\right)\\right]\\\\\n",
    "&= E_{Z, x}trace\\left[E_{\\epsilon}\\left(Z(Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ)^{-1}Z^T\\epsilon\\epsilon^T \\right)\\right]\\\\\n",
    "&= E_{Z, x}trace\\left[E_{\\epsilon}\\left(Z(Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ)^{-1}Z^T\\right)E_{\\epsilon}\\left(\\epsilon\\epsilon^T \\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z, x}trace\\left[\\left(Z(Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ)^{-1}Z^T\\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z, x}trace\\left[\\left((Z^TZ)^{-1}Z^TZ(Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)\\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z, x}trace\\left[\\left((Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)\\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z}E_xtrace\\left[\\left((Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)\\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z}traceE_x\\left[\\left((Z^TZ)^{-1}\\Phi(x)\\Phi^T(x)\\right)\\right]\\\\\n",
    "&= \\sigma^2E_{Z}trace\\left[E_x\\left[(Z^TZ)^{-1}\\right]E_x\\left[\\Phi(x)\\Phi^T(x)\\right]\\right]\\\\\n",
    "&= \\sigma^2E_{Z}trace\\left[(Z^TZ)^{-1}\\Sigma_{\\Phi}\\right]\\\\\n",
    "&= \\sigma^2trace\\left[\\Sigma_{\\Phi}E_{Z}(Z^TZ)^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{N}trace\\left[\\Sigma_{\\Phi}E_{Z}(\\frac{1}{N}Z^TZ)^{-1}\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) \n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{1}{N}Z^TZ &= \\frac{1}{N}\\sum^N_{n=1}\\Phi(x_n)\\Phi^T(x_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This is the in-sample estimate of $\\Sigma_{\\Phi}$, by the law of large numbers, we have $\\frac{1}{N}Z^TZ = \\Sigma_{\\Phi} + o(1)$. The $\\Sigma_{\\Phi}$ has a dimension of $(Q+1)\\times (Q+1)$ for a given $x$, so we have\n",
    "\n",
    "\\begin{align*}\n",
    "var &= \\frac{\\sigma^2}{N}trace\\left[\\Sigma_{\\Phi}E_{Z}(\\frac{1}{N}Z^TZ)^{-1}\\right]\\\\\n",
    "&\\approx \\frac{\\sigma^2}{N}trace\\left[\\Sigma_{\\Phi}(\\Sigma_{\\Phi}+o(1))^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{N}traceI_{(Q+1)\\times (Q+1)}\\\\\n",
    "&= \\frac{\\sigma^2 (Q+1)}{N}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So for the well specified linear model, the bias is zero and the variance is increasing as the model gets larger (i.e. $Q$ increases) but decreasing in $N$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.12\n",
    "\n",
    "* (a) \n",
    "\n",
    "\\begin{align*}\n",
    "w_{reg} &= \\left(Z^TZ + \\lambda I\\right)^{-1}Z^Ty\\\\\n",
    "&= \\left(Z^TZ + \\lambda I\\right)^{-1}Z^T\\left(Zw_f+\\epsilon\\right)\\\\\n",
    "&= \\left(Z^TZ + \\lambda I\\right)^{-1}Z^TZw_f+\\left(Z^TZ + \\lambda I\\right)^{-1}Z^T\\epsilon\\\\\n",
    "&= \\left(I - \\left(Z^TZ + \\lambda I\\right)^{-1}\\lambda \\right)w_f+\\left(Z^TZ + \\lambda I\\right)^{-1}Z^T\\epsilon\\\\\n",
    "&= w_f - \\lambda\\left(Z^TZ + \\lambda I\\right)^{-1}w_f +\\left(Z^TZ + \\lambda I\\right)^{-1}Z^T\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The average function $\\bar{g}$ is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\bar{g}(x) &= E_{\\mathcal{D}}[g(x)]\\\\\n",
    "&= E_{\\mathcal{D}}[w^T_{reg}x]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\left(w_f- \\lambda\\left(Z^TZ + \\lambda I\\right)^{-1}w_f +\\left(Z^TZ + \\lambda I\\right)^{-1}Z^T\\epsilon\\right)^Tx\\right]\\\\\n",
    "&= E_{Z,\\epsilon}\\left[w^T_fx - \\lambda w^T_f\\left(Z^TZ + \\lambda I\\right)^{-1}x + \\epsilon^T Z(Z^TZ+ \\lambda I)^{-1} x\\right]\\\\\n",
    "&= E_{Z,\\epsilon}[w^T_fx- \\lambda w^T_f\\left(Z^TZ + \\lambda I\\right)^{-1}x] + E_{Z,\\epsilon}[\\epsilon^T Z(Z^TZ+ \\lambda I)^{-1} x]\\\\\n",
    "&= w^T_fx - \\lambda E_Z\\left[ w^T_f\\left(Z^TZ + \\lambda I\\right)^{-1}\\right]x + E_{Z,\\epsilon}[\\epsilon^T]E_{Z,\\epsilon}[Z(Z^TZ+ \\lambda I)^{-1}]x\\\\\n",
    "&= w^T_fx- \\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}x\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note that $\\frac{1}{N}Z^TZ = \\frac{1}{N}\\sum^N_{n=1}\\Phi(x_n)\\Phi^T(x_n)$ is the in-sample estimate of $\\Sigma_{\\Phi} = E[\\Phi(x)\\Phi^T(x)] = I_{(d+1)\\times(d+1)}$, so by the law of large numbers, $\\frac{1}{N}Z^TZ = \\Sigma_{\\Phi} + o(1)$, so $\\frac{1}{N}(Z^TZ + \\lambda I) = \\Sigma_{\\Phi} + \\frac{1}{N}\\lambda I + o(1) = (1+\\frac{\\lambda}{N})I + o(1)$.\n",
    "\n",
    "Then when $N$ is large\n",
    "\n",
    "\\begin{align*}\n",
    "E_Z\\left(Z^TZ + \\lambda I\\right)^{-1} &= \\frac{1}{N}E_Z\\left(\\frac{1}{N}(Z^TZ + \\lambda I)\\right)^{-1} \\\\\n",
    "&= \\frac{1}{N}(1+\\frac{\\lambda}{N})^{-1}I \\\\\n",
    "&= \\frac{1}{\\lambda + N}I \\\\\n",
    "\\end{align*}\n",
    "\n",
    " \n",
    "\n",
    "We can compute the bias as: \n",
    "\n",
    "\\begin{align*}\n",
    "bias &= E_{\\mathcal{D},x}[\\left(\\bar{g}(x) - f(x)\\right)^2]\\\\\n",
    "&= E_{\\mathcal{D},x}\\left[\\left(w^T_f\\Phi(x) - \\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x)- w^T_f\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= E_{\\mathcal{D},x}\\left[\\left(\\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= \\lambda^2 E_{\\mathcal{D},x}\\left[\\Phi^T(x)E_Z\\left(Z^TZ + \\lambda I\\right)^{-T}w_f w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x)\\right]\\\\\n",
    "&= \\lambda^2 E_{\\mathcal{D},x}\\left[\\Phi^T(x)\\frac{1}{\\lambda + N}I w_f w^T_f\\frac{1}{\\lambda + N}I \\Phi(x)\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} E_{\\mathcal{D},x}\\left[\\Phi^T(x)w_f w^T_f\\Phi(x)\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2}  E_{\\mathcal{D},x}trace\\left[\\Phi^T(x)w_f w^T_f\\Phi(x)\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2}  E_{\\mathcal{D},x}trace\\left[\\Phi(x)\\Phi^T(x)w_f w^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} trace E_{\\mathcal{D},x}\\left[\\Phi(x)\\Phi^T(x)w_f w^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} trace \\left[E_{\\mathcal{D},x}[\\Phi(x)\\Phi^T(x)]w_f w^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} trace [Iw_f w^T_f]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} trace [w_f w^T_f]\\\\\n",
    "&= \\frac{\\lambda^2}{(\\lambda + N)^2} \\|w_f\\|^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have used the assumption that the input probability satisfies $E_{x} \\left[\\Phi(x)\\Phi^T(x)\\right]= I $\n",
    "\n",
    "Also note, that $w_f w^T_f$ is a $(d+1)\\times (d+1)$ matrix, its trace is just $\\|w_f\\|^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.12 (b)\n",
    "\n",
    "\\begin{align*}\n",
    "var &= E_{\\mathcal{D}, x}\\left[(g^{\\mathcal{D}}(x) - \\bar{g}(x))^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}, x}\\left[\\left(w^T_{reg}\\Phi(x) - (w^T_f\\Phi(x) - \\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x) \\right)^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}, x}\\left[\\left((w^T_f-w^T_f\\lambda(Z^TZ+\\lambda I)^{-1}+\\epsilon^TZ(Z^TZ+\\lambda I)^{-1})\\Phi(x) - (w^T_f\\Phi(x) - \\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x) \\right)^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}, x}\\left[\\left((-w^T_f\\lambda(Z^TZ+\\lambda I)^{-1}+\\epsilon^TZ(Z^TZ+\\lambda I)^{-1})\\Phi(x) - ( - \\lambda w^T_fE_Z\\left(Z^TZ + \\lambda I\\right)^{-1}\\Phi(x) \\right)^2\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "To the first oder in $\\frac{1}{N}$, we know from 4.12 (a) that $\\frac{1}{N}(Z^TZ + \\lambda I) = (1+\\frac{\\lambda}{N})I + o(1)$ and $E_Z\\left(Z^TZ + \\lambda I\\right)^{-1} = \\frac{1}{\\lambda + N}I$\n",
    "\n",
    "So we have\n",
    "\n",
    "\\begin{align*}\n",
    "var &= E_{\\mathcal{D}, x}\\left(\\epsilon^TZ(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\right)^2\\\\\n",
    "&= E_{\\mathcal{D}, x}\\left(\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\epsilon\\epsilon^TZ(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\right)\\\\\n",
    "&= E_{\\mathcal{D}, x}trace\\left(\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\epsilon\\epsilon^TZ(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\right)\\\\\n",
    "&= E_{\\mathcal{D}, x}trace\\left(Z(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\epsilon\\epsilon^T\\right)\\\\\n",
    "&= trace E_{\\mathcal{D}, x}\\left(Z(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\epsilon\\epsilon^T\\right)\\\\\n",
    "&= trace E_{\\mathcal{D}, x}\\left(Z(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\right)E_{\\mathcal{D}, x}\\left(\\epsilon\\epsilon^T\\right)\\\\\n",
    "&= \\sigma^2 trace E_{\\mathcal{D}, x}\\left(Z(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\right)\\\\\n",
    "&= \\sigma^2  E_{\\mathcal{D}, x}trace\\left(Z(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)(Z^TZ+\\lambda I)^{-1}Z^T\\right)\\\\\n",
    "&= \\sigma^2  E_{\\mathcal{D}, x}trace\\left((Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}\\Phi(x)\\Phi^T(x)\\right)\\\\\n",
    "&= \\sigma^2 trace E_{\\mathcal{D}}\\left((Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}\\right) E_{ x}\\left(\\Phi(x)\\Phi^T(x)\\right)\\\\\n",
    "&= \\sigma^2 trace E_{\\mathcal{D}}\\left((Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}\\right)\\\\\n",
    "&= \\sigma^2  E_{\\mathcal{D}}trace\\left((Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}\\right)\\\\\n",
    "&= \\sigma^2  E_{\\mathcal{D}}trace\\left(Z(Z^TZ+\\lambda I)^{-1}(Z^TZ+\\lambda I)^{-1}Z^T\\right)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "To first order in $\\frac{1}{N}$, we have $Z^TZ = NI$, take this into variance expression, we obtain\n",
    "\n",
    "\\begin{align*}\n",
    "var &= \\sigma^2  E_{\\mathcal{D}}trace\\left(Z(Z^TZ+\\lambda I)^{-1}(Z^TZ+\\lambda I)^{-1}Z^T\\right)\\\\\n",
    "&= \\frac{\\sigma^2}{N}  E_{\\mathcal{D}}trace\\left(Z(Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}Z^T\\right)\\\\\n",
    "&= \\frac{\\sigma^2}{N}  E_{\\mathcal{D}}[trace(H^2(\\lambda))]\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.13\n",
    "\n",
    "\\begin{align*}\n",
    "H(\\lambda) &= Z(Z^TZ+\\lambda I)^{-1}Z^T\\\\\n",
    "H^2(\\lambda) &= Z(Z^TZ+\\lambda I)^{-1}Z^TZ(Z^TZ+\\lambda I)^{-1}Z^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Also from exercise 3.3 (d), when $\\lambda=0$, $H(\\lambda) = H$, and we have $trace(H) = \\tilde{d}+1$, and $H^2=H$, so $trace(H^2) = \\tilde{d}+1$.\n",
    "\n",
    "* (a) When $\\lambda = 0$, we have $trace(H(\\lambda)) = trace\\left(Z(Z^TZ)^{-1}Z^T\\right) = trace(H) = \\tilde{d}+1$\n",
    "  * (i) $d_{eff}(\\lambda) = 2trace(H(\\lambda)) - trace(H^2(\\lambda)) = 2trace(H) - trace(H^2) = trace(H) = \\tilde{d} + 1$.\n",
    "  * (ii) $d_{eff}(\\lambda)= trace(H(\\lambda)) = trace(H) = \\tilde{d} + 1$\n",
    "  * (iii) $d_{eff}(\\lambda)= trace(H^2(\\lambda)) = trace(H) = \\tilde{d} + 1$\n",
    "  \n",
    "* (b) When $\\lambda \\gt 0$. Let the SVD of $Z=U\\Gamma V^T$, where $U$ is $n\\times (\\tilde{d}+1)$ matrix, $U^TU=I_{(\\tilde{d} + 1) \\times (\\tilde{d}+1)}$ and $V$ is $(\\tilde{d} + 1) \\times (\\tilde{d}+1)$, $V^TV=VV^T=I_{(\\tilde{d} + 1) \\times (\\tilde{d}+1)}$, and $\\Gamma$ is $(\\tilde{d} + 1) \\times (\\tilde{d}+1)$ diagonal matrix. We have\n",
    "\n",
    "\\begin{align*}\n",
    "Z^TZ &= V\\Gamma U^T U\\Gamma V^T\\\\\n",
    "&= V\\Gamma^2V^T\\\\\n",
    "H(\\lambda) &= Z(Z^TZ+\\lambda I)^{-1}Z^T\\\\\n",
    "&= U\\Gamma V^T\\left(V\\Gamma^2V^T+\\lambda I\\right)^{-1}V\\Gamma U^T\\\\\n",
    "&= U\\Gamma V^T\\left(V(\\Gamma^2+\\lambda I)V^T\\right)^{-1}V\\Gamma U^T\\\\\n",
    "&= U\\Gamma V^TV^{-T}(\\Gamma^2+\\lambda I)^{-1}V^{-1}V\\Gamma U^T\\\\\n",
    "&= U\\Gamma^2(\\Gamma^2+\\lambda I)^{-1}U^T\\\\\n",
    "&= U\\Sigma U^T\\\\\n",
    "H^2(\\lambda) &= (U\\Sigma U^T)^TU\\Sigma U^T\\\\\n",
    "&= U\\Sigma^2U^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $\\Sigma = \\Gamma^2(\\Gamma^2+\\lambda I)^{-1}$, let $\\sigma_1, \\sigma_2, \\dots, \\sigma_d$ be the eigenvalues of matrix $\\Gamma$.\n",
    "\n",
    "So the traces are\n",
    "\n",
    "\\begin{align*}\n",
    "trace(H(\\lambda)) &= trace(U\\Sigma U^T) \\\\\n",
    "&= trace(U^TU\\Sigma) \\\\\n",
    "&= trace(\\Sigma)\\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{\\sigma^2_i}{\\sigma^2_i+\\lambda}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "trace(H^2(\\lambda)) &= trace(U\\Sigma^2 U^T) \\\\\n",
    "&= trace(U^TU\\Sigma^2) \\\\\n",
    "&= trace(\\Sigma^2)\\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{\\sigma^4_i}{(\\sigma^2_i+\\lambda)^2}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * (i) \n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &= 2trace(H(\\lambda)) - trace(H^2(\\lambda)) \\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{2\\sigma^2_i}{\\sigma^2_i+\\lambda} - \\frac{\\sigma^4_i} {(\\sigma^2_i+\\lambda)^2}\\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{\\sigma^4_i+2\\lambda\\sigma^2_i} {(\\sigma^2_i+\\lambda)^2}\\\\\n",
    "&\\le \\sum^{\\tilde{d}+1}_{i=1}1\\\\\n",
    "&= \\tilde{d}+1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * (ii)\n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &=trace(H(\\lambda))\\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{\\sigma^2_i}{\\sigma^2_i+\\lambda}\\\\\n",
    "&\\le \\sum^{\\tilde{d}+1}_{i=1}1\\\\\n",
    "&= \\tilde{d}+1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "  * (iii)\n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &=trace(H^2(\\lambda))\\\\\n",
    "&= \\sum^{\\tilde{d}+1}_{i=1}\\frac{\\sigma^4_i} {(\\sigma^2_i+\\lambda)^2}\\\\\n",
    "&\\le \\sum^{\\tilde{d}+1}_{i=1}1\\\\\n",
    "&= \\tilde{d}+1\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Also, it's clear that they are all larger or equal to zero. So we have finished the proof."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.14 \n",
    "\n",
    "\\begin{align*}\n",
    "E_{in} &= \\frac{1}{N}\\|y-Xw\\|^2\\\\\n",
    "&= \\frac{1}{N}\\|y-H(\\lambda)y\\|^2\\\\\n",
    "&= \\frac{1}{N}\\|\\left(I-H(\\lambda)\\right)y\\|^2\\\\\n",
    "&= \\frac{1}{N}\\|\\left(I-H(\\lambda)\\right)(f+\\epsilon)\\|^2\\\\\n",
    "&= \\frac{1}{N}(f+\\epsilon)^T\\left(I-H(\\lambda)\\right)^T\\left(I-H(\\lambda)\\right)(f+\\epsilon)\\\\\n",
    "&= \\frac{1}{N}(f^T+\\epsilon^T)\\left(I-H(\\lambda)\\right)^2(f+\\epsilon)\\\\\n",
    "&= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}f^T\\left(I-H(\\lambda)\\right)^2\\epsilon +  \\frac{1}{N}\\epsilon^T\\left(I-H(\\lambda)\\right)^2f +  \\frac{1}{N}\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\epsilon\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take expectation w.r.t. $\\epsilon$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\epsilon}[E_{in}] &= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}E_{\\epsilon}\\left[\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\epsilon\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We need work out the expectation of the second term, we have\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\epsilon}\\left[\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\epsilon\\right] &= E_{\\epsilon}trace\\left[\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\epsilon\\right]\\\\\n",
    "&= E_{\\epsilon}trace\\left[\\epsilon\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\right]\\\\\n",
    "&= traceE_{\\epsilon}\\left[\\epsilon\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\right]\\\\\n",
    "&= traceE_{\\epsilon}[\\epsilon\\epsilon^T]E_{\\epsilon}\\left(I-H(\\lambda)\\right)^2\\\\\n",
    "&= trace\\sigma^2\\left(I-H(\\lambda)\\right)^2\\\\\n",
    "&= \\sigma^2trace\\left(I-H(\\lambda)\\right)^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So the final expectation of \n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\epsilon}[E_{in}] &= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}E_{\\epsilon}\\left[\\epsilon^T\\left(I-H(\\lambda)\\right)^2\\epsilon\\right]\\\\\n",
    "&= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}\\sigma^2trace\\left(I-H(\\lambda)\\right)^2\\\\\n",
    "&= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}\\sigma^2trace\\left(I-2H(\\lambda)+H^2(\\lambda)\\right)\\\\\n",
    "&= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\frac{1}{N}\\sigma^2\\left(N -\\left(2traceH(\\lambda)-trace(H^2(\\lambda))\\right)\\right)\\\\\n",
    "&= \\frac{1}{N} f^T \\left(I-H(\\lambda)\\right)^2 f +  \\sigma^2(1 -\\frac{d_{eff}}{N})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (a) If the noise was not overfit, the term involving $\\sigma^2$ should equal to $\\sigma^2$ because \n",
    "$E_{\\epsilon}[E_{in}]$ is an estimate of the out-of-sample error, which should have constant term $\\sigma^2$. When the noise was overfit, $E_{\\epsilon}[E_{in}]$ underestimates the out-of-sample error. \n",
    "\n",
    "* (b) Hence, the degree to which the noise has been overfit is $\\frac{\\sigma^2d_{eff}}{N}$. It says that when $d_{eff}$ increases, the overfit to noise increases, when $N$ increases, the overift decreases. $d_{eff}$ acts as an effective number of parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.15 \n",
    "\n",
    "* (a) Let $\\tilde{Z} = Z\\Gamma^{-1}$, then $Z=\\tilde{Z}\\Gamma$, also assume $\\Gamma$ is square and invertible, we have\n",
    "\n",
    "\\begin{align*}\n",
    "H(\\lambda) &= Z(Z^TZ+\\lambda \\Gamma^T\\Gamma)^{-1}Z^T\\\\\n",
    "&= \\tilde{Z}\\Gamma((\\tilde{Z}\\Gamma)^T\\tilde{Z}\\Gamma+\\lambda \\Gamma^T\\Gamma)^{-1}(\\tilde{Z}\\Gamma)^T\\\\\n",
    "&= \\tilde{Z}\\Gamma(\\Gamma^T\\tilde{Z}^T\\tilde{Z}\\Gamma+\\Gamma^T\\lambda \\Gamma)^{-1}\\Gamma^T\\tilde{Z}^T\\\\\n",
    "&= \\tilde{Z}\\Gamma\\left(\\Gamma^T(\\tilde{Z}^T\\tilde{Z}+\\lambda I) \\Gamma\\right)^{-1}\\Gamma^T\\tilde{Z}^T\\\\\n",
    "&= \\tilde{Z}\\Gamma\\Gamma^{-1}\\left(\\tilde{Z}^T\\tilde{Z}+\\lambda I \\right)^{-1}\\Gamma^{-T}\\Gamma^T\\tilde{Z}^T\\\\\n",
    "&= \\tilde{Z}\\left(\\tilde{Z}^T\\tilde{Z}+\\lambda I \\right)^{-1}\\tilde{Z}^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Now we can proceed similar with problem 4.13 (i), we let the SVD of $\\tilde{Z}=U\\Lambda V^T$, where $U$ is $n\\times (d+1)$ matrix, $U^TU=I_{(d + 1) \\times (d+1)}$ and $V$ is $(d + 1) \\times (d+1)$, $V^TV=VV^T=I_{(d + 1) \\times (d+1)}$, and $\\Lambda$ is $(d + 1) \\times (d+1)$ diagonal matrix with entries $s_0,s_1,\\dots,s_{d}$. We have\n",
    "\n",
    "\\begin{align*}\n",
    "\\tilde{Z}^T\\tilde{Z} &= V\\Lambda U^T U\\Lambda V^T\\\\\n",
    "&= V\\Lambda^2V^T\\\\\n",
    "H(\\lambda) &= \\tilde{Z}\\left(\\tilde{Z}^T\\tilde{Z}+\\lambda I \\right)^{-1}\\tilde{Z}^T\\\\\n",
    "&= U\\Lambda V^T\\left(V\\Lambda^2V^T+\\lambda \\right)^{-1}V\\Lambda U^T\\\\\n",
    "&= U\\Lambda V^T\\left(V(\\Lambda^2+\\lambda )V^T\\right)^{-1}V\\Lambda U^T\\\\\n",
    "&= U\\Lambda V^TV^{-T}(\\Lambda^2+\\lambda I)^{-1}V^{-1}V\\Lambda U^T\\\\\n",
    "&= U\\Lambda^2(\\Lambda^2+\\lambda I)^{-1}U^T\\\\\n",
    "&= U\\Sigma U^T\\\\\n",
    "H^2(\\lambda) &= (U\\Sigma U^T)^TU\\Sigma U^T\\\\\n",
    "&= U\\Sigma^2U^T\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $\\Sigma = \\Lambda^2(\\Lambda^2+\\lambda I)^{-1}$ with entries on diagonal of $\\frac{s^2_i}{s^2_i+\\lambda}$, the effective degree of freedom is then\n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &= 2trace(H(\\lambda)) - trace(H^2(\\lambda)) \\\\\n",
    "&= \\sum^{d}_{i=0}\\frac{s^4_i+2\\lambda s^2_i} {(s^2_i+\\lambda)^2}\\\\\n",
    "&= \\sum^{d}_{i=0}\\frac{(s^2_i+\\lambda)^2-\\lambda^2} {(s^2_i+\\lambda)^2}\\\\\n",
    "&= \\sum^{d}_{i=0}1 - \\frac{\\lambda^2} {(s^2_i+\\lambda)^2}\\\\\n",
    "&= d+1 - \\sum^{d}_{i=0}\\frac{\\lambda^2} {(s^2_i+\\lambda)^2}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (b) Similarly, we have\n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &= trace(H(\\lambda)) \\\\\n",
    "&= trace(U\\Sigma U^T) \\\\\n",
    "&= trace(U^TU\\Sigma) \\\\\n",
    "&= trace(\\Sigma) \\\\\n",
    "&= \\sum^{d}_{i=0}\\frac{s^2_i} {s^2_i+\\lambda}\\\\\n",
    "&= d+1- \\sum^{d}_{i=0}\\frac{\\lambda} {s^2_i+\\lambda}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) \n",
    "\n",
    "\\begin{align*}\n",
    "d_{eff}(\\lambda) &= trace(H^2(\\lambda)) \\\\\n",
    "&= \\sum^{d}_{i=0}\\frac{s^4_i} {(s^2_i+\\lambda)^2}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "In all cases, for $\\lambda \\ge 0$, we have $d_{eff}(\\lambda) \\ge 0$ because all the terms in the sum are non-negative. Also $d_{eff}(\\lambda) \\le d+1$ as shown in problem 4.13. \n",
    "\n",
    "When $\\lambda = 0$, we have $d_{eff}(0) = d+1$ as easily shown from above formulas. \n",
    "\n",
    "It's also easy to see that $d_{eff}$ is decreasing with $\\lambda$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.16\n",
    "\n",
    "For linear models and the general Tikhonov regularizer $\\Gamma$ with penalty term $\\frac{\\lambda}{N}w^T\\Gamma^T\\Gamma w$ in the augmented error, we have $E_{aug}(w) = E_{in}(w) + \\frac{\\lambda}{N}w^T\\Gamma^T\\Gamma w = \\frac{1}{N}\\|Zw-y\\|^2 + \\frac{\\lambda}{N}w^T\\Gamma^T\\Gamma w$. Take derivative w.r.t. $w$ and let it equal to $0$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{dE_{aug}}{dw} &= \\frac{1}{N}\\left(2Z^TZw - 2Z^Ty\\right) + \\frac{2\\lambda}{N}\\Gamma^T\\Gamma w\\\\\n",
    "&= \\frac{2}{N}\\left(Z^TZ + \\lambda \\Gamma^T\\Gamma\\right)w - \\frac{2}{N}Z^Ty\\\\\n",
    "&= 0\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We then have\n",
    "\n",
    "\\begin{align*}\n",
    "w_{reg} &= \\left(Z^TZ + \\lambda \\Gamma^T\\Gamma\\right)^{-1}Z^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (a) For the in-sample predictions, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y} &= Zw_{reg}\\\\\n",
    "&= Z\\left(Z^TZ + \\lambda \\Gamma^T\\Gamma\\right)^{-1}Z^Ty\\\\\n",
    "&= H(\\lambda)y\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (b) When $\\Gamma = Z$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "w_{reg} &=\\left(Z^TZ + \\lambda \\Gamma^T\\Gamma\\right)^{-1}Z^Ty\\\\\n",
    "&= \\left(Z^TZ + \\lambda Z^TZ\\right)^{-1}Z^Ty\\\\\n",
    "&= \\left((I+\\lambda)Z^TZ\\right)^{-1}Z^Ty\\\\\n",
    "&= (Z^TZ)^{-1}(I+\\lambda)^{-1}Z^Ty\\\\\n",
    "&= \\frac{1}{1+\\lambda}(Z^TZ)^{-1}Z^Ty\\\\\n",
    "&= \\frac{1}{1+\\lambda}w_{lin}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This is called uniform weight decay."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.17\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{E}_{in}(w) &= E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\frac{1}{N}\\sum^N_{n=1}\\left(w^T\\hat{x}_n - y_n\\right)^2\\right]\\\\\n",
    "&= E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\frac{1}{N}\\sum^N_{n=1}\\left(w^T(x_n+\\epsilon_n) - y_n\\right)^2\\right]\\\\\n",
    "&= E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\frac{1}{N}\\sum^N_{n=1}\\left((w^Tx_n-y_n)+w^T\\epsilon_n\\right)^2\\right]\\\\\n",
    "&= E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\frac{1}{N}\\sum^N_{n=1}(w^Tx_n-y_n)^2+2w^T\\epsilon_n(w^Tx_n-y_n) + (w^T\\epsilon_n)^2\\right]\\\\\n",
    "&= \\frac{1}{N}\\sum^N_{n=1}(w^Tx_n-y_n)^2 + \\frac{1}{N}E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\sum^N_{n=1}(w^T\\epsilon_n)^2\\right]\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[\\sum^N_{n=1}(w^T\\epsilon_n)^2\\right]\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}\\sum^N_{n=1}E_{\\epsilon_1,\\dots,\\epsilon_N}\\left[(w^T\\epsilon_n)^2\\right]\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}\\sum^N_{n=1}w^TE_{\\epsilon_n}[\\epsilon_n\\epsilon^T_n]w\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}\\sum^N_{n=1}w^T\\sigma^2_nIw\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}w^T\\left(\\sum^N_{n=1}\\sigma^2_nI\\right)w\\\\\n",
    "&= E_{in}(w) + \\frac{1}{N}w^T(\\sum^N_{n=1}\\sigma^2_n)w\\\\\n",
    "&= E_{in}(w) + \\frac{\\lambda}{N}w^T\\Gamma^T\\Gamma w\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $\\lambda = 1$, and $\\Gamma = \\begin{bmatrix}\\sigma_1\\\\\\sigma_2\\\\\\dots\\\\\\sigma_N\\end{bmatrix}$\n",
    "\n",
    "So the weights $\\hat{w}_{lin}$ resulting from minimizing $\\hat{E}_{in}$ are equivalent to the weights that would have been obtained by minimizing $E_{in}$ for the observed data with Tikhonov regularization using $\\Gamma$ and $\\lambda$ as above. \n",
    "\n",
    "We can intrepret this result as: regularization enforces a robustness to potential measurement errors (noise) in the observed inputs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.18\n",
    "\n",
    "* (a) The regularized weights are given by\n",
    "\n",
    "\\begin{align*}\n",
    "w_{reg} &= \\left(Z^TZ + \\lambda Z^TZ\\right)^{-1}Z^Ty\\\\\n",
    "&= \\left(Z^T(I+\\lambda I)Z\\right)^{-1}Z^Ty\\\\\n",
    "&= Z^{-1}(I+\\lambda I)^{-1}Z^{-T}Z^Ty\\\\\n",
    "&= \\frac{1}{1+\\lambda}Z^{-1}y\\\\\n",
    "&= \\frac{1}{1+\\lambda}Z^{-1}(Zw_f+\\epsilon)\\\\\n",
    "&= \\frac{1}{1+\\lambda}(w_f+Z^{-1}\\epsilon)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The average function $\\bar{g}$ is:\n",
    "\n",
    "\\begin{align*}\n",
    "\\bar{g}(x) &= E_{\\mathcal{D}}[g(x)]\\\\\n",
    "&= E_{\\mathcal{D}}[w^T_{reg}x]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\frac{1}{1+\\lambda}(w_f+Z^{-1}\\epsilon)^Tx\\right]\\\\\n",
    "&= \\frac{1}{1+\\lambda}E_{Z,\\epsilon}\\left[w^T_fx + \\epsilon^TZ^{-T}x\\right]\\\\\n",
    "&= \\frac{1}{1+\\lambda}\\left[E_{Z,\\epsilon}[w^T_fx] + E_{Z,\\epsilon}[\\epsilon^TZ^{-T}x]\\right]\\\\\n",
    "&= \\frac{1}{1+\\lambda}\\left[w^T_fx + E_{Z, \\epsilon}[\\epsilon^TZ^{-T}x]\\right]\\\\\n",
    "&= \\frac{1}{1+\\lambda}\\left[w^T_fx + E_{\\epsilon}[\\epsilon^T]E_{Z}[Z^{-T}x]\\right]\\\\\n",
    "&= \\frac{1}{1+\\lambda}w^T_fx\\\\\n",
    "&= \\frac{1}{1+\\lambda}f(x)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "We can compute the bias as: \n",
    "\n",
    "\\begin{align*}\n",
    "bias &= E_{\\mathcal{D},x}[\\left(\\bar{g}(x) - f(x)\\right)^2]\\\\\n",
    "&= E_{\\mathcal{D},x}\\left[\\left(\\frac{1}{1+\\lambda}w^T_f\\Phi(x)- w^T_f\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}E_{\\mathcal{D},x}\\left[\\left(w^T_f\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}E_{\\mathcal{D},x}trace\\left[\\left(w^T_f\\Phi(x)\\right)^2\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}E_{\\mathcal{D},x}trace\\left[\\Phi^T(x)w_fw^T_f\\Phi(x)\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}E_{\\mathcal{D},x}trace\\left[\\Phi(x)\\Phi^T(x)w_fw^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}traceE_{\\mathcal{D},x}\\left[\\Phi(x)\\Phi^T(x)w_fw^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}trace\\left[E_{\\mathcal{D},x}[\\Phi(x)\\Phi^T(x)]w_fw^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}trace\\left[w_fw^T_f\\right]\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}\\|w_f\\|^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have used the assumption that the input probability satisfies $E_{x} \\left[\\Phi(x)\\Phi^T(x)\\right]= I $\n",
    "\n",
    "Also note, that $w_f w^T_f$ is a $(d+1)\\times (d+1)$ matrix, its trace is just $\\|w_f\\|^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (b) \n",
    "\n",
    "\\begin{align*}\n",
    "var &= E_{\\mathcal{D}, x, \\epsilon}\\left[(g^{\\mathcal{D}}(x) - \\bar{g}(x))^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}, x, \\epsilon}\\left[\\left(w^T_{reg}\\Phi(x) - \\frac{1}{1+\\lambda}w^T_f\\Phi(x) \\right)^2\\right]\\\\\n",
    "&= E_{\\mathcal{D}, x, \\epsilon}\\left[\\left(\\frac{1}{1+\\lambda}(w_f+Z^{-1}\\epsilon)^T\\Phi(x) - \\frac{1}{1+\\lambda}w^T_f\\Phi(x) \\right)^2\\right]\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^2} E_{\\mathcal{D}, x, \\epsilon}\\left[(\\epsilon^TZ^{-T}\\Phi(x))^2\\right]\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^2}E_{\\mathcal{D}, x, \\epsilon}\\left[\\Phi^T(x)Z^{-1}\\epsilon\\epsilon^TZ^{-T}\\Phi(x)\\right]\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^2}E_{\\mathcal{D}, x, \\epsilon}trace\\left[\\Phi^T(x)Z^{-1}\\epsilon\\epsilon^TZ^{-T}\\Phi(x)\\right]\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^2}E_{\\mathcal{D}, x, \\epsilon}trace\\left[Z^{-T}\\Phi(x)\\Phi^T(x)Z^{-1}\\epsilon\\epsilon^T\\right]\\\\\n",
    "&=\\frac{1}{(1+\\lambda)^2} trace E_{\\mathcal{D}, x, \\epsilon}\\left[Z^{-T}\\Phi(x)\\Phi^T(x)Z^{-1}\\epsilon\\epsilon^T\\right]\\\\\n",
    "&=\\frac{1}{(1+\\lambda)^2} trace E_{\\mathcal{D}, x}\\left[Z^{-T}\\Phi(x)\\Phi^T(x)Z^{-1}\\right]E_{\\epsilon}\\left[\\epsilon\\epsilon^T\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}, x}\\left[Z^{-T}\\Phi(x)\\Phi^T(x)Z^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2}  E_{\\mathcal{D}, x}trace\\left[Z^{-T}\\Phi(x)\\Phi^T(x)Z^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2}  E_{\\mathcal{D}, x}trace\\left[Z^{-1}Z^{-T}\\Phi(x)\\Phi^T(x)\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}, x}\\left[Z^{-1}Z^{-T}\\Phi(x)\\Phi^T(x)\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}}\\left[Z^{-1}Z^{-T}\\right]E_{x}\\left[\\Phi(x)\\Phi^T(x)\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}}\\left[(Z^TZ)^{-1}\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note that $\\frac{1}{N}Z^TZ = \\frac{1}{N}\\sum^N_{n=1}\\Phi(x_n)\\Phi^T(x_n)$ is the in-sample estimate of $\\Sigma_{\\Phi} = E[\\Phi(x)\\Phi^T(x)] = I_{(d+1)\\times(d+1)}$, so by the law of large numbers, $\\frac{1}{N}Z^TZ = \\Sigma_{\\Phi} + o(1)$, so $\\frac{1}{N}Z^TZ = \\Sigma_{\\Phi} + o(1) = I + o(1)$.\n",
    "\n",
    "Then when $N$ is large\n",
    "\n",
    "\\begin{align*}\n",
    "var &= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}}\\left[(Z^TZ)^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace E_{\\mathcal{D}}\\left[(NI)^{-1}\\right]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} trace [\\frac{1}{N}I_{(d+1)\\times(d+1)}]\\\\\n",
    "&= \\frac{\\sigma^2}{(1+\\lambda)^2} \\frac{d+1}{N}\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (c) Combine part(a) and part(b),we have \n",
    "\n",
    "\\begin{align*}\n",
    "E[E_{out}] &= bias + var + \\sigma^2\\\\\n",
    "&= \\frac{\\lambda^2}{(1+\\lambda)^2}\\|w_f\\|^2 + \\frac{\\sigma^2}{(1+\\lambda)^2} \\frac{d+1}{N} + \\sigma^2\\\\\n",
    "&= \\frac{\\lambda^2\\|w_f\\|^2 +\\sigma^2\\frac{d+1}{N}}{(1+\\lambda)^2} + \\sigma^2\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Take derivative w.r.t. $\\lambda$, and set it equal to zero, we have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{dE[E_{out}]}{d\\lambda} &= \\frac{1}{(1+\\lambda)^4} \\left(2\\|w_f\\|^2\\lambda(1+\\lambda)^2 - 2(1+\\lambda)(\\|w_f\\|^2\\lambda^2+\\sigma^2\\frac{d+1}{N})\\right)\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^4} \\left(\\|w_f\\|^2\\lambda(1+\\lambda) - (\\|w_f\\|^2\\lambda^2+\\sigma^2\\frac{d+1}{N})\\right)\\\\\n",
    "&= \\frac{1}{(1+\\lambda)^4} \\left(\\|w_f\\|^2\\lambda - \\sigma^2\\frac{d+1}{N}\\right)\\\\\n",
    "&= 0\n",
    "\\end{align*}\n",
    "\n",
    "We find the optimal $\\lambda^* = \\frac{\\sigma^2(d+1)}{N\\|w_f\\|^2} = \\frac{\\frac{d+1}{N}}{\\frac{\\|w_f\\|^2}{\\sigma^2}}$\n",
    "\n",
    "* (d) From the optimal $\\lambda$, we see that\n",
    "  * If the model is complex, i.e. $d$ is large, we need more regularization, since it's easier to overfit, i.e. $\\lambda$ is large. \n",
    "  * More data, $N$ is large, requires less regularization\n",
    "  * With high variance of noise, it's easier to overfit, since it takes more efforts in weights to fit the noises, so need larger regularization.\n",
    "  * We can't control $w_f$, which is the true target function. But with larger $\\|w_f\\|$, it requires less regularization. Larger $\\|w_f\\|$ generally indicates more complex target function, we need reduce regularization to fit better the target."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.19\n",
    "\n",
    "* (a) Let's formulate the Lasso algorithm as a quadratic program.\n",
    "\n",
    "\\begin{align*}\n",
    "\\min E_{in}(w) &= \\frac{1}{N}\\|Xw-y\\|^2\\\\\n",
    "&= \\frac{1}{N}(w^TX^TXw -2y^TXw + y^Ty)\\\\\n",
    "\\text{ subject to} \\sum^d_{i=0}|w_i| &\\le C\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Assume we have $t_1,\\dots,t_d$ such that $|w_i| \\le t_i$ for $i=1,2,\\dots,d$. Then the above problem is equivalent to\n",
    "\n",
    "\\begin{align*}\n",
    "\\min E_{in}(w) &= \\frac{1}{N}\\|Xw-y\\|^2\\\\\n",
    "&= \\frac{1}{N}w^TX^TXw -2y^TXw + y^Ty\\\\\n",
    "\\text{ subject to} \\sum^d_{i=0}t_i &\\le C\\\\\n",
    "-t_i \\le &w_i \\le t_i \\text{ for } i=1,\\dots,d\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Note the last constraints indicate that $t_i \\ge 0$ for $i=1,\\dots,d$, otherwise there won't a solution $w_i$ for negative $t_i$.\n",
    "\n",
    "The above problem is a linearly constrained quadratic program. \n",
    "\n",
    "Now let's look at the optimization problem with $l_2$ constraints (The Ridge Algorithm).\n",
    "\n",
    "\\begin{align*}\n",
    "\\min E_{in}(w) &= \\frac{1}{N}\\|Xw-y\\|^2\\\\\n",
    "&= \\frac{1}{N}w^TX^TXw -2y^TXw + y^Ty\\\\\n",
    "\\text{ subject to} \\sum^d_{i=0}w^2_i &\\le C\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This is a quadratically constrained quadratic programm(QCQP)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Problem 4.19\n",
    "def run_one_experiment_419(Q, xs, ys, test_xs, test_ys, algo_name,reg_type, reg_param, solver=None):\n",
    "    \"\"\"Generate one set of training data for a given model and compute the test error\n",
    "    \"\"\"\n",
    "    lr = lm.LinearRegression(algo_name, reg_type, reg_param, poly_degree = Q, solver=solver)\n",
    "    lr.fit(xs, ys)\n",
    "    Eout = lr.calc_error(test_xs, test_ys)\n",
    "    return Eout, lr\n",
    "\n",
    "def calc_avg_Eout_419(num_exps, Q, N, sigma_square, normalized_aqs, \n",
    "                      test_xs, test_ys, algo_name, reg_type, reg_params, solver=None):\n",
    "    \n",
    "    all_Eouts = np.zeros((num_exps, len(reg_params)))\n",
    "    for i in np.arange(num_exps):#For each set of data, we change the regularization parameters\n",
    "        xs, ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "        for j, reg_param in enumerate(reg_params):\n",
    "            Eout, lr = run_one_experiment_419(Q, xs, ys, test_xs, test_ys, \n",
    "                                              algo_name, reg_type, reg_param, solver)\n",
    "            all_Eouts[i, j] = Eout\n",
    "            \n",
    "    avg_Eouts = np.zeros(len(reg_params))\n",
    "    for j, _ in enumerate(reg_params):\n",
    "        avg_Eouts[j] = np.mean(all_Eouts[:, j])\n",
    "    return avg_Eouts, all_Eouts\n",
    "\n",
    "def prepare_data(Qf, sigma_square, Q, N):\n",
    "    normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "\n",
    "    # Use for plotting\n",
    "    x_true = np.arange(-1, 1, 0.01).reshape(-1, 1)\n",
    "    # Noiseless y\n",
    "    y_true = data.calc_legendre_array(normalized_aqs, x_true)  \n",
    "\n",
    "    # Training data\n",
    "    xs, ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    \n",
    "    # Test data\n",
    "    test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "    \n",
    "    # Sorted for plotting\n",
    "    xs_sorted = xs.flatten()\n",
    "    ys_sorted = ys.flatten()\n",
    "    xs_ids = np.argsort(xs_sorted)\n",
    "    xs_sorted = xs_sorted[xs_ids]\n",
    "    ys_sorted = ys_sorted[xs_ids]\n",
    "    \n",
    "    return x_true, y_true, xs, ys, test_xs, test_ys"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lasso: Compare with scikit-learn LinearRegression with big $C$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.33624621e-07 6.15415252e-06 1.13987953e-06 2.44560916e-06\n",
      " 2.58522872e-06 2.12605433e-06 2.39707024e-06 2.42647636e-06\n",
      " 1.98811250e-06 2.76511140e-06 1.69386595e-06 3.06777513e-06\n",
      " 1.48181070e-06 3.34693263e-06 1.31943478e-06 3.62647852e-06]\n"
     ]
    }
   ],
   "source": [
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15 # Hypothesis set\n",
    "N = 30 # Training set size\n",
    "\n",
    "#Fix training data\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "algo_name = 'lasso'\n",
    "reg_type = 'Ivanov'\n",
    "\n",
    "C = 1e+16 # Huge C means no regularization\n",
    "lambda_t = 0 # No regularization \n",
    "X = data.polynomial_transform(Q, xs)\n",
    "w_sklearn_lasso = lm.sklearn_linear_regression(X, ys)\n",
    "w_lasso = lm.lasso_fit_ivanov(X, ys, C)\n",
    "relative_diff = np.abs((w_sklearn_lasso.flatten()-w_lasso.flatten())/w_sklearn_lasso.flatten())\n",
    "print(relative_diff)\n",
    "assert(np.allclose(relative_diff, np.zeros(len(relative_diff)), rtol = 1.0e-5, atol=1.0e-5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lasso: Out-of-Sample error vs. regularization parameter C\n",
    "\n",
    "Expect the out-of-sample error to increase with increasing $C$, as larger $C$ indicates less regularization, thus larger generalization error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C =  1 Eout:  [0.92886592]\n",
      "C =  100 Eout:  [0.58907259]\n",
      "C =  10000 Eout:  [0.91605687]\n",
      "C =  1000000 Eout:  [53.02694773]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x720 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15 # Hypothesis set\n",
    "N = 30 # Training set size\n",
    "\n",
    "#Fix training data\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "f, axarr = plt.subplots(2, 2, figsize=(12, 10))\n",
    "\n",
    "algo_name = 'lasso'\n",
    "reg_type = 'Ivanov'\n",
    "solver= None\n",
    "Cs = [1, 100, 10000, 1000000]\n",
    "for idx, C in enumerate(Cs):\n",
    "    i = int(idx / 2)\n",
    "    j = int(idx - i*2)\n",
    "    #print(idx, i, j)\n",
    "    lr = lm.LinearRegression(algo_name, reg_type, reg_param = C, poly_degree = Q, solver=solver)\n",
    "    lr.fit(xs, ys)\n",
    "    Eout = lr.calc_error(test_xs, test_ys)\n",
    "    print('C = ', C, 'Eout: ', Eout)\n",
    "    ytrain_pred = lr.predict(x_true)\n",
    "\n",
    "    axarr[i,j].plot(xs, ys, '.', label='Training Data')\n",
    "    axarr[i,j].plot(x_true, ytrain_pred, label='Prediction')\n",
    "    axarr[i,j].plot(x_true, y_true, label='True Target')\n",
    "    axarr[i,j].legend()\n",
    "    axarr[i,j].set_title(\"C = \"+str(C))\n",
    "    \n",
    "# Fine-tune figure; make subplots farther from each other.    \n",
    "f.subplots_adjust(hspace=0.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lasso vs. Ridge Ivanov Regularization: Expected out-of-sample errors versus regularization parameter $C$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "#### Problem 4.19 (a)\n",
    "\n",
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "\n",
    "Q = 15\n",
    "N = 30\n",
    "num_exps = 1000\n",
    "\n",
    "max_C1, max_C2, max_C3 = 1, 10, 20\n",
    "Cs1 = np.arange(0, max_C1+1, 0.05)\n",
    "Cs2 = np.arange(0, max_C2+1, 1)\n",
    "#Cs3 = np.arange(0, max_C3+1, 100)\n",
    "Cs = np.unique(np.concatenate([Cs1[1:], Cs2[1:]]))#, Cs3[1:]]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### N.B. When C is too large, some randome set of data will generate huge E_{out} for the test data set.\n",
    "In the order of 10^9. This defeats the purpose of comparing E_{out} vs. C with Ridge.\n",
    "so we limited the range of C here to be less than 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "algo_name = 'lasso'\n",
    "reg_type = 'Ivanov'\n",
    "\n",
    "lasso_avg_Eouts, lasso_all_Eouts = calc_avg_Eout_419(num_exps, Q, N, sigma_square, normalized_aqs, \n",
    "                              test_xs, test_ys, algo_name, reg_type, Cs)\n",
    "\n",
    "df = pd.DataFrame({'lambda':Cs, 'avg_Eout':lasso_avg_Eouts})\n",
    "df.to_csv('./lasso_ivanov_Eout_C', index=False)\n",
    "np.savetxt(\"lasso_ivanov_all_Eouts.csv\", lasso_all_Eouts, delimiter=\",\", fmt='%12.8f')    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "algo_name = 'ridge'\n",
    "reg_type = 'Ivanov'\n",
    "\n",
    "ridge_avg_Eouts, ridge_all_Eouts = calc_avg_Eout_419(num_exps, Q, N, sigma_square, normalized_aqs, \n",
    "                              test_xs, test_ys, algo_name, reg_type, Cs)\n",
    "\n",
    "df = pd.DataFrame({'lambda':Cs, 'avg_Eout':ridge_avg_Eouts})\n",
    "df.to_csv('./ridge_ivanov_Eout_C', index=False)\n",
    "np.savetxt(\"ridge_ivanov_all_Eouts.csv\", ridge_all_Eouts, delimiter=\",\", fmt='%12.8f')    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(Cs, lasso_avg_Eouts, label='lasso: Average E_out')\n",
    "plt.plot(Cs, ridge_avg_Eouts, label='ridge: Average E_out')\n",
    "plt.legend()\n",
    "plt.title(\"Lasso vs Ridge with Ivanov Regularization: Average E_out vs. C\")\n",
    "#plt.ylim(0, 10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.19 (a)\n",
    "\n",
    "From the graph above, we see that the expected out-of-sample errors are larger from lasso most of the time. Except at the very small $C$ values, both errors increase with the increasing $C$, which is expected, as larger $C$ indicates smaller regularization. If $C$ is too small, training data gives worse fit, so the out-of-sample errors increase instead. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.19 Ridge Regression\n",
    "##### Validation with scikit-learn Ridge Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15\n",
    "N = 30\n",
    "\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "X = data.polynomial_transform(Q, xs)\n",
    "lambda_t = 0.1\n",
    "w_sklearn_ridge = lm.sklearn_ridge(X, ys, lambda_t, False, 'cholesky')\n",
    "w_ridge = lm.ridge_fit_tikhonov(X, ys, lambda_t)\n",
    "assert(np.allclose(w_sklearn_ridge.flatten(), w_ridge.flatten(), rtol = 1.0e-12, atol=1.0e-12))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ridge: Out-of-sample error versus regularization parameter $\\lambda$\n",
    "\n",
    "Check the general trend of out-of-sample error with $\\lambda$. It should decrease with increasing $\\lambda$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lambda =  0.001 Eout:  [2.22575402]\n",
      "lambda =  0.01 Eout:  [2.25472121]\n",
      "lambda =  0.1 Eout:  [1.67967964]\n",
      "lambda =  1 Eout:  [1.3233053]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x720 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Ridge Regression: Expected out-of-sample error vs. lambda\n",
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15 #hypothesis model\n",
    "N = 30 # training set size\n",
    "    \n",
    "#Fix training data\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "algo_name = 'ridge'\n",
    "reg_type = 'Tikhonov'\n",
    "lambdas = [0.001, 0.01, 0.1, 1]\n",
    "f, axarr = plt.subplots(2, 2, figsize=(12, 10))\n",
    "for idx, lambda_t in enumerate(lambdas):\n",
    "    i = int(idx / 2)\n",
    "    j = int(idx - i*2)\n",
    "    lambda_t = lambda_t\n",
    "    lr = lm.LinearRegression(algo_name, reg_type, reg_param = lambda_t, poly_degree = Q)\n",
    "    lr.fit(xs, ys)\n",
    "    Eout = lr.calc_error(test_xs, test_ys)\n",
    "    print('lambda = ', lambda_t, 'Eout: ', Eout)\n",
    "    ytrain_pred = lr.predict(x_true)\n",
    "\n",
    "    axarr[i,j].plot(xs, ys, '.', label='Training Data')\n",
    "    axarr[i,j].plot(x_true, ytrain_pred, label='Prediction')\n",
    "    axarr[i,j].plot(x_true, y_true, label='True Target')\n",
    "    axarr[i,j].set_ylim(-5,5)\n",
    "    axarr[i,j].legend()\n",
    "    axarr[i,j].set_title(\"lambda = \"+str(lambda_t))\n",
    "    \n",
    "# Fine-tune figure; make subplots farther from each other.    \n",
    "f.subplots_adjust(hspace=0.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ridge: Expected out-of-sample errors versus the change of regularization parameter $\\lambda$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ridge with Tikhonov regularization\n",
    "max_lambda1, max_lambda2, max_lambda3 = 0.01,0.1,10\n",
    "lambdas1 = np.arange(0, max_lambda1+1, 0.001)\n",
    "lambdas2 = np.arange(0, max_lambda2+1, 0.01)\n",
    "lambdas3 = np.arange(0, max_lambda3+1, 0.2)\n",
    "lambdas = np.unique(np.concatenate([lambdas1[1:], lambdas2[1:], lambdas3[1:]]))\n",
    "\n",
    "algo_name = 'ridge'\n",
    "reg_type = 'Tikhonov'\n",
    "Q = 15\n",
    "N = 30\n",
    "\n",
    "num_exps = 1000\n",
    "avg_Eouts, all_Eouts = calc_avg_Eout_419(num_exps, Q, N, sigma_square, normalized_aqs, \n",
    "                                         test_xs, test_ys, algo_name, reg_type, lambdas)\n",
    "\n",
    "df = pd.DataFrame({'lambda':lambdas, 'avg_Eout':avg_Eouts})\n",
    "df.to_csv('./ridge_tikhonov_Eout_lambda', index=False)\n",
    "np.savetxt(\"ridge_tikhonov_all_Eouts.csv\", all_Eouts, delimiter=\",\", fmt='%12.8f')    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "stop_idx = 1025\n",
    "plt.plot(lambdas[:stop_idx], avg_Eouts[:stop_idx], label='Average E_out')\n",
    "plt.legend()\n",
    "plt.title(\"Ridge Tikhonov: Average E_out vs. lambda\")\n",
    "plt.ylim(1.2, 2.0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ridge Ivanov regularization vs. scikit-learn Ridge"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.41508527e-07 7.78314827e-07 7.93856740e-07 8.70244579e-07\n",
      " 6.70303323e-07 1.04070218e-06 5.31795165e-07 1.17941685e-06\n",
      " 4.56799707e-07 1.29426398e-06 4.16114736e-07 1.40503021e-06\n",
      " 3.92486871e-07 1.52480311e-06 3.77694769e-07 1.66109489e-06]\n"
     ]
    }
   ],
   "source": [
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15\n",
    "N = 30\n",
    "\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "X = data.polynomial_transform(Q, xs)\n",
    "lambda_t = 0\n",
    "w_sklearn_ridge = lm.sklearn_ridge(X, ys, lambda_t, False, 'cholesky')\n",
    "C = 1e+16\n",
    "#print(w_sklearn_ridge)\n",
    "w_ridge_ivanov = lm.ridge_fit_ivanov(X, ys, C)\n",
    "#print(w_ridge_ivanov)\n",
    "relative_diff = np.abs((w_sklearn_ridge.flatten()-w_ridge_ivanov.flatten())/w_sklearn_ridge.flatten())\n",
    "print(relative_diff)\n",
    "assert(np.allclose(relative_diff, np.zeros(len(relative_diff)), rtol = 1.0e-5, atol=1.0e-5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ridge Regression with Ivanov regularization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C =  1 Eout:  [0.73860002]\n",
      "C =  100 Eout:  [1.31425145]\n",
      "C =  10000 Eout:  [11.89738451]\n",
      "C =  100000 Eout:  [38.84795292]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x720 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Ridge Regression: Expected out-of-sample error vs. lambda\n",
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 15 #hypothesis model\n",
    "N = 30 # training set size\n",
    "    \n",
    "#Fix training data\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "algo_name = 'ridge'\n",
    "reg_type = 'Ivanov'\n",
    "Cs = [1, 100, 10000, 100000]\n",
    "f, axarr = plt.subplots(2, 2, figsize=(12, 10))\n",
    "for idx, C in enumerate(Cs):\n",
    "    i = int(idx / 2)\n",
    "    j = int(idx - i*2)\n",
    "    lr = lm.LinearRegression(algo_name, reg_type, reg_param = C, poly_degree = Q)\n",
    "    lr.fit(xs, ys)\n",
    "    Eout = lr.calc_error(test_xs, test_ys)\n",
    "    print('C = ', C, 'Eout: ', Eout)\n",
    "    ytrain_pred = lr.predict(x_true)\n",
    "\n",
    "    axarr[i,j].plot(xs, ys, '.', label='Training Data')\n",
    "    axarr[i,j].plot(x_true, ytrain_pred, label='Prediction')\n",
    "    axarr[i,j].plot(x_true, y_true, label='True Target')\n",
    "    axarr[i,j].set_ylim(-5,5)\n",
    "    axarr[i,j].legend()\n",
    "    axarr[i,j].set_title(\"C = \"+str(C))\n",
    "    \n",
    "# Fine-tune figure; make subplots farther from each other.    \n",
    "f.subplots_adjust(hspace=0.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.19 (b) \n",
    " \n",
    " The augmented error is $E_{aug} = \\frac{1}{N}\\|Xw-y\\|^2 + \\lambda_C w^Tw$, where $\\lambda_C$ depends on the value of $C$. This is more convenient to optimize than the Ivanov regularization because it's an optimization problem without any constraint. \n",
    " \n",
    "#### Problem 4.19 (c) \n",
    "\n",
    "Let $d=5$, and $N=3$ see below comparision of the weights between lasso and ridge (quadratic penalty). From the picture below, we see that in the lasso algorithm there are 4 out of 6 $w$ that are close to zero, while for ridge, none of them are close to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------ Lasso ------\n",
      "[[-2.91276038e-01]\n",
      " [ 7.08723907e-01]\n",
      " [ 2.63687406e-09]\n",
      " [ 3.43743114e-09]\n",
      " [ 1.82215649e-09]\n",
      " [ 1.35377456e-09]]\n",
      "------ Ridge ------\n",
      "[[-0.47689469]\n",
      " [ 0.80239241]\n",
      " [ 0.21666993]\n",
      " [ 0.23556808]\n",
      " [ 0.1313591 ]\n",
      " [ 0.09510258]]\n"
     ]
    }
   ],
   "source": [
    "Qf = 15 #target function\n",
    "sigma_square = 0.5\n",
    "test_N = 100 #number of test samples\n",
    "Q = 5 # Hypothesis set\n",
    "N = 3 # Training set size\n",
    "C = 1 \n",
    "\n",
    "#Fix training data\n",
    "x_true, y_true, xs, ys, test_xs, test_ys = prepare_data(Qf, sigma_square, Q, N)\n",
    "\n",
    "X = data.polynomial_transform(Q, xs)\n",
    "w_lasso = lm.lasso_fit_ivanov(X, ys, C)\n",
    "w_ridge_ivanov = lm.ridge_fit_ivanov(X, ys, C)\n",
    "print('------ Lasso ------')\n",
    "print(w_lasso)\n",
    "print('------ Ridge ------')\n",
    "print(w_ridge_ivanov)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.20\n",
    "\n",
    "* (a) For transformed problem, we have $Z = XA^T$\n",
    "\n",
    "\\begin{align*}\n",
    "E_{in}(w) &= \\frac{1}{N}\\|Zw - \\bar{y}\\|^2 \\\\\n",
    "&= \\frac{1}{N}(Zw - \\bar{y})^T(Zw - \\bar{y})\\\\\n",
    "&= \\frac{1}{N}(w^TZ^T - \\bar{y}^T)(Zw - \\bar{y})\\\\\n",
    "&= \\frac{1}{N}(w^TZ^TZw - w^TZ^T\\bar{y} - \\bar{y}^TZw + \\bar{y}^T\\bar{y})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "The optimal weights are \n",
    "\n",
    "\\begin{align*}\n",
    "\\bar{w} &= (Z^TZ)^{-1}Z^T\\bar{y}\\\\\n",
    "&= \\left((XA^T)^TXA^T\\right)^{-1}(XA^T)^T\\alpha y\\\\\n",
    "&= \\left(AX^TXA^T\\right)^{-1}AX^T\\alpha y\\\\\n",
    "&= (A^T)^{-1}(X^TX)^{-1}A^{-1}AX^T\\alpha y\\\\\n",
    "&= \\alpha(A^T)^{-1}(X^TX)^{-1}X^Ty\\\\\n",
    "&= \\alpha(A^T)^{-1}w\\\\\n",
    "\\end{align*}\n",
    "\n",
    "where we have used the fact that $A$ is invertible.\n",
    "\n",
    "* (b) The regularized solution for original data is:\n",
    "\n",
    "\\begin{align*}\n",
    "w_{reg}(\\lambda) &= (X^TX+\\lambda X^TX)^{-1}X^Ty\\\\\n",
    "&= (X^T(1+\\lambda)X)^{-1}X^Ty\\\\\n",
    "&= \\frac{1}{1+\\lambda}(X^TX)^{-1}X^Ty\\\\\n",
    "&= \\frac{1}{1+\\lambda}w\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So for the transformed data, the regularized solution is: \n",
    "\n",
    "\\begin{align*}\n",
    "\\tilde{w}_{reg}(\\lambda) &= (Z^TZ+\\lambda Z^TZ)^{-1}Z^T\\bar{y}\\\\\n",
    "&=\\frac{1}{1+\\lambda}\\bar{w}\\\\\n",
    "&=\\frac{1}{1+\\lambda}\\alpha(A^T)^{-1}w\\\\\n",
    "&=\\alpha(A^T)^{-1}\\frac{1}{1+\\lambda}w\\\\\n",
    "&=\\alpha(A^T)^{-1}w_{reg}(\\lambda)\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.21 TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.22 \n",
    "\n",
    "For $N=100$ data points, we have $M=100$ hypothesis sets (models) $\\mathcal{H}_m$, where $m=1,2,\\dots, M$. All models have VC dimension $d = 10$. \n",
    "\n",
    "If we set aside $K=25$ points for validation and select the model $m^*$ (whose corresponding final hypothesis for this validation data is $g^-_{m^*}$), which produced minimum validation error of $E_{val}(g^-_{m^*}) = 0.25$. \n",
    "\n",
    "The out-of-sample error for this selected function $E_{out}(g^-_{m^*})$ is then bounded by equation (4.11):\n",
    "\n",
    "\\begin{align*}\n",
    "E_{out}(g^-_{m^*}) &\\le E_{val}(g^-_{m^*}) + O\\left(\\sqrt{\\frac{\\ln M}{K}}\\right)\\\\\n",
    "&= 0.25 + O\\left(\\sqrt{\\frac{\\ln 100}{25}}\\right)\\\\\n",
    "&= 0.25 + 0.4292\\\\\n",
    "&= 0.6792\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Now instead we train each model on all the data and selected the function with minimum in-sample error. Our hypothesis set is actually the union of the 100 hypothesis sets. According to the bound in problem 2.14, the VC dimension of the union is bounded by:\n",
    "\n",
    "\\begin{align*}\n",
    "d_{VC}(\\mathcal{H}) &= O\\left(max(d_{VC}, K)\\log_2 max(d_{VC}, K)\\right)\\\\\n",
    "&= O\\left(max(10, 100)\\log_2 max(10, 100)\\right)\\\\\n",
    "&= O\\left(100\\log_2 100\\right)\\\\\n",
    "&\\approx 664\n",
    "\\end{align*}\n",
    "\n",
    "Then the growth function of the union is $m_{\\mathcal{H}}(N) \\le N^{d_{VC}} + 1$ according to formula (2.10).\n",
    "\n",
    "The resulting in-sample error $E_{in}(g_{m^*}) = 0.15$. The out-of-sample error in this case is then bounded by the VC generalization bound, i.e. formula (2.12) and formula (3.1):\n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "E_{out}(g_{m^*}) &\\le E_{in}(g_{m^*}) + \\sqrt{\\frac{8}{N}\\ln\\frac{4m_{\\mathcal{H}}(2N)}{\\delta}}\\\\\n",
    "&\\le E_{in}(g_{m^*}) + O\\left(\\sqrt{\\frac{d_{VC}(\\mathcal{H})}{N}\\ln N}\\right)\\\\\n",
    "&= 0.15 + \\sqrt{\\frac{664}{100}\\ln 100}\\\\\n",
    "&\\approx 5.680\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This is a much higher upper bound than the one obtained from validation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.23\n",
    "\n",
    "* (a) $E_{cv} = \\frac{1}{N}\\sum^N_{n=1}e_n = \\frac{1}{N}\\sum^N_{n=1} e\\left(g^-_n(x_n), y_n\\right)$, then by the formula of variance-covariance: $Var(ax+by) = a^2Var(x) + b^2Var(y) + 2abCov(x,y)$. We have\n",
    "\n",
    "\\begin{align*}\n",
    "Var_{\\mathcal{D}}[E_{cv}] &= Var_{\\mathcal{D}}\\left[\\frac{1}{N}\\sum^N_{n=1}e_n\\right]\\\\\n",
    "&= \\frac{1}{N^2}\\sum^N_{n=1}Var_{\\mathcal{D}}(e_n) + \\frac{1}{N^2}\\sum^N_{m\\ne n}Cov_{\\mathcal{D}}(e_m, e_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (b) \n",
    "\n",
    "For a given $n$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D}}[e_n] &= E_{\\mathcal{D}}[e\\left(g^-_n(x_n), y_n\\right)]\\\\\n",
    "&= E_{\\mathcal{D}}[e\\left(g^{N-2}(x_n)+\\delta_n, y_n\\right)]\\\\\n",
    "&= E_{\\mathcal{D}}[e\\left(g^{N-2}(x_n), y_n\\right) + k(\\delta_n)]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $k(\\delta_n)$ is a function in the power of $\\delta_n$. We have also used the assumption that for well behaved models, the learning process is 'stable', and so the change in the learned hypothesis should be small, $O(\\frac{1}{N})$. If a new data point $n$ is added to the data set of size $N$, so we have $g^-_n(x_n) = g^{N-2}(x_n) + \\delta_n$, we assume that any terms involving $\\delta_n$ are $O(\\frac{1}{N})$, we have $E[e\\left(g^{N-2}(x_n) + \\delta_n, y_n\\right)] = E[e(g^{N-2}(x_n), y_n)] + O(\\frac{1}{N})$\n",
    "\n",
    "\\begin{align*}\n",
    "E_{\\mathcal{D}}[e_n] &= E_{\\mathcal{D}}[e\\left(g^{N-2}(x_n), y_n\\right)+ k(\\delta_n)]\\\\\n",
    "&= E_{\\mathcal{D}_{N-1}}E_{(x_n,y_n)}[e\\left(g^{N-2}(x_n), y_n\\right)+ k(\\delta_n)]\\\\\n",
    "&= E_{\\mathcal{D}_{N-1}}\\left[E_{out}(g^{N-2})+O(\\frac{1}{N})\\right]\\\\\n",
    "&= E_{\\mathcal{D}_{N-2}}E_{out}(g^{N-2})+O(\\frac{1}{N})\\\\\n",
    "&= \\bar{E}_{out}(g^{N-2})+O(\\frac{1}{N})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The last step is because $g^{N-2}$ only depends on $\\mathcal{D}_{N-2}$.\n",
    "\n",
    "To compute the covariance, we have\n",
    "\n",
    "\\begin{align*}\n",
    "Cov_{\\mathcal{D}}(e_m, e_n) &= E_{\\mathcal{D}}\\left[\\left(e_m-E[e_m]\\right)\\left(e_n - E[e_n]\\right)\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\left(e_m-\\bar{E}_{out}(g^{N-2})-O(\\frac{1}{N})\\right)\\left(e_n - \\bar{E}_{out}(g^{N-2})-O(\\frac{1}{N})\\right)\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)-O(\\frac{1}{N})\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)-O(\\frac{1}{N})\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)+O(\\frac{1}{N^2})\\right]\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)\\right] \\\\\n",
    "&- O(\\frac{1}{N})E_{\\mathcal{D}}\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)-O(\\frac{1}{N})E_{\\mathcal{D}}\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)+O(\\frac{1}{N^2})\\\\\n",
    "&= E_{\\mathcal{D}}\\left[\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)\\right] +O(\\frac{1}{N^2})\\\\\n",
    "&= E_{\\mathcal{D}^{N-2}}E_{(x_m,y_m),(x_n,y_n)}\\left[\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)\\right] +O(\\frac{1}{N^2})\\\\\n",
    "&= E_{\\mathcal{D}^{N-2}}\\left[E_{(x_m,y_m)}\\left(e_m-\\bar{E}_{out}(g^{N-2})\\right)E_{(x_n,y_n)}\\left(e_n - \\bar{E}_{out}(g^{N-2})\\right)\\right] +O(\\frac{1}{N^2})\\\\\n",
    "&= E_{\\mathcal{D}^{N-2}}\\left[\\left(E_{out}(g^{N-2})+O(\\frac{1}{N})-\\bar{E}_{out}(g^{N-2})\\right)\\left(E_{out}(g^{N-2})+O(\\frac{1}{N}) - \\bar{E}_{out}(g^{N-2})\\right)\\right]\\\\\n",
    "&= Var_{\\mathcal{D}^{N-2}}[E_{out}(g^{N-2})]+O(\\frac{1}{N^2})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (c) If we assume that terms involving $\\delta_n$, $\\delta_m$ are $O(\\frac{1}{N})$, then we have $Cov_{\\mathcal{D}}(e_m, e_n) \\approx  Var_{\\mathcal{D}}[E_{out}(g)]$\n",
    "\n",
    "\\begin{align*}\n",
    "Var_{\\mathcal{D}}[E_{cv}] &= \\frac{1}{N^2}\\sum^N_{n=1}Var_{\\mathcal{D}}(e_n) + \\frac{1}{N^2}\\sum^N_{m\\ne n}Cov_{\\mathcal{D}}(e_m, e_n)\\\\\n",
    "&= \\frac{1}{N}Var_{\\mathcal{D}}(e_1) + \\frac{1}{N^2}\\sum^N_{m\\ne n}Var_{\\mathcal{D}^{N-2}}[E_{out}(g^{N-2})]+O(\\frac{1}{N^2})\\\\\n",
    "&= \\frac{1}{N}Var_{\\mathcal{D}}(e_1) + (1-\\frac{1}{N})Var_{\\mathcal{D}}[E_{out}(g)]+O(\\frac{1}{N^2})\\\\\n",
    "&\\approx \\frac{1}{N}Var_{\\mathcal{D}}(e_1) +  Var_{\\mathcal{D}}[E_{out}(g)]+O(\\frac{1}{N})\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Since $e_1 = e(g^-_1(x_1), y_1)$, \n",
    "\n",
    "$Var_{\\mathcal{D}}[e_1] = E_{\\mathcal{D}}[e_1 -E_{\\mathcal{D}}[e_1]]^2 =  E_{\\mathcal{D}}[e_1 -\\bar{E}_{out}(g^{N-1})]^2 \\approx E_{\\mathcal{D}}[e_1 -\\bar{E}_{out}(g)]^2 $ when $N$ is large.\n",
    "\n",
    "$Var_{\\mathcal{D}}[E_{out}(g)] = E_{\\mathcal{D}}[E_{out}(g) -E_{\\mathcal{D}}[E_{out}(g)]]^2 = E_{\\mathcal{D}}[E_{out}(g) -\\bar{E}_{out}(g)]^2$\n",
    "\n",
    "\n",
    "$Var_{\\mathcal{D}}[e_1]$ doesn't decay to zero with $N$, because regardless of the value of $N$, we always evaluate $e_1$ on a single data point.\n",
    "$Var_{\\mathcal{D}}[E_{out}(g)]$ however, decays to zero, because as $N$ increases, the $g$ becomes better approximation to the target function, $E_{out}(g) = E_x[e]$ becomes closer to $\\bar{E}_{out}(g)$\n",
    "\n",
    "so $Var_{\\mathcal{D}}[E_{cv}]$ decays to 0 with $N$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.23 (d)\n",
    "\n",
    "From the picture below, we plot the log-log-plot of $Var_{\\mathcal{D}}[E_{cv}]/Var_{\\mathcal{D}}[e_1]$ vs. $N$, where $N=50, 55, \\dots, 200$. One can see that the decay rate is close to -1. This means that as $N$ goes large, we have $Var_{\\mathcal{D}}[E_{cv}] \\approx \\frac{1}{N}Var_{\\mathcal{D}}[e_1]$. This is because $Var_{\\mathcal{D}}[E_{out}(g)] \\approx 0$ as we conclude from problem (c). This says that if $N$ is large, the correlations between $e_n$ and $e_m$ can be ignored. We can treat $e_n$s as independent. This shows the reduction of variance of cross-validation from each variance of $e_n$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 354,
   "metadata": {},
   "outputs": [],
   "source": [
    "#### Problem 4.23 (d)\n",
    "\n",
    "import libs.data_util as data\n",
    "\n",
    "def run_one_crossvalidation_experiment(N, sigma_square, normalized_aqs, Q):\n",
    "    \"\"\"Generate one set of training data for a given model and compute the cross validation error\n",
    "    normalized_aqs: coefficients of the target function\n",
    "    N: The number of points in training set\n",
    "    sigma_square: variance of noise of data\n",
    "    Q: The degree of freedom of the hypothesis model\n",
    "    \"\"\"\n",
    "        \n",
    "    xs, ys = data.generate_data_set(N, normalized_aqs, sigma_square)\n",
    "    ens = [] #List of validation errors with each data removed, so it has e_1, e_2, ...\n",
    "    for idx in np.arange(len(ys)):\n",
    "        x = xs[idx,:].reshape(1,-1)\n",
    "        y = ys[idx]\n",
    "        \n",
    "        #build a data set without (x,y)\n",
    "        xs_minus = np.vstack((xs[:idx, :], xs[idx+1:,:]))\n",
    "        ys_minus = np.vstack((ys[:idx, :], ys[idx+1:,:]))\n",
    "        \n",
    "        #Train on the LOO (leave-one-out) data set\n",
    "        # Hypothesis with degree Q\n",
    "        Z = data.polynomial_transform(Q, xs_minus)\n",
    "        w = np.matmul(np.linalg.inv(np.matmul(Z.transpose(), Z)), np.matmul(Z.transpose(), ys_minus))\n",
    "\n",
    "        # Calculate out-of-sample error e_n on (x,y)\n",
    "        Eout = data.calc_Eout(w, x, y)\n",
    "        ens.append(Eout)\n",
    "\n",
    "    Ecv = np.mean(ens)\n",
    "    e1, e2 = ens[0], ens[1]\n",
    "    var_e = np.var(ens)\n",
    "    return Ecv, e1, e2, var_e\n",
    "\n",
    "\n",
    "def calc_avg_Ecv(num_exps, N, sigma_square, normalized_aqs, Q):\n",
    "    Ecvs = []\n",
    "    e1s, e2s = [], []\n",
    "    var_es = [] # \n",
    "    for ei in np.arange(num_exps):\n",
    "        res = run_one_crossvalidation_experiment(N, sigma_square, normalized_aqs, Q)\n",
    "        Ecv, e1, e2, var_e = res\n",
    "        Ecvs.append(Ecv)\n",
    "        e1s.append(e1)\n",
    "        e2s.append(e2)\n",
    "        var_es.append(var_e)\n",
    "    \n",
    "    mean_Ecv = np.mean(Ecvs)\n",
    "    mean_e1 = np.mean(e1s)\n",
    "    mean_e2 = np.mean(e2s)\n",
    "    mean_vare = np.mean(var_es)\n",
    "    \n",
    "    std_Ecv = np.std(Ecvs)\n",
    "    std_e1 = np.std(e1s)\n",
    "    std_e2 = np.std(e2s)\n",
    "    std_vare = np.std(var_es)\n",
    "    return mean_Ecv, std_Ecv, mean_e1, std_e1, mean_e2, std_e2, mean_vare, std_vare\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 358,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Working on N =  50\n",
      "Working on N =  55\n",
      "Working on N =  60\n",
      "Working on N =  65\n",
      "Working on N =  70\n",
      "Working on N =  75\n",
      "Working on N =  80\n",
      "Working on N =  85\n",
      "Working on N =  90\n",
      "Working on N =  95\n",
      "Working on N =  100\n",
      "Working on N =  105\n",
      "Working on N =  110\n",
      "Working on N =  115\n",
      "Working on N =  120\n",
      "Working on N =  125\n",
      "Working on N =  130\n",
      "Working on N =  135\n",
      "Working on N =  140\n",
      "Working on N =  145\n",
      "Working on N =  150\n",
      "Working on N =  155\n",
      "Working on N =  160\n",
      "Working on N =  165\n",
      "Working on N =  170\n",
      "Working on N =  175\n",
      "Working on N =  180\n",
      "Working on N =  185\n",
      "Working on N =  190\n",
      "Working on N =  195\n",
      "Working on N =  200\n"
     ]
    }
   ],
   "source": [
    "sigma_square = 0.25\n",
    "max_N = 200\n",
    "Ns = np.arange(50, max_N+1, 5)\n",
    "Qf = 10 #target function\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "\n",
    "Q = 2 #hypothesis model degree of freedom\n",
    "\n",
    "num_exps = 1000\n",
    "\n",
    "var_Ecvs = []\n",
    "var_e1s = []\n",
    "for N in Ns:\n",
    "    print('Working on N = ', N)\n",
    "    res = calc_avg_Ecv(num_exps, N, sigma_square, normalized_aqs, Q)\n",
    "    mean_Ecv, std_Ecv, mean_e1, std_e1, mean_e2, std_e2, mean_vare, std_vare = res\n",
    "    var_Ecvs.append(std_Ecv**2)\n",
    "    var_e1s.append(std_e1**2)\n",
    "print('done')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 360,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Coefficients: \n",
      " [-0.97622869]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "from sklearn import datasets, linear_model\n",
    "from sklearn.metrics import mean_squared_error, r2_score\n",
    "regr = linear_model.LinearRegression()\n",
    "\n",
    "ratios = [var_ecv/var_e1 for var_ecv, var_e1 in zip(var_Ecvs, var_e1s)]\n",
    "# Train the model using the training sets\n",
    "X_train = np.log(Ns).reshape(-1, 1)\n",
    "y_train = np.log(ratios)\n",
    "regr.fit(X_train, y_train)\n",
    "\n",
    "# Make predictions using the testing set\n",
    "y_pred = regr.predict(X_train)\n",
    "print('Coefficients: \\n', regr.coef_)\n",
    "\n",
    "\n",
    "#plt.plot(np.log(Ns), np.log(ratios), label='Ratio of Var(E_cv)/Var(E_1)')\n",
    "plt.scatter(X_train, y_train,  color='black', label='Ratio of Var(E_cv)/Var(E_1)')\n",
    "plt.plot(X_train, y_pred, color='blue', linewidth=3)\n",
    "plt.legend()\n",
    "#plt.yscale('log')\n",
    "#plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.24"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 371,
   "metadata": {},
   "outputs": [],
   "source": [
    "import libs.data_util as data\n",
    "\n",
    "def run_one_crossvalidation_experiment_normal(N, d, lambda_t, noise_sigma):\n",
    "    \"\"\"Generate one set of training data for a given model and compute the cross validation error\n",
    "    normalized_aqs: coefficients of the target function\n",
    "    N: The number of points in training set\n",
    "    sigma_square: variance of noise of data\n",
    "    Q: The degree of freedom of the hypothesis model\n",
    "    \"\"\"\n",
    "    \n",
    "    mu, sigma = 0, 0.1 # mean and standard deviation\n",
    "    X0 = np.random.normal(mu, sigma, (N, d))\n",
    "    ones = np.ones((N, 1)) \n",
    "    xs = np.hstack((ones, X0))# N x (d+1)\n",
    "    \n",
    "    \n",
    "    wf = np.random.normal(mu, sigma, (d+1, 1))\n",
    "    epsilons = np.random.normal(mu, sigma*noise_sigma, (N, 1))\n",
    "    ys = np.matmul(xs, wf) + epsilons\n",
    "    #print('xs, ys: ', xs.shape, ys.shape)\n",
    "    \n",
    "    ens = [] #List of validation errors with each data removed, so it has e_1, e_2, ...\n",
    "    for idx in np.arange(len(ys)):\n",
    "        x = xs[idx,:].reshape(1,-1)\n",
    "        y = ys[idx]\n",
    "        \n",
    "        #build a data set without (x,y)\n",
    "        xs_minus = np.vstack((xs[:idx, :], xs[idx+1:,:]))\n",
    "        ys_minus = np.vstack((ys[:idx, :], ys[idx+1:,:]))\n",
    "        \n",
    "        #Train on the LOO (leave-one-out) data set\n",
    "        inv_z = np.linalg.inv(np.matmul(xs_minus.transpose(), xs_minus)+lambda_t)\n",
    "        w = np.matmul(inv_z, np.matmul(xs_minus.transpose(), ys_minus))\n",
    "\n",
    "        # Calculate out-of-sample error e_n on (x,y)\n",
    "        pred_y = np.matmul(x, w)\n",
    "        test_err = (pred_y - y)\n",
    "        en = np.matmul(test_err.transpose(), test_err).flatten()/x.shape[0]        \n",
    "        ens.append(en)\n",
    "\n",
    "    Ecv = np.mean(ens)\n",
    "    e1, e2 = ens[0], ens[1]\n",
    "    var_e = np.var(ens)\n",
    "    return Ecv, e1, e2, var_e\n",
    "\n",
    "\n",
    "def calc_avg_Ecv_norm(num_exps, d, N, lambda_t, noise_sigma):\n",
    "    Ecvs = []\n",
    "    e1s, e2s = [], []\n",
    "    var_es = [] # \n",
    "    for ei in np.arange(num_exps):\n",
    "        res = run_one_crossvalidation_experiment_normal(N, d, lambda_t, noise_sigma)\n",
    "        Ecv, e1, e2, var_e = res\n",
    "        Ecvs.append(Ecv)\n",
    "        e1s.append(e1)\n",
    "        e2s.append(e2)\n",
    "        var_es.append(var_e)\n",
    "    \n",
    "    mean_Ecv = np.mean(Ecvs)\n",
    "    mean_e1 = np.mean(e1s)\n",
    "    mean_e2 = np.mean(e2s)\n",
    "    mean_vare = np.mean(var_es)\n",
    "    \n",
    "    std_Ecv = np.std(Ecvs)\n",
    "    std_e1 = np.std(e1s)\n",
    "    std_e2 = np.std(e2s)\n",
    "    std_vare = np.std(var_es)\n",
    "    return mean_Ecv, std_Ecv, mean_e1, std_e1, mean_e2, std_e2, mean_vare, std_vare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 372,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Working on N =  18\n",
      "Working on N =  28\n",
      "Working on N =  38\n",
      "Working on N =  48\n",
      "Working on N =  58\n",
      "Working on N =  68\n",
      "Working on N =  78\n",
      "Working on N =  88\n",
      "Working on N =  98\n",
      "Working on N =  108\n",
      "Working on N =  118\n",
      "done\n"
     ]
    }
   ],
   "source": [
    "noise_sigma = 0.5\n",
    "d = 3\n",
    "max_N = d+115\n",
    "Ns = np.arange(d+15, max_N+1, 10)\n",
    "\n",
    "num_exps = 100000\n",
    "\n",
    "var_Ecvs, var_e1s, var_e2s = [], [], []\n",
    "mean_Ecvs, mean_e1s, mean_e2s = [], [], []\n",
    "\n",
    "for N in Ns:\n",
    "    print('Working on N = ', N)    \n",
    "    lambda_t = 0.05/N\n",
    "    res = calc_avg_Ecv_norm(num_exps, d, N, lambda_t, noise_sigma)\n",
    "    \n",
    "    mean_Ecv, std_Ecv, mean_e1, std_e1, mean_e2, std_e2, mean_vare, std_vare = res\n",
    "    var_Ecvs.append(std_Ecv**2)\n",
    "    var_e1s.append(std_e1**2)\n",
    "    var_e2s.append(std_e2**2)\n",
    "    mean_Ecvs.append(mean_Ecv)\n",
    "    mean_e1s.append(mean_e1)\n",
    "    mean_e2s.append(mean_e2)\n",
    "    \n",
    "print('done')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.24\n",
    "(b) I would expect the average of the $e_1$'s to approach the average of the $E_{cv}$'s. The same holds for the average of the $e_2$'s. See the graph below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 376,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SEhJ4++23z+hzTDGzdKmrgWzb5mohZ5BETpyAt992/SCpqa5G0rdv0U8iYImkUNh+JEV7P5L4+HhGjBjBunXraNq0KaGhoXz88cdn9HmmiDt+HAYNgg4d4IUX3AJXwcE+Xar6135Vs2a5obyjR59xS1hAs/1IjMkn9jNXTEVHu02n6taFDz+Eiy7y+dI1a9xw3vh41w9y002B1Q+SX/uRWI3EGGMyc/w4DBniJhQOHgzTpvmcRPbsgcceg9at3cTC1avdivGBlETyk0+JRETaichGEYkTkdOm9YpIORGZ7J1fKiI10p0b5B3fKCJtvWPlRSRKRFaJSKyIvJCu/Ode2bUi8omIlMn7Y5rx48ef0tQVGhpKz549/R1WwGnevPlp36c1a9b4OyxT2JYvh2bNXJVi5Uo3ydCHLHD8OLz5JlxxBZQr5xZWfOIJKFPcf4vlND4YCAK2ALWAssAqoH6GMo8DY7zXnYHJ3uv6XvlyQE3vPkGAABW8MmWApUAL7/3N3nkBJgGP5RRjVvNIUlNT82GktTE5S01NtXkkxcHx46pDh6oGB6tOnKjq4++Q1FQ3jaR2bdX27VU3bCjgOPMJhTiPJByIU9WtACISAXQA1qUr0wEY5r2eCrwvbrxoByBCVY8D20QkDghX1cVAole+jPelXmL75uRNRSQKqOZDjKcpX748+/bto1KlSpkOXTUmv6gq+/btO2WYtimCVq2Cbt2genVXC7n4Yp8uW7nSjb7au9d1oeRhr6oiy5dEUhXYnu59PNA8qzKqmiwiCUAl7/iSDNdWBRCRICAGuAwYraqnbE3nNWl1BU4f7uSDatWqER8fz549e3JzuTFnpHz58mkTRU0Rk5Tkthp8/3144w23Y5QPf3z+8YfrQpk5080LeeQRKF1CZ+b58tiZfUczDvXKqkyW16pqChAqIucDX4pIQ1Vdm67cB8BPqvpzpkGJ9AB6AFxyySWnnS9Tpgw1a9bM7FJjjHHWrHG1kIsucv0iPvwxcOyYW5PxzTfhoYdcP8j55xdCrAHMl872eKB6uvfVgB1ZlRGR0kBFYL8v16rqQWA+kLbji4g8DwQD/bIKSlXHqmqYqoYF+zie2xhjALdK4ogR0KoV9OrlNvvIIYmowpQpbkLhsmVubuIbb1gSAd9qJMuAOiJSE/gd15l+b4YykUA3YDHQEZirqioikcB/ReQt4GKgDhAlIsFAkqoeFJGzgBuA1wBE5BGgLdBaVVPz/ITGGJNebKybF/K3v7laSPXqOV4SHe1WiE9MhPHjIcO82RIvxxqJqiYDvYDvgPXAFFWNFZEXReQ2r9g4oJLXmd4PGOhdGwtMwXXMfwv09Jq0qgDzRGQ1LlHNVtWZ3r3GABcBi0VkpYjkvFCTMcbkJDkZXn0VrrsOevSAb7/NMYn8/rtr+br1Vrc+Y0yMJZHMFNuZ7cYYk2b9elcLOfdcGDcOLr002+JHjrg+kHfecTln0CA477zCCbUw2cx2Y4zJSUqK68j4xz9clWL27GyTSGoqfP656wdZu9Y1ab3ySvFMIvmphA5WM8YUexs3uuRRrhxERUEOoziXLHH9IMnJLpn4uJWOwWokxpjiJiUF3noLrr4a7r0X5szJNon89psr1rGjWx8rKsqSyJmyGokxpvjYvNnVQkqVcuNza9fOsmhiIrz+ulvSvWdPGDsWKlQoxFiLEauRGGOKvtRU1zP+97/DPfe4TT+ySCKpqTBhAlx+OWzZAitWwIsvWhLJC6uRGGOKti1b3BTzlBS3j3qdOlkWXbDA9YOULg1Tp7q8Y/LOaiTGmKIpNdW1SzVv7nYu/PHHLJPItm2uonLvvW6jqUWLLInkJ6uRGGOKnm3bXC3k2DFYuNC1U2Xi0CE3fHfsWFcT+fRTOPvswg21JLAaiTGm6EhNdWu1h4e7nQsXLMg0iaSkwMcfu/kgO3a4HQqfe86SSEGxGokxpmj49Vd4+GH480/46Se3DWEm5s1z+4NUqABffeU2OjQFy2okxpjApgoffQRhYXDDDa4pK5MkEhcHd9zhWryefRZ+/tmSSGGxGokxJnBt3+52jNq3zw3pbdDgtCIJCfDSS25V3qeegkmTwDarLFxWIzHGBB5Vt7hi06Zw7bVu/ZIMSSQ5GcaMcV0kBw64tbEGDbIk4g9WIzHGBJb4ePjnP2HXLre8SePGpxWZPdv1gwQHu9XgQ0P9EKdJYzUSY0xgUHVTzps2dZM8li49LYls2ADt27s1sV56CebOtSQSCKxGYozxvx073MYf8fHw/fenZYf9++GFF9yqvAMHwrRpblFfExisRmKM8R9VmDjRJY4rr3RL76ZLIklJ8N57bj7IiROwbh30729JJNBYjcQY4x9//AGPPupmqX/7rWvS8qjCrFluFFa1aq6rpFEjP8ZqsuVTjURE2onIRhGJE5GBmZwvJyKTvfNLRaRGunODvOMbRaStd6y8iESJyCoRiRWRF9KV7+WVVxGpnPdHNMYEFFU3RjckxGWHZctOSSKxsXDTTa4z/Y03XEuXJZHAlmONRESCgNFAGyAeWCYikaq6Ll2xh4EDqnqZiHQGXgM6iUh9oDPQALgY+EFE6gLHgVaqmigiZYAFIjJLVZcAC4GZwPx8e0pjTGDYtcv1lG/cCF9/7SYZevbuheefhylTYMgQV6xsWT/GanzmS40kHIhT1a2qegKIADpkKNMBmOC9ngq0FhHxjkeo6nFV3QbEAeHqJHrly3hfCqCqK1T1l7w8lDEmAE2Z4mohl18Oy5enJZETJ9yGhldcAUFBbmRWnz6WRIoSX/pIqgLb072PB5pnVUZVk0UkAajkHV+S4dqqkFbTiQEuA0ar6tLcPIAxJsDt2eO2IFyzxi1+1dz9+lCFyEjXeV6nTrbLZ5kA50uNRDI5pj6WyfJaVU1R1VCgGhAuIg19iOWvDxTpISLRIhK9Z8+eM7nUGFNYpk1zc0Fq1HBbEXpJZPVqt2zWoEFuVNY331gSKcp8SSTxQPV076sBO7IqIyKlgYrAfl+uVdWDuP6QdmcQN6o6VlXDVDUsODj4TC41xhS0vXuhc2e3euL06W5z9PLl2bXLTRdp0wbuvNMllHZn9C/fBCJfEskyoI6I1BSRsrjO88gMZSKBbt7rjsBcVVXveGdvVFdNoA4QJSLBInI+gIicBdwAbMj74xhj/O7LL10tpGpVWLkS/v53jh93uaRBA7e8+4YNrrWrtE1AKBZy/N/o9Xn0Ar4DgoBPVDVWRF4EolU1EhgHTBSROFxNpLN3bayITAHWAclAT1VNEZEqwASvn6QUMEVVZwKISG9gAPB/wGoR+UZVH8nn5zbG5Lf9+6F3b7e0yZQpcM01qML0afD0024I76JFULeuvwM1+U1cxaFoCwsL0+joaH+HYUzJFRnpxuvefTe8/DKcfTbLl7u5IAcOwKhR0Lq1v4M0GYlIjKqG5Vwye1axNMbk3oEDbqzuwoVukuE//sHOnTC4l+tAf/FFt6lhUJC/AzUFydbaMsbkztdfu/aqihVh9WqONvsHI0ZAw4ZuefeNG13HuiWR4s9qJMaYM3PwIPTr53YsnDgRve56Jk92q/KGhbl1F2vX9neQpjBZIjHG+O7bb101o317WL2aqHUV6HsNHD3qthK59lp/B2j8wRKJMSZnCQluKd4ffoDx44m/vDWDHnOr8o4YAQ88YE1YJZn1kRhjsjd7tpsXEhTE4cWrGfZza0JC4NJLXT9I9+6WREo6q5EYYzL3559uAsisWaT++yP+u/dGBoXD1Ve7NRcvvdTfAZpAYYnEGHO6OXPcuN0bbmDpR6t5YkhFACIiXCIxJj1LJMaYvyQmwjPPQGQku4aPpc+3N7HwYXjlFbj3XihljeEmE/ZjYYxx5s+Hxo1JSjjC8HvWUP+pm6hXz62Ldf/9lkRM1qxGYkxJd/gwDBqETp/ODx3/Tbcpt9C6Naxa5fZLNyYnlkiMKcl+/hm6d+eP2lfR6W9rSIq6gBkzIDzc34GZosQqq8aUREeOQN++JHfsxGsXjqT5hv/w2LMXsHChJRFz5iyRGFPSLFxISuNQVny3m3on1pB8Swc2bHD7UElme5oakwNr2jKmpDh6lNTBz3H0k//Si9HInXfw00tw8cX+DswUdZZIjCkJlizh8D0P8vOhUEZfsZph71fmyiv9HZQpLiyRGFOcHTvG/t7PI/+ZwJCK79Fq3N1E3mlNWCZ/WSIxppg6NGcZR+7pRlRifX4dsJqRgy+kfHl/R2WKI58620WknYhsFJE4ERmYyflyIjLZO79URGqkOzfIO75RRNp6x8qLSJSIrBKRWBF5IV35mt49Nnv3LJv3xzSm5Eg+fJyYts9yvE17IkOfp/mvX/DEcEsipuDkmEhEJAgYDdwE1Ae6iEj9DMUeBg6o6mXAKOA179r6QGegAdAO+MC733GglaqGAKFAOxFp4d3rNWCUqtYBDnj3Nsb4YPF70WyrdCVHl69n9+xV9JjTiYv+z9qxTMHypUYSDsSp6lZVPQFEAB0ylOkATPBeTwVai4h4xyNU9biqbgPigHB1Er3yZbwv9a5p5d0D75635/LZjCkx1q86waTLnqNO31tIePxZrt41nQat/8/fYZkSwpdEUhXYnu59vHcs0zKqmgwkAJWyu1ZEgkRkJbAbmK2qS71rDnr3yOqz8K7vISLRIhK9Z88eHx7DmOJn3z54vcsKUq5sRni5VZy3ZSVhb92LlLJaiCk8viSSzH4i1ccyWV6rqimqGgpUA8JFpKGPn4V3/VhVDVPVsODg4CyDN6Y4SkqC90ae4JNLhvHYV2259N3+1F77FWUvreLv0EwJ5MuorXigerr31YAdWZSJF5HSQEVgvy/XqupBEZmP60MZCZwvIqW9Wklmn2VMiaUKX38NH/Vaxcj9D3LRlRdz7qQVUDXTirsxhcKXGskyoI43mqosrvM8MkOZSKCb97ojMFdV1Tve2RvVVROoA0SJSLCInA8gImcBNwAbvGvmeffAu+dXuX88Y4qPtWvh5jZJbO0+nCkHbqD2O30498eZlkSM3+VYI1HVZBHpBXwHBAGfqGqsiLwIRKtqJDAOmCgicbiaSGfv2lgRmQKsA5KBnqqaIiJVgAneCK5SwBRVnel95DNAhIi8BKzw7m1MibVnDwwdCrGT1zL1nG5UvvJCSn28wtZ4NwFDXCWgaAsLC9Po6Gh/h2FMvjp+HN57D958NZmP677OzZtGUeq1V+Ghh2xquskXIhKjqmF5vY/NbDcmwKjCV19B//5wY7V1bKvSjbPOOR+Wx8All/g7PGNOY4nEmACyciX07Qv7dyfzzbUjqRv5JowYAf/8p9VCTMCy/UiMCQB//AGPPALt2sGj125gZYVrqPvL97BsGfToYUnEBDRLJMb40bFj8Mor0LAhXHBeCtt6vknn969Buj0As2dDjRr+DtGYHFnTljF+oApTp8KAARAaCjGTNnHp8w9C2bIQFQW1avk7RGN8ZonEmEIWHe36Qf78Ez75KIXr17wLXUbAsGHw+ONQyhoKTNFiicSYQvL77zB4MHz3HQwfDt1bxhH0SHd3cskSuOwy/wZoTC7Znz7GFLD9+10TVqNGUKUKbFyfyiNH3iXo6hZw113w44+WREyRZjUSYwrI4cPwzjvw1lvQsSOsWQNVj2+FDt0hORkWLYK6df0dpjF5ZjUSY/LZiRPwwQdQpw6sXu3yxZjRKVSd9i40bw4dOsBPP1kSMcWG1UiMySepqRARAc8955LIzJnQtCmwfDm0eBTOOQcWLIDLL/d3qMbkK0skxuSRKsyaBYMGwVlnwccfw/XXA4mJ0G8ofP45vPYadOtmEwtNsWSJxJg8WLjQJZC9e+Hll12rlQhusawnnoDWrSE2FipX9neoxhQYSyTG5MKaNW4o76pV8MIL0LUrBAUB27dD796wbh1MmOBVTYwp3qyz3ZgzsG2bSxo33ACtWsHGjfDggxBEihui1aSJm6q+erUlEVNiWI3EGB/s2gUvvQT//a9rsdq8Gc47zzsZEwOPPgrnnuvauqwz3ZQwViMxJhsJCW4UVv36ULo0bNjgVjI57zzcGidPPgk33+yyy9y5lkRMiWSJxJhMHD0KI0e6Ybzx8W4E76hREBzsFfjqK2jDZCKoAAAZaElEQVTQwGWa2FgbkWVKNJ8SiYi0E5GNIhInIgMzOV9ORCZ755eKSI105wZ5xzeKSFvvWHURmSci60UkVkT6pCsfIiKLRWSNiPxPRM7L+HnGFJTkZBg3zs0VXLAA5s2D8ePh0ku9Atu3wx13wDPPwH/+407aiCxTwuWYSEQkCBgN3ATUB7qISP0MxR4GDqjqZcAo4DXv2vpAZ6AB0A74wLtfMvCUql4BtAB6prvnx8BAVW0EfAk8nbdHNCZnqjBtmtsXZOJE+OIL+PJLV+kAXIZ5+23Xmd6kiRuudd11/gzZmIDhS2d7OBCnqlsBRCQC6ACsS1emAzDMez0VeF9ExDseoarHgW0iEgeEq+piYCeAqv4pIuuBqt49Lwd+8u41G/gOeC7XT2hMDubMgYEDIcUbeHXjjRlaqWJi3C6FFStaZ7oxmfClaasqsD3d+3jvWKZlVDUZSAAq+XKt1wzWBFjqHVoL3Oa9vhuonllQItJDRKJFJHrPnj0+PIYxp1q2DNq0gX/9C/r3d/uEtG2bLomc7Ey/5Rbo08dlHEsixpzGl0SSWQ+i+lgm22tFpAIwDXhSVQ95hx/CNXXFAOcCJzILSlXHqmqYqoYFp/WAGpOzDRvcary33+7+u24ddOqUYT+pGTPcUK1Dh2DtWnjgAetMNyYLvjRtxXNqraAasCOLMvEiUhqoCOzP7loRKYNLIp+r6vSTBVR1A3CjV6YucMsZPI8xWdq+3c1C/+orePpp11d+9tkZCv32mxvKu2kTfPYZXHutX2I1pijxpUayDKgjIjVFpCyu8zwyQ5lIoJv3uiMwV1XVO97ZG9VVE6gDRHn9J+OA9ar6VvobiciF3n9LAUOAMbl7NGOcfftc01VoKFx4ocsRAwZkSCLJyW58b9OmEBYGK1daEjHGRznWSFQ1WUR64Tq9g4BPVDVWRF4EolU1EpcUJnqd6ftxyQav3BRcJ3oy0FNVU0TkGqArsEZEVnof9ayqfoMbFdbTOzYdGJ9vT2tKlMREN9Dq7bfhnntcC1WVKpkUjI52nekXXGCbTRmTC+IqDkVbWFiYRkdH+zsMEyBOnICxY2HECLfc1YsvZrGT7aFDbtr65Mnwxhtw//3WD2JKFBGJUdWwvN7HZrabYiMlxXVr1KsH33zjvv7730ySiOpfk0QSE93M9K5dLYkYk0u2aKMp8lTh66/h2WehQgU32TzL7g3rTDcm31mNxBRpP/8MLVu6CYUvveTmC2aaG5KT4a23XGd6s2bWmW5MPrIaiSmSVq1yNZB169yQ3vvu8zaWysyyZa4zvVIlWLzYrcRojMk3ViMxRcqWLS5ptG0L7dq5yYUPPJBFEjl0yO1WeOut8NRTMHu2JRFjCoAlElMk7NwJPXtC8+auM33zZtfVUa5cJoVVYfp0NzP9yBHXmW4jsowpMNa0ZQLawYNuZO6YMW5L2w0bcli1/ddfoVcvV3WZNMl1oBhjCpTVSExAOnrUJZC6deGPP2DFCrfRVJZJJDnZFbjySmjRwnWmWxIxplBYjcQElORkN3z3hRdcM9aPP8IVV+RwUVSU2zO9cmVYsiSL2YfGmIJiicQEhNRUt7HUkCFQtarr4ggPz+GihAR3wdSp8OabcO+91g9ijB9YIjF+peoGUw0a5HLA++/DDTfkkA9Odqb36QM33+w60//2t0KL2RhzKkskxm9WroR+/eD33926WHfd5UOF4mRn+tatEBEB11xTKLEaY7Jmne3GL8aPd7sTdu7sKhQdO+aQRNJ3pv/976733ZKIMQHBaiSmUJ044XavnTsXfvrJh450gKVLXWf6hRdaZ7oxAcgSiSk0O3bA3XdDcLAbaHXeeTlckJAAgwe7XviRI6FLF+tMNyYAWdOWKRQLF7pRWDfd5PrJs00iqm4kVoMGrgoTG2sjsowJYFYjMQVKFT780M0L+fRTl0iy9csvrjN92zbrTDemiPCpRiIi7URko4jEicjATM6XE5HJ3vmlIlIj3blB3vGNItLWO1ZdROaJyHoRiRWRPunKh4rIEhFZKSLRIpLTbAIToI4ehYcecsubLFqUQxJJSnJT2cPC4OqrrTPdmCIkxxqJiAQBo4E2QDywTEQiVXVdumIPAwdU9TIR6Qy8BnQSkfq4/dsbABcDP4hIXdz+7U+p6nIROReIEZHZ3j1fB15Q1VkicrP3/rr8emBTOH79Fe680y1xsngxnHNONoWXLHGd6f/3f65jvXbtQovTGJN3vtRIwoE4Vd2qqieACKBDhjIdgAne66lAaxER73iEqh5X1W1AHBCuqjtVdTmAqv4JrAeqetcrcLIFvSKwI3ePZvxlzhy3vMl997mtbrNMIgkJbknfO+5wO1N9+60lEWOKIF8SSVVge7r38fz1S/+0MqqaDCQAlXy51msGawIs9Q49CbwhItuBN4FBPsRoAoCqa526/3638G6/fln0j6emuokk9eq5Jq1162xEljFFmC+d7Zn961Yfy2R7rYhUAKYBT6rqIe/wY0BfVZ0mIvcA44AbTgtKpAfQA+CSSy7J6RlMAUtMhIcfdhPOly6FLP+XREW5jURKlYLISLftrTGmSPOlRhIPVE/3vhqnNzellRGR0rgmqf3ZXSsiZXBJ5HNVnZ6uTDfg5PsvcE1rp1HVsaoapqphwcHBPjyGKShxcW6yeYUKbg/1TJPIrl2u5/32211z1sKFlkSMKSZ8SSTLgDoiUlNEyuI6zyMzlInEJQCAjsBcVVXveGdvVFdNoA4Q5fWfjAPWq+pbGe61A7jWe90K2HymD2UKz9dfu0FWPXvCxx9D+fIZCiQlwVtvuTkhlSr9tTduKZvCZExxkWPTlqomi0gv4DsgCPhEVWNF5EUgWlUjcUlhoojE4Woinb1rY0VkCrAON1Krp6qmiMg1QFdgjYis9D7qWVX9Bvgn8I5XszmG13xlAktqKrz0EowdCzNmuBrJaWbPdnumX3opLFjg+kSMMcWOuIpD0RYWFqbR0dH+DqPESEiArl1h/3744guoUiVDga1b4amnYPVqePttaN/eOtKNCUAiEqOqYXm9j7UvmDMSG+u6Ni691C28eEoSOXwYnnvOrYUSHu4K33qrJRFjijlLJMZnU6fCdde5dRTfew/KlvVOqMLkyW4p3y1b3EYjgwZl0mFijCmObK0tk6OUFJc8IiLgu++gadN0J1evdv0gBw/C559Dy5Z+i9MY4x9WIzHZ2rcP2rWDmBiIjk6XRPbvd4srntydKibGkogxJZQlEpOlFSvcGopNm8KsWVC5Mq56MmbMXztSrV8P//oXBAX5NVZjjP9Y05bJ1MSJbomTDz5wm1EBbrZh795uM5Hvv4eQEL/GaIwJDJZIzCmSktzI3VmzYN48aNgQ+P13ePppNxfkjTfgnntsJJYxJo01bZk0f/wBrVq5PaWWLYOGlx2DV15xNY/atV0zVqdOlkSMMaewRGIAt2dIs2bQujV8NUM5/+f/uerI0qVuocXhw3PYVMQYU1JZ01YJp+qWOXnuOfjkE2hfZyO0f9JVS0aPhrZt/R2iMSbAWY2kBDt0yK2f+N57sPi7Q7T/8Wm3AmObNm5+iCURY4wPLJGUUEuXQpMmcM5ZqcT0+Q+121/hJo2sXeuGa6VNWzfGmOxZ01YJk5ICr78O/x51hC/v+A9NfnwbVp8P06e7/XGNMeYMWSIpQX7/HZ68Zwcdtr/PlpSPCPrjKje58NprbSSWMSbXLJGUEPPfWs6ewaP4DzMp9/D9lOqzCOrU8XdYxphiwBJJcZaSwvFpM/m17yjq7tpCjUef4KyX3oULLvB3ZMaYYsQSSXGUmAiffsrx199m096/MS+0H91W3cXFlcv4OzJjTDFko7aKk+3bYcAAtEYNtnw8jzsPTWD5B0t5YmFnKloSMcYUEJ8SiYi0E5GNIhInIgMzOV9ORCZ755eKSI105wZ5xzeKSFvvWHURmSci60UkVkT6pCs/WURWel+/pNvT3WQlKgq6dIGQEI4eSqJH6DI6lZ7G28uuptuDYv3oxpgClWMiEZEgYDRwE1Af6CIi9TMUexg4oKqXAaOA17xr6wOdgQZAO+AD737JwFOqegXQAuh58p6q2klVQ1U1FJgGTM/7YxZDKSkwbZqbQHjPPdCsGT9+uo06M0dxQdOaLLK+dGNMIfGlRhIOxKnqVlU9AUQAHTKU6QBM8F5PBVqLiHjHI1T1uKpuA+KAcFXdqarLAVT1T2A9UDX9Db3r7wEm5e7RiqlDh2DUKLjsMhg5Evr2JWl9HIP29OPexyoyfrybJ2LzCY0xhcWXRFIV2J7ufTwZfumnL6OqyUACUMmXa71msCbA0gz3bAnsUtXNPsRY/B05Ai+/DDVrumnpERGwaBFbmnTk6mtLs2aN2yq9TRt/B2qMKWl8SSSZtbCrj2WyvVZEKuCar55U1UMZynUhm9qIiPQQkWgRid6zZ09WxYq+1FS3y1S9em7LwpNJpHlzJk6EFi2ga1f43/8gONjfwRpjSiJfhv/GA9XTva8G7MiiTLyIlAYqAvuzu1ZEyuCSyOeqeko/iHePO4ErswpKVccCYwHCwsIyJrbiYf58t8tUmTIwaZLrD8G1bj3+OCxfDnPmQOPG/g3TGFOy+VIjWQbUEZGaIlIW13kemaFMJNDNe90RmKuq6h3v7I3qqgnUAaK8/o9xwHpVfSuTz7wB2KCq8Wf+SMXAxo3QoQN07w4DBrjNQrwkcnKxxQoVIDrakogxxv9yTCRen0cv4Dtcp/gUVY0VkRdF5Dav2DigkojEAf2Agd61scAUYB3wLdBTVVOAq4GuQKt0Q31vTvexnSmJnex79kCvXnDNNdCy5Wk7En74Idx2m9vtdswYOPtsP8drjDGAuIpD0RYWFqbR0dH+DiP3jh2Dd95xGeLee2HoUKhcOe10cjI8+STMnQuRkW7AljHG5JWIxKhqWF7vY0uk+JOq6zgfNMi1Vy1aBHXrnlLkwAFXKQkKci1cFSv6KVZjjMmCJRJ/WbDAdaSnpMCECW4p9ww2b4b27eHmm11lpbT93zLGBCBba8sfRo1yTVi9e7vlTTJJInPnuq6S/v1dcUsixphAZb+eCtvChfDaay6BXHJJpkXGjIFhw2DyZLjuukKNzhhjzpglksK0Z49bXPGTTzJNIsnJ0Lcv/PCDa/myTnVjTFFgiaSwpKa6Kej33ec6PTI42aleqhQsWWKd6saYosP6SArLK6+49bKGDz/t1ObNbqmT+vVh5kxLIsaYosVqJIVh/nx4/303FT1Dr/ncua61a/hw6NHDP+EZY0xeWI2koO3a5ZqzJkyAqqcumjxmjBu8FRFhScQYU3RZjaQgpaS4TPHww3DjjWmHDx92cxBnz7ZOdWNM0WeJpACkpsIff0Dy4Bcps1MZy/PEdYWtW93XwYPQtq2bqX7++f6O1hhj8sYSST45ftw1UX34IaxeDbeUm80Hhz9m2K0xVCKI1q1d81WtWlClihudZYwxxYElkjzatcv1dYwZ45Z0f+45uLbODipc+wBM/y+jr/8/f4dojDEFyhJJbsyYwe+LfmXKyrpMWHI5LTpdyg8/BNGgAW5WYavO0LMnXH+9vyM1xpgCZ4nkTCQm8ucDPTn4/VLm0Jrbq3xN74obCfpsNyyq5VbuTUpyG4U8+6y/ozXGmEJhicRHJ5Ys58/2nZmV2JI9Q2LoOeAcypb1Th45Alu2uJ0Nf/sNHnjAOkGMMSWGJZKcqLKp5ztUGvsyn4a+Q8dpXbj00gxlzj4bGjVyX8YYU8JYIsnK/v3w2Wfse+0jEveew28fLOGpHrX8HZUxxgQcn9pfRKSdiGwUkTgRGZjJ+XIiMtk7v1REaqQ7N8g7vlFE2nrHqovIPBFZLyKxItInw/2e8MrHisjreXvEM6AK8+a5mei1apG0YCk9jr5L6ajF3GBJxBhjMpVjjUREgoDRQBsgHlgmIpGqui5dsYeBA6p6mYh0Bl4DOolIfaAz0AC4GPhBROoCycBTqrpcRM4FYkRktqquE5HrgQ5AY1U9LiIX5uPzZm3nTrcqb1IS/POf8O67vPpBJcqXgcYhhRKBMcYUSb40bYUDcaq6FUBEInC/6NMnkg7AMO/1VOB9ERHveISqHge2iUgcEK6qi4GdAKr6p4isB6p693wMeNW7BlXdnbdH9MHOnW6o7v33w+DBIMK+ffDOO25Jd2OMMVnzpWmrKrA93ft471imZVQ1GUgAKvlyrdcM1gRY6h2qC7T0msh+FJFmvjxIbixYACP7e0mka1cYMgREAHj1VejY0dbBMsaYnPhSI5FMjqmPZbK9VkQqANOAJ1X1ULqYLgBaAM2AKSJSS1VP+UwR6QH0ALgkiy1rc3LF+Tu56O3r2d3rAS4c/Ne8j99/h3HjYM2aXN3WGGNKFF9qJPFA9XTvqwE7siojIqWBisD+7K4VkTK4JPK5qk7PcK/p6kQBqUDljEGp6lhVDVPVsODgYB8e43SVTuzk91t68OCmUycPDh/uFuytmrHeZYwx5jS+JJJlQB0RqSkiZXGd55EZykQC3bzXHYG5Xg0iEujsjeqqCdQBorz+k3HAelV9K8O9ZgCtALyO+bLA3jN/NB80bcpVX/QjLg5mzXKH4uJg6lQYeNrYNGOMMZnJMZF4fR69gO+A9cAUVY0VkRdF5Dav2DigkteZ3g8Y6F0bC0zBdaJ/C/RU1RTgaqAr0EpEVnpfJzcy/wSoJSJrgQigW8ZmrfxUtiyMHAlPPeUGbD3/PPTpA5UqFdQnGmNM8SIF+Du60ISFhWl0dHSur1d1+07Vrg0zZrg91M89Nx8DNMaYACQiMaoaltf72Mx23ECtt96C0FBXO7EkYowxvrNE4mnUyE1qb9HC35EYY0zRYokknX/8w98RGGNM0WNrnRtjjMkTSyTGGGPyxBKJMcaYPLFEYowxJk8skRhjjMkTSyTGGGPyxBKJMcaYPLFEYowxJk+KxVpbIrIH+DUfb1mZglpxuGBYvAWvqMVs8Ras4hLvpaqau3040ikWiSS/iUh0fixkVlgs3oJX1GK2eAuWxXsqa9oyxhiTJ5ZIjDHG5IklksyN9XcAZ8jiLXhFLWaLt2BZvOlYH4kxxpg8sRqJMcaYPCmWiURE2onIRhGJE5GBmZwvJyKTvfNLRaRGunODvOMbRaStd6y6iMwTkfUiEisifTLc7wmvfKyIvB7I8Xr3Wel9/SIiK880Xj/EHCoiS7yYo0UkPMDjDRGRxSKyRkT+JyLnBUC85UUkSkRWefG+kK58Te8em717lg3weHt55VVEKp9prH6K+XOv7FoR+UREygR4vOO846tFZKqIVMg2OFUtVl9AELAFqAWUBVYB9TOUeRwY473uDEz2Xtf3ypcDanr3CQKqAE29MucCm07eE7ge+AEo572/MJDjzXDfkcDQIvA9/h64yXt9MzA/wONdBlzrvX4IGB4A8QpQwStTBlgKtPDeTwE6e6/HAI8FeLxNgBrAL0DlAPo9kV3MN3vnBZhUBL7H56W771vAwOziK441knAgTlW3quoJIALokKFMB2CC93oq0FpExDseoarHVXUbEAeEq+pOVV0OoKp/AuuBqt71jwGvqupx7/zuAI8XAO/6e3A/1GeqsGNW4ORf9RWBHQEe7+XAT97r2cBdARCvqmqiV76M96XeNa28e+Dd8/ZAjRdAVVeo6i9nGKO/Y/7GO69AFFAtwOM9BGm/J846eTwrxTGRVAW2p3sfT4ZfounLqGoykABU8uVar7rYBJe9AeoCLb2q5I8i0izA4z2pJbBLVTefYbz+iPlJ4A0R2Q68CQwK8HjXArd5r+8GqgdCvCISJK4pczcwW1WXetcc9O6R1WcFUrz5xS8xe01aXYFvAz1eERkP/AHUA97LLrjimEgkk2MZs2lWZbK91msnnAY8eTJj4/a9vwBoATwNTPGyeKDGe1IXclcbyS4eX8rkJubHgL6qWh3oC4wL8HgfAnqKSAyu2etEIMSrqimqGor7azhcRBr6+Fk5Kcx484u/Yv4A+ElVfw70eFW1O3AxrrbdKbvgimMiiefUvwCrcXpTSFoZESmNay7Zn9213l8S04DPVXV6hntN96qJUUAqbl2bQI335D3uBCafQZz+jLkbcPL9F7hqfsDGq6obVPVGVb0Sl6y3BEK86eI7CMwH2uHWXzrfu0dWnxVI8eaXQo9ZRJ4HgoF+RSFe73gK7vdE9s2zWXWeFNUvXA1hK65T6WSnVIMMZXpyaqfUFO91A07tlNrKX51S/wHezuTz/gW86L2ui6tCSqDG613XDvixCH2P1wPXea9bAzEBHu+F3n9LeWUeCoB4g4HzvTJnAT8D7b33X3BqZ/vjgRxvunv+Qu472wv7e/wIsAg4K9Dj9X62L/OOC645+c1s48vNQwX6F26ExCbcX4KDvWMvArd5r8t7/3jicB1ftdJdO9i7biN/jRS6BlcVXA2s9L5u9s6VBT7DtYsvB1oFcrze+U+BfxWh7/E1QIz3j2EpcGWAx9vH+6xNwKucwR8WBRhvY2CFF+9a0o3Ww40EivLu9QXeCMQAjrc37q/sZNxf1h8HyM9wdjEne+VP/qzkZrRkocSL+wNoIbDGO/456UZxZfZlM9uNMcbkSXHsIzHGGFOILJEYY4zJE0skxhhj8sQSiTHGmDyxRGKMMSZPLJEYY4zJE0skxhhj8sQSiTHGmDz5f8FFibIeKm55AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(mean_Ecvs, mean_e1s, color='blue', linewidth=1, label='Mean E_cv vs. Mean e_1')\n",
    "plt.plot(mean_Ecvs, mean_e2s, color='red', linewidth=1, label='Mean E_cv vs. Mean e_2')\n",
    "plt.legend()\n",
    "#plt.yscale('log')\n",
    "#plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.24 (c) -(e)\n",
    "\n",
    "* (c) The contributors to the variance of the $e_1$'s are point $(x_1, y_1)$, and all the other data points which are used to find the best hypothesis $g^{-}$.\n",
    "\n",
    "* (d) If the cross validation errors were truly independent, the variance of the $e_1$'s should be $N$ times the variance of the $E_{cv}$'s, i.e. $E_{cv} = \\frac{1}{N}\\sum^N_{n=1}e_n$, $Var_{\\mathcal{D}}E_{cv} = \\frac{1}{N^2}\\sum^N_{n=1}Var_{\\mathcal{D}}e_n = \\frac{1}{N}Var_{\\mathcal{D}}e_n$\n",
    "\n",
    "* (e) As shown in problem (d), when the cross validation errors were truly independent, the effective number of fresh examples $N_{eff} = N$. As the cross validation errors are not independent (as shown in problem (c), all of them are depending on all the data points), there are covariance terms with each pair of $e_n$ and $e_m$. But as shown in problem 4.23, when $N$ is large, the covariance terms decay to zero and the variance of $E_{cv}$ approaches the case when cross validation errors are truly independent. \n",
    "  If we treat each cross validation error similar to each other, then the $E_{cv}$ can be thought of as a sum of $N_{eff}$ fresh examples where the fresh examples are independent of each other.\n",
    "  \n",
    "  The graph below shows that $N_{eff}$ is close to $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 384,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "e1_Ecv_ratios = np.array(var_e1s)/np.array(var_Ecvs)\n",
    "e1_Ecv_ratios = e1_Ecv_ratios/np.array(Ns)\n",
    "plt.plot(Ns, e1_Ecv_ratios, color='red', linewidth=1, label='Effective Number of Fresh Examples vs. N')\n",
    "plt.legend()\n",
    "#plt.yscale('log')\n",
    "#plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.24 (f)\n",
    "If we increase the amount of regularization, we reduce the impact of data points with large noise. This should reduce the correlation of the $e_n$'s and in turn make $N_{eff}$ closer to $N$. See the graph below, it is at a higher level (closer to 1) than the graph in problem (e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 385,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Working on N =  18\n",
      "Working on N =  28\n",
      "Working on N =  38\n",
      "Working on N =  48\n",
      "Working on N =  58\n",
      "Working on N =  68\n",
      "Working on N =  78\n",
      "Working on N =  88\n",
      "Working on N =  98\n",
      "Working on N =  108\n",
      "Working on N =  118\n",
      "done\n"
     ]
    }
   ],
   "source": [
    "#### Problem 4.24 (f) with lambda_t = 2.5/N\n",
    "\n",
    "noise_sigma = 0.5\n",
    "d = 3\n",
    "max_N = d+115\n",
    "Ns = np.arange(d+15, max_N+1, 10)\n",
    "\n",
    "num_exps = 100000\n",
    "\n",
    "var_Ecvs, var_e1s, var_e2s = [], [], []\n",
    "mean_Ecvs, mean_e1s, mean_e2s = [], [], []\n",
    "\n",
    "for N in Ns:\n",
    "    print('Working on N = ', N)    \n",
    "    lambda_t = 2.5/N\n",
    "    res = calc_avg_Ecv_norm(num_exps, d, N, lambda_t, noise_sigma)\n",
    "    \n",
    "    mean_Ecv, std_Ecv, mean_e1, std_e1, mean_e2, std_e2, mean_vare, std_vare = res\n",
    "    var_Ecvs.append(std_Ecv**2)\n",
    "    var_e1s.append(std_e1**2)\n",
    "    var_e2s.append(std_e2**2)\n",
    "    mean_Ecvs.append(mean_Ecv)\n",
    "    mean_e1s.append(mean_e1)\n",
    "    mean_e2s.append(mean_e2)\n",
    "    \n",
    "print('done')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 386,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "e1_Ecv_ratios = np.array(var_e1s)/np.array(var_Ecvs)\n",
    "e1_Ecv_ratios = e1_Ecv_ratios/np.array(Ns)\n",
    "plt.plot(Ns, e1_Ecv_ratios, color='red', linewidth=1, label='Effective Number of Fresh Examples vs. N (lambda_t = 2.5/N)')\n",
    "plt.legend()\n",
    "#plt.yscale('log')\n",
    "#plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.25\n",
    "\n",
    "* (a) We should not select the learner with minimum validation error. This is because each learner $m$ uses a different validation set, with different number of points $K_m$.  So for each learner $m$, by the VC-bound formula for a finite model with one hypothesis in it (formula 4.9), we have $E_{out}(g_m) \\le E_{in}(g_m) + O\\left(\\sqrt{\\frac{1}{K_m}}\\right)$. Lowest $E_{in}(g_m)$ doesn't necessarily end up with the lowest sum because of the $K_m$ term. \n",
    "\n",
    "\n",
    "* (b) If all models are validated on the same validation set as described in the text, it is okeay to select the learner with the lowest validation error because they have the same amount of data $K$, as from above formula, selecting the learner with lowest validation error guarantees the lowest bound.\n",
    "\n",
    "* (c) After selecting learner $m^*$, we have that \n",
    "\n",
    "\\begin{align*}\n",
    "P\\left[E_{out}(m^*) \\gt E_{val}(m^*) + \\epsilon\\right] &= P[E_{out}(m_1) \\gt E_{val}(m_1) + \\epsilon\\\\\n",
    "&\\text{or } E_{out}(m_2) \\gt E_{val}(m_2) + \\epsilon \\\\\n",
    "&\\dots\\\\\n",
    "&\\text{or } E_{out}(m_M) \\gt E_{val}(m_M) + \\epsilon]\\\\\n",
    "&\\le \\sum^M_{m=1}P[E_{out}(m) \\gt E_{val}(m) + \\epsilon]\\\\\n",
    "&\\le \\sum^M_{m=1}e^{-2\\epsilon^2K_m}\\\\\n",
    "&= Me^{-2\\epsilon^2k(\\epsilon)} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where $k(\\epsilon) = -\\frac{1}{2\\epsilon^2}\\ln \\left(\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K_m}\\right)$\n",
    "\n",
    "Note we can't apply VC-Bound here because of the various number of validation sets.\n",
    "\n",
    "* (d) For any $\\epsilon^*$, we have \n",
    "\n",
    "\\begin{align*}\n",
    "P\\left[E_{out}(m^*) \\le E_{val}(m^*) + \\epsilon\\right] &= 1 - P\\left[E_{out}(m^*) \\gt E_{val}(m^*) + \\epsilon\\right] \\\\\n",
    "&\\ge 1 - Me^{-2(\\epsilon^*)^2k(\\epsilon^*)}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "To have a probability at least $1-\\delta$, let $Me^{-2(\\epsilon^*)^2k(\\epsilon^*)} \\le \\delta$, we have\n",
    "\n",
    "$\\epsilon^* \\ge \\sqrt{\\frac{\\ln (M/\\delta)}{2k(\\epsilon)}}$\n",
    "\n",
    "* (e) Let $K^{min} = \\min_m K_m$, we have $e^{-2\\epsilon^2K_m} \\le e^{-2\\epsilon^2K^{min}}$\n",
    "\n",
    "\\begin{align*}\n",
    "k(\\epsilon) &= -\\frac{1}{2\\epsilon^2}\\ln \\left(\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K_m}\\right)\\\\\n",
    "&\\ge -\\frac{1}{2\\epsilon^2}\\ln \\left(\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K^{min}}\\right)\\\\\n",
    "&= -\\frac{1}{2\\epsilon^2}\\ln e^{-2\\epsilon^2K^{min}}\\\\\n",
    "&= K^{min}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "On the other hand, by the Jesen's inequality of convex function, here $e^x$, we also have\n",
    "\n",
    "\\begin{align*}\n",
    "\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K_m} &\\ge e^{\\frac{1}{M}\\sum^M_{m=1}-2\\epsilon^2K_m}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So we have \n",
    "\\begin{align*}\n",
    "\\ln\\left(\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K_m}\\right) &\\ge \\frac{1}{M}\\sum^M_{m=1}-2\\epsilon^2K_m\\\\\n",
    "-\\frac{1}{2\\epsilon^2}\\ln \\left(\\frac{1}{M}\\sum^M_{m=1}e^{-2\\epsilon^2K_m}\\right) &\\le \\frac{1}{M}\\sum^M_{m=1}K_m\\\\\n",
    "k(\\epsilon) &\\le \\frac{1}{M}\\sum^M_{m=1}K_m\\\\\n",
    "\\end{align*}\n",
    "\n",
    "This bound is worse than the bound when all models use the same validation set size which equal to the average validation set size $\\frac{1}{M}\\sum^M_{m=1}K_m$ because $k(\\epsilon) \\le \\frac{1}{M}\\sum^M_{m=1}K_m$, so the bound is larger than the bound with the same validation set size. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.26\n",
    "\n",
    "* (a) We have $z_n = \\Phi(x_n)$ is of dimension $d \\times 1$, then $z_nz^T_n$ has a dimension of $d\\times d$. Let $z_n = \\begin{bmatrix}z_{n1}\\\\z_{n2}\\\\\\dots\\\\z_{nd}\\end{bmatrix}$, then we have\n",
    "\n",
    "\\begin{align*}\n",
    "Z &= \\begin{bmatrix}z_1\\\\z_2\\\\\\dots\\\\z_N\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}\n",
    "z_{11}&z_{12}&\\dots&z_{1d}\\\\\n",
    "z_{21}&z_{22}&\\dots&z_{2d}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "z_{N1}&z_{N2}&\\dots&z_{Nd}\\\\\n",
    "\\end{bmatrix}_{N\\times d}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "Z^TZ &= \n",
    "\\begin{bmatrix}\n",
    "z_{11}&z_{21}&\\dots&z_{N1}\\\\\n",
    "z_{12}&z_{22}&\\dots&z_{N2}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "z_{1d}&z_{2d}&\\dots&z_{Nd}\\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "z_{11}&z_{12}&\\dots&z_{1d}\\\\\n",
    "z_{21}&z_{22}&\\dots&z_{2d}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "z_{N1}&z_{N2}&\\dots&z_{Nd}\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}\n",
    "\\sum^N_{n=1}z_{n1}z_{n1}&\\sum^N_{n=1}z_{n1}z_{n2}&\\dots&\\sum^N_{n=1}z_{n1}z_{nd}\\\\\n",
    "\\sum^N_{n=1}z_{n2}z_{n1}&\\sum^N_{n=1}z_{n2}z_{n2}&\\dots&\\sum^N_{n=1}z_{n2}z_{nd}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "\\sum^N_{n=1}z_{nd}z_{n1}&\\sum^N_{n=1}z_{nd}z_{n2}&\\dots&\\sum^N_{n=1}z_{nd}z_{nd}\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&= \\sum^N_{n=1}\\begin{bmatrix}\n",
    "z_{n1}z_{n1}&z_{n1}z_{n2}&\\dots&z_{n1}z_{nd}\\\\\n",
    "z_{n2}z_{n1}&z_{n2}z_{n2}&\\dots&z_{n2}z_{nd}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "z_{nd}z_{n1}&z_{nd}z_{n2}&\\dots&z_{nd}z_{nd}\\\\\n",
    "\\end{bmatrix}_{d\\times d}\\\\\n",
    "&= \\sum^N_{n=1}\\begin{bmatrix}z_{n1}\\\\z_{n2}\\\\\\dots\\\\z_{nd}\\end{bmatrix}\n",
    "\\begin{bmatrix}z_{n1}&z_{n2}&\\dots&z_{nd}\\end{bmatrix}\\\\\n",
    "&= \\sum^N_{n=1}z_nz^T_n\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have $y = \\begin{bmatrix}y_1\\\\y_2\\\\\\dots\\\\y_N\\end{bmatrix}$. \n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "Z^Ty &= \\begin{bmatrix}\n",
    "z_{11}&z_{21}&\\dots&z_{N1}\\\\\n",
    "z_{12}&z_{22}&\\dots&z_{N2}\\\\\n",
    "\\dots&\\dots&\\dots&\\dots\\\\\n",
    "z_{1d}&z_{2d}&\\dots&z_{Nd}\\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}y_1\\\\y_2\\\\\\dots\\\\y_N\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}\n",
    "\\sum^N_{n=1}z_{n1}y_n\\\\\n",
    "\\sum^N_{n=1}z_{n2}y_n\\\\\n",
    "\\dots\\\\\n",
    "\\sum^N_{n=1}z_{nd}y_n\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&= \\sum^N_{n=1}\\begin{bmatrix}\n",
    "z_{n1}y_n\\\\\n",
    "z_{n2}y_n\\\\\n",
    "\\dots\\\\\n",
    "z_{nd}y_n\\\\\n",
    "\\end{bmatrix}\\\\\n",
    "&=\\sum^N_{n=1}z_ny_n\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Now let's look at $H(\\lambda)$, we have\n",
    "\n",
    "\\begin{align*}\n",
    "H(\\lambda) &= Z\\left( Z^TZ+\\lambda\\Gamma^T\\Gamma \\right)^{-1}Z^T\\\\\n",
    "&= ZA^{-1}(\\lambda)Z^T\\\\\n",
    "&= ZA^{-1}(\\lambda) \n",
    "\\begin{bmatrix}z_1&z_2&\\dots&z_N\\end{bmatrix}\\\\\\\\\n",
    "&= Z \n",
    "\\begin{bmatrix}A^{-1}(\\lambda)z_1&A^{-1}(\\lambda)z_2&\\dots&A^{-1}(\\lambda)z_N\\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}z_1\\\\z_2\\\\\\dots&\\\\z_N\\end{bmatrix}\n",
    "\\begin{bmatrix}A^{-1}(\\lambda)z_1&A^{-1}(\\lambda)z_2&\\dots&A^{-1}(\\lambda)z_N\\end{bmatrix}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "It is easy to see that $H_{nm}(\\lambda) = z^T_nA^{-1}(\\lambda)z_m$.\n",
    "\n",
    "Clearly, we see that when $(z_n, y_n)$ is left out, we have $Z^TZ \\to Z^TZ - z_nz^T_n$ and $Z^Ty \\to Z^Ty - z_ny_n$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* (b) Let's compute $w^-_n$ the weight vector learned when the $n$th data point is left out, \n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    "w^-_n &= \\left(Z_-^TZ_-+\\lambda \\Gamma^T\\Gamma\\right)^{-1}\n",
    "Z_-^Ty_-\\\\\n",
    "&= A_-^{-1}Z_-^Ty_-\\\\\n",
    "&= \\left(A - z_nz^T_n\\right)^{-1}(Z^Ty - z_ny_n)\\\\\n",
    "&= \\left(A^{-1} + \\frac{A^{-1}z_nz^T_nA^{-1}}{1-z^T_nA^{-1}z_n}\\right)(Z^Ty - z_ny_n)\\\\\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "* (c) From (a) and (b) we have\n",
    "\n",
    "\\begin{align*}\n",
    "w^-_n &= \\left(A^{-1} + \\frac{A^{-1}z_nz^T_nA^{-1}}{1-z^T_nA^{-1}z_n}\\right)(Z^Ty - z_ny_n)\\\\\n",
    "&= A^{-1}Z^Ty - A^{-1}z_ny_n + \\frac{A^{-1}z_nz^T_nA^{-1}}{1-z^T_nA^{-1}z_n}Z^Ty - \\frac{A^{-1}z_nz^T_nA^{-1}}{1-z^T_nA^{-1}z_n}z_ny_n\\\\\n",
    "&= w - A^{-1}z_ny_n + \\frac{A^{-1}z_nz^T_nA^{-1}}{1-H_{nn}}Z^Ty - \\frac{A^{-1}z_nz^T_nA^{-1}}{1-H_{nn}}z_ny_n\\\\\n",
    "&= w + A^{-1}z_n \\left[- y_n + \\frac{z^T_nA^{-1}}{1-H_{nn}}Z^Ty - \\frac{H_{nn}y_n}{1-H_{nn}}\\right]\\\\\n",
    "&= w + A^{-1}z_n \\left[\\frac{z^T_nA^{-1}}{1-H_{nn}}Z^Ty - \\frac{y_n}{1-H_{nn}}\\right]\\\\\n",
    "&= w + A^{-1}z_n \\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Where we have used the fact \n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{y}_n &= H_{n}y \\\\\n",
    "&= \\begin{bmatrix}z^T_n A^{-1}z_1&z^T_n A^{-1}z_2&\\dots&z^T_n A^{-1}z_N\\end{bmatrix}y\\\\\n",
    "&= z^T_nA^{-1}Z^Ty\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (d) \n",
    "\n",
    "\\begin{align*}\n",
    "z^T_nw^-_n &= z^T_n \\left[w + A^{-1}z_n \\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right]\\\\\n",
    "&= \\hat{y}_n + z^T_nA^{-1}z_n \\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\\\\n",
    "&= \\hat{y}_n + H_{nn} \\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\\\\n",
    "&= \\frac{\\hat{y}_n - H_{nn}y_n}{1-H_{nn}}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "* (e) \n",
    "\n",
    "\\begin{align*}\n",
    "e_n &= \\left(z^T_nw^-_n - y_n\\right)^2\\\\\n",
    "&= \\left(\\frac{\\hat{y}_n - H_{nn}y_n}{1-H_{nn}} - y_n\\right)^2\\\\\n",
    "&= \\left(\\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right)^2\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.27\n",
    "\n",
    "* (a) Because $var(e_1)$ is approximately $N$ times the variance of $E_{cv}$ from problem 4.23, 4.24 when $N$ is large. So we divide by $\\sqrt{N}$ to make the $\\sigma_{cv}$ scale to the standard deviation of $E_{cv}$.\n",
    "\n",
    "* (b) From problem 4.26 we have $e_n = \\left(\\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right)^2$. So\n",
    "\n",
    "\\begin{align*}\n",
    "var(e_1,\\dots,e_N) &= \\frac{\\sum^N_{n=1}(e_n-E(e_n))^2}{N}\\\\\n",
    "&= \\frac{\\sum^N_{n=1}(e_n-E_{cv})^2}{N}\\\\\n",
    "&= \\frac{\\sum^N_{n=1}(e^2_n-2e_nE_{cv}+E^2_{cv})}{N}\\\\\n",
    "&= \\sum^N_{n=1}\\frac{e^2_n}{N}-E^2_{cv}\\\\\n",
    "&= \\frac{1}{N}\\sum^N_{n=1}\\left(\\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right)^4-E^2_{cv}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "So we have $N\\sigma^2_{CV} \\approx \\frac{1}{N}\\sum^N_{n=1}\\left(\\frac{\\hat{y}_n - y_n}{1-H_{nn}}\\right)^4-E^2_{cv}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#### Problem 4.27 (c)\n",
    "\n",
    "def find_w(xs, ys, Q, lambda_t):\n",
    "    Z = data.polynomial_transform(Q, xs)\n",
    "    inv_z = np.linalg.inv(np.matmul(Z.transpose(), Z)+lambda_t)\n",
    "    w = np.matmul(inv_z, np.matmul(Z.transpose(), ys))\n",
    "    return w\n",
    "    \n",
    "def run_one_crossvalidation_experiment2(xs, ys, Q, lambda_t):\n",
    "    \"\"\"Generate one set of training data for a given model and compute the cross validation error\n",
    "    N: The number of points in training set\n",
    "    Q: The degree of freedom of the hypothesis model\n",
    "    \"\"\"\n",
    "    \n",
    "    ens = [] #List of validation errors with each data removed, so it has e_1, e_2, ...\n",
    "    for idx in np.arange(len(ys)):\n",
    "        x = xs[idx,:].reshape(1,-1)\n",
    "        y = ys[idx]\n",
    "        \n",
    "        #build a data set without (x,y)\n",
    "        xs_minus = np.vstack((xs[:idx, :], xs[idx+1:,:]))\n",
    "        ys_minus = np.vstack((ys[:idx, :], ys[idx+1:,:]))\n",
    "\n",
    "        #Train on the LOO (leave-one-out) data set\n",
    "        # Hypothesis with degree Q        \n",
    "        w = find_w(xs_minus, ys_minus, Q, lambda_t)\n",
    "        \n",
    "        # Calculate out-of-sample error e_n on (x,y)\n",
    "        en = data.calc_Eout(w, x, y).flatten()[0]\n",
    "        ens.append(en)\n",
    "    #print(ens)\n",
    "    Ecv = np.mean(ens)\n",
    "    var_cv = np.var(ens)\n",
    "    return Ecv, var_cv\n",
    "\n",
    "\n",
    "def calc_mean_std_cv(lambda_ts, N, sigma_square, normalized_aqs, Q):\n",
    "    E_cvs, Sigma_cvs = [], []\n",
    "    xs, ys = data.generate_data_set(N, normalized_aqs, sigma_square, tol=0.1)\n",
    "    for lambda_t in lambda_ts:\n",
    "        res = run_one_crossvalidation_experiment2(xs, ys, Q, lambda_t)\n",
    "        E_cv, var_cv = res\n",
    "        sigma_cv = np.sqrt(var_cv/N)\n",
    "        E_cvs.append(E_cv)\n",
    "        Sigma_cvs.append(sigma_cv)\n",
    "    return E_cvs, Sigma_cvs, xs, ys\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------ Working on N =  40 ------\n",
      "Traditional CV-Index with minimum E_cv:  98 69.6088219045686 64.77072690701813\n",
      "One-Sigma Approach bounds:  4.838094997550471 134.37954881158674\n",
      "Index of optimal pairs (index:lambda)  98 : 4.95 \t 98 : 4.95 \t 98 : 4.95\n",
      "------ Working on N =  60 ------\n",
      "Traditional CV-Index with minimum E_cv:  99 18.481176501354913 12.295262964219248\n",
      "One-Sigma Approach bounds:  6.185913537135665 30.77643946557416\n",
      "Index of optimal pairs (index:lambda)  99 : 5.0 \t 99 : 5.0 \t 99 : 5.0\n",
      "------ Working on N =  80 ------\n",
      "Traditional CV-Index with minimum E_cv:  69 1.4465773037024756 0.25877121406568987\n",
      "One-Sigma Approach bounds:  1.1878060896367857 1.7053485177681655\n",
      "Index of optimal pairs (index:lambda)  69 : 3.5 \t 99 : 5.0 \t 69 : 3.5\n",
      "------ Working on N =  100 ------\n",
      "Traditional CV-Index with minimum E_cv:  92 9.431358378544695 8.2157917144709\n",
      "One-Sigma Approach bounds:  1.2155666640737959 17.647150093015597\n",
      "Index of optimal pairs (index:lambda)  92 : 4.65 \t 99 : 5.0 \t 92 : 4.65\n",
      "------ Working on N =  120 ------\n",
      "Traditional CV-Index with minimum E_cv:  99 1.0188954190987558 0.13336474252276395\n",
      "One-Sigma Approach bounds:  0.8855306765759918 1.1522601616215198\n",
      "Index of optimal pairs (index:lambda)  99 : 5.0 \t 99 : 5.0 \t 99 : 5.0\n",
      "------ Working on N =  140 ------\n",
      "Traditional CV-Index with minimum E_cv:  98 1.1924668308694149 0.13935239173117425\n",
      "One-Sigma Approach bounds:  1.0531144391382405 1.3318192226005892\n",
      "Index of optimal pairs (index:lambda)  98 : 4.95 \t 99 : 5.0 \t 98 : 4.95\n",
      "------ Working on N =  160 ------\n",
      "Traditional CV-Index with minimum E_cv:  87 1.0747808955342308 0.15053857164023232\n",
      "One-Sigma Approach bounds:  0.9242423238939985 1.2253194671744632\n",
      "Index of optimal pairs (index:lambda)  87 : 4.4 \t 99 : 5.0 \t 87 : 4.4\n",
      "------ Working on N =  180 ------\n",
      "Traditional CV-Index with minimum E_cv:  88 1.0737912099577245 0.11249737068720082\n",
      "One-Sigma Approach bounds:  0.9612938392705237 1.1862885806449253\n",
      "Index of optimal pairs (index:lambda)  88 : 4.45 \t 99 : 5.0 \t 88 : 4.45\n",
      "------ Working on N =  200 ------\n",
      "Traditional CV-Index with minimum E_cv:  81 0.9810912003217455 0.09042666505125815\n",
      "One-Sigma Approach bounds:  0.8906645352704873 1.0715178653730035\n",
      "Index of optimal pairs (index:lambda)  81 : 4.1 \t 99 : 5.0 \t 81 : 4.1\n"
     ]
    }
   ],
   "source": [
    "sigma_square = 1\n",
    "Qf = 15 #target function\n",
    "Q = 20\n",
    "max_N = 10*Q\n",
    "test_N = 100\n",
    "Ns = np.arange(2*Q, max_N+1, Q)\n",
    "normalized_aqs = data.generate_target_coefficients(Qf)\n",
    "test_xs, test_ys = data.generate_data_set(test_N, normalized_aqs, sigma_square)\n",
    "lambda_ts = np.arange(0.05, 5+0.01, 0.05)\n",
    "\n",
    "E_outs1 = [] #Traiditional cross validation approach\n",
    "E_outs2 = [] #One-Sigma approach\n",
    "E_outs3 = [] #Bound minimizing approach\n",
    "\n",
    "for N in Ns:\n",
    "    print('------ Working on N = ', N, '------')\n",
    "    E_cvs, Sigma_cvs, xs, ys = calc_mean_std_cv(lambda_ts, N, sigma_square, normalized_aqs, Q)\n",
    "                    \n",
    "    #Traditional cross validation to select the optimal lambda\n",
    "    min_idx1 = np.argmin(E_cvs)\n",
    "    lambda1 = lambda_ts[min_idx1]\n",
    "    w_opt1 = find_w(xs, ys, Q, lambda1)\n",
    "    Eout1 = data.calc_Eout(w_opt1, test_xs, test_ys)\n",
    "    E_outs1.append(Eout1)\n",
    "    \n",
    "    best_Ecv = E_cvs[min_idx1]\n",
    "    print('Traditional CV-Index with minimum E_cv: ', min_idx1, best_Ecv, Sigma_cvs[min_idx1])\n",
    "    \n",
    "    \n",
    "    # Use one-sigma approach\n",
    "    # Find the simplest model within one sigma of Sigma_cvs[min_idx1]\n",
    "    ub = best_Ecv + Sigma_cvs[min_idx1]\n",
    "    lb = best_Ecv - Sigma_cvs[min_idx1]\n",
    "    print('One-Sigma Approach bounds: ', lb, ub)\n",
    "    \n",
    "#     for idx in np.arange(min_idx1+1):\n",
    "#         current_Ecv = E_cvs[idx]\n",
    "#         if current_Ecv >= lb and current_Ecv <= ub:\n",
    "#             min_idx2 = idx\n",
    "#             break\n",
    "    \n",
    "    min_idx2 = None\n",
    "    # The simplest model has the largest lambda within bounds, so we go in reverse order\n",
    "    for idx in range(len(E_cvs)-1, -1, -1):\n",
    "        Ecv = E_cvs[idx]\n",
    "        if lb <= Ecv <= ub:\n",
    "            min_idx2 = idx\n",
    "            break\n",
    "            \n",
    "    lambda2 = lambda_ts[min_idx2]\n",
    "    w_opt2 = find_w(xs, ys, Q, lambda2)            \n",
    "    Eout2 = data.calc_Eout(w_opt2, test_xs, test_ys)\n",
    "    E_outs2.append(Eout2)\n",
    "    \n",
    "    # Use bound minimizing approach\n",
    "    bounds = np.array(E_cvs) + np.array(Sigma_cvs)\n",
    "    min_idx3 = np.argmin(bounds)\n",
    "    \n",
    "    lambda3 = lambda_ts[min_idx3]\n",
    "    w_opt3 = find_w(xs, ys, Q, lambda3)\n",
    "    Eout3 = data.calc_Eout(w_opt3, test_xs, test_ys)\n",
    "    E_outs3.append(Eout3)\n",
    "    \n",
    "    print('Index of optimal pairs (index:lambda) ', min_idx1, ':', lambda1, '\\t', min_idx2, ':', lambda2, '\\t', min_idx3, ':', lambda3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(Ns, E_outs1, '.', color = 'red', label='Traidtional Cross Validation')\n",
    "plt.plot(Ns, E_outs2, color='blue', linewidth=1, label='One-Sigma Approach')\n",
    "plt.plot(Ns, E_outs3, color='green', linewidth=1, label='Bound Minimizing Approach')\n",
    "plt.legend()\n",
    "plt.yscale('log')\n",
    "#plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(Ns, np.abs(np.array(E_outs2)-np.array(E_outs1)), '.', color = 'red', label='One-Sigma - Traditional')\n",
    "plt.plot(Ns, np.abs(np.array(E_outs3)-np.array(E_outs1)), '.', color='blue', label='Bound Minimizing - Traditional')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 4.27 (c)\n",
    "\n",
    "* From the experiment, we see that the traditional cross validation agree with the bound minimizing approach most of the time on the optimal lambda. \n",
    "* We see that the conservative one-sigma approach usually will select the maximum lambda (from all lambdas) as the optimal lambda. Which is the simplest in the available $\\lambda$ range. Most of the time, the out-of-sample errors are close to and sometimes better than what we get from traditional cross-validation. \n",
    "* From the graph of out-of-sample error, we do see that all three apporaches are very close to each other in most of the out-of-sample errors(Except when $N=80$, $N=100$), even when the bound minimizing approach may select quite different $\\lambda$ as the other two approachs. \n",
    "* The conclusion is that traditional cross validation works pretty well compared with other two approaches. So we don't need spend extra computation efforts on the $\\sigma_{cv}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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    "name": "ipython",
    "version": 3
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