07_logistic_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "## Logistic Regression\n",
    "\n",
    "We saw in the previous lesson how supervised machine learning allows us to build predictive models of linear relationships in our data. What if our data cannot be modeled well by a linear function? For example, we might wish to model the probability of a borrower defaulting on their credit card balance as a function of the size of the balance relative to their income. The dataset for this problem might look like the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "balance_as_pct_of_income = np.array([5, 10, 15, 20, 25, 30, 35, 40, 45, 50])\n",
    "probability_of_default = np.array([0.05, 0.07, 0.10, 0.15, 0.25, 0.5, 0.75, 0.90, 0.95,0.99])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "The problem for us is that this data has a decidely non-linear shape:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "plt.scatter(balance_as_pct_of_income, probability_of_default)\n",
    "plt.xlabel('Balance as % of Income')\n",
    "plt.ylabel('Probability of Default')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "The following effects cause the graph to have this distinctive S-shape:\n",
    "\n",
    "1. The probability of default is relatively insensitive to the balance as long as the balance is well within the range that the customer can pay, so it increases slowly at the beginning. \n",
    "2. As the customer approaches the limits of their ability to make payments, the probability of default rises rapidly. \n",
    "3. Eventually, the curve starts to bend as we move along the x-axis because the probability can never exceed 1, as that indicates absolute certainty of default."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "### Modeling Probabilities with Logistic Curves\n",
    "\n",
    "When our data looked like a line, we used machine learning to build a linear function to model it. When our data looks like the graph above, what type of function should we build to model it? It turns out that data with the above pattern is modeled well by a **logistic curve**, which is a fuction of the form: \n",
    "\n",
    "$$f(x) = \\frac{1}{1 + e^{-k(x - x_0)}}$$\n",
    "\n",
    "Do not worry excessively about the math involved in this definition - the important fact to grasp is that, just as we could adjust the slope and the intercept of our linear model in the linear regression lesson, we can adjust the following parameters of our logistic curve model:\n",
    "\n",
    "1. The steepness, $k$\n",
    "2. The midpoint, $x_0$\n",
    "\n",
    "Let's experiment with different values of these parameters to see if we can come up with a model that can accurately predict our probability values. First, we will vary the midpoint value: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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": [
    "import numpy as np\n",
    "\n",
    "midpoints = [20,30,40]\n",
    "steepness = 0.5\n",
    "x = np.arange(0, 50)\n",
    "for midpoint in midpoints:\n",
    "    y = 1/(1 + np.exp(-steepness*(x - midpoint)))\n",
    "    plt.plot(x,y,label='midpoint=' + str(midpoint))\n",
    "plt.scatter(balance_as_pct_of_income, probability_of_default, label='raw data')\n",
    "plt.xlabel('Balance as % of Income')\n",
    "plt.ylabel('Probability of Default')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "It looks like a value of `30` is the best value for the midpoint. Now let's try to find a good value for the steepness:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": [
    "midpoint = 30\n",
    "steepness_values = [0.25, 0.50, 0.75]\n",
    "for steepness in steepness_values:\n",
    "    y = 1/(1 + np.exp(-steepness*(x - midpoint)))\n",
    "    plt.plot(x,y,label='steepness=' + \"{0:.2f}\".format(steepness))\n",
    "plt.scatter(balance_as_pct_of_income, probability_of_default, label='raw data')\n",
    "plt.xlabel('Balance as % of Income')\n",
    "plt.ylabel('Probability of Default')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "A steepness value of `0.25` seems very close to correct. So our final guess for the appropriate model parameters is: \n",
    "\n",
    "1. Steepness = 0.25\n",
    "2. Midpoint = 30"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "### Building Logistic Curves with the Logistic Regression Method\n",
    "\n",
    "The above process of finding the appropriate model parameters is obviously cumbersome; instead of finding the parameters manually, we can invoke the machine learning method of **logistic regression** to automatically build a logistic curve with the correct parameter values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Since we are going to build a real machine learning model, let's begin our analysis by using a more realistic dataset. Usually, we will not begin with an array containing *probabilities*, but rather with an array of *individual default events*. So instead of the `probability_of_default` array above, we will start from a larger array containing default data for *individual customers*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Balance as % of Income</th>\n",
       "      <th>Defaulted? (0/1 = no/yes)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>30</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>45</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>20</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>15</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Balance as % of Income  Defaulted? (0/1 = no/yes)\n",
       "0                       5                          0\n",
       "1                      30                          1\n",
       "2                      45                          1\n",
       "3                      20                          0\n",
       "4                      15                          1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "raw_default_data_df = pd.read_csv(\"data/raw_default_data.csv\")\n",
    "raw_default_data_df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "With this data in hand, we can perform logistic regression to obtain a logistic curve model of our data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,\n",
       "                   intercept_scaling=1, l1_ratio=None, max_iter=100,\n",
       "                   multi_class='warn', n_jobs=None, penalty='l2',\n",
       "                   random_state=None, solver='lbfgs', tol=0.0001, verbose=0,\n",
       "                   warm_start=False)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LogisticRegression\n",
    "\n",
    "logistic = LogisticRegression(solver='lbfgs')\n",
    "x = np.array(raw_default_data_df['Balance as % of Income'])\n",
    "X = x.reshape(-1,1)\n",
    "y = np.array(raw_default_data_df['Defaulted? (0/1 = no/yes)'])\n",
    "logistic.fit(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Note that we use the keyword argument `solver='lbfgs'` to control *how* the machine learning module builds our model - no matter what solver we use, the *type* of model we get out at the end is always the same: a logistic curve."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Let's see how accurate our new model is by *predicting* our output variable, the probability of default:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.98313341, 0.01686659],\n",
       "       [0.96151194, 0.03848806],\n",
       "       [0.91458178, 0.08541822],\n",
       "       [0.82107623, 0.17892377],\n",
       "       [0.6629358 , 0.3370642 ],\n",
       "       [0.45739192, 0.54260808],\n",
       "       [0.26539804, 0.73460196],\n",
       "       [0.13408075, 0.86591925],\n",
       "       [0.06223389, 0.93776611],\n",
       "       [0.0276564 , 0.9723436 ]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predicted_probs = logistic.predict_proba(balance_as_pct_of_income.reshape(-1,1))\n",
    "predicted_probs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Notice that the array above has two columns: the probability of NOT defaulting, and the probability of defaulting. We want the second column. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Let's plot our results and see what they look like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": [
    "X = balance_as_pct_of_income.reshape(-1,1)\n",
    "plt.scatter(X, predicted_probs[:,1], label='predicted')\n",
    "plt.scatter(X, probability_of_default, label='raw')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Not too bad! How close is this model to the model we came up with through trial and error? The `sklearn` model labels its parameters slightly differently, but some simple math allows us to convert them to steepness and midpoint values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.16945027864782097\n",
      "28.991759281578272\n"
     ]
    }
   ],
   "source": [
    "model_steepness = logistic.coef_[0][0]\n",
    "model_midpoint = -1 * logistic.intercept_[0]/model_steepness\n",
    "print(model_steepness)\n",
    "print(model_midpoint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "These values seem reasonable when compared to our manually chosen values above. Based on our graph and our examination of the model parameters, we now know that our training process has produced a qualitatively reasonable machine learning model. We still need to perform a rigorous *quantitative*  assessment of our model's accuracy, but performing this assessment is actually a surprisingly nuanced task - we will therefore defer this process until the next lesson. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "<span style=\"color:blue;font-weight:bold\">Exercise</span>: Predicting Credit Card Default. \n",
    "\n",
    "Suppose that the probability of customer defaulting on a particular credit card decreases rapidly with the number of months that they have successfully made prior payments. Intuitively, this makes sense: someone who has been making payments on-time for years is much less likely to miss future payments than someone with no prior history. In this exercise, we will use logistic regression to model the probability of default as a function of the number of months in which the borrower has successfully made their payments. The following $x$ and $y$ values give the relationship between the number of months that the borrower has been in good standing and their probability of default:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "months_in_good_standing = np.array([3, 6, 12, 18, 24, 36, 48])\n",
    "probabilities_by_months = np.array([0.35, 0.33, 0.20, 0.07, 0.03, 0.02, 0.01])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "run_control": {
     "frozen": true
    }
   },
   "source": [
    "Perform the following tasks:\n",
    "\n",
    "1. Load the dataset from the file `data/credit_card_exercise.csv` and store it in the DataFrame variable `df`\n",
    "2. Use this data to construct a logistic regression model with `'Months in Good Standing'` as your $x$ variable and `'Defaulted? (0/1 = no/yes)'` as your $y$ variable - you must store this model in the variable `logistic`, and pass `solver='lbfgs'` and `random_state=0` to the `LogisticRegression` function call\n",
    "3. Use your model to predict the probability of default for each `months_in_good_standing` value - store this result in the variable `predicted_probs`\n",
    "4. Graph your predicted probabilities vs. the actual probabilities - does your model look qualitatively correct?\n",
    "5. Print the model parameters - What important thing has changed about the model paramaters in this example compared to the one above?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true,
    "tags": [
     "exercise"
    ]
   },
   "outputs": [],
   "source": [
    "logistic = LogisticRegression(solver=\"lbfgs\")\n",
    "df = pd.read_csv(\"data/credit_card_exercise.csv\")\n",
    "logistic.fit(df[[\"Months in Good Standing\"]], df[\"Defaulted? (0/1 = no/yes)\"])\n",
    "\n",
    "predicted_probs = logistic.predict_proba(months_in_good_standing.reshape(-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "deletable": false,
    "editable": false,
    "strategicml": {
     "execution_time": 1584776848402,
     "exercise_id": "e7b4608e-dde4-4292-a5a2-618c29b90c1f",
     "points": 3
    },
    "tags": [
     "test"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"background-color:#e8fcef\n",
       ";color:#333333;padding:10px;border-style:solid;border-width:2px;border-color:#038c33\">\n",
       "<div style=\"font-weight:bold\">Correct!</div>\n",
       "</div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df_hint = \"Check that you are loading the dataset into a dataframe correctly\"\n",
    "correct_df = pd.read_csv(\"data/credit_card_exercise.csv\")\n",
    "assert isinstance(df, pd.DataFrame), df_hint\n",
    "assert correct_df.equals(df), df_hint\n",
    "\n",
    "import math\n",
    "pred_probs_hint = \"Double-check your <code>predicted_probs</code> variable value.\"\n",
    "assert isinstance(predicted_probs, np.ndarray), pred_probs_hint\n",
    "assert predicted_probs.shape == (7,2), pred_probs_hint\n",
    "assert math.isclose(predicted_probs[6,0], 0.997, abs_tol=1e-3), pred_probs_hint\n",
    "\n",
    "logistic_hint = \"Check the value of your <code>logistic</code> model object\"\n",
    "assert isinstance(logistic, LogisticRegression), logistic_hint\n",
    "assert math.isclose(logistic.coef_, -0.115, abs_tol=1e-3), logistic_hint\n",
    "assert math.isclose(logistic.intercept_, -0.174, abs_tol=1e-3), logistic_hint\n",
    "success()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.62654312, 0.37345688],\n",
       "       [0.70293063, 0.29706937],\n",
       "       [0.82477611, 0.17522389],\n",
       "       [0.90350604, 0.09649396],\n",
       "       [0.94904712, 0.05095288],\n",
       "       [0.98661397, 0.01338603],\n",
       "       [0.99658303, 0.00341697]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predicted_probs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f045a429eb8>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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5NXCkqnYnuXTU61kgXlJVh5M8Ffhskm/03tnPc8Ur+Pn3vSRPB+j+fWTE65lXSc6iE/ePV9WnusOLek+q6kHgduDFwLIkj114rQQOj2xh8+sSYH2S+4Db6Lw08z4W735QVYe7fx+hcwFwEQM+Vwz8/NsBvKF7+w3AP4xwLfOq+5rq3wD3VtW7e+5adHuSZLx75U6SpcAr6LwncTvw2u60RbEXAFV1XVWtrKrVwEbg81X1ehbpfiT52SRPeuw28OvA1xnwueI3WedQkluBS+n8mM/vAW8DtgPbgPPo/Ljk11XV9Ddim5TkJcC/Ant5/HXWP6LzOvyi2pMkL6DzJtkYnQutbVV1Y5JfoHMFey6wB7iyqn48upXOv+5LNH9QVa9erPvR/Xd/unu4BPhEVd2U5CkM8Fwx8JLUKF+ikaRGGXhJapSBl6RGGXhJapSBl6RGGXhJapSBl6RG/T9k6XVVIH7cUQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(df[[\"Months in Good Standing\"]], df[\"Defaulted? (0/1 = no/yes)\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD4CAYAAAANbUbJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAe2klEQVR4nO3df3BV9Z3/8eerEAv+KCimXSUo6Zai/IhEb1kttRVZhA4KTkWqlV264y5tR8f99gcKXReF3U6xzNSWLc7KbF2crQrRqoW1W7Ria3VrJQiCoGhQKwnuGkGorFEDvL9/3JN4ExNyE5LcJOf1mMncez7nc27e9zB53cPnnPs5igjMzKxv+0ihCzAzs67nsDczSwGHvZlZCjjszcxSwGFvZpYC/QtdQHMnn3xyDB8+vNBlmJn1Khs3bnwzIopbW9/jwn748OFUVlYWugwzs15F0h+PtN7DOGZmKeCwNzNLAYe9mVkK9LgxezPrOvX19VRXV/Puu+8WuhTroAEDBlBSUkJRUVG7tnPYm6VIdXU1J5xwAsOHD0dSocuxdooI9uzZQ3V1NaWlpe3a1sM4Ziny7rvvMmTIEAd9LyWJIUOGdOh/Zg57s5Rx0PduHf33c9ibmaVA3wz7LRVw6xi4eXD2cUtFoSsyM2Dfvn3cdtttHd7+Rz/6Ee+8806b/X7zm99w8cUXH7HP5s2b+eUvf9nhWnqbvhf2Wypg7XWwfxcQ2ce11znwzXqA7gr7fDjse7tHF0N9XdO2+rpsu5m1y4ObapiwZD2l8x9iwpL1PLip5qheb/78+ezcuZNx48Yxb948AJYuXcpnPvMZysrKuOmmmwD4v//7P6ZNm8ZZZ53FmDFjWL16NcuWLWP37t1MnDiRiRMnfui1f/WrX3HGGWdw9tlnc//99ze2P/3005x33nmUl5fz2c9+lh07dvD++++zcOFCVq9ezbhx41i9enWL/fqSvnfp5f7q9rWbWYse3FTDgvu3Uld/CICafXUsuH8rAJeWD+3Qay5ZsoTnnnuOzZs3A/Dwww/z0ksv8fTTTxMRTJ8+nccff5za2lpOPfVUHnroIQD279/PoEGD+OEPf8hjjz3GySef3OR13333Xf7u7/6O9evX86lPfYovf/nLjevOOOMMfve739G/f39+/etf893vfpef//znLF68mMrKSn7yk58A8Kc//anFfn1F3wv7QSXJEE4L7WaWt6XrdjQGfYO6+kMsXbejw2Hf3MMPP8zDDz9MeXk5AAcOHOCll17i/PPP59vf/jY33HADF198Meeff/4RX+eFF16gtLSUESNGADB79mxWrFgBZD8o5syZw0svvYQk6uvrW3yNfPv1Vn1vGGfSQiga2LStaGC23czytntfXbvaOyIiWLBgAZs3b2bz5s1UVVVx9dVX8+lPf5pnnnmGsWPHcuONN7J4cceHYf/xH/+RiRMn8txzz7F27dpWr1HPt19v1ffCvmwWXLIMBg0DlH28ZFm23czydurgge1qz8cJJ5zA22+/3bg8ZcoU7rjjDg4cOABATU0Nb7zxBrt37+bYY49l9uzZzJs3j2eeeabF7RucccYZvPrqq+zcuROAe+65p3Hd/v37GTo0+z+RlStXtlpLa/36ir4X9pAN9m8+Bzfvyz466M3abd6UkQws6tekbWBRP+ZNGdnh1xwyZAgTJkxgzJgxzJs3j4suuoivfOUrnHfeeYwdO5aZM2fy9ttvs3XrVsaPH8+4ceNYtGgRN954IwBz585l6tSpHzpBO2DAAFasWMG0adM4++yz+fjHP9647vrrr2fBggWUl5dz8ODBxvaJEyeyffv2xhO0rfXrKxQRha6hiUwmE755iVnXeP755znzzDPz7v/gphqWrtvB7n11nDp4IPOmjOy08XrruJb+HSVtjIhMa9v0vRO0ZtZpLi0f6nDvI/rmMI6ZmTXhsDczS4G8wl7SVEk7JFVJmt/C+q9L2ipps6QnJI1K2odLqkvaN0v6185+Ay3p7G/9mZn1dm2O2UvqBywHJgPVwAZJayJie063uyPiX5P+04EfAlOTdTsjYlznlt26rvjWn5lZb5fPkf14oCoiXo6I94FVwIzcDhHxp5zF44CCXeJzpG/9mZmlVT5hPxTInX+gOmlrQtI1knYCPwCuy1lVKmmTpN9KavE7z5LmSqqUVFlbW9uO8j+sO771Z2Y9x/HHHw/A7t27mTlz5hH7dmTWzHymS+6ohtq7Q6edoI2I5RHx58ANwI1J8+vAaRFRDnwLuFvSx1rYdkVEZCIiU1xcfFR1dMW3/sysex06dKjtTs2ceuqp3HfffUfs05lTJOcjIjh8+HC3/b4jySfsa4BhOcslSVtrVgGXAkTEexGxJ3m+EdgJfLpjpeanK771Z5ZanXwjoFdffZUzzjiDq666ijPPPJOZM2c2hu/w4cO54YYbOPvss7n33nvZuXMnU6dO5ZxzzuH888/nhRdeAOCVV15p/MZtwzdrG157zJgxQPbD4jvf+Q5jxoyhrKyMf/mXf2lxiuSHH36Y8847j7PPPpvLL7+8cdqG1qZLzrVy5UpmzJjBBRdcwIgRI1i0aFFjHSNHjuSv//qvGTNmDLt27eKee+5h7NixjBkzhhtuuKHJ63zzm99k9OjRTJo0iYaRjWXLljFq1CjKysq44oorjmqfN4qII/6QPYn7MlAKHAM8C4xu1mdEzvNLgMrkeTHQL3n+SbIfEicd6fedc845cbQeeKY6Pvv9R2P4Df8Zn/3+o/HAM9VH/ZpmfcH27dvz7/zs6oh//kTETR/74OefP5Ft76BXXnklgHjiiSciIuJv/uZvYunSpRERcfrpp8ctt9zS2PfCCy+MF198MSIinnrqqZg4cWJERFxyySVx5513RkTET37ykzjuuOMaX3v06NEREXHbbbfFZZddFvX19RERsWfPnsbfUVtbGxERtbW1cf7558eBAwciImLJkiWxaNGiqKuri5KSknjxxRfj8OHDcfnll8e0adM+9F7+/d//Pf7sz/4s3nzzzXjnnXdi9OjRsWHDhnjllVdCUvz+97+PiIiampoYNmxYvPHGG1FfXx8TJ06MBx54ICIigPjZz34WERGLFi2Ka665JiIiTjnllHj33XcjIuKtt9760O9u6d+xIXdb+2nzyD4iDgLXAuuA54GKiNgmaXFy5Q3AtZK2SdpMdrhmTtL+eWBL0n4f8PWI2HsUn015ubR8KE/Ov5BXlkzjyfkX+iocs47oohsBDRs2jAkTJgDZqYifeOKJxnUN89AfOHCA//7v/+byyy9n3LhxfO1rX+P1118H4Mknn+TKK68E4K/+6q9a/B2//vWv+drXvkb//tkLDk866aQP9XnqqafYvn07EyZMYNy4cdx555388Y9/bDJdsiRmz57d6nuZPHkyQ4YMYeDAgXzpS19qfC+nn3465557LgAbNmzgggsuoLi4mP79+3PVVVfx+OOPA/CRj3yk8T3n7ouysjKuuuoqfvaznzW+h6OV16tExC+BXzZrW5jz/O9b2e7nQN+Z/d8sTbroRkCSWl0+7rjjADh8+DCDBw9uvMlJW6/RERHB5MmTm8yQCbT6O/Opo2G54X20V8P2Dz30EI8//jhr167le9/7Hlu3bj3q0Pc3aM2sZa3d8OcobwT02muv8fvf/x6Au+++m8997nMf6vOxj32M0tJS7r33XiAbzM8++ywAEyZMYNWqVQDcddddLf6OyZMnc/vttzfOXrl3b3ZAIXda43PPPZcnn3ySqqoqIHsrxBdffPGI0yU398gjj7B3717q6up48MEHG//Hkmv8+PH89re/5c033+TQoUPcc889fOELXwCyH2oNJ5Ub9sXhw4fZtWsXEydO5JZbbmH//v2N5xKOhsPezFrWRTcCGjlyJMuXL+fMM8/krbfe4hvf+EaL/e666y5++tOfctZZZzF69Gh+8YtfAPDjH/+Y5cuXM3bsWGpqWr5W5G//9m857bTTKCsr46yzzuLuu+8Gmk6RXFxczMqVK7nyyispKyvjvPPO44UXXjjidMnNjR8/nssuu4yysjIuu+wyMpkPTzp5yimnsGTJEiZOnMhZZ53FOeecw4wZ2a8qHXfccTz99NOMGTOG9evXs3DhQg4dOsTs2bMZO3Ys5eXlXHfddQwePLhd+7glnuI4D57m1fqK9k5xzJaK7Bj9/ursEf2khUd1f4hXX32Viy++mOeee67Dr9FTrFy5ssk9bLuTpzjuAp5+wVKtbJZv/tNHeBinDZ5+wazzDB8+vE8c1QN89atfLchRfUc57Nvg6Resr+lpQ7fWPh3993PYt8HTL1hfMmDAAPbs2ePA76Uigj179jBgwIB2b+sx+zbMmzKyyZg9ePoF671KSkqorq7maCcctMIZMGAAJSXtv/zVYd+GhpOwvhrH+oKioiJKS0sLXYYVgMM+D77pspn1dh6zNzNLAYe9mVkKOOzNzFLAYW9mlgIOezOzFHDYm5mlgMPezCwF8gp7SVMl7ZBUJWl+C+u/LmmrpM2SnpA0KmfdgmS7HZKmdGbxZmaWnzbDXlI/YDnwRWAUcGVumCfujoixETEO+AHww2TbUcAVwGhgKnBb8npmZtaN8jmyHw9URcTLEfE+sAqYkdshIv6Us3gc0DDL0gxgVUS8FxGvAFXJ6/UuWyrg1jFw8+Ds45aKQldkZtYu+UyXMBTYlbNcDfxF806SrgG+BRwDXJiz7VPNtv3QvAOS5gJzAU477bR86u4+Wypg7XVQn0xpvH9Xdhl8Uwcz6zU67QRtRCyPiD8HbgBubOe2KyIiExGZ4uLiziqpczy6+IOgb1Bfl203M+sl8gn7GmBYznJJ0taaVcClHdy259lf3b52M7MeKJ+w3wCMkFQq6RiyJ1zX5HaQNCJncRrwUvJ8DXCFpI9KKgVGAE8ffdndaFAr80a31m5m1gO1GfYRcRC4FlgHPA9URMQ2SYslTU+6XStpm6TNZMft5yTbbgMqgO3Ar4BrIuLQh35JTzZpIRQ1uytV0cBsu5lZL6GednuyTCYTlZWVhS6jqS0V2TH6/dXZI/pJC31y1sx6FEkbIyLT2nrfvCQfZbMc7mbWq3m6BDOzFHDYm5mlgMPezCwFHPZmZingsDczSwGHvZlZCjjszcxSwGFvZpYCDnszsxRw2JuZpYDD3swsBRz2ZmYp4LA3M0sBh72ZWQo47M3MUsBhb2aWAnmFvaSpknZIqpI0v4X135K0XdIWSY9KOj1n3SFJm5OfNc23NTOzrtfmnaok9QOWA5OBamCDpDURsT2n2yYgExHvSPoG8APgy8m6uogY18l1m5lZO+RzZD8eqIqIlyPifWAVMCO3Q0Q8FhHvJItPASWdW6aZmR2NfMJ+KLArZ7k6aWvN1cB/5SwPkFQp6SlJl7a0gaS5SZ/K2traPEoyM7P26NQbjkuaDWSAL+Q0nx4RNZI+CayXtDUiduZuFxErgBUAmUwmOrMmMzPL78i+BhiWs1yStDUh6S+BfwCmR8R7De0RUZM8vgz8Big/inrNzKwD8gn7DcAISaWSjgGuAJpcVSOpHLidbNC/kdN+oqSPJs9PBiYAuSd2zcysG7Q5jBMRByVdC6wD+gF3RMQ2SYuByohYAywFjgfulQTwWkRMB84Ebpd0mOwHy5JmV/GYmVk3UETPGiLPZDJRWVlZ6DLMzHoVSRsjItPaen+D1swsBRz2ZmYp4LA3M0sBh72ZWQo47M3MUsBhb2aWAg57M7MUcNibmaWAw97MLAUc9mZmKeCwNzNLAYe9mVkKOOzNzFLAYW9mlgIOezOzFHDYm5mlgMPezCwF8gp7SVMl7ZBUJWl+C+u/JWm7pC2SHpV0es66OZJeSn7mdGbxBg9uqmHCkvWUzn+ICUvW8+CmD90L3sys7bCX1A9YDnwRGAVcKWlUs26bgExElAH3AT9Itj0JuAn4C2A8cJOkEzuv/HR7cFMNC+7fSs2+OgKo2VfHgvu3OvDN7EPyObIfD1RFxMsR8T6wCpiR2yEiHouId5LFp4CS5PkU4JGI2BsRbwGPAFM7p3Rbum4HdfWHmrTV1R9i6bodBarIzHqqfMJ+KLArZ7k6aWvN1cB/tWdbSXMlVUqqrK2tzaMkA9i9r65d7WaWXp16glbSbCADLG3PdhGxIiIyEZEpLi7uzJL6tFMHD2xXu5mlVz5hXwMMy1kuSdqakPSXwD8A0yPivfZsax0zb8pIBhb1a9I2sKgf86aMLFBFZtZT5RP2G4ARkkolHQNcAazJ7SCpHLidbNC/kbNqHXCRpBOTE7MXJW3WCS4tH8r3vzSWoYMHImDo4IF8/0tjubT8SKNsZpZG/dvqEBEHJV1LNqT7AXdExDZJi4HKiFhDdtjmeOBeSQCvRcT0iNgr6Z/IfmAALI6IvV3yTlLq0vKhDncza5MiotA1NJHJZKKysrLQZZiZ9SqSNkZEprX1/gatmVkKOOzNzFLAYW9mlgIO+95uSwXcOgZuHpx93FJR6IrMrAdq82oc68G2VMDa66A++cbs/l3ZZYCyWYWry8x6HB/Z92aPLv4g6BvU12XbzcxyOOx7s/3V7Ws3s9Ry2Pdmg0ra125mqeWw780mLYSiZpOeFQ3MtpuZ5XDY92Zls+CSZTBoGKDs4yXLfHLWzD7EV+P0dmWzHO5m1iYf2ZuZpYDD3swsBRz2ZmYp4LA3M0sBh72ZWQrkFfaSpkraIalK0vwW1n9e0jOSDkqa2WzdIUmbk581zbc1M7Ou1+all5L6AcuByUA1sEHSmojYntPtNeCrwHdaeIm6iBjXCbWamVkH5XOd/XigKiJeBpC0CpgBNIZ9RLyarDvcBTWamdlRymcYZyiwK2e5OmnL1wBJlZKeknRpu6ozM7NO0R3foD09ImokfRJYL2lrROzM7SBpLjAX4LTTTuuGkszM0iWfI/saYFjOcknSlpeIqEkeXwZ+A5S30GdFRGQiIlNcXJzvS5uZWZ7yCfsNwAhJpZKOAa4A8rqqRtKJkj6aPD8ZmEDOWL+ZmXWPNsM+Ig4C1wLrgOeBiojYJmmxpOkAkj4jqRq4HLhd0rZk8zOBSknPAo8BS5pdxWNmZt1AEVHoGprIZDJRWVlZ6DLMzHoVSRsjItPaen+D1swsBRz2ZmYp4LA3M0sBh72ZWQo47M3MUsBhb2aWAg57M7MUcNibmaWAw97MLAUc9mZmKeCwNzNLAYe9mVkKOOzNzFLAYW9mlgIOezOzFHDYm5mlgMPezCwF8gp7SVMl7ZBUJWl+C+s/L+kZSQclzWy2bo6kl5KfOZ1VuJmZ5a/NsJfUD1gOfBEYBVwpaVSzbq8BXwXubrbtScBNwF8A44GbJJ149GWbmVl75HNkPx6oioiXI+J9YBUwI7dDRLwaEVuAw822nQI8EhF7I+It4BFgaifUbb3Flgq4dQzcPDj7uKWi0BWZpVI+YT8U2JWzXJ205SOvbSXNlVQpqbK2tjbPl7Yeb0sFrL0O9u8CIvu49joHvlkB9IgTtBGxIiIyEZEpLi4udDnWWR5dDPV1Tdvq67LtZtat8gn7GmBYznJJ0paPo9nWerv91e1rN7Muk0/YbwBGSCqVdAxwBbAmz9dfB1wk6cTkxOxFSZulwaCS9rWbWZdpM+wj4iBwLdmQfh6oiIhtkhZLmg4g6TOSqoHLgdslbUu23Qv8E9kPjA3A4qTN0mDSQiga2LStaGC23cy6lSKi0DU0kclkorKystBlWGfZUpEdo99fnT2in7QQymYVuiqzPkfSxojItLa+f3cWYylUNsvhbtYD9IircczMrGs57M3MUsBhb2aWAg57M7MUcNibmaWAw97MLAV86aV1qQc31bB03Q5276vj1MEDmTdlJJeW5zuPnpl1Foe9dZkHN9Ww4P6t1NUfAqBmXx0L7t8K4MA362YexrEus3Tdjsagb1BXf4il63YUqCKz9HLYW5fZva+uXe1m1nUc9tZlTh08sF3tZtZ1HPbWZeZNGcnAon5N2gYW9WPelJEFqsgsvXyC1rpMw0lYX41jVngOe+tSl5YPdbib9QAexjEzSwGHvZlZCuQV9pKmStohqUrS/BbWf1TS6mT9HyQNT9qHS6qTtDn5+dfOLd/sKG2pgFvHwM2Ds49bKgpdkVmXaHPMXlI/YDkwGagGNkhaExHbc7pdDbwVEZ+SdAVwC/DlZN3OiBjXyXWbHb0tFbD2OqhPrvvfvyu7DL67lvU5+RzZjweqIuLliHgfWAXMaNZnBnBn8vw+YJIkdV6ZZl3g0cUfBH2D+rpsu1kfk0/YDwV25SxXJ20t9omIg8B+YEiyrlTSJkm/lXR+S79A0lxJlZIqa2tr2/UGzDpsf3X72s16sa4+Qfs6cFpElAPfAu6W9LHmnSJiRURkIiJTXFzcxSWZJQaVtK/drBfLJ+xrgGE5yyVJW4t9JPUHBgF7IuK9iNgDEBEbgZ3Ap4+2aLNOMWkhFDWbuqFoYLbdrI/JJ+w3ACMklUo6BrgCWNOszxpgTvJ8JrA+IkJScXKCF0mfBEYAL3dO6WZHqWwWXLIMBg0DlH28ZJlPzlqf1ObVOBFxUNK1wDqgH3BHRGyTtBiojIg1wE+B/5BUBewl+4EA8HlgsaR64DDw9YjY2xVvxKxDymY53C0VFBGFrqGJTCYTlZWVhS7DzKxXkbQxIjKtrfc3aM3MUsAToVmq+R65lhYOe0st3yPX0sTDOJZavkeupYnD3lLL98i1NHHYW2r5HrmWJg57Sy3fI9fSxCdoLbV8j9yj4yuZeheHvaWa75HbMb6SqffxMI6ZtZuvZOp9fGRvZu22e18d0z/yBNf3r+BUvcnuOJkfHJzF2n2fK3Rp1gqHvZm125zjn+b6+n/jWL0PQIneZEnRv3FS0THAtMIWZy1y2JtZu11ftJpjD77fpO1Yvc/1RauBRYUpqpfp7hPcDnsza7dj6/6nXe3WVCFOcPsErZm1n2/peFQKcYLbYW9m7edbOh6VhhPcTxxzHS9/9Cs8ccx1TP/IE106VYeHccys/Rru7vXoYthfnT2in7TQd/3KUyFOcOd1ZC9pqqQdkqokzW9h/UclrU7W/0HS8Jx1C5L2HZKmdF7pZlZQZbPgm8/Bzfuyjw76vF1ftLox6Bt8cIK7a7QZ9skNw5cDXwRGAVdKGtWs29XAWxHxKeBW4JZk21Fk70c7GpgK3NZwA3Izs7QqxAnufI7sxwNVEfFyRLwPrAJmNOszA7gzeX4fMEmSkvZVEfFeRLwCVCWvZ2aWXgU4wZ1P2A8FduUsVydtLfaJiIPAfmBIntsiaa6kSkmVtbW1+VdvZtYbFeAEd4+4GiciVkREJiIyxcXFhS7HzKxrlc2CS5bBoGGAso+XLOvS8x75XI1TAwzLWS5J2lrqUy2pPzAI2JPntmZm6VM2q1tPaudzZL8BGCGpVNIxZE+4rmnWZw0wJ3k+E1gfEZG0X5FcrVMKjACe7pzSzcwsX20e2UfEQUnXAuuAfsAdEbFN0mKgMiLWAD8F/kNSFbCX7AcCSb8KYDtwELgmIg61+IvMzKzLKHsA3nNkMpmorKwsdBlmZr2KpI0RkWltfY84QWtmZl3LYW9mlgIOezOzFHDYm5mlgMPezCwFHPZmZinQ4y69lFQL/LHQdRTYycCbhS6iB/B++ID3RZb3Q1ZL++H0iGh1vpkeF/YGkiqPdL1sWng/fMD7Isv7Iasj+8HDOGZmKeCwNzNLAYd9z7Si0AX0EN4PH/C+yPJ+yGr3fvCYvZlZCvjI3swsBRz2ZmYp4LAvMEl3SHpD0nM5bSdJekTSS8njiYWssTtIGibpMUnbJW2T9PdJe6r2haQBkp6W9GyyHxYl7aWS/iCpStLq5EZCfZ6kfpI2SfrPZDmt++FVSVslbZZUmbS162/DYV94K4GpzdrmA49GxAjg0WS5rzsIfDsiRgHnAtdIGkX69sV7wIURcRYwDpgq6VzgFuDWiPgU8BZwdQFr7E5/Dzyfs5zW/QAwMSLG5Vxf366/DYd9gUXE42Tv7pVrBnBn8vxO4NJuLaoAIuL1iHgmef422T/woaRsX0TWgWSxKPkJ4ELgvqS9z+8HAEklwDTg35JlkcL9cATt+ttw2PdMn4iI15Pn/wN8opDFdDdJw4Fy4A+kcF8kQxebgTeAR4CdwL6IOJh0qSb7QdjX/Qi4HjicLA8hnfsBsh/4D0vaKGlu0tauv40270FrhRURISk118dKOh74OfD/IuJP2YO5rLTsi+Q+zeMkDQYeAM4ocEndTtLFwBsRsVHSBYWupwf4XETUSPo48IikF3JX5vO34SP7nul/JZ0CkDy+UeB6uoWkIrJBf1dE3J80p3JfAETEPuAx4DxgsKSGg7MSoKZghXWPCcB0Sa8Cq8gO3/yY9O0HACKiJnl8g+wBwHja+bfhsO+Z1gBzkudzgF8UsJZukYzH/hR4PiJ+mLMqVftCUnFyRI+kgcBksucvHgNmJt36/H6IiAURURIRw4ErgPURcRUp2w8Ako6TdELDc+Ai4Dna+bfhb9AWmKR7gAvITln6v8BNwINABXAa2emeZ0VE85O4fYqkzwG/A7bywRjtd8mO26dmX0gqI3uyrR/Zg7GKiFgs6ZNkj3BPAjYBsyPivcJV2n2SYZzvRMTFadwPyXt+IFnsD9wdEd+TNIR2/G047M3MUsDDOGZmKeCwNzNLAYe9mVkKOOzNzFLAYW9mlgIOezOzFHDYm5mlwP8HJAy6yBclmD8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(months_in_good_standing, probabilities_by_months, label=\"test data\")\n",
    "plt.scatter(months_in_good_standing, predicted_probs[:,1], label=\"predicted probs\")\n",
    "plt.legend();"
   ]
  },
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   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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