rls-variable-directional-forgetting.ipynb

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   "source": [
    "## Recursive Least Squares : Variable Directional Forgetting "
   ]
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    "- This notebook contains the implementation of the paper [Recursive Least Squares with Variable-Direction Forgetting -- Compensating for the loss of persistency](https://arxiv.org/abs/2003.03523)\n",
    "- Generated a synthetic dataset with two parameters (beta, gamma) for test the algorithm convergence [Present as example_data_gen() method in the RLS_VDF class\n",
    "- The algorithm is stable over increase in time steps\n",
    "- Estimated parameters are quite close to the ones used for generation of the synthetic dataset"
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    "### Use cases\n",
    "\n",
    "- Adaptive filtering: Enables adaptive filters to adapt to changing conditions by updating parameters based on relevant and recent information while forgetting outdated data\n",
    "\n",
    "- Signal processing: Widely used in noise cancellation, echo cancellation, equalization, and channel estimation to accurately estimate and track time-varying parameters\n",
    "\n",
    "- Non-stationary environments: Enhances adaptability in scenarios where input signal properties change over time or exhibit non-stationary behavior\n",
    "\n",
    "- System identification: Useful for estimating parameters of unknown systems based on observed input-output data, particularly for tracking time-varying parameters and capturing changes in system dynamics\n",
    "\n",
    "- Robustness to outliers: Improves robustness by adaptively adjusting the forgetting factor to assign less weight to outliers, reducing their impact on parameter estimation\n",
    "\n",
    "- Online learning: Suitable for real-time learning scenarios where data arrives sequentially, allowing continuous adaptation and utilization of new information\n",
    "\n",
    "- Resource optimization: Optimizes computational resources by allocating effort based on parameter adaptability, leading to improved efficiency in memory usage and computational complexity"
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   "source": [
    "import numpy as np\n",
    "import plotly.graph_objs as go\n",
    "import matplotlib.pyplot as plt"
   ]
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    "# Λk is a positive definite (and thus symmetric) matrix\n",
    "# theta_0 : initial estimate of theta\n",
    "\n",
    "\n",
    "class RLS_VDF():\n",
    "    def __init__(self,n_samples,lamda , e,parameters =2):\n",
    "        self.parameters = parameters\n",
    "        self.theta_0 = np.random.uniform(-1, 1, self.parameters)\n",
    "        self.theta =  self.theta_0#The vector of unknown parameters   \n",
    "        #replacing the scalar forgetting factor m by a data-dependent forgetting matrix Λ\n",
    "        self.lamda = lamda\n",
    "        self.e = e\n",
    "        self.P = np.eye(self.parameters)\n",
    "        self.A = np.linalg.inv(np.eye(self.parameters,self.parameters))\n",
    "        self.theta_vals = []\n",
    "        self.n_samples = n_samples\n",
    "        self.pkcondition_number = []\n",
    "        self.losses = []\n",
    "        \n",
    "        \n",
    "    def example_data_gen(self, beta,gamma):\n",
    "        '''Used for generating time series data for Example 1 [Equation 6] '''\n",
    "        phi =np.zeros((self.n_samples, 2))\n",
    "        y = np.array([0.00]*self.n_samples)\n",
    "        x= []\n",
    "        x.append(0.12)\n",
    "        for i in range(1,self.n_samples) :\n",
    "                v=x[i-1] +x[i-1]*(beta*(1-x[i-1]) - gamma) + 8e-5*np.random.normal(0, 1)\n",
    "                x.append(v)\n",
    "        x = np.array(x).reshape(self.n_samples,1)\n",
    "        for i in range(self.n_samples-1) :\n",
    "            phi[i][0] = (1 - x[i][0]) * x[i][0]\n",
    "            phi[i][1] = -x[i][0]\n",
    "            y[i]= (x[i+1][0] - x[i][0]) \n",
    "\n",
    "        return phi,y,x\n",
    "        \n",
    "    \n",
    "    def update(self,phi,y):\n",
    "        for i in range(self.n_samples-1):\n",
    "            U, Sigma, VT = np.linalg.svd(self.P)\n",
    "            psi_k = phi @ U\n",
    "            column_norms = np.linalg.norm(psi_k, axis=0)\n",
    "            lamda_bar = np.sqrt(self.lamda) * np.eye(self.parameters,self.parameters)\n",
    "            for i in range(len(column_norms)):\n",
    "                if column_norms[i] <= self.e: \n",
    "                    lamda_bar[i, i] = 1 \n",
    "            lamda_k = U@lamda_bar@U.T   \n",
    "            \n",
    "            pbar_k = np.linalg.inv(lamda_k)@self.P@np.linalg.inv(lamda_k)\n",
    "            self.P = (pbar_k - pbar_k@(phi.T)@(np.linalg.inv(np.eye(phi.shape[0]) + phi @ pbar_k @ phi.T))@phi@pbar_k)\n",
    "            self.theta = self.theta + self.P@(phi.T)@(y - phi@self.theta) + self.P@((lamda_k@self.A@lamda_k - self.A)@(self.theta_0 - self.theta))\n",
    "            self.A = lamda_k@self.A@lamda_k + phi.T@phi\n",
    "            self.theta_vals.append(self.theta)\n",
    "            e = y - phi @ self.theta\n",
    "                    \n",
    "            loss = np.linalg.norm(e)\n",
    "            \n",
    "            self.losses.append(loss)\n",
    "            \n",
    "            self.pkcondition_number.append(np.linalg.cond(self.P))\n",
    "       \n",
    "    \n",
    "        return self.theta_vals , self.P, self.pkcondition_number ,self.losses\n",
    "    \n",
    "    def fit(self,phi,y):\n",
    "        return self.update(phi,y)\n",
    "            \n",
    "\n",
    "    def plot_parameters(self):\n",
    "        ''' After model training, it can be used to plot the parameters against time'''\n",
    "        b_vals  = [list(i)[0] for i in self.theta_vals]\n",
    "        g_vals  = [list(i)[1] for i in self.theta_vals]\n",
    "        if len(b_vals) == self.n_samples-1 :\n",
    "                data = [\n",
    "                    go.Scatter(x=list(range(len(b_vals))),y=b_vals,mode='lines',name = 'Beta Values : Infection Rate'),\n",
    "                    go.Scatter(x=list(range(len(g_vals))),y=g_vals,mode='lines',name  = 'Gamma Values : Recovery Rate')]\n",
    "\n",
    "                layout = go.Layout(title='Infection and Recovery Rates over time (RLS- vdf)',xaxis=dict(title='Time step'),yaxis=dict(title='Value'))\n",
    "                fig = go.Figure(data=data, layout=layout)\n",
    "                fig.update_layout( plot_bgcolor='rgba(226, 237, 223, 90)')\n",
    "                         \n",
    "                fig.show()\n",
    "        else : \n",
    "            print('Error : The model is not yet trained on the time series data')"
   ]
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    "def loss_plot(model, loss, time_steps=359, data='Epidemic Data'):\n",
    "    fig = go.Figure()\n",
    "\n",
    "    #layout\n",
    "    fig.update_layout(\n",
    "        title=f'Least Square Error vs Time Steps for {model} on {data}',\n",
    "        xaxis=dict(title='Time steps'),\n",
    "        yaxis=dict(title='Least Squares Error'),\n",
    "        template='plotly_white'  # Adjust template as needed\n",
    "    )\n",
    "\n",
    "    # Adding trace for the loss plot\n",
    "    fig.add_trace(go.Scatter(x=list(range(time_steps)), y=loss, mode='lines'))\n",
    "\n",
    "    fig.show()\n",
    "    del fig"
   ]
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     "text": [
      "Estimated Beta (RLS-vdf): 0.6379599937328685\n",
      "Estimated Gamma (RLS-vdf): 0.3391435780511553\n",
      "\n"
     ]
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   "source": [
    "rls_vdf = RLS_VDF(360,0.99991,1e-5)\n",
    "phi_1,y_1,x_1 = rls_vdf.example_data_gen(0.65,0.35)\n",
    "theta_valss_1, P_k, cond_pk_3, loss= rls_vdf.fit(phi_1,y_1)\n",
    "\n",
    "print(f'Estimated Beta (RLS-vdf): {theta_valss_1[-1][0]}\\nEstimated Gamma (RLS-vdf): {theta_valss_1[-1][1]}\\n')\n",
    "loss_plot('RLS-VDF', loss,359,'Epidemic Data')\n",
    "rls_vdf.plot_parameters()"
   ]
  },
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    "tags": []
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   "source": [
    "## Observation \n",
    "#### - L2 Norm decreases with time\n",
    "#### - Parameters converge near to the true values"
   ]
  }
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