{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# L1 and L2 Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The goal of this notebook is to validate the optimization code in `macaw`.\n",
    "For that, we will compare results against sklearn's linear regression example on the `diabetes` dataset.\n",
    "See sklearn's example here: http://scikit-learn.org/stable/auto_examples/linear_model/plot_ols.html#sphx-glr-auto-examples-linear-model-plot-ols-py"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from macaw.models import LinearModel\n",
    "from macaw.objective_functions import L1Norm, L2Norm\n",
    "from sklearn import datasets\n",
    "from sklearn.metrics import mean_squared_error, r2_score\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Load the diabetes dataset\n",
    "diabetes = datasets.load_diabetes()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Use only one feature\n",
    "diabetes_X = diabetes.data[:, np.newaxis, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Split the data into training/testing sets\n",
    "diabetes_X_train = diabetes_X[:-20]\n",
    "diabetes_X_test = diabetes_X[-20:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Split the targets into training/testing sets\n",
    "diabetes_y_train = diabetes.target[:-20]\n",
    "diabetes_y_test = diabetes.target[-20:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Create linear model object\n",
    "model = LinearModel(diabetes_X_train.reshape(-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Train the model using the training sets\n",
    "l1norm = L1Norm(y=diabetes_y_train.reshape(-1), model=model)\n",
    "res_l1 = l1norm.fit(x0=[0., 0.])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Train the model using the training sets\n",
    "l2norm = L2Norm(y=diabetes_y_train.reshape(-1), model=model)\n",
    "res_l2 = l2norm.fit(x0=[0., 0.])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Make predictions using the testing set\n",
    "diabetes_y_pred_l1 = LinearModel(diabetes_X_test).evaluate(*res_l1.x)\n",
    "diabetes_y_pred_l2 = LinearModel(diabetes_X_test).evaluate(*res_l2.x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(L1 Norm) Coefficients: \n",
      " [ 1038.68262063   146.84722666]\n",
      "(L2 Norm) Coefficients: \n",
      " [ 938.23786125  152.91885622]\n"
     ]
    }
   ],
   "source": [
    "print('(L1 Norm) Coefficients: \\n', res_l1.x)\n",
    "print('(L2 Norm) Coefficients: \\n', res_l2.x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(L1 Norm) Mean squared error: 2296.18\n",
      "(L2 Norm) Mean squared error: 2548.07\n"
     ]
    }
   ],
   "source": [
    "print(\"(L1 Norm) Mean squared error: %.2f\"\n",
    "      % mean_squared_error(diabetes_y_test, diabetes_y_pred_l1))\n",
    "print(\"(L2 Norm) Mean squared error: %.2f\"\n",
    "      % mean_squared_error(diabetes_y_test, diabetes_y_pred_l2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(L1 Norm) Variance score: 0.52\n",
      "(L2 Norm) Variance score: 0.47\n"
     ]
    }
   ],
   "source": [
    "print('(L1 Norm) Variance score: %.2f' % r2_score(diabetes_y_test, diabetes_y_pred_l1))\n",
    "print('(L2 Norm) Variance score: %.2f' % r2_score(diabetes_y_test, diabetes_y_pred_l2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10b05a710>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10afb91d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(diabetes_X_test, diabetes_y_test, color='black')\n",
    "plt.plot(diabetes_X_test, diabetes_y_pred_l1, color='blue')\n",
    "plt.plot(diabetes_X_test, diabetes_y_pred_l2, color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Note that the L1 norm estimator for this particular linear model and dataset presents a smaller mean squared error when compared against the L2 norm estimator. That is because the L1 norm finds the \"median model\", and therefore is more robust against outliers, whereas the L2 norm finds the \"mean model\" which is less robust to the presence of outliers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}