{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Homework 1: Lasso and Ridge regression on the Advertising data set"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this homework you will perform lasso and ridge regression on the advertising data set.\n",
    "The following cell imports the data set (adjust the path as necessary)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "\n",
    "adv = pd.read_csv('./datasets/Advertising.csv', index_col=0)\n",
    "X = adv.values[:,0:3]\n",
    "y = adv.values[:,3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task**: Perform both lasso and ridge regression on the data.\n",
    "Read the documentation for the functions `RidgeCV` and `LassoCV` from `sklearn.linear_model`.\n",
    "These functions select the regularization parameter by cross-validation, as we did by hand in Problem 8.\n",
    "Use 5-fold cross-validation and the following range for $\\alpha$: `np.logspace(-4,4,100)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LassoCV(alphas=array([1.00000e-04, 1.20450e-04, ..., 8.30218e+03, 1.00000e+04]),\n",
       "    copy_X=True, cv=5, eps=0.001, fit_intercept=True, max_iter=1000,\n",
       "    n_alphas=100, n_jobs=None, normalize=False, positive=False,\n",
       "    precompute='auto', random_state=None, selection='cyclic', tol=0.0001,\n",
       "    verbose=False)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from sklearn.linear_model import LassoCV, RidgeCV\n",
    "\n",
    "adv = pd.read_csv('./datasets/Advertising.csv', index_col=0)\n",
    "X = adv.values[:,0:3]\n",
    "y = adv.values[:,3]\n",
    "\n",
    "Alpha = np.logspace(-4,4,100)\n",
    "\n",
    "ridge_poly = RidgeCV(cv=5,alphas=Alpha)\n",
    "ridge_poly.fit(X,y)\n",
    "\n",
    "lasso_poly = LassoCV(cv=5, alphas=Alpha)\n",
    "lasso_poly.fit(X,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task**: Compute the $R^2$-scores for both lasso and ridge regression. Which coefficients are selected by the lasso?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4.57629772e-02,  1.84966340e-01, -1.75482797e-04])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ridge_poly.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8971206766074686"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ridge_poly.score(X,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.04557567, 0.17930643, 0.        ])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lasso_poly.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8965664034716233"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lasso_poly.score(X,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task**: At the end of Exercise 3 and in the lecture, we noticed that there may exist interactions or higher order variations among the predictor variables.\n",
    "Use the function `PolynomialFeatures` from `sklearn.preprocessing` to add quadratic terms as well as interaction terms between the predictors. How does this improve your $R^2$-score?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "deg2 = PolynomialFeatures(degree=2)\n",
    "X2 = deg2.fit_transform(X)\n",
    "\n",
    "ridge_poly = RidgeCV(cv=5,alphas=Alpha)\n",
    "ridge_poly.fit(X2,y)\n",
    "\n",
    "lasso_poly = LassoCV(cv=5, alphas=Alpha)\n",
    "lasso_poly.fit(X2,y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00,  4.96893182e-02,  9.42785581e-03,  4.92727481e-03,\n",
       "       -1.05025820e-04,  1.12424510e-03, -4.32094721e-05,  2.66284411e-04,\n",
       "        1.08818342e-04,  2.06637263e-05])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ridge_poly.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.986396878676749"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ridge_poly.score(X2,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00,  5.01397308e-02,  0.00000000e+00,  0.00000000e+00,\n",
       "       -1.07173978e-04,  1.12719843e-03, -3.85449940e-05,  4.07309225e-04,\n",
       "        1.71195821e-04,  4.70557602e-05])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lasso_poly.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9862333741123677"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lasso_poly.score(X2,y)"
   ]
  }
 ],
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