CompareRegressionMethods.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example of Regression Analysis Using the Boston Housing Data Set\n",
    "http://facweb.cs.depaul.edu/mobasher/classes/CSC478/Data/housing-dscr.txt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\ProgramData\\Anaconda3\\lib\\site-packages\\sklearn\\cross_validation.py:44: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n",
      "  \"This module will be removed in 0.20.\", DeprecationWarning)\n"
     ]
    }
   ],
   "source": [
    "from sklearn.linear_model import LinearRegression, Lasso, Ridge, ElasticNet, SGDRegressor\n",
    "from sklearn.cross_validation import KFold\n",
    "from sklearn.datasets import load_boston\n",
    "\n",
    "import numpy as np\n",
    "import pylab as pl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "boston = load_boston()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'DESCR': \"Boston House Prices dataset\\n===========================\\n\\nNotes\\n------\\nData Set Characteristics:  \\n\\n    :Number of Instances: 506 \\n\\n    :Number of Attributes: 13 numeric/categorical predictive\\n    \\n    :Median Value (attribute 14) is usually the target\\n\\n    :Attribute Information (in order):\\n        - CRIM     per capita crime rate by town\\n        - ZN       proportion of residential land zoned for lots over 25,000 sq.ft.\\n        - INDUS    proportion of non-retail business acres per town\\n        - CHAS     Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\\n        - NOX      nitric oxides concentration (parts per 10 million)\\n        - RM       average number of rooms per dwelling\\n        - AGE      proportion of owner-occupied units built prior to 1940\\n        - DIS      weighted distances to five Boston employment centres\\n        - RAD      index of accessibility to radial highways\\n        - TAX      full-value property-tax rate per $10,000\\n        - PTRATIO  pupil-teacher ratio by town\\n        - B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town\\n        - LSTAT    % lower status of the population\\n        - MEDV     Median value of owner-occupied homes in $1000's\\n\\n    :Missing Attribute Values: None\\n\\n    :Creator: Harrison, D. and Rubinfeld, D.L.\\n\\nThis is a copy of UCI ML housing dataset.\\nhttp://archive.ics.uci.edu/ml/datasets/Housing\\n\\n\\nThis dataset was taken from the StatLib library which is maintained at Carnegie Mellon University.\\n\\nThe Boston house-price data of Harrison, D. and Rubinfeld, D.L. 'Hedonic\\nprices and the demand for clean air', J. Environ. Economics & Management,\\nvol.5, 81-102, 1978.   Used in Belsley, Kuh & Welsch, 'Regression diagnostics\\n...', Wiley, 1980.   N.B. Various transformations are used in the table on\\npages 244-261 of the latter.\\n\\nThe Boston house-price data has been used in many machine learning papers that address regression\\nproblems.   \\n     \\n**References**\\n\\n   - Belsley, Kuh & Welsch, 'Regression diagnostics: Identifying Influential Data and Sources of Collinearity', Wiley, 1980. 244-261.\\n   - Quinlan,R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243, University of Massachusetts, Amherst. Morgan Kaufmann.\\n   - many more! (see http://archive.ics.uci.edu/ml/datasets/Housing)\\n\",\n",
       " 'data': array([[  6.32000000e-03,   1.80000000e+01,   2.31000000e+00, ...,\n",
       "           1.53000000e+01,   3.96900000e+02,   4.98000000e+00],\n",
       "        [  2.73100000e-02,   0.00000000e+00,   7.07000000e+00, ...,\n",
       "           1.78000000e+01,   3.96900000e+02,   9.14000000e+00],\n",
       "        [  2.72900000e-02,   0.00000000e+00,   7.07000000e+00, ...,\n",
       "           1.78000000e+01,   3.92830000e+02,   4.03000000e+00],\n",
       "        ..., \n",
       "        [  6.07600000e-02,   0.00000000e+00,   1.19300000e+01, ...,\n",
       "           2.10000000e+01,   3.96900000e+02,   5.64000000e+00],\n",
       "        [  1.09590000e-01,   0.00000000e+00,   1.19300000e+01, ...,\n",
       "           2.10000000e+01,   3.93450000e+02,   6.48000000e+00],\n",
       "        [  4.74100000e-02,   0.00000000e+00,   1.19300000e+01, ...,\n",
       "           2.10000000e+01,   3.96900000e+02,   7.88000000e+00]]),\n",
       " 'feature_names': array(['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD',\n",
       "        'TAX', 'PTRATIO', 'B', 'LSTAT'], \n",
       "       dtype='<U7'),\n",
       " 'target': array([ 24. ,  21.6,  34.7,  33.4,  36.2,  28.7,  22.9,  27.1,  16.5,\n",
       "         18.9,  15. ,  18.9,  21.7,  20.4,  18.2,  19.9,  23.1,  17.5,\n",
       "         20.2,  18.2,  13.6,  19.6,  15.2,  14.5,  15.6,  13.9,  16.6,\n",
       "         14.8,  18.4,  21. ,  12.7,  14.5,  13.2,  13.1,  13.5,  18.9,\n",
       "         20. ,  21. ,  24.7,  30.8,  34.9,  26.6,  25.3,  24.7,  21.2,\n",
       "         19.3,  20. ,  16.6,  14.4,  19.4,  19.7,  20.5,  25. ,  23.4,\n",
       "         18.9,  35.4,  24.7,  31.6,  23.3,  19.6,  18.7,  16. ,  22.2,\n",
       "         25. ,  33. ,  23.5,  19.4,  22. ,  17.4,  20.9,  24.2,  21.7,\n",
       "         22.8,  23.4,  24.1,  21.4,  20. ,  20.8,  21.2,  20.3,  28. ,\n",
       "         23.9,  24.8,  22.9,  23.9,  26.6,  22.5,  22.2,  23.6,  28.7,\n",
       "         22.6,  22. ,  22.9,  25. ,  20.6,  28.4,  21.4,  38.7,  43.8,\n",
       "         33.2,  27.5,  26.5,  18.6,  19.3,  20.1,  19.5,  19.5,  20.4,\n",
       "         19.8,  19.4,  21.7,  22.8,  18.8,  18.7,  18.5,  18.3,  21.2,\n",
       "         19.2,  20.4,  19.3,  22. ,  20.3,  20.5,  17.3,  18.8,  21.4,\n",
       "         15.7,  16.2,  18. ,  14.3,  19.2,  19.6,  23. ,  18.4,  15.6,\n",
       "         18.1,  17.4,  17.1,  13.3,  17.8,  14. ,  14.4,  13.4,  15.6,\n",
       "         11.8,  13.8,  15.6,  14.6,  17.8,  15.4,  21.5,  19.6,  15.3,\n",
       "         19.4,  17. ,  15.6,  13.1,  41.3,  24.3,  23.3,  27. ,  50. ,\n",
       "         50. ,  50. ,  22.7,  25. ,  50. ,  23.8,  23.8,  22.3,  17.4,\n",
       "         19.1,  23.1,  23.6,  22.6,  29.4,  23.2,  24.6,  29.9,  37.2,\n",
       "         39.8,  36.2,  37.9,  32.5,  26.4,  29.6,  50. ,  32. ,  29.8,\n",
       "         34.9,  37. ,  30.5,  36.4,  31.1,  29.1,  50. ,  33.3,  30.3,\n",
       "         34.6,  34.9,  32.9,  24.1,  42.3,  48.5,  50. ,  22.6,  24.4,\n",
       "         22.5,  24.4,  20. ,  21.7,  19.3,  22.4,  28.1,  23.7,  25. ,\n",
       "         23.3,  28.7,  21.5,  23. ,  26.7,  21.7,  27.5,  30.1,  44.8,\n",
       "         50. ,  37.6,  31.6,  46.7,  31.5,  24.3,  31.7,  41.7,  48.3,\n",
       "         29. ,  24. ,  25.1,  31.5,  23.7,  23.3,  22. ,  20.1,  22.2,\n",
       "         23.7,  17.6,  18.5,  24.3,  20.5,  24.5,  26.2,  24.4,  24.8,\n",
       "         29.6,  42.8,  21.9,  20.9,  44. ,  50. ,  36. ,  30.1,  33.8,\n",
       "         43.1,  48.8,  31. ,  36.5,  22.8,  30.7,  50. ,  43.5,  20.7,\n",
       "         21.1,  25.2,  24.4,  35.2,  32.4,  32. ,  33.2,  33.1,  29.1,\n",
       "         35.1,  45.4,  35.4,  46. ,  50. ,  32.2,  22. ,  20.1,  23.2,\n",
       "         22.3,  24.8,  28.5,  37.3,  27.9,  23.9,  21.7,  28.6,  27.1,\n",
       "         20.3,  22.5,  29. ,  24.8,  22. ,  26.4,  33.1,  36.1,  28.4,\n",
       "         33.4,  28.2,  22.8,  20.3,  16.1,  22.1,  19.4,  21.6,  23.8,\n",
       "         16.2,  17.8,  19.8,  23.1,  21. ,  23.8,  23.1,  20.4,  18.5,\n",
       "         25. ,  24.6,  23. ,  22.2,  19.3,  22.6,  19.8,  17.1,  19.4,\n",
       "         22.2,  20.7,  21.1,  19.5,  18.5,  20.6,  19. ,  18.7,  32.7,\n",
       "         16.5,  23.9,  31.2,  17.5,  17.2,  23.1,  24.5,  26.6,  22.9,\n",
       "         24.1,  18.6,  30.1,  18.2,  20.6,  17.8,  21.7,  22.7,  22.6,\n",
       "         25. ,  19.9,  20.8,  16.8,  21.9,  27.5,  21.9,  23.1,  50. ,\n",
       "         50. ,  50. ,  50. ,  50. ,  13.8,  13.8,  15. ,  13.9,  13.3,\n",
       "         13.1,  10.2,  10.4,  10.9,  11.3,  12.3,   8.8,   7.2,  10.5,\n",
       "          7.4,  10.2,  11.5,  15.1,  23.2,   9.7,  13.8,  12.7,  13.1,\n",
       "         12.5,   8.5,   5. ,   6.3,   5.6,   7.2,  12.1,   8.3,   8.5,\n",
       "          5. ,  11.9,  27.9,  17.2,  27.5,  15. ,  17.2,  17.9,  16.3,\n",
       "          7. ,   7.2,   7.5,  10.4,   8.8,   8.4,  16.7,  14.2,  20.8,\n",
       "         13.4,  11.7,   8.3,  10.2,  10.9,  11. ,   9.5,  14.5,  14.1,\n",
       "         16.1,  14.3,  11.7,  13.4,   9.6,   8.7,   8.4,  12.8,  10.5,\n",
       "         17.1,  18.4,  15.4,  10.8,  11.8,  14.9,  12.6,  14.1,  13. ,\n",
       "         13.4,  15.2,  16.1,  17.8,  14.9,  14.1,  12.7,  13.5,  14.9,\n",
       "         20. ,  16.4,  17.7,  19.5,  20.2,  21.4,  19.9,  19. ,  19.1,\n",
       "         19.1,  20.1,  19.9,  19.6,  23.2,  29.8,  13.8,  13.3,  16.7,\n",
       "         12. ,  14.6,  21.4,  23. ,  23.7,  25. ,  21.8,  20.6,  21.2,\n",
       "         19.1,  20.6,  15.2,   7. ,   8.1,  13.6,  20.1,  21.8,  24.5,\n",
       "         23.1,  19.7,  18.3,  21.2,  17.5,  16.8,  22.4,  20.6,  23.9,\n",
       "         22. ,  11.9])}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(506, 13)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston.data.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(506,)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston.target.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD',\n",
       "       'TAX', 'PTRATIO', 'B', 'LSTAT'], \n",
       "      dtype='<U7')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston.feature_names"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "CRIM      per capita crime rate by town\n",
    "ZN        proportion of residential land zoned for lots over 25,000 sq.ft.\n",
    "INDUS     proportion of non-retail business acres per town\n",
    "CHAS      Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
    "NOX       nitric oxides concentration (parts per 10 million)\n",
    "RM        average number of rooms per dwelling\n",
    "AGE       proportion of owner-occupied units built prior to 1940\n",
    "DIS       weighted distances to five Boston employment centres\n",
    "RAD       index of accessibility to radial highways\n",
    "TAX      full-value property-tax rate per $10,000\n",
    "PTRATIO  pupil-teacher ratio by town\n",
    "B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town\n",
    "LSTAT    % lower status of the population\n",
    "MEDV     Median value of owner-occupied homes in $1000's"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.set_printoptions(precision=2, linewidth=120, suppress=True, edgeitems=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   0.01,   18.  ,    2.31,    0.  , ...,  296.  ,   15.3 ,  396.9 ,    4.98],\n",
       "       [   0.03,    0.  ,    7.07,    0.  , ...,  242.  ,   17.8 ,  396.9 ,    9.14],\n",
       "       [   0.03,    0.  ,    7.07,    0.  , ...,  242.  ,   17.8 ,  392.83,    4.03],\n",
       "       [   0.03,    0.  ,    2.18,    0.  , ...,  222.  ,   18.7 ,  394.63,    2.94],\n",
       "       ..., \n",
       "       [   0.05,    0.  ,   11.93,    0.  , ...,  273.  ,   21.  ,  396.9 ,    9.08],\n",
       "       [   0.06,    0.  ,   11.93,    0.  , ...,  273.  ,   21.  ,  396.9 ,    5.64],\n",
       "       [   0.11,    0.  ,   11.93,    0.  , ...,  273.  ,   21.  ,  393.45,    6.48],\n",
       "       [   0.05,    0.  ,   11.93,    0.  , ...,  273.  ,   21.  ,  396.9 ,    7.88]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston.data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# In order to do multiple regression we need to add a column of 1s for x0\n",
    "x = np.array([np.concatenate((v,[1])) for v in boston.data])\n",
    "y = boston.target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   0.01,   18.  ,    2.31,    0.  ,    0.54,    6.58,   65.2 ,    4.09,    1.  ,  296.  ,   15.3 ,  396.9 ,\n",
       "           4.98,    1.  ],\n",
       "       [   0.03,    0.  ,    7.07,    0.  ,    0.47,    6.42,   78.9 ,    4.97,    2.  ,  242.  ,   17.8 ,  396.9 ,\n",
       "           9.14,    1.  ],\n",
       "       [   0.03,    0.  ,    7.07,    0.  ,    0.47,    7.18,   61.1 ,    4.97,    2.  ,  242.  ,   17.8 ,  392.83,\n",
       "           4.03,    1.  ],\n",
       "       [   0.03,    0.  ,    2.18,    0.  ,    0.46,    7.  ,   45.8 ,    6.06,    3.  ,  222.  ,   18.7 ,  394.63,\n",
       "           2.94,    1.  ],\n",
       "       [   0.07,    0.  ,    2.18,    0.  ,    0.46,    7.15,   54.2 ,    6.06,    3.  ,  222.  ,   18.7 ,  396.9 ,\n",
       "           5.33,    1.  ],\n",
       "       [   0.03,    0.  ,    2.18,    0.  ,    0.46,    6.43,   58.7 ,    6.06,    3.  ,  222.  ,   18.7 ,  394.12,\n",
       "           5.21,    1.  ],\n",
       "       [   0.09,   12.5 ,    7.87,    0.  ,    0.52,    6.01,   66.6 ,    5.56,    5.  ,  311.  ,   15.2 ,  395.6 ,\n",
       "          12.43,    1.  ],\n",
       "       [   0.14,   12.5 ,    7.87,    0.  ,    0.52,    6.17,   96.1 ,    5.95,    5.  ,  311.  ,   15.2 ,  396.9 ,\n",
       "          19.15,    1.  ],\n",
       "       [   0.21,   12.5 ,    7.87,    0.  ,    0.52,    5.63,  100.  ,    6.08,    5.  ,  311.  ,   15.2 ,  386.63,\n",
       "          29.93,    1.  ],\n",
       "       [   0.17,   12.5 ,    7.87,    0.  ,    0.52,    6.  ,   85.9 ,    6.59,    5.  ,  311.  ,   15.2 ,  386.71,\n",
       "          17.1 ,    1.  ]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First 10 elements of the data\n",
    "x[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 24. ,  21.6,  34.7,  33.4,  36.2,  28.7,  22.9,  27.1,  16.5,  18.9])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First 10 elements of the response variable\n",
    "y[:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create linear regression object\n",
    "linreg = LinearRegression()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Train the model using the training sets\n",
    "linreg.fit(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 30.01,  25.03,  30.57,  28.61,  27.94,  25.26,  23.  ,  19.53,  11.52,  18.92])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let's see predictions for the first 10 instances\n",
    "linreg.predict(x[:10])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6.01,  3.43,  4.13,  4.79,  8.26,  3.44,  0.1 ,  7.57,  4.98,  0.02])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute RMSE on training data\n",
    "# p = np.array([linreg.predict(xi) for xi in x])\n",
    "p = linreg.predict(x)\n",
    "# Now we can constuct a vector of errors\n",
    "err = abs(p-y)\n",
    "\n",
    "# Let's see the error on the first 10 predictions\n",
    "err[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.6795063006355164"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Dot product of error vector with itself gives us the sum of squared errors\n",
    "total_error = np.dot(err,err)\n",
    "# Compute RMSE\n",
    "rmse_train = np.sqrt(total_error/len(p))\n",
    "rmse_train"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Regression Coefficients: \n",
      " [ -0.11   0.05   0.02   2.69 -17.8    3.8    0.    -1.48   0.31  -0.01  -0.95   0.01  -0.53   0.  ]\n"
     ]
    }
   ],
   "source": [
    "# We can view the regression coefficients\n",
    "print('Regression Coefficients: \\n', linreg.coef_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xsFpDBSBXq4ol+hy31xMtoJLM6ltxHvXt0DNOc+oEHToS/Wi79PeZk0ehLi+Z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\nGGPKnAUCY4wpc/8PLwJ+/dSlWncAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x14e0b619c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot outputs\n",
    "%matplotlib inline\n",
    "pl.plot(p, y,'ro')\n",
    "pl.plot([0,50],[0,50], 'g-')\n",
    "pl.xlabel('predicted')\n",
    "pl.ylabel('real')\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.8819250724301071"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now let's compute RMSE using 10-fold x-validation\n",
    "kf = KFold(len(x), n_folds=10)\n",
    "xval_err = 0\n",
    "for train,test in kf:\n",
    "    linreg.fit(x[train],y[train])\n",
    "    # p = np.array([linreg.predict(xi) for xi in x[test]])\n",
    "    p = linreg.predict(x[test])\n",
    "    e = p-y[test]\n",
    "    xval_err += np.dot(e,e)\n",
    "    \n",
    "rmse_10cv = np.sqrt(xval_err/len(x))\n",
    "rmse_10cv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Method: Simple Linear Regression\n",
      "RMSE on training: 4.6857\n",
      "RMSE on 10-fold CV: 5.8819\n"
     ]
    }
   ],
   "source": [
    "method_name = 'Simple Linear Regression'\n",
    "print('Method: %s' %method_name)\n",
    "print('RMSE on training: %.4f' %rmse_train)\n",
    "print('RMSE on 10-fold CV: %.4f' %rmse_10cv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ridge Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Create linear regression object with a ridge coefficient 0.5\n",
    "ridge = Ridge(fit_intercept=True, alpha=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ridge(alpha=0.5, copy_X=True, fit_intercept=True, max_iter=None,\n",
       "   normalize=False, random_state=None, solver='auto', tol=0.001)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Train the model using the training set\n",
    "ridge.fit(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.6857079166774627"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute RMSE on training data\n",
    "# p = np.array([ridge.predict(xi) for xi in x])\n",
    "p = ridge.predict(x)\n",
    "err = p-y\n",
    "total_error = np.dot(err,err)\n",
    "rmse_train = np.sqrt(total_error/len(p))\n",
    "rmse_train"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.8428362247708749"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute RMSE using 10-fold x-validation\n",
    "kf = KFold(len(x), n_folds=10)\n",
    "xval_err = 0\n",
    "for train,test in kf:\n",
    "    ridge.fit(x[train],y[train])\n",
    "    p = ridge.predict(x[test])\n",
    "    e = p-y[test]\n",
    "    xval_err += np.dot(e,e)\n",
    "rmse_10cv = np.sqrt(xval_err/len(x))\n",
    "rmse_10cv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Method: Ridge Regression\n",
      "RMSE on training: 4.6857\n",
      "RMSE on 10-fold CV: 5.8428\n"
     ]
    }
   ],
   "source": [
    "method_name = 'Ridge Regression'\n",
    "print('Method: %s' %method_name)\n",
    "print('RMSE on training: %.4f' %rmse_train)\n",
    "print('RMSE on 10-fold CV: %.4f' %rmse_10cv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### try different values of alpha and observe the impact on x-validation RMSE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ridge Regression\n",
      "alpha\t RMSE_train\t RMSE_10cv\n",
      "\n",
      "0.010\t 4.6795\t\t 5.8806\n",
      "0.418\t 4.6842\t\t 5.8467\n",
      "0.826\t 4.6921\t\t 5.8319\n",
      "1.234\t 4.7000\t\t 5.8234\n",
      "1.642\t 4.7070\t\t 5.8175\n",
      "2.050\t 4.7133\t\t 5.8126\n",
      "2.458\t 4.7187\t\t 5.8082\n",
      "2.866\t 4.7234\t\t 5.8041\n",
      "3.274\t 4.7276\t\t 5.8000\n",
      "3.682\t 4.7313\t\t 5.7960\n",
      "4.090\t 4.7346\t\t 5.7920\n",
      "4.498\t 4.7375\t\t 5.7880\n",
      "4.906\t 4.7402\t\t 5.7840\n",
      "5.313\t 4.7426\t\t 5.7800\n",
      "5.721\t 4.7448\t\t 5.7760\n",
      "6.129\t 4.7469\t\t 5.7720\n",
      "6.537\t 4.7488\t\t 5.7680\n",
      "6.945\t 4.7505\t\t 5.7641\n",
      "7.353\t 4.7522\t\t 5.7602\n",
      "7.761\t 4.7537\t\t 5.7563\n",
      "8.169\t 4.7552\t\t 5.7524\n",
      "8.577\t 4.7565\t\t 5.7485\n",
      "8.985\t 4.7578\t\t 5.7447\n",
      "9.393\t 4.7591\t\t 5.7410\n",
      "9.801\t 4.7603\t\t 5.7372\n",
      "10.209\t 4.7614\t\t 5.7335\n",
      "10.617\t 4.7625\t\t 5.7298\n",
      "11.025\t 4.7635\t\t 5.7262\n",
      "11.433\t 4.7646\t\t 5.7226\n",
      "11.841\t 4.7655\t\t 5.7190\n",
      "12.249\t 4.7665\t\t 5.7155\n",
      "12.657\t 4.7674\t\t 5.7120\n",
      "13.065\t 4.7683\t\t 5.7086\n",
      "13.473\t 4.7692\t\t 5.7052\n",
      "13.881\t 4.7700\t\t 5.7018\n",
      "14.289\t 4.7708\t\t 5.6985\n",
      "14.697\t 4.7717\t\t 5.6952\n",
      "15.104\t 4.7724\t\t 5.6919\n",
      "15.512\t 4.7732\t\t 5.6887\n",
      "15.920\t 4.7740\t\t 5.6856\n",
      "16.328\t 4.7747\t\t 5.6824\n",
      "16.736\t 4.7755\t\t 5.6793\n",
      "17.144\t 4.7762\t\t 5.6762\n",
      "17.552\t 4.7769\t\t 5.6732\n",
      "17.960\t 4.7776\t\t 5.6702\n",
      "18.368\t 4.7783\t\t 5.6672\n",
      "18.776\t 4.7790\t\t 5.6643\n",
      "19.184\t 4.7797\t\t 5.6614\n",
      "19.592\t 4.7804\t\t 5.6585\n",
      "20.000\t 4.7811\t\t 5.6557\n"
     ]
    }
   ],
   "source": [
    "print('Ridge Regression')\n",
    "print('alpha\\t RMSE_train\\t RMSE_10cv\\n')\n",
    "alpha = np.linspace(.01,20,50)\n",
    "t_rmse = np.array([])\n",
    "cv_rmse = np.array([])\n",
    "\n",
    "for a in alpha:\n",
    "    ridge = Ridge(fit_intercept=True, alpha=a)\n",
    "    \n",
    "    # computing the RMSE on training data\n",
    "    ridge.fit(x,y)\n",
    "    p = ridge.predict(x)\n",
    "    err = p-y\n",
    "    total_error = np.dot(err,err)\n",
    "    rmse_train = np.sqrt(total_error/len(p))\n",
    "\n",
    "    # computing RMSE using 10-fold cross validation\n",
    "    kf = KFold(len(x), n_folds=10)\n",
    "    xval_err = 0\n",
    "    for train, test in kf:\n",
    "        ridge.fit(x[train], y[train])\n",
    "        p = ridge.predict(x[test])\n",
    "        err = p - y[test]\n",
    "        xval_err += np.dot(err,err)\n",
    "    rmse_10cv = np.sqrt(xval_err/len(x))\n",
    "    \n",
    "    t_rmse = np.append(t_rmse, [rmse_train])\n",
    "    cv_rmse = np.append(cv_rmse, [rmse_10cv])\n",
    "    print('{:.3f}\\t {:.4f}\\t\\t {:.4f}'.format(a,rmse_train,rmse_10cv))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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MGtaRTaEgOdHW0UlTSzuNLe00tXTQ1NpOc3rYGszLHDa3Bo17U8Z4c9jwN7d2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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x14e0bbf5080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pl.plot(alpha, t_rmse, label='RMSE-Train')\n",
    "pl.plot(alpha, cv_rmse, label='RMSE_XVal')\n",
    "pl.legend( ('RMSE-Train', 'RMSE_XVal') )\n",
    "pl.ylabel('RMSE')\n",
    "pl.xlabel('Alpha')\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}