{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " # Machine Learning LAB 1 (course 2022/23, F. Chiariotti) \n",
    "\n",
    "The notebook contains some simple tasks to be performed about classification and regression. Complete all the required code sections and answer to all the questions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1) Classification of NBA players role\n",
    "\n",
    "## IMPORTANT: make sure to rerun all the code from the beginning to obtain the results for the final version of your notebook, since this is the way we will do it before evaluting your notebook!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Place your name and ID number. Also recall to save the file as Surname_Name_LAB1.ipynb\n",
    "\n",
    "Student name: <br>\n",
    "ID Number:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dataset description\n",
    "\n",
    "We will be working with a dataset of NBA basketball players data (you can get from https://www.kaggle.com/jacobbaruch/nba-player-of-the-week the full dataset).\n",
    "\n",
    "The provided data is a subset of the Kaggle dataset containing the players that have the role of Center and of Point Guard. For each player the dataset contains 3 features, the height, the weight and the age.\n",
    "\n",
    "From Wikipedia (if you are not a basketball fan!!):\n",
    "\n",
    "The Center (C), also known as the five, or the big man, is one of the five positions in a regular basketball game. The center is normally the tallest player on the team, and often has a great deal of strength and body mass as well. In the NBA, the center is usually 6' 10\" (2.08 m) or taller and usually weighs 240 lbs (109 kg) or more. \n",
    "\n",
    "Point Guards (PG, a.k.a. as \"play maker\") are expected to run the team's offense by controlling the ball and making sure that it gets to the right players at the right time. In the NBA, point guards are usually about 6' 3\" (1.93 m) or shorter, and average about 6' 2\" (1.88 m). Having above-average size (height, muscle) is considered advantageous, although size is secondary to situational awareness, speed, quickness, and ball handling skills. Shorter players tend to be better dribblers since they are closer to the floor, and thus have better control of the ball while dribbling. \n",
    "\n",
    "\n",
    "As it is clear from the description, the height and weight of the player are good hints to predict their role and in this lab we will exploit this features to estimate the role.\n",
    "\n",
    "\n",
    "### Three features are considered for this dataset:\n",
    "\n",
    "\n",
    "1) Height in cm\n",
    "\n",
    "2) Weight in kg\n",
    "\n",
    "3) Age in years"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first import all the packages that are needed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import csv\n",
    "\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import sklearn as sl\n",
    "from scipy import stats\n",
    "from sklearn import datasets\n",
    "from sklearn import linear_model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Perceptron\n",
    "Firstly we will implement the perceptron algorithm and use it to learn a halfspace."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** Set the random seed (you can use your ID (matricola) or any other number!)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "IDnumber = 235245 #replace with your ID or try some random values\n",
    "np.random.seed(IDnumber)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the dataset and then split in training set and test set (the training set is typically larger, you can use a 70% tranining 30% test split) after applying a random permutation to the datset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A) Load dataset and perform permutation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Position', 'Height cm', 'kg', 'Age']\n",
      "260\n"
     ]
    }
   ],
   "source": [
    "#load the dataset\n",
    "filename = 'data/NBA.csv'\n",
    "NBA = csv.reader(open(filename, newline=''), delimiter=',')\n",
    "\n",
    "header = next(NBA) #skip first line\n",
    "print(header)\n",
    "\n",
    "dataset = list(NBA)\n",
    "for i in range(len(dataset)):\n",
    "    dataset[i] = [int(x) for x in dataset[i]]\n",
    "    \n",
    "dataset = np.asarray(dataset)\n",
    "\n",
    "X = dataset[:,1:4] #columns 1,2,3 contain the features\n",
    "Y = dataset[:,0]  # column 0: labels\n",
    "\n",
    "Y = Y*2-1  # set labels to -1, 1 as required by perceptron implementation\n",
    "\n",
    "m = dataset.shape[0]\n",
    "print(m)\n",
    "permutation = np.random.permutation(m) # random permurtation\n",
    "\n",
    "X = X[permutation]\n",
    "Y = Y[permutation]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are going to classify class \"1\" (Center) vs class \"-1\" (Point Guard)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "B) **TO DO** Divide the data into training set and test set (70% of the data in the first set, 30% in the second one)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1  1 -1 -1 -1  1  1 -1 -1 -1  1 -1 -1 -1 -1 -1 -1 -1  1  1  1 -1  1 -1\n",
      "  1 -1  1  1  1  1 -1 -1  1  1  1 -1 -1  1  1 -1  1  1  1 -1  1 -1  1  1\n",
      "  1  1 -1 -1 -1  1  1  1  1  1 -1  1  1 -1  1  1  1  1 -1  1 -1 -1  1  1\n",
      "  1  1 -1 -1 -1  1 -1 -1 -1  1 -1 -1  1  1  1  1  1  1 -1 -1  1  1  1  1\n",
      " -1  1  1 -1 -1  1  1  1 -1  1 -1  1 -1  1  1  1 -1 -1  1 -1  1  1  1  1\n",
      " -1 -1 -1  1  1  1 -1  1 -1  1 -1  1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1\n",
      " -1 -1  1 -1  1  1  1  1  1  1  1  1  1 -1  1 -1 -1  1  1  1  1 -1  1  1\n",
      " -1 -1 -1  1  1  1  1  1  1 -1 -1  1  1 -1]\n",
      "78\n",
      "Shape of training set: (182, 3)\n",
      "Shape of test set: (78, 3)\n"
     ]
    }
   ],
   "source": [
    "#Divide in training and test: make sure that your training set\n",
    "#contains at least 10 elements from class 1 and at least 10 elements\n",
    "#from class -1! If it does not, modify the code so to apply more random\n",
    "#permutations (or the same permutation multiple times) until this happens.\n",
    "#IMPORTANT: do not change the random seed.\n",
    "\n",
    "#m_training needs to be the number of samples in the training set\n",
    "m_training = int(X.shape[0]*0.7)\n",
    "\n",
    "#m_test needs to be the number of samples in the test set\n",
    "m_test = int(X.shape[0]-(X.shape[0]*0.7))\n",
    "\n",
    "#X_training = instances for training set\n",
    "X_training = X[:m_training,:]\n",
    "#Y_training = labels for the training set\n",
    "Y_training = Y[:m_training]\n",
    "\n",
    "#X_test = instances for test set\n",
    "X_test =  X[m_training:,]\n",
    "#Y_test = labels for the test set\n",
    "Y_test = Y[m_training:]\n",
    "\n",
    "print(Y_training) #to make sure that Y_training contains both 1 and -1\n",
    "print(m_test)\n",
    "\n",
    "print(\"Shape of training set: \" + str(X_training.shape))\n",
    "print(\"Shape of test set: \" + str(X_test.shape))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** Now add a 1 in front of each sample so that we can use a vector in homogeneous coordinates to describe all the coefficients of the model. You can use the function $hstack$ in $numpy$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#add a 1 to each sample (homogeneous coordinates)\n",
    "X_training = np.hstack((np.ones((m_training,1)),X_training))\n",
    "X_test = np.hstack((np.ones((m_test,1)),X_test))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** Now complete the function *perceptron*. Since the perceptron does not terminate if the data is not linearly separable, your implementation should return the desired output (see below) if it reached the termination condition seen in class or if a maximum number of iterations have already been run, where one iteration corresponds to one update of the perceptron weights. In case the termination is reached because the maximum number of iterations have been completed, the implementation should return **the best model** seen up to now.\n",
    "\n",
    "The input parameters to pass are:\n",
    "- $X$: the matrix of input features, one row for each sample\n",
    "- $Y$: the vector of labels for the input features matrix X\n",
    "- $max\\_num\\_iterations$: the maximum number of iterations for running the perceptron\n",
    "\n",
    "The output values are:\n",
    "- $best\\_w$: the vector with the coefficients of the best model\n",
    "- $best\\_error$: the *fraction* of misclassified samples for the best model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def perceptron_update(current_w, x, y):\n",
    "    # Place in this function the update rule of the perceptron algorithm\n",
    "    new_w = current_w + y*x\n",
    "    return new_w\n",
    "\n",
    "def perceptron(X, Y, max_num_iterations):\n",
    "    \n",
    "    #init the algorith with w=0, use a best_w variable to keep track of the best solution\n",
    "    curr_w = np.zeros(X.shape[1])\n",
    "    best_w = curr_w\n",
    "    num_samples = X.shape[0]\n",
    "    best_error = num_samples+1\n",
    "    #now scan the entire dataset, and while we do it we compute the empirical\n",
    "    #risk (0-1 loss) for the curr_w and also save 1 point that is misclassified\n",
    "    #by curr_w\n",
    "    #in output produce the best_w and the corresponding best_error\n",
    "    index_misclassified = 1\n",
    "    num_misclassified = 0\n",
    "    \n",
    "    \n",
    "    \n",
    "    num_iter = 0\n",
    "    #main loop continue until all samples correctly classified or max # iterations reached\n",
    "    while ((index_misclassified != -1) and (num_iter < max_num_iterations)):\n",
    "      \n",
    "        index_misclassified = -1\n",
    "        num_misclassified = 0\n",
    "        permutation = np.random.permutation(num_samples)\n",
    "        for i in range(num_samples):\n",
    "            #let's generate a random permutation so to check the elements in a random order\n",
    "            \n",
    "            x = X[permutation[i],:]\n",
    "            y = Y[permutation[i]]\n",
    "            prediction = np.sign(np.dot(curr_w,x))\n",
    "            if (prediction != y):\n",
    "                index_misclassified = permutation[i]\n",
    "                num_misclassified += 1\n",
    "        if num_misclassified < best_error:\n",
    "            best_error = num_misclassified\n",
    "            best_w = curr_w\n",
    "        num_iter += 1\n",
    "        #call update function using a misclassifed sample\n",
    "        curr_w = perceptron_update(curr_w, X[index_misclassified,:],Y[index_misclassified])\n",
    "    best_error = float(best_error)/float(num_samples)\n",
    "    \n",
    "    return best_w, best_error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we use the implementation above of the perceptron to learn a model from the training data using 100 iterations and print the error of the best model we have found."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training Error of perpceptron (100 iterations): 0.06593406593406594\n"
     ]
    }
   ],
   "source": [
    "#now run the perceptron for 100 iterations\n",
    "w_found, error = perceptron(X_training,Y_training, 100)\n",
    "print(\"Training Error of perpceptron (100 iterations): \" + str(error))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** use the best model $w\\_found$ to predict the labels for the test dataset and print the fraction of misclassified samples in the test set (the test error that is an estimate of the true loss)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test Error of perpceptron (100 iterations): 0.10256410256410256\n"
     ]
    }
   ],
   "source": [
    "#now use the w_found to make predictions on test dataset\n",
    "\n",
    "num_errors = 0\n",
    "\n",
    "for i in range(m_test):\n",
    "    x = X_test[i,:]\n",
    "    y = Y_test[i]\n",
    "    predicted = np.sign(np.dot(w_found,x))\n",
    "    if (predicted != y):\n",
    "        num_errors += 1\n",
    "\n",
    "true_loss_estimate = num_errors/m_test  # error rate on the test set\n",
    "#NOTE: you can avoid using num_errors if you prefer, as long as true_loss_estimate is correct\n",
    "print(\"Test Error of perpceptron (100 iterations): \" + str(true_loss_estimate))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** **[Answer the following]** What about the difference betweeen the training error and the test error  in terms of fraction of misclassified samples)? Explain what you observe. [Write the answer in this cell]\n",
    "\n",
    "**ANSWER QUESTION 1**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** Copy the code from the last 2 cells above in the cell below and repeat the training with 3000 iterations. Then print the error in the training set and the estimate of the true loss obtained from the test set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.06593406593406594\n",
      "0.10256410256410256\n"
     ]
    }
   ],
   "source": [
    "#now run the perceptron for 5000 iterations here!\n",
    "\n",
    "w_found, error = perceptron(X_training,Y_training, 3000)\n",
    "\n",
    "print(error)\n",
    "\n",
    "num_errors = 0.\n",
    "\n",
    "for i in range(m_test):\n",
    "    x = X_test[i,:]\n",
    "    y = Y_test[i]\n",
    "    predicted = np.sign(np.dot(w_found,x))\n",
    "    if (predicted != y):\n",
    "        num_errors += 1\n",
    "\n",
    "true_loss_estimate = num_errors/m_test\n",
    "print(true_loss_estimate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** [Answer the following] What about the difference betweeen the training error and the test error  in terms of fraction of misclassified samples) when running for a larger number of iterations ? Explain what you observe and compare with the previous case. [Write the answer in this cell]\n",
    "\n",
    "**ANSWER QUESTION 2**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Logistic Regression\n",
    "Now we use logistic regression, exploiting the implementation in Scikit-learn, to predict labels. We will also plot the decision region of logistic regression.\n",
    "\n",
    "We first load the dataset again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Position', 'Height cm', 'kg', 'Age']\n"
     ]
    }
   ],
   "source": [
    "filename = 'data/NBA.csv'\n",
    "NBA = csv.reader(open(filename, newline=''), delimiter=',')\n",
    "\n",
    "header = next(NBA)\n",
    "print(header)\n",
    "\n",
    "dataset = list(NBA)\n",
    "for i in range(len(dataset)):\n",
    "    dataset[i] = [int(x) for x in dataset[i]]\n",
    "    \n",
    "dataset = np.asarray(dataset)\n",
    "\n",
    "X = dataset[:,1:]\n",
    "Y = dataset[:,0]\n",
    "\n",
    "Y = Y*2-1  # set labels to -1, 1 as required by perceptron implementation\n",
    "\n",
    "m = dataset.shape[0]\n",
    "permutation = np.random.permutation(m)\n",
    "\n",
    "X = X[permutation]\n",
    "Y = Y[permutation]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** As for the previous part, divide the data into training and test (70%-30%) and add a 1 as first component to each sample."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Divide in training and test: make sure that your training set\n",
    "#contains at least 10 elements from class 1 and at least 10 elements\n",
    "#from class -1! If it does not, modify the code so to apply more random\n",
    "#permutations (or the same permutation multiple times) until this happens.\n",
    "#IMPORTANT: do not change the random seed.\n",
    "\n",
    "m_training = int(X.shape[0]*0.7)\n",
    "m_test = int(X.shape[0]-(X.shape[0]*0.7))\n",
    "\n",
    "X_training = X[:m_training,:]\n",
    "Y_training = Y[:m_training]\n",
    "\n",
    "X_test = X[m_training:,]\n",
    "Y_test = Y[m_training:]\n",
    "\n",
    "#add a 1 as first component to each sample\n",
    "#X_training = np.hstack((np.ones((m_training,1)),X_training))\n",
    "#X_test = np.hstack((np.ones((m_test,1)),X_test))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To define a logistic regression model in Scikit-learn use the instruction\n",
    "\n",
    "$linear\\_model.LogisticRegression(C=1e5)$\n",
    "\n",
    "($C$ is a parameter related to *regularization*, a technique that\n",
    "we will see later in the course. Setting it to a high value is almost\n",
    "as ignoring regularization, so the instruction above corresponds to the\n",
    "logistic regression you have seen in class.)\n",
    "\n",
    "To learn the model you need to use the $fit(...)$ instruction and to predict you need to use the $predict(...)$ function. See the Scikit-learn documentation for how to use it.\n",
    "\n",
    "**TO DO** Define the logistic regression model, then learn the model using the training set and predict on the test set. Then print the fraction of samples misclassified in the training set and in the test set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept: [-19.41146172]\n",
      "Coefficients: [[-0.02864034  0.23066252  0.0599345 ]]\n",
      "Error rate on training set: 0.06593406593406594\n",
      "Error rate on test set: 0.0641025641025641\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Anay Deshpande\\Anaconda3\\lib\\site-packages\\sklearn\\linear_model\\logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.\n",
      "  FutureWarning)\n"
     ]
    }
   ],
   "source": [
    "#part on logistic regression for 2 classes\n",
    "logreg = linear_model.LogisticRegression(C=1e5) #C very large to ignore regularization\n",
    "\n",
    "#learn from training set\n",
    "\n",
    "logreg.fit(X_training, Y_training)\n",
    "print(\"Intercept:\" , logreg.intercept_)\n",
    "print(\"Coefficients:\" , logreg.coef_)\n",
    "\n",
    "#predict on training set\n",
    "\n",
    "predicted_training = logreg.predict(X_training)\n",
    "\n",
    "#print the error rate = fraction of misclassified samples\n",
    "error_rate_training = 0\n",
    "for i in range(len(Y_training)):\n",
    "    if predicted_training[i]!=Y_training[i]:\n",
    "        error_rate_training += 1\n",
    "error_rate_training = error_rate_training / m_training\n",
    "print(\"Error rate on training set: \"+str(error_rate_training))\n",
    "\n",
    "#predict on test set\n",
    "\n",
    "predicted_test = logreg.predict(X_test)\n",
    "\n",
    "#print the error rate = fraction of misclassified samples\n",
    "error_rate_test = 0\n",
    "for i in range(len(Y_test)):\n",
    "    if predicted_test[i]!=Y_test[i]:\n",
    "        error_rate_test += 1\n",
    "error_rate_test = error_rate_test / m_test\n",
    "print(\"Error rate on test set: \" + str(error_rate_test))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** Now pick two features and restrict the dataset to include only two features, whose indices are specified in the $feature$ vector below. Then split into training and test. Which features are you going to select ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#to make the plot we need to reduce the data to 2D, so we choose two features\n",
    "features_list = ['height', 'weight', 'age']\n",
    "labels_list = ['Center', 'Point guard']\n",
    "\n",
    "index_feature1 = 0 # we choose height\n",
    "index_feature2 = 1 # and weight (age of course is not very meaningful)\n",
    "features = [index_feature1, index_feature2]\n",
    "\n",
    "feature_name0 = features_list[features[0]]\n",
    "feature_name1 = features_list[features[1]]\n",
    "\n",
    "X_reduced = X[:,features]\n",
    "\n",
    "X_training = X_reduced[:m_training,]\n",
    "Y_training = Y[:m_training]\n",
    "\n",
    "X_test = X_reduced[m_training:,]\n",
    "Y_test = Y[m_training:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now learn a model using the training data and measure the performances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error rate on test set: 0.0641025641025641\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Anay Deshpande\\Anaconda3\\lib\\site-packages\\sklearn\\linear_model\\logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.\n",
      "  FutureWarning)\n"
     ]
    }
   ],
   "source": [
    "# learning from training data\n",
    "logreg.fit(X_training, Y_training)\n",
    "\n",
    "#print the error rate = fraction of misclassified samples\n",
    "error_rate_test = 0\n",
    "for i in range(len(Y_test)):\n",
    "    if predicted_test[i]!=Y_test[i]:\n",
    "        error_rate_test += 1\n",
    "error_rate_test = error_rate_test / m_test\n",
    "print(\"Error rate on test set: \" + str(error_rate_test))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**TO DO** [Answer the following] Which features did you select and why ? Compare the perfromances with the ones of the case with all the 3 features and comment about the results. [Write the answer in this cell]\n",
    "\n",
    "**ANSWER QUESTION 3**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If everything is ok, the code below uses the model in $logreg$ to plot the decision region for the two features chosen above, with colors denoting the predicted value. It also plots the points (with correct labels) in the training set. It makes a similar plot for the test set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        fig.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\" width=\"288\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the decision boundary. For that, we will assign a color to each\n",
    "# point in the mesh [x_min, x_max]x[y_min, y_max].\n",
    "h = .02  # step size in the mesh\n",
    "x_min, x_max = X_reduced[:, 0].min() - .5, X_reduced[:, 0].max() + .5\n",
    "y_min, y_max = X_reduced[:, 1].min() - .5, X_reduced[:, 1].max() + .5\n",
    "xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))\n",
    "\n",
    "Z = logreg.predict(np.c_[xx.ravel(), yy.ravel()])\n",
    "\n",
    "# Put the result into a color plot\n",
    "Z = Z.reshape(xx.shape)\n",
    "\n",
    "plt.figure(1, figsize=(4, 3))\n",
    "plt.pcolormesh(xx, yy, Z, cmap=plt.cm.Paired)\n",
    "\n",
    "# Plot also the training points\n",
    "plt.scatter(X_training[:, 0], X_training[:, 1], c=Y_training, edgecolors='k', cmap=plt.cm.Paired)\n",
    "plt.xlabel(feature_name0)\n",
    "plt.ylabel(feature_name1)\n",
    "\n",
    "plt.xlim(xx.min(), xx.max())\n",
    "plt.ylim(yy.min(), yy.max())\n",
    "plt.xticks(())\n",
    "plt.yticks(())\n",
    "plt.title('Training set')\n",
    "\n",
    "plt.show()\n",
    "\n",
    "# Put the result into a color plot\n",
    "Z = Z.reshape(xx.shape)\n",
    "plt.figure(1, figsize=(4, 3))\n",
    "plt.pcolormesh(xx, yy, Z, cmap=plt.cm.Paired)\n",
    "\n",
    "# Plot also the test points \n",
    "plt.scatter(X_test[:, 0], X_test[:, 1], c=Y_test, edgecolors='k', cmap=plt.cm.Paired, marker='s')\n",
    "plt.xlabel(feature_name0)\n",
    "plt.ylabel(feature_name1)\n",
    "\n",
    "plt.xlim(xx.min(), xx.max())\n",
    "plt.ylim(yy.min(), yy.max())\n",
    "plt.xticks(())\n",
    "plt.yticks(())\n",
    "plt.title('Test set')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2) Linear Regression on the Boston House Price dataset\n",
    "\n",
    "Dataset description\n",
    "The Boston House Price Dataset involves the prediction of a house price in thousands of dollars given details about the house and its neighborhood.\n",
    "\n",
    "The dataset contains a total of 500 observations, which relate 13 input features to an output variable (house price).\n",
    "\n",
    "The variable names are as follows:\n",
    "\n",
    "CRIM: per capita crime rate by town.\n",
    "\n",
    "ZN: proportion of residential land zoned for lots over 25,000 sq.ft.\n",
    "\n",
    "INDUS: proportion of nonretail business acres per town.\n",
    "\n",
    "CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).\n",
    "\n",
    "NOX: nitric oxides concentration (parts per 10 million).\n",
    "\n",
    "RM: average number of rooms per dwelling.\n",
    "\n",
    "AGE: proportion of owner-occupied units built prior to 1940.\n",
    "\n",
    "DIS: weighted distances to five Boston employment centers.\n",
    "\n",
    "RAD: index of accessibility to radial highways.\n",
    "\n",
    "TAX: full-value property-tax rate per $10,000.\n",
    "\n",
    "PTRATIO: pupil-teacher ratio by town.\n",
    "\n",
    "B: 1000*(Bk – 0.63)2 where Bk is the proportion of blacks by town.\n",
    "\n",
    "LSTAT: % lower status of the population.\n",
    "\n",
    "MEDV: Median value of owner-occupied homes in $1000s.\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "#needed if you get the IPython/javascript error on the in-line plots\n",
    "%matplotlib nbagg  \n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "DescribeResult(nobs=500, minmax=(array([6.3200e-03, 0.0000e+00, 4.6000e-01, 0.0000e+00, 3.8500e-01,\n",
      "       3.5610e+00, 2.9000e+00, 1.1296e+00, 1.0000e+00, 1.8700e+02,\n",
      "       1.2600e+01, 3.2000e-01, 1.7300e+00, 5.0000e+01]), array([ 88.9762, 100.    ,  27.74  ,   1.    ,   0.871 ,   8.78  ,\n",
      "       100.    ,  12.1265,  24.    , 711.    ,  22.    , 396.9   ,\n",
      "        37.97  , 500.    ])), mean=array([3.65578576e+00, 1.15000000e+01, 1.11317400e+01, 7.00000000e-02,\n",
      "       5.54451400e-01, 6.28297000e+00, 6.84246000e+01, 3.81193180e+00,\n",
      "       9.64200000e+00, 4.09624000e+02, 1.84286000e+01, 3.56208040e+02,\n",
      "       1.26987400e+01, 2.25680000e+02]), variance=array([7.47252195e+01, 5.48905812e+02, 4.76198228e+01, 6.52304609e-02,\n",
      "       1.35838329e-02, 4.97618330e-01, 7.99314163e+02, 4.46304043e+00,\n",
      "       7.59617595e+01, 2.85605197e+04, 4.67663531e+00, 8.41657137e+03,\n",
      "       5.13361942e+01, 8.52982926e+03]), skewness=array([ 5.17851447,  2.19924065,  0.29469001,  3.37060575,  0.72944382,\n",
      "        0.40691344, -0.58266003,  0.99061328,  0.9886802 ,  0.65098333,\n",
      "       -0.79127942, -2.85972639,  0.89051925,  1.09625625]), kurtosis=array([36.3384518 ,  3.88460431, -1.25254394,  9.3609831 , -0.10223624,\n",
      "        1.84311394, -0.99523636,  0.43299159, -0.90519243, -1.16696682,\n",
      "       -0.30903922,  7.00864873,  0.44527324,  1.4266838 ]))\n",
      "Shape of data array: (500, 14)\n"
     ]
    }
   ],
   "source": [
    "#Import Data: Load the data from a .csv file\n",
    "\n",
    "filename = \"data/house.csv\"\n",
    "Data = np.genfromtxt(filename, delimiter=';',skip_header=1)\n",
    "\n",
    "#A quick overview of data, to inspect the data you can use the method describe()\n",
    "\n",
    "dataDescription = stats.describe(Data)\n",
    "print(dataDescription)\n",
    "print (\"Shape of data array: \" + str(Data.shape))\n",
    "\n",
    "\n",
    "#for more interesting visualization: use Panda!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Split data in training, validation and test sets\n",
    "\n",
    "\n",
    "\n",
    "Given $m$ total data, denote with $m_{tv}$ the part used for training and validation. Keep $m_t$ data as training data, $m_{val}:=m_{tv}-m_t$ as validation data and $m_{test}:=m - m_{val} - m_t = m-m_{tv}$. For instance one can take $m_t=0.6m$ of the data as training, $m_{val}=0.2m$  validation and $m_{test}=0.2m$ as testing. Let us define as define\n",
    "\n",
    "$\\bullet$ $S_{t}$ the training data set\n",
    "\n",
    "$\\bullet$ $S_{val}$ the validation data set\n",
    "\n",
    "$\\bullet$ $S_{test}$ the testing data set\n",
    "\n",
    "\n",
    "The reason for this splitting is as follows:\n",
    "\n",
    "TRAINING DATA: The training data are used to compute the empirical loss\n",
    "$$\n",
    "L_S(h) = \\frac{1}{m_t} \\sum_{z_i \\in S_{t}} \\ell(h,z_i)\n",
    "$$\n",
    "which is used to estimate $h$ in a given model class ${\\cal H}$.\n",
    "i.e. \n",
    "$$\n",
    "\\hat{h} = {\\rm arg\\; min}_{h \\in {\\cal H}} \\, L_S(h)\n",
    "$$\n",
    "\n",
    "VALIDATION DATA: When different model classes are present (e.g. of different complexity such as linear regression which uses a different number $d_j$ of regressors $x_1$,...$x_{d_j}$), one has to choose which one is the \"best\" complexity. In this simple example the validation set is not needed, but it is better to get used with it.\n",
    "Let ${\\cal H}_{d_j}$ be the space of models as a function of the complexity $d_j$ and let \n",
    "$$\n",
    "\\hat h_{d_j} = {\\rm arg\\; min}_{h \\in {\\cal H}_{d_j}} \\, L_S(h)\n",
    "$$\n",
    "\n",
    "One can estimate the generalization error for model $\\hat h_{d_j}$ as follows:\n",
    "$$\n",
    "L_{{\\cal D}}(\\hat h_{d_j}) \\simeq \\frac{1}{m_{val}} \\sum_{ z_i \\in S_{val}} \\ell(\\hat h_{d_j},z_i)\n",
    "$$\n",
    "and then choose the complexity which achieves the best estimate of the generalization error\n",
    "$$\n",
    "\\hat d_j: = {\\rm arg\\; min}_{d_j} \\,\\frac{1}{m_{val}} \\sum_{ z_i \\in S_{val}} \\ell(\\hat h_{d_j},z_i)\n",
    "$$\n",
    "\n",
    "TESTING DATA: Last, the test data set can be used to estimate the performance of the final estimated model\n",
    "$\\hat h_{\\hat d_j}$ using:\n",
    "$$\n",
    "L_{{\\cal D}}(\\hat h_{\\hat d_j}) \\simeq \\frac{1}{m_{test}} \\sum_{ z_i \\in S_{test}} \\ell(\\hat h_{\\hat d_j},z_i)\n",
    "$$\n",
    "\n",
    "\n",
    "**TO DO**: split the data in training, validation and test sets (60%-20%-20%)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total number of samples:  500\n",
      "Size of each chunk of data:  125\n",
      "Training input data size:  (250, 13)\n",
      "Training output data size:  (250,)\n",
      "Validation input data size:  (125, 13)\n",
      "Validation output data size:  (125,)\n",
      "Test input data size:  (125, 13)\n",
      "Test output data size:  (125,)\n"
     ]
    }
   ],
   "source": [
    "#get number of total samples\n",
    "num_total_samples = Data.shape[0]\n",
    "\n",
    "print (\"Total number of samples: \", num_total_samples)\n",
    "\n",
    "#size of each chunk of data (1/4 each): 2 of them for training, 1 for validation, 1 for testing\n",
    "size_chunk = int(num_total_samples/4)\n",
    "\n",
    "print (\"Size of each chunk of data: \", size_chunk)\n",
    "\n",
    "#shuffle the data\n",
    "np.random.shuffle(Data)\n",
    "\n",
    "#training data \n",
    "\n",
    "X_training = Data[:size_chunk*2,:-1]\n",
    "Y_training = Data[:size_chunk*2,-1]\n",
    "print (\"Training input data size: \", X_training.shape)\n",
    "print (\"Training output data size: \", Y_training.shape)\n",
    "\n",
    "#validation data, to be used to choose among different models\n",
    "X_validation = Data[size_chunk*2:3*size_chunk,:-1]\n",
    "Y_validation = Data[size_chunk*2:3*size_chunk,-1]\n",
    "print (\"Validation input data size: \", X_validation.shape)\n",
    "print (\"Validation output data size: \", Y_validation.shape)\n",
    "\n",
    "#test data, to be used to estimate the true loss of the final model(s)\n",
    "X_test = Data[3*size_chunk:,:-1]\n",
    "Y_test = Data[3*size_chunk:,-1]\n",
    "print (\"Test input data size: \", X_test.shape)\n",
    "print (\"Test output data size: \", Y_test.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Data Normalization\n",
    "It is common practice in Statistics and Machine Learning to scale the data (= each variable) so that it is centered (zero mean) and has standard deviation equal to 1. This helps in terms of numerical conditioning of the (inverse) problems of estimating the model (the coefficients of the linear regression in this case), as well as to give the same scale to all the coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean of the training input data: [-6.09290396e-16  5.37347944e-17  1.72128978e-15  1.04805054e-16\n",
      "  3.11262127e-15  1.10089715e-15  9.42801393e-16 -1.07425180e-15\n",
      "  1.32338585e-16  1.14797061e-16  1.68123293e-14  3.13238324e-15\n",
      "  6.51478871e-16]\n",
      "Std of the training input data: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      "Mean of the validation input data: [-0.14740298  0.23256696 -0.0603621   0.11790127 -0.13369781 -0.02312501\n",
      " -0.22724021  0.09525032 -0.04878219 -0.03344884  0.01564353  0.15917195\n",
      " -0.0709213 ]\n",
      "Std of the validation input data: [0.52284193 1.23946282 1.02243448 1.19288672 1.02752349 0.82968376\n",
      " 1.12815333 0.98833995 0.97532068 1.01102089 0.9507612  0.77793535\n",
      " 0.95364664]\n",
      "Mean of the test input data: [-0.05482774  0.07407825 -0.02709736  0.05052912 -0.11041248  0.04318083\n",
      " -0.14229709  0.07970127  0.10492772  0.03507466  0.14726495  0.09011\n",
      " -0.14800238]\n",
      "Std of the test input data: [0.68886425 1.04390324 0.94057338 1.08843045 0.93677478 1.08424565\n",
      " 1.11452094 1.10229731 1.02476287 1.02418115 0.92374582 0.85852195\n",
      " 0.91370802]\n"
     ]
    }
   ],
   "source": [
    "#scale the data\n",
    "\n",
    "# standardize the input matrix\n",
    "from sklearn import preprocessing\n",
    "# the transformation is computed on training data and then used on all the 3 sets\n",
    "scaler = preprocessing.StandardScaler().fit(X_training) \n",
    "\n",
    "X_training = scaler.transform(X_training)\n",
    "print (\"Mean of the training input data:\", X_training.mean(axis=0))\n",
    "print (\"Std of the training input data:\",X_training.std(axis=0))\n",
    "\n",
    "X_validation = scaler.transform(X_validation) # use the same transformation on validation data\n",
    "print (\"Mean of the validation input data:\", X_validation.mean(axis=0))\n",
    "print (\"Std of the validation input data:\", X_validation.std(axis=0))\n",
    "\n",
    "X_test = scaler.transform(X_test) # use the same transformation on test data\n",
    "print (\"Mean of the test input data:\", X_test.mean(axis=0))\n",
    "print (\"Std of the test input data:\", X_test.std(axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Model Training \n",
    "\n",
    "The model is trained (= estimated) minimizing the empirical error\n",
    "$$\n",
    "L_S(h) := \\frac{1}{m_t} \\sum_{z_i \\in S_{t}} \\ell(h,z_i)\n",
    "$$\n",
    "When the loss function is the quadratic loss\n",
    "$$\n",
    "\\ell(h,z) := (y - h(x))^2\n",
    "$$\n",
    "we define  the Residual Sum of Squares (RSS) as\n",
    "$$\n",
    "RSS(h):= \\sum_{z_i \\in S_{t}} \\ell(h,z_i) = \\sum_{z_i \\in S_{t}} (y_i - h(x_i))^2\n",
    "$$ so that the training error becomes\n",
    "$$\n",
    "L_S(h) = \\frac{RSS(h)}{m_t}\n",
    "$$\n",
    "\n",
    "We recal that, for linear models we have $h(x) = <w,x>$ and the Empirical error $L_S(h)$ can be written\n",
    "in terms of the vector of parameters $w$ in the form\n",
    "$$\n",
    "L_S(w) = \\frac{1}{m_t} \\|Y - X w\\|^2\n",
    "$$\n",
    "where $Y$ and $X$ are the matrices whose $i-$th row are, respectively, the output data $y_i$ and the input vectors $x_i^\\top$.\n",
    "\n",
    "\n",
    " **TO DO:** compute the linear regression coefficients using np.linalg.lstsq from scikitlear \n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LS coefficients with numpy lstsq: [221.196      -11.02638719  16.00682038   1.02525834   7.33426532\n",
      " -17.01526507  34.16175547  -2.23461839 -32.279903    25.13312978\n",
      " -27.01491023 -20.04235464   9.11790379 -29.30084985]\n",
      "RSS with numpy lstsq:  [357573.04897754]\n",
      "Empirical risk with numpy lstsq: [1430.29219591]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Anay Deshpande\\Anaconda3\\lib\\site-packages\\ipykernel_launcher.py:18: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n",
      "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n"
     ]
    }
   ],
   "source": [
    "#compute linear regression coefficients for training data\n",
    "\n",
    "#add a 1 at the beginning of each sample for training, validation, and testing (use homogeneous coordinates)\n",
    "m_training = X_training.shape[0]\n",
    "X_trainingH = np.hstack((np.ones((m_training,1)),X_training)) # H: in homogeneous coordinates\n",
    "#print X_training[0,:]\n",
    "\n",
    "m_validation = X_validation.shape[0]\n",
    "X_validationH = np.hstack((np.ones((m_validation,1)),X_validation))  # H: in homogeneous coordinates\n",
    "#print X_validation[0,:]\n",
    "\n",
    "m_test = X_test.shape[0]\n",
    "X_testH = np.hstack((np.ones((m_test,1)),X_test))  # H: in homogeneous coordinates\n",
    "#print X_test[0,:]  \n",
    "\n",
    "\n",
    "# Compute the least-squares coefficients using linalg.lstsq\n",
    "w_np, RSStr_np, rank_Xtr, sv_Xtr =  np.linalg.lstsq(X_trainingH, Y_training) # rcond=None avoids warning message\n",
    "\n",
    "\n",
    "print(\"LS coefficients with numpy lstsq:\", w_np)\n",
    "\n",
    "\n",
    "# compute Residual sums of squares \n",
    "Y_trainingR = Y_training.reshape(250,1)  # recall that in Python 1D arraymof size n and 2D matrix nx1 are different\n",
    "RSStr_hand = np.linalg.norm(Y_trainingR-np.dot(X_trainingH,  w_np))**2 # compute error\n",
    "\n",
    "print(\"RSS with numpy lstsq: \", RSStr_np)\n",
    "print(\"Empirical risk with numpy lstsq:\", RSStr_np/m_training)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data prediction \n",
    "\n",
    "Compute the output predictions on both training and validation set.and compute the Residual Sum of Sqaures (RSS). \n",
    "\n",
    "**TO DO**: Compute these quantities on  training, validation and test sets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSS on validation data: 182243.4933877363\n",
      "Loss estimated from validation data: 1457.9479471018903\n",
      "RSS on test data: 596107.8003100309\n",
      "Loss estimated from test data: 4768.8624024802475\n"
     ]
    }
   ],
   "source": [
    "#compute predictions on training and validation\n",
    "\n",
    "#prediction_training \n",
    "prediction_training = np.dot(X_trainingH, w_np)\n",
    "prediction_validation = np.dot(X_validationH, w_np)\n",
    "prediction_test = np.dot(X_testH, w_np)\n",
    "\n",
    "#what about the loss for points in the validation data?\n",
    "RSS_validation = np.linalg.norm(Y_validation-np.dot(X_validationH, w_np))**2\n",
    "RSS_test = np.linalg.norm(Y_test-np.dot(X_testH, w_np))**2\n",
    "\n",
    "print(\"RSS on validation data:\",  RSS_validation)\n",
    "print(\"Loss estimated from validation data:\", RSS_validation/m_validation)\n",
    "\n",
    "print(\"RSS on test data:\",  RSS_test)\n",
    "print(\"Loss estimated from test data:\", RSS_test/m_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### QUESTION 4: Comment on the results you get and on the difference between the train, validation and test errors.\n",
    "\n",
    "Insert your answer here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ordinary Least-Squares using scikit-learn\n",
    "Another fast way to compute the LS estimate is through sklearn.linear_model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept: 221.1960000000003\n",
      "Least-Squares Coefficients: [-11.02638719  16.00682038   1.02525834   7.33426532 -17.01526507\n",
      "  34.16175547  -2.23461839 -32.279903    25.13312978 -27.01491023\n",
      " -20.04235464   9.11790379 -29.30084985]\n",
      "Measure on training data: 0.16304104759657623\n"
     ]
    }
   ],
   "source": [
    "from sklearn import linear_model\n",
    "LinReg = linear_model.LinearRegression()  # build the object LinearRegression\n",
    "LinReg.fit(X_training, Y_training)  # estimate the LS coefficients\n",
    "print(\"Intercept:\", LinReg.intercept_)\n",
    "print(\"Least-Squares Coefficients:\", LinReg.coef_)\n",
    "prediction_training = LinReg.predict(X_training)  # predict output values on training set\n",
    "prediction_test = LinReg.predict(X_test)  # predict output values on test set\n",
    "print(\"Measure on training data:\", 1-LinReg.score(X_training, Y_training))"
   ]
  }
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