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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "preamble": true
+   },
+   "source": [
+    "(Defining latex commands: not to be shown...)\n",
+    "$$\n",
+    "\\newcommand{\\norm}[1]{\\left \\| #1 \\right \\|}\n",
+    "\\DeclareMathOperator{\\minimize}{minimize}\n",
+    "\\newcommand{\\real}{\\mathbb{R}}\n",
+    "\\newcommand{\\normal}{\\mathcal{N}}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#  Gaussian Algebra (25 Points)\n",
+    "\n",
+    "Prove that the product of two univariate (scalar) Gaussian distributions is a Gaussian again, i.e. show, by explicitly performing the required arithmetic transformations, that\n",
+    "\n",
+    "\\begin{equation}\n",
+    "   \\normal(x;\\mu,\\sigma^2)\\normal(x;m,s^2) = \\normal[x; (\\frac 1{\\sigma^2}+\\frac 1{s^2})^{-1}(\\frac \\mu{\\sigma^2}+\\frac m{s^2}),(\\frac 1{\\sigma^2}+\\frac 1{s^2})^{-1}]\\normal[m,\\mu,\\sigma^2+s^2].\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#  Maximum Likelihood Estimator of Simple Linear Regression (25 Points)\n",
+    "\n",
+    "Derive the formula $\\mathbf{w}_{MLE} = (X^TX)^{-1}X^T\\mathbf{y}$ from the lecture, by calculating the derivative of $p(\\mathbf{y}\\,|X,\\mathbf{w}) = \\normal(\\mathbf{y}\\,|X\\mathbf{w}, \\sigma^2I)$ with respect to $\\mathbf{w}$, setting it to zero and solving it for $\\mathbf{w}$.\n",
+    "\n",
+    "\n",
+    "Note: _To refresh your linear algebra you might find it useful to have a look in [here](http://webdav.tuebingen.mpg.de/lectures/ei-SS2015/pdfs/Murray_cribsheet.pdf)._"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#  Linear regression (50 Points)\n",
+    "\n",
+    "In this exercise you will perform a regression analysis on a toy dataset. You will implement ridge regression and learn how to find a good model through a comparative performance analysis.\n",
+    "\n",
+    "1) Download the [training set](http://webdav.tuebingen.mpg.de/lectures/ei-SS2015/data/ex1_train.csv)! <br>\n",
+    "2) Implement $\\mathbf{w}_{RIDGE}$ as a function of a given $X, \\mathbf{y}$ array and a regularization parameter $\\lambda$!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# Loading the required packages\n",
+    "%matplotlib inline\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from IPython.html.widgets import interact"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "def wRidge(X,y,lamb):\n",
+    "  # Change the following line and implement the ridge regression estimator wRidge\n",
+    "  w = np.zeros(X.shape[-1])\n",
+    "  return w"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "3) Load \"ex1_train.csv\" into a numpy array! The first column in the csv file is $X$ and the second column is $\\mathbf{y}$, assign them to each variable!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "# Read ex1_train.csv and assign the first column and \n",
+    "# second column to variables x and y respectively."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "4) Plot the training data with appropriate labels on each axes!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "# Plot the input data here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "5) Implement a function which constructs features upto a input polynomial degree $d$!<br>\n",
+    "Note: _Constructing higher polynomial features is similar to what you implemented in Exercise 3 (SVM) of the previous exercise sheet._"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 68,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def construct_poly(x,d):\n",
+    "  ## Implement a method which given an array of size N, \n",
+    "  ## returns an array of dimension (N,d)\n",
+    "  return np.zeros((x.shape[0],d+x.shape[1]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "6) Implement the Mean Squared Error Loss (MSE) as a function of the predicted and true values of the target variable! <br>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def MSE(y_predict,y_true):\n",
+    "  ## Implement mean squared error for a given input y and its predictions.\n",
+    "  return 0.0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "7) By comparing the MSE find the degree $d$ for the polynomial that fits the training data best! You might find it useful to use the code below to interactively change the variable $d$, set $\\lambda = 1$ and keep it fixed. Plot the error as a function of different values of $d$!<br>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "##This function provides an interactive mode to change polynomial degree. \n",
+    "@interact(n=[1,16])\n",
+    "def plot(n):\n",
+    "  X = construct_poly(x,n)\n",
+    "  w = wRidge(X,y,1.0)\n",
+    "  plt.plot(x,X.dot(w))\n",
+    "  plt.title(\"MSE %f\" % MSE(X.dot(w),y))\n",
+    "  plt.plot(x,y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "8) Apply models with different values of $d$ after being trained on the training dataset, to the test data available [here](http://webdav.tuebingen.mpg.de/lectures/ei-SS2015/data/ex1_test.csv). Compare the errors on the test data to the ones from the training by plotting the error curves as functions of the polynomial degree in a single plot! What do you conclude? <br>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 82,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[  1.00000000e+02,   1.00000000e+01,   1.00000000e+00],\n",
+       "       [  1.00000000e+01,   1.00000000e+00,   1.00000000e-01],\n",
+       "       [  1.00000000e+00,   1.00000000e-01,   1.00000000e-02]])"
+      ]
+     },
+     "execution_count": 82,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A.T.dot(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "import scipy\n",
+    "from scipy import linalg\n",
+    "import numpy as np"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "1\n",
+      "[[  9.80101960e-03   9.80101960e-04   9.80101960e-05]\n",
+      " [  9.80101960e-04   9.80101960e-05   9.80101960e-06]\n",
+      " [  9.80101960e-05   9.80101960e-06   9.80101960e-07]]\n",
+      "True\n"
+     ]
+    }
+   ],
+   "source": [
+    "A = np.array([[10,1,0.1],[10,1,0.2],[10,1.1,0.7]])\n",
+    "a = np.array([[10,1,0.1]])\n",
+    "A = a.T.dot(a)\n",
+    "\n",
+    "B,rank = linalg.pinv(A,return_rank=True)\n",
+    "print rank\n",
+    "print B\n",
+    "print np.allclose(A,A.dot(B.dot(A)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 67,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "## Read test data here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "9) With a fixed optimal $d$, change the value of $\\lambda$ to one of the following values $[0.1, 1.0, 10.0]$ and find the minimum MSE!<br>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Hand in printed copy of completed notebook."
+   ]
+  }
+ ],
+ "metadata": {
+  "annotations": {
+   "author": "",
+   "categories": [
+    "intelligent-systems-1-2015"
+   ],
+   "date": "2015-04-30",
+   "location": "Beginning of next lecture",
+   "parent": "IS_SS2015",
+   "submission_date": "2015-05-07",
+   "subtitle": "Exercise Sheet 3, Linear Regression",
+   "tags": [
+    "IntelligenSystems",
+    "Course"
+   ],
+   "title": "Intelligent Systems 1 - Summer Semester 2015"
+  },
+  "celltoolbar": "Edit Metadata",
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}