{
 "metadata": {
  "name": "week4_tutorial"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Psych216A Tutorial 4: Model Fitting"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This tutorial will give an introduction to linear and non-linear fitting\n",
      "proceedures. It covers the basic algebra of linear regression and\n",
      "compares the solutions from ordinary least squares regression to the\n",
      "solutions obtained from a non-linear fitting proceedure.\n",
      "\n",
      "Written by Jason D. Yeatman jyeatman@stanford.edu\n",
      "\n",
      "Translated to Python by Michael Waskom"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import utils\n",
      "import seaborn\n",
      "seaborn.set()\n",
      "colors = seaborn.color_palette()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Linear Models"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As was emphasized in class, linear models can be fit using simple\n",
      "algebra.  This tutorial will briefly review the math underlying the\n",
      "linear model.  The optimal weights to minimize the squared difference\n",
      "between the model prediction and the measured data can be calculated\n",
      "with he following formula:\n",
      "\n",
      "$$w = (X^TX)^{-1}X^Ty$$\n",
      "\n",
      "Where w is the weights or model parameters to be estimated, X is the\n",
      "regressor matrix and y is the dependent variable.  The weights obtained\n",
      "from this formula provide the best prediction of y as a weighted sum of\n",
      "the regressors in X:\n",
      "\n",
      "$$y = Xw$$\n",
      "\n",
      "Understanding the ordinary least squares formula requires understanding\n",
      "matrix inversion.  The inverse of matrix, A, will reverse the effects of\n",
      "multiplying by A."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# For example if we have some data X\n",
      "X1 = randn(100, 2)\n",
      "plot(X1[:, 0], X1[:, 1], \"o\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
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AwDDCFwAAwwhfAAAMI3wBADCM8AUAwDDCFwAAwwhfAAAM4zpfGBG22xACQD5y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\n"
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# And we multiple that data by a matrix A\n",
      "A = array([[1, .5], [.5, 1]])\n",
      "X2 = dot(X1, A)\n",
      "plot(X2[:, 0], X2[:, 1], \"o\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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+WFu3bs2ZeKVT3yhuQZUft+oyR6zMECdzxMoMcTLj+i0FFy5cqCNHjmj8+PEa\nPny4br/99lJfCgCAWCl55vvRRx/ZHAcijo5PAPA1OlzBdXR8AoBMnDyE6+j4BACZSL5wHR2fACAT\nyReuoz80AGTi1x9cR39oAMhEwRVcR39oAMhE8oUn6A8NAF9j2RkAAI+RfAEA8BjJFwAAj7HnWyba\nJgIAikXyzcEkqdI2EQBQitAlXy9mmqZJNV/bRJIvACCXUO35diTF9MBJcs6fqPTASVq2cq02p1JW\n38e0FzFtEwEApQhV8vWqQb9pUqVtIgCgFKFKE17NNE2TKm0TAQClCFXy9WqmaZpUx9XXq3H+Dapq\naVLF502qammibSIAoKBQFVzNnH5tRiGUdCopNs6fYfV9iulFTNtEAECxQpV8vWzQT1IFALglVMlX\nIikCAMIvVHu+AABEQehmvkFFm0kAgCmSrwW0mQQAFINlZwu8av4BAIgGkq8FtJkEABSD5GsBbSYB\nAMUgPVhAm0kAQDEouLLAy+YfAIDwI/laQvMPAIAplp0BAPAYyRcAAI+RfAEA8BjJFwAAj5F8AQDw\nGMkXAACPkXwBAPAYyRcAAI+RfAEA8BjJFwAAj5F8AQDwGMkXAACPkXwBAPAYyRcAAI+VnHzvu+8+\nXXbZZRo2bJhmz56t1tZWm+MCACCySk6+d911l95991298847GjRokB599FGb4wIAILJKTr69e/eW\nJKXTaR09elRnnXWWtUEBABBlFY7jOKX+43vvvVfLly/XxRdfrC1btqhbt25nPKe93VE63VbWIKOu\nsjIpScTJALEyQ5zMESszxMlMZWVSiURFweflTb4TJkzQvn37znj8/vvv1+TJkyVJX375pe69915J\n0tKlS0sdLwAAsVHWzLfDrl27dMstt+jNN9+0MSYAACKt5D3fjz76SNKpPd/nnntO06ZNszYoAACi\nrOTke88992jo0KH6/ve/r3Q6rVtuucXmuAAAiKySk++6deu0a9cu/fGPf9RDDz2kc845J+vzOA9s\nbvHixaqtrdXll1+uRYsW6dixY34PKZCef/551dXVKZlMaufOnX4PJ3C2bt2q2tpaDRo0SI899pjf\nwwmsuXPnasCAARo6dKjfQwm0Tz/9VGPHjlVdXZ3GjBmj1atX+z2kQDp+/LhGjhypYcOGadSoUYVr\noByX/f3vf+/8/5///OfOfffd5/ZbhtbGjRudtrY2p62tzZk/f76zcuVKv4cUSLt373Y++OADZ8yY\nMc6f/vRevIDoAAADI0lEQVQnv4cTOMOGDXNSqZSzd+9e5+KLL3b279/v95ACaevWrc7OnTudIUOG\n+D2UQGtpaXHefvttx3EcZ//+/c63vvWtjN/r+NrRo0cdx3Gc48ePO3V1dc5HH32U87mut5fkPLC5\nCRMmKJFIKJFI6KqrrlIqlfJ7SIE0ePBgfec73/F7GIF0+PBhSdLo0aN14YUXqqGhQdu3b/d5VMF0\n5ZVX5lyxw9dqamo0bNgwSVK/fv1UV1enHTt2+DyqYOrZs6ck6ciRI0qn0+revXvO53rS2/nee+9V\nTU2Ntm3bpjvvvNOLtwy9FStWdB7nAky99dZbGjx4cOefL7nkEk4hwJqPP/5Yzc3NuuKKK/weSiC1\nt7frsssu04ABA3THHXfoggsuyPlcK8l3woQJGjp06Bn//fa3v5Uk/fKXv9Sf//xnXXHFFbr77rtt\nvGVoFYqVJC1ZskS9e/fWj370Ix9H6i+TOAHwzhdffKEZM2Zo6dKl6tWrl9/DCaREIqF3331XH3/8\nsZ544gm9/fbbOZ9baeMNN23aVPA5PXv21Ny5c2NfFV0oVk899ZR+97vf6fXXX/doRMFkck3hTCNG\njNDixYs7/9zc3KyJEyf6OCJEwcmTJzV9+nTNnj1bU6ZM8Xs4gXfRRRfp6quv1vbt2zV8+PCsz3F9\n2ZnzwOaampr08MMP66WXXmJv3JBTfo+YSKmurpZ0quJ579692rRpk0aOHOnzqBBmjuNo3rx5GjJk\niBYtWuT3cALrwIEDOnTokCSptbVVGzduzPtBxUqHq3x++MMf6oMPPlCPHj00ZswY3XPPPRQ55DBo\n0CCdOHFCffv2lSR973vf0xNPPOHzqILnxRdfVGNjow4cOKDq6moNHz5cr776qt/DCoxUKqXbbrtN\nJ0+eVGNjoxobG/0eUiDNmjVLqVRKra2t6t+/v5YsWaKbb77Z72EFzrZt2zR69Ghdeumlqqg41bP4\ngQceYEWli127dmnOnDlqa2tTTU2NbrzxRt100005n+968gUAAJk8qXYGAABfI/kCAOAxki8AAB4j\n+QIA4DGSLwAAHiP5AgDgsf8PPrwE33ZWMz8AAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Multiplying by the inverse of matrix A will undue the\n",
      "# linear transformation of X that was obtained by multiplying by A\n",
      "Ainv = inv(A)\n",
      "X1b = dot(X2, Ainv)\n",
      "plot(X1b[:, 0], X1b[:, 1], \"o\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# We can check this assertion formally in Python with the caveat that there\n",
      "# is some very small numerical rounding errors.\n",
      "assert all(X1 - X1b < 1e-5), \"Matrix 1 does not equal matrix 2\"\n",
      "\n",
      "# We can also get built-in assertion with consideration of floating point precision\n",
      "from numpy.testing import assert_array_almost_equal\n",
      "assert_array_almost_equal(X1, X1b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Another way to say this is that the matrix obtained by multiplying A with\n",
      "# its inverse is a matrix that will not change the values of X. This matrix\n",
      "# is known as the identity matrix and contains 1s on the diagonal and 0s in\n",
      "# every other entry.\n",
      "I = dot(A, inv(A))\n",
      "assert all(I == eye(I.shape[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The formula $w = (X^TX)^{-1}X^Ty$ is used for fitting an ordinary least\n",
      "fsquares regression in any statistics package. We can show that this\n",
      "formula obtains the same results as the function ``lstsq`` (which\n",
      "lives in the ``numpy.linalg`` namespace normally."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fabricate a dataset with 4 variables\n",
      "X = randn(100, 4)\n",
      "# Fabricate a variable y that depends on these four variables and some noise\n",
      "noise = randn(100) * 10\n",
      "y = 1 * X[:, 0] + 7 * X[:, 1] + 3 * X[:, 2] + 5 * X[:, 3] + 7 + noise\n",
      "\n",
      "# To capture the constant term, we will add a column of 1s to our regressor matrix\n",
      "X = column_stack((X, ones(100)))\n",
      "\n",
      "# Fit the model based on the ordinary last squares (OLS) formula\n",
      "w_ols = dot(dot(inv(dot(X.T, X)), X.T), y) \n",
      "\n",
      "# Note how the lack of builtin matrix multiplication for arrays is a bit annoying\n",
      "# We can also use matrix objects\n",
      "X_mtx = asmatrix(X)\n",
      "y_mtx = asmatrix(y).T\n",
      "w_ols2 = inv(X_mtx.T * X_mtx) * X_mtx.T * y_mtx\n",
      "\n",
      "# We see these are the same (once we account for the 2D nature of the matrix\n",
      "assert_array_almost_equal(w_ols.reshape(w_ols2.shape), w_ols2)\n",
      "\n",
      "# Use numpy's lstsq function to fit the model\n",
      "# lstsq returns a 4-tuple\n",
      "w_lstsq, residue, rnk, s = lstsq(X, y)\n",
      "\n",
      "# You can also do\n",
      "w_lstsq = lstsq(X, y)[0]\n",
      "\n",
      "# If you are following the MATLAB tutorial, note that the argument order for lstsq is backward relative to regress\n",
      "\n",
      "# These are also identical\n",
      "assert_array_almost_equal(w_ols, w_lstsq)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see how well the model describes the data be computing the mean\n",
      "squared difference between our model's prediction and our data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# First we compute our models prediction\n",
      "modelprediction = dot(X, w_ols)\n",
      "\n",
      "# Visualize this prediction by plotting it against our data\n",
      "# The imperfections arise from the noise we added to the data\n",
      "plot(modelprediction, y, \"o\", color=colors[1])\n",
      "xlabel(\"model prediction\")\n",
      "ylabel(\"original data\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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8j+5kt27dOmRkZFhtG9tKp9PhpZdecj1CIpKF1Mv1lMAb68yvv4B4MPEBUe/N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\nfGQTYQxCAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Then we subract the model prediction from the measured data, square that difference and take the mean\n",
      "mse_ols = mean((y - modelprediction) ** 2)\n",
      "print \"Our mean squared error is %.4f using ordinary least squares regression\" % mse_ols"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Our mean squared error is 78.7821 using ordinary least squares regression\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now to convince you that this formula did in fact find the weights that\n",
      "best predict y as a weighted sum of the variables in X we can use a\n",
      "non-linear fitting proceedure to search for the parameters that minimize\n",
      "the squared difference between the model prediction and the data.  The\n",
      "fitting function we will use is lsqnonlin.  The following code will walk\n",
      "you through specifying a model to be fit with lsqnonlin.\n",
      "\n",
      "First we need to define our cost function.  lsqnonlin will search for the\n",
      "parameters that minimize the squared value of this function for each\n",
      "datapoint.  In this case the cost function we will use is y-X*w meaning\n",
      "that we want the algorithm to search for the weights, w, that minimize the\n",
      "difference between our data, y, and our regressors, X, multiplied by our\n",
      "weights.  Notice that we are defining this function in terms of the\n",
      "variable containing our regressors, X and the variable containing our\n",
      "dependent measure, y."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Various solvers live in the optimize package\n",
      "from scipy import optimize\n",
      "\n",
      "# Define an anonymous function that returns the mean squared error between our data\n",
      "# and the result of multiplying the predictor matrix by a weight vector\n",
      "costfunc = lambda w: mean(square(y - dot(X, w)))\n",
      "\n",
      "# Start the search with all weights at 0\n",
      "seed = zeros(5)\n",
      "\n",
      "# Fit the model by searching over a space of cost function values and finding the lowest\n",
      "# Set the limit on function evaluations to infinity\n",
      "optsol = optimize.fmin(costfunc, seed, maxfun=inf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 78.782052\n",
        "         Iterations: 633\n",
        "         Function evaluations: 1019\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that in the report we see the final function value is identical to the mean squared error from the OLS fit\n",
      "\n",
      "Plot the results to see the correspondence in the resulting weight vectors"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "axes(aspect=\"equal\")\n",
      "plot(w_ols, optsol, \"D\", color=colors[2], markersize=7)\n",
      "xlabel(\"OLS solution\")\n",
      "ylabel(\"Optimize solution\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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2bNigRjYiIVxZ0gAulzXAsiZVuC3q0aNHo66uDtu3b4ckSSgoKECPHp1/KRSRGpyVdBvH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\nIiLBsaiJiATHoiYiEtz/B1RAsuTNlvBQAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To verify the correspondence in mean squared error values, compute the MSE for the nonlinear solution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mse_optim = mean(square(y - dot(X, optsol)))\n",
      "print \"OLS MSE: %.4f\" % mse_ols\n",
      "print \"Optimized MSE: %.4f \" % mse_optim"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "OLS MSE: 78.7821\n",
        "Optimized MSE: 78.7821 \n"
       ]
      }
     ],
     "prompt_number": 13
    }
   ],
   "metadata": {}
  }
 ]
}