notebooks/02.1_ModelFitting.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Fitting Models to Data I: Frequentist Models"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This morning we will introduce concepts of model fitting, particularly the maximum likelihood method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "from scipy import stats\n",
      "\n",
      "# use seaborn plotting defaults\n",
      "import seaborn as sns; sns.set()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Simple Model Fitting: Straight Line Fit"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "One of the most archetypical model fits you'll encounter is that of fitting a line to data.\n",
      "We'll use the following randomly-generated dataset and discuss how we might fit this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def linear_data_sample(N=40, rseed=0, m=3, b=-2):\n",
      "    rng = np.random.RandomState(rseed)\n",
      "\n",
      "    x = 10 * rng.rand(N)\n",
      "    dy = m / 2 * (1 + rng.rand(N))\n",
      "    y = m * x + b + dy * rng.randn(N)\n",
      "\n",
      "    return (x, y, dy)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y, dy = linear_data_sample()\n",
      "plt.errorbar(x, y, dy, fmt='o');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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TSM/4ReB9QPn2+1uAnw8hPB1C+GII4acyq04aQO2aRUrpeL1ZzarbM44xPhZC2Fyx6xng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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1099a2150>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You can see these points have errors; for now let's ignore those errors and just look at the points themselves:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y, 'o');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x109992fd0>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's think about what it takes to fit a straight line to this data."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Defining the Model"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The first thing we must do is define our model. The *model* is usually some functional form which, as a function of various parameters, specifies what we **think** the data should look like.\n",
      "In this case, we'll use a standard linear model defined by a slope *m* and an intercept *b*:\n",
      "\n",
      "$$\n",
      "M(x;m,b) = m x + b\n",
      "$$\n",
      "\n",
      "For clarity, we'll try to write our models in this way: to the left of the semicolon is our data, and to the right of the semicolon are our model parameters.\n",
      "\n",
      "Now, examining our data above, we can quickly see that this model will **not** perfectly fit the data. So generally, what we're saying is that the data is distributed in some way around the model. For the most common assumption of normality, we're saying that\n",
      "\n",
      "$$\n",
      "y_i \\sim \\mathcal{N}[M(x;m,b), \\sigma]\n",
      "$$\n",
      "\n",
      "but we'll get into this later."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Defining a Cost Function\n",
      "\n",
      "In order to fit a model to data, we need some way of *quantitatively* measuring how well a model with given parameters fits the data. That is, given a set of data $\\{x_i, y_i\\}$ and some parameters $\\theta = \\{m, b\\}$, how well does the model fit? This is usually denoted via **Cost Function**, which tells us how far our model is from being ideal.\n",
      "\n",
      "Specify the cost function & minimize it with respect to the model parameters, and you have your best fit! The cost function will look like this:\n",
      "\n",
      "$$\n",
      "C(\\theta;\\{x_i, y_i\\}) = ??\n",
      "$$\n",
      "\n",
      "**Question: What are some cost functions we might use?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Some ideas:\n",
      "\n",
      "#### Absolute deviation\n",
      "\n",
      "$$\n",
      "C_{AD} = \\sum_i \\left|y_i - M(x_i;\\theta)\\right|\n",
      "$$\n",
      "\n",
      "#### Square deviation\n",
      "\n",
      "$$\n",
      "C_{SD} = \\sum_i [y_i - M(x_i;\\theta)]^2\n",
      "$$\n",
      "\n",
      "#### Absolute N-th power?\n",
      "\n",
      "$$\n",
      "C_{N} = \\sum_i \\left|y_i - M(x_i;\\theta)\\right|^N\n",
      "$$\n",
      "\n",
      "#### Hyperbolic cosine?\n",
      "\n",
      "$$\n",
      "C_{HC} = \\sum_i \\cosh[y_i - M(x_i;\\theta)]\n",
      "$$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We could literally use **any cost function we want**, though it's true that some have better properties than others, and are better motivated than others.\n",
      "Probably the most commonly-used cost functions are the first two here: least absolute deviation and least squares."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Trying it Out\n",
      "\n",
      "Let's quickly code these up and see what happens:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def model(theta, x):\n",
      "    b, m = theta\n",
      "    return m * x + b\n",
      "\n",
      "def abs_deviation(theta, x, y):\n",
      "    return np.sum(abs(model(theta, x) - y))\n",
      "\n",
      "def square_deviation(theta, x, y):\n",
      "    return np.sum((model(theta, x) - y) ** 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.optimize import fmin\n",
      "theta_guess = [0, 1]\n",
      "\n",
      "theta_LAD = fmin(abs_deviation, theta_guess, args=(x, y))\n",
      "theta_LS = fmin(square_deviation, theta_guess, args=(x, y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 62.478260\n",
        "         Iterations: 110\n",
        "         Function evaluations: 205\n",
        "Optimization terminated successfully.\n",
        "         Current function value: 150.032670\n",
        "         Iterations: 83\n",
        "         Function evaluations: 156\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xfit = np.linspace(0, 10)\n",
      "\n",
      "plt.plot(x, y, 'o')\n",
      "plt.plot(xfit, model(theta_LAD, xfit), label='Least Abs. Dev.')\n",
      "plt.plot(xfit, model(theta_LS, xfit), label='Least Sq.')\n",
      "plt.legend(loc='best');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GgeIjrD23mTp3PT0t3UmMewzjhu2UHl8FBgPRU58ievJT6M1mf5cLwKsr0jmT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\n+py8zMKTdRjdGiHKQOJSF2GO7+LvEoU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       "text": [
        "<matplotlib.figure.Figure at 0x109bb4250>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The results of both cost functions are similar! But this doesn't mean they're equal.\n",
      "In particular, you may have noticed that **least squares** is used much more often than **least absolute deviation**. There are several reasons for this, but we'll see below that part of it is the equivalence with the likelihood."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Motivating our Cost Function: the Likelihood"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "One way we can motivate the choice of cost function is to think of the problem probabilistically."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Likelihood of a single point\n",
      "\n",
      "Consider just a single point in our data, $(x, y)$. What are we **actually** trying to figure out?\n",
      "\n",
      "Well, one way of looking at it is this: we know that for each $x$, there exists some \"true\" $y$ which we'll call $y_{true}$. But in our observing process, we have some probability distribution which describes how **likely** we are to observe a particular $y$ given the true $y$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ydiff = np.linspace(-4, 4, 1000)\n",
      "plt.plot(ydiff, stats.norm.pdf(ydiff, 0, 1))\n",
      "plt.xlabel('$y - y_{true}$', size=18)\n",
      "plt.ylabel('$p(y|y_{true})$', size=18);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MZYzxAl8A1gNjwBPW2uYpxx8F/hBwgG9aa/92unNEkt3k6PgtKwIuJ0lsm1cEaGrto66p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\nzX/61a3cu3UR7ZeH+YtvHOSfX2xgbDz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       "text": [
        "<matplotlib.figure.Figure at 0x109e39950>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Most commonly, we'll use a **normal (Gaussian) distribution** for this error.\n",
      "This can be justified in a number of ways (Central limit theorem, maximum entropy, etc.). So in effect, what we're saying is that $y$ is a random variable, which is distributed like\n",
      "\n",
      "$$\n",
      "y \\sim \\mathcal{N}[y_{true}, \\sigma^2]\n",
      "$$\n",
      "\n",
      "Given this (assumed) model for a single value of $y$, we can compute the probability of observing this point as follows:\n",
      "\n",
      "$$\n",
      "p(y|y_{true},\\sigma) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[\\frac{-(y - y_{true})^2}{2\\sigma^2}\\right]\n",
      "$$\n",
      "\n",
      "Here this should be read \"the probability of $y$ given $y_{true}$ and $\\sigma$ is...\"\n",
      "\n",
      "With this error distribution, we see that if our single observed point is far from the (unknown) true value, the probability is very low. If our single observed point is close to the true value, the probability is high."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### From Single Points to Two Points\n",
      "\n",
      "Above we've computed the likelihood for a single point: this is just a probability that we'd see this point given an (unknown) true value. How do we then combine this likelihood for multiple points?\n",
      "\n",
      "Essentially, we're asking **what is the joint probability of our points $y_1$ and $y_2$ given the true values $y_{1, true}$ and $y_{2, true}$?**\n",
      "In symbols, we want:\n",
      "\n",
      "$$\n",
      "p(y_1, y_2~|~I)\n",
      "$$\n",
      "\n",
      "where $I$ is shorthand for the information we're using (here the true values & errors).\n",
      "This is difficult to evaluate in general, but we can make one simplifying assumption that will change everything:\n",
      "\n",
      "**Assumption: $y_1$ is *independent* of $y_2$.**\n",
      "\n",
      "By the definition of independence, we can write\n",
      "\n",
      "$$\n",
      "p(y_1, y_2~|~I) = p(y_1~|~I) p(y_2~|~I)\n",
      "$$\n",
      "\n",
      "That is, independence lets us consider each measurement individually, and then multiply their probabilities together."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### From Two Points to $N$ Points\n",
      "\n",
      "You might see where this is going: if every single draw from our model is *independent* (that is, the random error in one point does not affect the random error in any other point), then we can compute the likelihood of the entire dataset very easily!\n",
      "\n",
      "$$\n",
      "p(\\{y_i\\}~|~I) = \\prod_{i=1}^N p(y_i~|~y_{i, true},\\sigma)\n",
      "$$\n",
      "\n",
      "This product of probabilities is known as the **Likelihood**. If we take the logarithm of this expression, the product turns into a sum, and we have the **log-Likelihood**, which is often denoted as follows:\n",
      "\n",
      "$$\n",
      "\\log L(\\{y_i\\}~|~I) = \\sum_{i=1}^N \\log p(y_i~|~y_{i, true},\\sigma)\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### So What?\n",
      "\n",
      "What we've created here is an expression, the **likelihood**, which depends on the data and our model for the true data.\n",
      "\n",
      "Above we've been talking about the \"true\" $y$.\n",
      "The problem is that we never actually know the true y! The true y (i.e. the best-fit model) is what we're trying to find.\n",
      "Given this formalism, it can be motivated that **the model which maximizes this likelihood is the model which best fits the data**. What this is saying is, essentially, the model which gives our observed data the highest probability is the most likely model.\n",
      "\n",
      "So: we need to maximize our likelihood (or equivalently, maximize our log-likelihood) by plugging-in the expression:\n",
      "\n",
      "$$\n",
      "y_{i, true} = m x_i + b\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Maximizing our Likelihood\n",
      "\n",
      "To do this, we'll start by plugging our Gaussian model into the log-likelihood. Doing this, we get the following:\n",
      "\n",
      "$$\n",
      "\\log L(\\{x_i, y_i\\}~|~\\theta, \\sigma) = \\frac{-N}{2}\\log(2\\pi\\sigma^2) - \\sum_{i=1}^N [(y_i - M(x_i;\\theta))]^2\n",
      "$$\n",
      "\n",
      "OK... now we notice something. The second term here is **exactly** the least-squares cost function we proposed before!\n",
      "\n",
      "$$\n",
      "\\log L(\\{x_i, y_i\\}~|~\\theta, \\sigma) = \\frac{-N}{2}\\log(2\\pi\\sigma^2) - C_{LS}(\\{x_i, y_i\\}~|~\\theta)\n",
      "$$\n",
      "\n",
      "And notice something else: the first term is basically a constant, which means that *maximizing the likelihood is **identical** to minimizing the sum of squares!*\n",
      "\n",
      "And this is one reason that least-squares fitting is so common: it follows from several assumptions:\n",
      "\n",
      "1. The error distribution of the observed points is Gaussian.\n",
      "2. The errors in observed points are independently drawn.\n",
      "3. The model which is most likely to have led to our data is the optimal model.\n",
      "\n",
      "**Question: what probability distribution for the data errors would lead to the least absolute deviation cost function?**\n",
      "\n",
      "**Exercise: use the maximum likelihood to derive the expression for determining the mean of a set of data points**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Closed-form Solution for Least Squares\n",
      "\n",
      "Above we've been using an iterative solver for our least squares/maximum likelihood solution, but there is a better way! It turns out that using matrix methods, the least squares / Gaussian maximum likelihood problem can be solved in closed-form.\n",
      "\n",
      "To do this, let's start by expressing our model as a matrix equation. We'll define the following two matrices:\n",
      "\n",
      "$$\n",
      "X = \\left[\n",
      "\\begin{array}{ll}\n",
      "  1 & x_1\\\\\n",
      "  1 & x_2\\\\\n",
      "  1 & x_3\\\\\n",
      "  \\vdots & \\vdots\\\\\n",
      "  1 & x_N\n",
      "\\end{array}\n",
      "\\right]~~~~\n",
      "y = \\left[\n",
      "\\begin{array}{l}\n",
      "  y_1\\\\\n",
      "  y_2\\\\\n",
      "  y_3\\\\\n",
      "  \\vdots\\\\\n",
      "  y_N\n",
      "\\end{array}\n",
      "\\right]\n",
      "$$\n",
      "\n",
      "The $X$ matrix here is sometimes known as the *design matrix*. This might look a bit strange, but if we now define the parameter vector $\\theta = [b, m]$, we see that the model can simply be expressed as a simple matrix-vector product:\n",
      "\n",
      "$$\n",
      "y_{model} = X\\theta\n",
      "$$\n",
      "\n",
      "we can write the likelihood as follows:\n",
      "\n",
      "$$\n",
      "L(x, y, dy~|~\\theta) = (2\\pi\\sigma^2)^{-1/2}\\exp\\left(-\\frac{1}{2\\sigma^2}\\|y - X\\theta\\|^2\\right)\n",
      "$$\n",
      "\n",
      "minimizing this with respect to $\\theta$ (by setting the derivative with respect to $\\theta$ to zero) gives:\n",
      "\n",
      "$$\n",
      "X^T(y - X\\hat{\\theta}) = 0\n",
      "$$\n",
      "\n",
      "which yields the maximum likelihood solution:\n",
      "\n",
      "$$\n",
      "\\hat{\\theta} = [X^TX]^{-1}X^Ty\n",
      "$$\n",
      "\n",
      "Let's give this a try:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.vstack([np.ones_like(x), x]).T\n",
      "\n",
      "theta_hat = np.linalg.solve(np.dot(X.T, X),\n",
      "                            np.dot(X.T, y))\n",
      "print(theta_hat)\n",
      "print(theta_LS)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-2.85195862  3.16960077]\n",
        "[-2.8519853   3.16960679]\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that the result is the same as our previous least squares solution... it's just **much** faster to compute!\n",
      "For good measure, let's again visualize this fit:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xfit = np.linspace(0, 10)\n",
      "\n",
      "plt.plot(x, y, 'o')\n",
      "plt.plot(xfit, model(theta_hat, xfit), label='Least Sq.')\n",
      "plt.legend(loc='best');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dCmNjTClwa9vXe4DbgliTiISh4qpDLDWFlDecpm+sh29kz+XGgRM0QUukG2hM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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ad72ed0>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Model Fitting with Errors\n",
      "\n",
      "The maximum likelihood formalism above gives us a natural means of including measurement errors in the dependent variables (i.e. $y$). We simply change the width of each individual Gaussian to reflect the (known) error in our data points!\n",
      "\n",
      "That is, we use\n",
      "\n",
      "$$\n",
      "\\log L(\\{y_i\\}~|~I) = \\sum_{i=1}^N \\log p(y_i~|~y_{i, true},\\sigma_i)\n",
      "$$\n",
      "\n",
      "note the $\\sigma_i$ at the end of the expression. From this, we can derive the weighted least-squares cost function:\n",
      "\n",
      "$$\n",
      "C_{LS}(\\{x_i, y_i, \\sigma_i\\}~|~\\theta) = \\sum_i \\frac{[y_i - M(x_i;\\theta)]^2}{2\\sigma_i^2}\n",
      "$$\n",
      "\n",
      "So already we've gained some insight about how to extend least-squares recipe into another regime!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But let's skip straight toward the closed form here. Based on some arguments we won't cover here, we'll start by defining an **error covariance matrix**:\n",
      "\n",
      "$$\n",
      "\\Sigma = \\left[\n",
      "\\begin{array}{lllll}\n",
      "\\sigma_1^2 & 0 & 0 & \\cdots & 0\\\\\n",
      "0 & \\sigma_2^2 & 0 & \\cdots & 0\\\\\n",
      "0 & 0 & \\sigma_3^2 & \\cdots & 0\\\\\n",
      "\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
      "0 & 0 & 0 & \\cdots & \\sigma_N^2\n",
      "\\end{array}\n",
      "\\right]\n",
      "$$\n",
      "\n",
      "we can write the likelihood as follows:\n",
      "\n",
      "$$\n",
      "L(x, y, dy~|~\\theta) = (2\\pi|\\Sigma|)^{-1/2}\\exp\\left[-\\frac{1}{2}(y - X\\theta)^T\\Sigma^{-1}(y - X\\theta)\\right]\n",
      "$$\n",
      "\n",
      "again minimizing this with respect to $\\theta$ gives:\n",
      "\n",
      "$$\n",
      "X^T\\Sigma^{-1}(y - X\\hat{\\theta}) = 0\n",
      "$$\n",
      "\n",
      "which yields the maximum likelihood solution:\n",
      "\n",
      "$$\n",
      "\\hat{\\theta} = [X^T\\Sigma^{-1}X]^{-1}X^T\\Sigma^{-1}y\n",
      "$$\n",
      "\n",
      "So this is very similar to our previous closed-form solution, except with some diagonal error matrices thrown in. Let's compute this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "XT_Sigma_X = np.dot(X.T / dy ** 2, X)\n",
      "XT_Sigma_y = np.dot(X.T / dy ** 2, y)\n",
      "theta_LS_err = np.linalg.solve(XT_Sigma_X, XT_Sigma_y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's use this with our full data + errors to find a fit to the data:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.errorbar(x, y, dy, fmt='o')\n",
      "plt.plot(xfit, model(theta_LAD, xfit), label='Least Abs. Dev.')\n",
      "plt.plot(xfit, model(theta_LS, xfit), label='Least Sq.')\n",
      "plt.plot(xfit, model(theta_LS_err, xfit), label='Fit w/ errors')\n",
      "plt.legend(loc='best');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Z8Zex75+70A0mNgf0QzF5GZJkYcLMSQQFW/1dsmgHCWMhhOjGvLqXbad7C5c5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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109fbbad0>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "All three results are very similar.\n",
      "\n",
      "But what if some of our data points have very large errors?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def linear_data_sample_big_errs(N=40, rseed=0, m=3, b=-2):\n",
      "    \"\"\"Generate data with some large errors\"\"\"\n",
      "    rng = np.random.RandomState(rseed)\n",
      "\n",
      "    x = 10 * rng.rand(N)\n",
      "    dy = m / 2 * (1 + rng.rand(N))\n",
      "    dy[20:25] *= 10\n",
      "    y = m * x + b + dy * rng.randn(N)\n",
      "\n",
      "    return (x, y, dy)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y, dy = linear_data_sample_big_errs()\n",
      "plt.errorbar(x, y, dy, fmt='o');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a98dc90>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's redo the fit now with and without the errorbars:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta_LAD = fmin(abs_deviation, theta_guess, args=(x, y))\n",
      "\n",
      "X = np.vstack([np.ones_like(x), x]).T\n",
      "\n",
      "theta_LS = np.linalg.solve(np.dot(X.T, X),\n",
      "                           np.dot(X.T, y))\n",
      "\n",
      "theta_LS_err = np.linalg.solve(np.dot(X.T / dy ** 2, X),\n",
      "                               np.dot(X.T / dy ** 2, y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 173.046330\n",
        "         Iterations: 120\n",
        "         Function evaluations: 220\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.errorbar(x, y, dy, fmt='o')\n",
      "plt.ylim(-20, 40)\n",
      "plt.plot(xfit, model(theta_LAD, xfit), label='Least Abs. Dev.')\n",
      "plt.plot(xfit, model(theta_LS, xfit), label='Least Sq.')\n",
      "plt.plot(xfit, model(theta_LS_err, xfit), label='Least Sq. w/ errors')\n",
      "plt.legend(loc='best');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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mY1zfgMJiNXKouZwfD0yyYlR46uG9m9aezbLeGelCiI1bd0Crqvqf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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ad5c3d0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice something interesting: the errors have a much larger effect on the least squares solution than on the Least Absolute Deviation solution! This is reflective of the well-known fact that the LAD solution is more robust to outliers than is the least squares solution."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Going Further: Generalized Linear Models\n",
      "\n",
      "One of the powerful applications of the above formalism is its use in **generalized linear models**. Essentially, we can modify the $X$ matrix at will to quickly find closed-form solutions to a large number of problems.  For example, we could fit a third-order polynomial model such as the following:\n",
      "\n",
      "$$\n",
      "y_{model} = \\theta_0 + \\theta_1 x + \\theta_2 x^2 + \\theta_3 x^2\n",
      "$$\n",
      "\n",
      "All that takes is to compute the above solution, but use the following $X$ matrix:\n",
      "\n",
      "$$\n",
      "X = \\left[\n",
      "\\begin{array}{llll}\n",
      "  1 & x_1 & x_1^2 & x_1^3\\\\\n",
      "  1 & x_2 & x_2^2 & x_2^3\\\\\n",
      "  1 & x_3 & x_3^2 & x_3^3\\\\\n",
      "  \\vdots & \\vdots & \\vdots & \\vdots\\\\\n",
      "  1 & x_N & x_N^2 & x_N^3\n",
      "\\end{array}\n",
      "\\right]\n",
      "$$\n",
      "\n",
      "We could, of course, add any number of columns to make the polynomial as high as we'd like.\n",
      "\n",
      "Similarly, we could fit a periodic model to data such as this one:\n",
      "\n",
      "$$\n",
      "y_{model} = \\theta_0 + \\theta_1\\sin(\\omega x) + \\theta_2\\cos(\\omega x)\n",
      "$$\n",
      "\n",
      "For this, we use the following $X$ matrix:\n",
      "\n",
      "$$\n",
      "X = \\left[\n",
      "\\begin{array}{lll}\n",
      "  1 & \\sin(\\omega x_1) & \\cos(\\omega x_1)\\\\\n",
      "  1 & \\sin(\\omega x_2) & \\cos(\\omega x_2)\\\\\n",
      "  1 & \\sin(\\omega x_3) & \\cos(\\omega x_3)\\\\\n",
      "  \\vdots & \\vdots & \\vdots \\\\\n",
      "  1 & \\sin(\\omega x_N) & \\cos(\\omega x_N)\\\\\n",
      "\\end{array}\n",
      "\\right]\n",
      "$$\n",
      "\n",
      "Note that for the model to be linear, we must **specify** $\\omega$. We could use the maximum likelihood approach to fit for $\\omega$ itself, but it will no longer be a linear model."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the breakout session next, we'll explore the construction of some of these generalized linear models."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}