{ "metadata": { "name": "", "signature": "sha256:d132a55f4fd06bd9bcfd3fce4e6a16df11818a010d327f7ad0a80eff9c24a96d" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Inroduction to Gaussian Processes\n", "\n", "## Gaussian Process Road Show, Genoa, Italy\n", "### 19th or 20th January 2015\n", "### Neil D. Lawrence\n", "\n", "When we form a Gaussian process we assume data is *jointly Gaussian* with a particular mean and covariance,\n", "$$\n", "\\mathbf{y}|\\mathbf{X} \\sim \\mathcal{N}(\\mathbf{m}(\\mathbf{X}), \\mathbf{K}(\\mathbf{X})),\n", "$$\n", "where the conditioning is on the inputs $\\mathbf{X}$ which are used for computing the mean and covariance. For this reason they are known as mean and covariance functions.\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import numpy as np\n", "import scipy as sp\n", "import matplotlib.pyplot as plt\n", "import pods" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 86 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Marginal Likelihood\n", "\n", "To understand the Gaussian process we're going to build on our understanding of the marginal likelihood for Bayesian regression. In the session on [Bayesian regression](./week7.ipynb) we sampled directly from the weight vector, $\\mathbf{w}$ and applied it to the basis matrix $\\boldsymbol{\\Phi}$ to obtain a sample from the prior and a sample from the posterior. It is often helpful to think of modeling techniques as *generative* models. To give some thought as to what the process for obtaining data from the model is. From the perspective of Gaussian processes, we want to start by thinking of basis function models, where the parameters are sampled from a prior, but move to thinking about sampling from the marginal likelihood directly.\n", "\n", "## Sampling from the Prior\n", "\n", "The first thing we'll do is to set up the parameters of the model, these include the parameters of the prior, the parameters of the basis functions and the noise level. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "# set prior variance on w\n", "alpha = 4.\n", "# set the order of the polynomial basis set\n", "degree = 5\n", "# set the noise variance\n", "sigma2 = 0.01" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 87 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we have the variance, we can sample from the prior distribution to see what form we are imposing on the functions *a priori*. \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now compute a range of values to make predictions at, spanning the *new* space of inputs," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def polynomial(x, degree, loc, scale):\n", " degrees = np.arange(degree+1)\n", " return ((x-loc)/scale)**degrees" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 88 }, { "cell_type": "markdown", "metadata": {}, "source": [ "now let's build the basis matrices. First we load in the data\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "data = pods.datasets.olympic_marathon_men()\n", "x = data['X']\n", "y = data['Y']" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 89 }, { "cell_type": "code", "collapsed": false, "input": [ "scale = np.max(x) - np.min(x)\n", "loc = np.min(x) + 0.5*scale\n", "\n", "num_data = x.shape[0]\n", "num_pred_data = 100 # how many points to use for plotting predictions\n", "x_pred = np.linspace(1880, 2030, num_pred_data)[:, None] # input locations for predictions\n", "Phi_pred = polynomial(x_pred, degree=degree, loc=loc, scale=scale)\n", "Phi = polynomial(x, degree=degree, loc=loc, scale=scale)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 90 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Weight Space View\n", "\n", "To generate typical functional predictions from the model, we need a set of model parameters. We assume that the parameters are drawn independently from a Gaussian density,\n", "$$\n", "\\mathbf{w} \\sim \\mathcal{N}(\\mathbf{0}, \\alpha\\mathbf{I}),\n", "$$\n", "then we can combine this with the definition of our prediction function $f(\\mathbf{x})$,\n", "$$\n", "f(\\mathbf{x}) = \\mathbf{w}^\\top \\boldsymbol{\\phi}(\\mathbf{x}).\n", "$$\n", "We can now sample from the prior density to obtain a vector $\\mathbf{w}$ using the function `np.random.normal` and combine these parameters with our basis to create some samples of what $f(\\mathbf{x})$ looks like," ] }, { "cell_type": "code", "collapsed": false, "input": [ "num_samples = 10\n", "K = degree+1\n", "for i in xrange(num_samples):\n", " z_vec = np.random.normal(size=(K, 1))\n", " w_sample = z_vec*np.sqrt(alpha)\n", " f_sample = np.dot(Phi_pred,w_sample)\n", " plt.plot(x_pred, f_sample)\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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AicFAidFIhkazKPJfhaNhJgITjPvHmQxMMhmYxBlwUm4tZ0Pxhll9Lyn0Esnl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\naWnJSSoh7InJSz3Nv+12w+uvw4svwiu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"text": [ "" ] } ], "prompt_number": 91 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Function Space View\n", "\n", "The process we have used to generate the samples is a two stage process. To obtain each function, we first generated a sample from the prior,\n", "$$\n", "\\mathbf{w} \\sim \\mathcal{N}(\\mathbf{0}, \\alpha \\mathbf{I})\n", "$$\n", "then if we compose our basis matrix, $\\boldsymbol{\\Phi}$ from the basis functions associated with each row then we get,\n", "$$\n", "\\mathbf{\\Phi} = \\begin{bmatrix}\\boldsymbol{\\phi}(\\mathbf{x}_1) \\\\ \\vdots \\\\ \\boldsymbol{\\phi}(\\mathbf{x}_n)\\end{bmatrix}\n", "$$\n", "then we can write down the vector of function values, as evaluated at\n", "$$\n", "\\mathbf{f} = \\begin{bmatrix} f_1 \\\\ \\vdots \\\\ f_n\\end{bmatrix}\n", "$$\n", "in the form\n", "$$\n", "\\mathbf{f} = \\boldsymbol{\\Phi} \\mathbf{w}.\n", "$$\n", "\n", "Now we can use standard properties of multivariate Gaussians to write down the probability density that is implied over $\\mathbf{f}$. In particular we know that if $\\mathbf{w}$ is sampled from a multivariate normal (or multivariate Gaussian) with covariance $\\alpha \\mathbf{I}$ and zero mean, then assuming that $\\boldsymbol{\\Phi}$ is a deterministic matrix (i.e. it is not sampled from a probability density) then the vector $\\mathbf{f}$ will also be distributed according to a zero mean multivariate normal as follows,\n", "$$\n", "\\mathbf{f} \\sim \\mathcal{N}(\\mathbf{0},\\alpha \\boldsymbol{\\Phi} \\boldsymbol{\\Phi}^\\top).\n", "$$\n", "\n", "The question now is, what happens if we sample $\\mathbf{f}$ directly from this density, rather than first sampling $\\mathbf{w}$ and then multiplying by $\\boldsymbol{\\Phi}$. Let's try this. First of all we define the covariance as\n", "$$\n", "\\mathbf{K} = \\alpha \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top.\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = alpha*np.dot(Phi_pred, Phi_pred.T)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 92 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can use the `np.random.multivariate_normal` command for sampling from a multivariate normal with covariance given by $\\mathbf{K}$ and zero mean," ] }, { "cell_type": "code", "collapsed": false, "input": [ "for i in np.arange(10):\n", " f_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), f_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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5H2aY/2GG46gDGWsyYLzZCNPNJqRVpH3Ad8FJKhij6Je33iLbu5ei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\nCmeXKhCk0yXOlUNKCP2Q3483LBa8brHgTYsFOSo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"text": [ "" ] } ], "prompt_number": 93 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The samples appear very similar to those which we obtained indirectly. That is no surprise because they are effectively drawn from the same mutivariate normal density. However, when sampling $\\mathbf{f}$ directly we created the covariance for $\\mathbf{f}$. We can visualise the form of this covaraince in an image in python with a colorbar to show scale." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(K, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 94, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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M7BuSHpT0qthgTIQAgKK4+02SnhN8+lDQ59LU8fIvjWqyNNokJaLJziqpO8s02ZGl7RJr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\nFmtNpKZaDFzaqWSriO31dfy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"text": [ "" ] } ], "prompt_number": 94 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This image is the covariance expressed between different points on the function. In regression we normally also add independent Gaussian noise to obtain our observations $\\mathbf{y}$,\n", "$$\n", "\\mathbf{y} = \\mathbf{f} + \\boldsymbol{\\epsilon}\n", "$$\n", "where the noise is sampled from an independent Gaussian distribution with variance $\\sigma^2$,\n", "$$\n", "\\epsilon \\sim \\mathcal{N}(\\mathbf{0}, \\sigma^2 \\mathbf{I}).\n", "$$\n", "we can use properties of Gaussian variables, i.e. the fact that sum of two Gaussian variables is also Gaussian, and that it's covariance is given by the sum of the two covariances, whilst the mean is given by the sum of the means, to write down the marginal likelihood,\n", "$$\n", "\\mathbf{y} \\sim \\mathcal{N}(\\mathbf{0}, \\alpha \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top + \\sigma^2\\mathbf{I}).\n", "$$\n", "Sampling directly from this density gives us the noise corrupted functions," ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = alpha*np.dot(Phi_pred, Phi_pred.T) + sigma2*np.eye(x_pred.size)\n", "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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dkXYvrWi7LF2GwC8D8bjRY9g2tYXne54ImB0AOx07BC8JVpltVCWKHEWVmf7e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"text": [ "" ] } ], "prompt_number": 95 }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the effect of our noise term is to roughen the sampled functions, we can also increase the variance of the noise to see a different effect," ] }, { "cell_type": "code", "collapsed": false, "input": [ "sigma2 = 1.\n", "K = alpha*np.dot(Phi_pred, Phi_pred.T) + sigma2*np.eye(x_pred.size)\n", "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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k5lo+36G3P45EfPGNru12O6tXr8ZqHUAQtAiCGo0mj+7uTqqrqzn33HPZt28f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\nZfw6gPAL2GCDDTZcDAYO2CYIBCIrIJQ2aLVa7c7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gV0ZQfSzEL9+tDDBULOL/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\n0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j+E2WwDVcVRLGI/cIBXpNN8fGGB+2ae5pboVR3pMGJ1U8HCoWIRnXmVrG6SMf8YlAw0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\nVmv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"text": [ "" ] } ], "prompt_number": 96 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Function Space Reflection\n", "\n", "How do you include the noise term when sampling in the weight space point of view?\n", "\n", "## Gaussian Process\n", "\n", "In our [session on Bayesian regression](./bayesian approach to regression.ipynb) we sampled from the prior over parameters. Through the properties of multivariate Gaussian densities this prior over parameters implies a particular density for our data observations, $\\mathbf{y}$. In this session we sampled directly from this distribution for our data, avoiding the intermediate weight-space representation. This is the approach taken by *Gaussian processes*. In a Gaussian process you specify the *covariance function* directly, rather than *implicitly* through a basis matrix and a prior over parameters. Gaussian processes have the advantage that they can be *nonparametric*, which in simple terms means that they can have *infinite* basis functions. In the lectures we introduced the *exponentiated quadratic* covariance, also known as the RBF or the Gaussian or the squared exponential covariance function. This covariance function is specified by\n", "$$\n", "k(\\mathbf{x}, \\mathbf{x}^\\prime) = \\alpha \\exp\\left( -\\frac{\\left\\Vert \\mathbf{x}-\\mathbf{x}^\\prime\\right\\Vert^2}{2\\ell^2}\\right).\n", "$$\n", "where $\\left\\Vert\\mathbf{x} - \\mathbf{x}^\\prime\\right\\Vert^2$ is the squared distance between the two input vectors \n", "$$\n", "\\left\\Vert\\mathbf{x} - \\mathbf{x}^\\prime\\right\\Vert^2 = (\\mathbf{x} - \\mathbf{x}^\\prime)^\\top (\\mathbf{x} - \\mathbf{x}^\\prime) \n", "$$\n", "Let's build a covariance matrix based on this function. First we define the form of the covariance function," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def exponentiated_quadratic(x, x_prime, variance, lengthscale):\n", " squared_distance = ((x-x_prime)**2).sum()\n", " return variance*np.exp((-0.5*squared_distance)/lengthscale**2)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 97 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use this to compute *directly* the covariance for $\\mathbf{f}$ at the points given by `x_pred`. Let's define a new function `K()` which does this," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def compute_kernel(X, X2, kernel, **kwargs):\n", " K = np.zeros((X.shape[0], X2.shape[0]))\n", " for i in np.arange(X.shape[0]):\n", " for j in np.arange(X2.shape[0]):\n", " K[i, j] = kernel(X[i, :], X2[j, :], **kwargs)\n", " \n", " return K" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 98 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can image the resulting covariance," ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = compute_kernel(x_pred, x_pred, exponentiated_quadratic, variance=1., lengthscale=10.)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 99 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To visualise the covariance between the points we can use the `imshow` function in matplotlib." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(K, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 100, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 100 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can sample functions from the marginal likelihood." ] }, { "cell_type": "code", "collapsed": false, "input": [ "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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U/MmTxh23sJC1CIyIYDXn9SE3N5c6duxI4eHh1Ebfi3QhlRL5+LDMrrg4opYtWZaWoUkA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\nG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"text": [ "" ] } ], "prompt_number": 101 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise: Moving Parameters\n", "\n", "Have a play with the parameters for this covariance function (the lengthscale and the variance) and see what effects the parameters have on the types of functions you observe." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 101 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gaussian Process\n", "\n", "The Gaussian process perspective takes the marginal likelihood of the data to be a joint Gaussian density with a covariance given by $\\mathbf{K}$. So the model likelihood is of the form,\n", "$$\n", "p(\\mathbf{y}|\\mathbf{X}) = \\frac{1}{(2\\pi)^{\\frac{n}{2}}|\\mathbf{K}|^{\\frac{1}{2}}} \\exp\\left(-\\frac{1}{2}\\mathbf{y}^\\top \\left(\\mathbf{K}+\\sigma^2 \\mathbf{I}\\right)^{-1}\\mathbf{y}\\right)\n", "$$\n", "where the input data, $\\mathbf{X}$, influences the density through the covariance matrix, $\\mathbf{K}$ whose elements are computed through the covariance function, $k(\\mathbf{x}, \\mathbf{x}^\\prime)$.\n", "\n", "This means that the negative log likelihood (the objective function) is given by,\n", "$$\n", "E(\\boldsymbol{\\theta}) = \\frac{1}{2} \\log |\\mathbf{K}| + \\frac{1}{2} \\mathbf{y}^\\top \\left(\\mathbf{K} + \\sigma^2\\mathbf{I}\\right)^{-1}\\mathbf{y}\n", "$$\n", "where the *parameters* of the model are also embedded in the covariance function, they include the parameters of the kernel (such as lengthscale and variance), and the noise variance, $\\sigma^2$.\n", "\n", "Let's create a class in python for storing these variables." ] }, { "cell_type": "code", "collapsed": false, "input": [ "class GP():\n", " def __init__(self, X, y, sigma2, kernel, **kwargs):\n", " self.K = compute_kernel(X, X, kernel, **kwargs)\n", " self.X = X\n", " self.y = y\n", " self.sigma2 = sigma2\n", " self.kernel = kernel\n", " self.kernel_args = kwargs\n", " self.update_inverse()\n", " \n", " def update_inverse(self):\n", " # Preompute the inverse covariance and some quantities of interest\n", " ## NOTE: This is not the correct *numerical* way to compute this! It is for ease of use.\n", " self.Kinv = np.linalg.inv(self.K+self.sigma2*np.eye(self.K.shape[0]))\n", " # the log determinant of the covariance matrix.\n", " self.logdetK = np.linalg.det(self.K+self.sigma2*np.eye(self.K.shape[0]))\n", " # The matrix inner product of the inverse covariance\n", " self.Kinvy = np.dot(self.Kinv, self.y)\n", " self.yKinvy = (self.y*self.Kinvy).sum()\n", "\n", " \n", " def log_likelihood(self):\n", " # use the pre-computes to return the likelihood\n", " return -0.5*(self.K.shape[0]*np.log(2*np.pi) + self.logdetK + self.yKinvy)\n", " \n", " def objective(self):\n", " # use the pre-computes to return the objective function \n", " return -self.log_likelihood()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 102 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Making Predictions\n", "\n", "We now have a probability density that represents functions. How do we make predictions with this density? The density is known as a process because it is *consistent*. By consistency, here, we mean that the model makes predictions for $\\mathbf{f}$ that are unaffected by future values of $\\mathbf{f}^*$ that are currently unobserved (such as test points). If we think of $\\mathbf{f}^*$ as test points, we can still write down a joint probability density over the training observations, $\\mathbf{f}$ and the test observations, $\\mathbf{f}^*$. This joint probability density will be Gaussian, with a covariance matrix given by our covariance function, $k(\\mathbf{x}_i, \\mathbf{x}_j)$. \n", "$$\n", "\\begin{bmatrix}\\mathbf{f} \\\\ \\mathbf{f}^*\\end{bmatrix} \\sim \\mathcal{N}\\left(\\mathbf{0}, \\begin{bmatrix} \\mathbf{K} & \\mathbf{K}_\\ast \\\\ \\mathbf{K}_\\ast^\\top & \\mathbf{K}_{\\ast,\\ast}\\end{bmatrix}\\right)\n", "$$\n", "where here $\\mathbf{K}$ is the covariance computed between all the training points, $\\mathbf{K}_\\ast$ is the covariance matrix computed between the training points and the test points and $\\mathbf{K}_{\\ast,\\ast}$ is the covariance matrix computed betwen all the tests points and themselves. To be clear, let's compute these now for our example, using `x` and `y` for the training data (although `y` doesn't enter the covariance) and `x_pred` as the test locations." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# set covariance function parameters\n", "variance = 16.0\n", "lengthscale = 32\n", "# set noise variance\n", "sigma2 = 0.05\n", "\n", "K = compute_kernel(x, x, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "K_star = compute_kernel(x, x_pred, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "K_starstar = compute_kernel(x_pred, x_pred, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 103 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we use this structure to visualise the covariance between test data and training data. This structure is how information is passed between trest and training data. Unlike the maximum likelihood formalisms we've been considering so far, the structure expresses *correlation* between our different data points. However, we now have a *joint density* between some variables of interest. In particular we have the joint density over $p(\\mathbf{f}, \\mathbf{f}^*)$. The joint density is *Gaussian* and *zero mean*. It is specified entirely by the *covariance matrix*, $\\mathbf{K}$. That covariance matrix is, in turn, defined by a covariance function. Now we will visualise the form of that covariance in the form of the matrix,\n", "$$\n", "\\begin{bmatrix} \\mathbf{K} & \\mathbf{K}_\\ast \\\\ \\mathbf{K}_\\ast^\\top & \\mathbf{K}_{\\ast,\\ast}\\end{bmatrix}\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(np.vstack([np.hstack([K, K_star]), np.hstack([K_star.T, K_starstar])]), interpolation='none')\n", "# Add lines for separating training and test data\n", "ax.axvline(x.shape[0]-1, color='w')\n", "ax.axhline(x.shape[0]-1, color='w')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 104, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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QZKpuF0IIISqNmd1jZk0z+ybN/Qcze8zMvmFmnzKzQveQFkQhhBDlcUUJ/9Zz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\nAZPZZHiUu2TUj1yF/UduAQC8Drfh//nQvf26tPW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"text": [ "" ] } ], "prompt_number": 104 }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are four blocks to this color plot. The upper left block is the covariance of the training data with itself, $\\mathbf{K}$. We see some structure here due to the missing data from the first and second world wars. Alongside this covariance (to the right and below) we see the cross covariance between the training and the test data ($\\mathbf{K}_*$ and $\\mathbf{K}_*^\\top$). This is giving us the covariation between our training and our test data. Finally the lower right block The banded structure we now observe is because some of the training points are near to some of the test points. This is how we obtain 'communication' between our training data and our test data. If there is no structure in $\\mathbf{K}_*$ then our belief about the test data simply matches our prior.\n", "\n", "## Conditional Density\n", "\n", "Just as in naive Bayes, we first defined the joint density (although there it was over both the labels and the inputs, $p(\\mathbf{y}, \\mathbf{X})$ and now we need to define *conditional* distributions that answer particular questions of interest. In particular we might be interested in finding out the values of the function for the prediction function at the test data given those at the training data, $p(\\mathbf{f}_*|\\mathbf{f})$. Or if we include noise in the training observations then we are interested in the conditional density for the prediction function at the test locations given the training observations, $p(\\mathbf{f}^*|\\mathbf{y})$. \n", "\n", "As ever all the various questions we could ask about this density can be answered using the *sum rule* and the *product rule*. For the multivariate normal density the mathematics involved is that of *linear algebra*, with a particular emphasis on the *partitioned inverse* or [*block matrix inverse*](http://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion), but they are beyond the scope of this course, so you don't need to worry about remembering them or rederiving them. We are simply writing them here because it is this *conditional* density that is necessary for making predictions.\n", "\n", "The conditional density is also a multivariate normal,\n", "$$\n", "\\mathbf{f}^* | \\mathbf{y} \\sim \\mathcal{N}(\\boldsymbol{\\mu}_f,\\mathbf{C}_f)\n", "$$\n", "with a mean given by\n", "$$\n", "\\boldsymbol{\\mu}_f = \\mathbf{K}_*^\\top \\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1} \\mathbf{y}\n", "$$\n", "and a covariance given by \n", "$$\n", "\\mathbf{C}_f = \\mathbf{K}_{*,*} - \\mathbf{K}_*^\\top \\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1} \\mathbf{K}_\\ast.\n", "$$\n", "Let's compute what those posterior predictions are for the olympic marathon data." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def posterior_f(self, X_test):\n", " K_star = compute_kernel(self.X, X_test, self.kernel, **self.kernel_args)\n", " A = np.dot(self.Kinv, K_star)\n", " mu_f = np.dot(A.T, y)\n", " C_f = K_starstar - np.dot(A.T, K_star)\n", " return mu_f, C_f\n", "\n", "# attach the new method to class GP():\n", "GP.posterior_f = posterior_f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 105 }, { "cell_type": "code", "collapsed": false, "input": [ "model = GP(x, y, sigma2, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "mu_f, C_f = model.posterior_f(x_pred)\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 106 }, { "cell_type": "markdown", "metadata": {}, "source": [ "where for convenience we've defined\n", "\n", "$$\\mathbf{A} = \\left[\\mathbf{K} + \\sigma^2\\mathbf{I}\\right]^{-1}\\mathbf{K}_*.$$ \n", "\n", "We can visualize the covariance of the *conditional*," ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(C_f, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 107, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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CAKxSgqkJzTwAACiQYJoPAOhcgqlJT5p51NHGKbQVR+pzl3XLNt1oazupdMmVYnxsWV3E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\nBAAYoBTHg0zvjAA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"text": [ "" ] } ], "prompt_number": 107 }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we can plot the mean of the conditional" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(x, y, 'rx')\n", "plt.plot(x_pred, mu_f, 'b-')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 108, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 108 }, { "cell_type": "markdown", "metadata": {}, "source": [ "as well as the associated error bars. These are given (similarly to the Bayesian parametric model from the last lab) by the standard deviations of the marginal posterior densities. The marginal posterior variances are given by the diagonal elements of the posterior covariance," ] }, { "cell_type": "code", "collapsed": false, "input": [ "var_f = np.diag(C_f)[:, None]\n", "std_f = np.sqrt(var_f)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 109 }, { "cell_type": "markdown", "metadata": {}, "source": [ "They can be added to the underlying mean function to give the error bars," ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(x, y, 'rx')\n", "plt.plot(x_pred, mu_f, 'b-')\n", "plt.plot(x_pred, mu_f+2*std_f, 'b--')\n", "plt.plot(x_pred, mu_f-2*std_f, 'b--')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 110, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 110 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This gives us a prediction from the Gaussian process. Remember machine learning is \n", "$$\n", "\\text{data} + \\text{model} \\rightarrow \\text{prediction}.\n", "$$\n", "Here our data is from the olympics, and our model is a Gaussian process with two parameters. The assumptions about the world are encoded entirely into our Gaussian process covariance. The GP covariance assumes that the function is highly smooth, and that correlation falls off with distance (scaled according to the length scale, $\\ell$). The model sustains the uncertainty about the function, this means we see an increase in the size of the error bars during periods like the 1st and 2nd World Wars when no olympic marathon was held. \n", "\n", "## Exercises\n", "\n", "Now try changing the parameters of the covariance function (and the noise) to see how the predictions change.\n", "\n", "Now try sampling from this conditional density to see what your predictions look like. What happens if you sample from the conditional density in regions a long way into the future or the past? How does this compare with the results from the polynomial model?\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Importance of the Covariance Function\n", "\n", "The covariance function encapsulates our assumptions about the data. The equations for the distribution of the prediction function, given the training observations, are highly sensitive to the covariation between the test locations and the training locations as expressed by the matrix $\\mathbf{K}_*$. We defined a matrix $\\mathbf{A}$ which allowed us to express our conditional mean in the form,\n", "$$\n", "\\boldsymbol{\\mu}_f = \\mathbf{A}^\\top \\mathbf{y},\n", "$$\n", "where $\\mathbf{y}$ were our *training observations*. In other words our mean predictions are always a linear weighted combination of our *training data*. The weights are given by computing the covariation between the training and the test data ($\\mathbf{K}_*$) and scaling it by the inverse covariance of the training data observations, $\\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1}$. This inverse is the main computational object that needs to be resolved for a Gaussian process. It has a computational burden which is $O(n^3)$ and a storage burden which is $O(n^2)$. This makes working with Gaussian processes computationally intensive for the situation where $n>10,000$. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo('ewJ3AxKclOg')" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", " \n", " " ], "metadata": {}, "output_type": "pyout", "prompt_number": 111, "text": [ "" ] } ], "prompt_number": 111 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Improving the Numerics\n", "\n", "In practice we shouldn't be using matrix inverse directly to solve the GP system. One more stable way is to compute the *Cholesky decomposition* of the kernel matrix. The log determinant of the covariance can also be derived from the Cholesky decomposition." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def update_inverse(self):\n", " # Perform Cholesky decomposition on matrix\n", " self.R = sp.linalg.cholesky(self.K + self.sigma2*self.K.shape[0])\n", " # compute the log determinant from Cholesky decomposition\n", " self.logdetK = 2*np.log(np.diag(self.R)).sum()\n", " # compute y^\\top K^{-1}y from Cholesky factor\n", " self.Rinvy = sp.linalg.solve_triangular(self.R, self.y)\n", " self.yKinvy = (self.Rinvy**2).sum()\n", " \n", " # compute the inverse of the upper triangular Cholesky factor\n", " self.Rinv = sp.linalg.solve_triangular(self.R, np.eye(self.K.shape[0]))\n", " self.Kinv = np.dot(self.Rinv, self.Rinv.T)\n", "\n", "GP.update_inverse = update_inverse" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 112 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Capacity Control\n", "\n", "Gaussian processes are sometimes seen as part of a wider family of methods known as kernel methods. Kernel methods are also based around covariance functions, but in the field they are known as Mercer kernels. Mercer kernels have interpretations as inner products in potentially infinite dimensional Hilbert spaces. This interpretation arises because, if we take $\\alpha=1$, then the kernel can be expressed as\n", "$$\n", "\\mathbf{K} = \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top \n", "$$\n", "which imples the elements of the kernel are given by,\n", "$$\n", "k(\\mathbf{x}, \\mathbf{x}^\\prime) = \\boldsymbol{\\phi}(\\mathbf{x})^\\top \\boldsymbol{\\phi}(\\mathbf{x}^\\prime).\n", "$$\n", "So we see that the kernel function is developed from an inner product between the basis functions. Mercer's theorem tells us that any valid *positive definite function* can be expressed as this inner product but with the caveat that the inner product could be *infinite length*. This idea has been used quite widely to *kernelize* algorithms that depend on inner products. The kernel functions are equivalent to covariance functions and they are parameterized accordingly. In the kernel modeling community it is generally accepted that kernel parameter estimation is a difficult problem and the normal solution is to cross validate to obtain parameters. This can cause difficulties when a large number of kernel parameters need to be estimated. In Gaussian process modelling kernel parameter estimation (in the simplest case proceeds) by maximum likelihood. This involves taking gradients of the likelihood with respect to the parameters of the covariance function. \n", "\n", "## Gradients of the Likelihood\n", "\n", "The easiest conceptual way to obtain the gradients is a two step process. The first step involves taking the gradient of the likelihood with respect to the covariance function, the second step involves considering the gradient of the covariance function with respect to its parameters. The relevant terms of the negative log likelihood are given by\n", "$$\n", "E(\\boldsymbol{\\theta}) = \\frac{1}{2}\\log |\\hat{\\mathbf{K}}| + \\frac{1}{2} \\mathbf{y}^\\top \\hat{\\mathbf{K}}^{-1} \\mathbf{y}\n", "$$\n", "where $\\hat{\\mathbf{K}} = \\mathbf{K} + \\sigma^2 \\mathbf{I}$ is the noise corrupted covariance matrix. The gradient with respect to that matrix can be computed as\n", "$$\n", "\\frac{\\text{d}E(\\boldsymbol{\\theta})}{\\text{d}\\hat{\\mathbf{K}}} = \\frac{1}{2}\\hat{\\mathbf{K}}^{-1} - \\frac{1}{2}\\hat{\\mathbf{K}}^{-1}\\mathbf{y}\\mathbf{y}^\\top \\hat{\\mathbf{K}}^{-1}\n", "$$\n", "The can then be combined with gradients of the covariance function with respect to any parameters, $\\frac{\\text{d}\\hat{\\mathbf{K}}}{\\text{d}\\theta}$ using the chain rule. Mathematically that is written as\n", "$$\n", "\\frac{\\text{d}E(\\boldsymbol{\\theta})}{\\text{d}\\theta} = \\text{tr}\\left(\\frac{\\text{d}E(\\boldsymbol{\\theta})}{\\text{d}\\hat{\\mathbf{K}}}\\frac{\\text{d}\\hat{\\mathbf{K}}}{\\text{d}\\theta}\\right),\n", "$$\n", "where the two gradient matrices are multiplied together. In in implementation, however, that is inefficient. It is more efficient to perform an element by element multiplication of the two matrices.\n", "\n", "### Overall Process Scale\n", "\n", "In general we won't be able to find parameters of the covariance function through fixed point equations, we will need to do graident based optimization, however there is one parameter that does have a fixed point update. Imagine the covariance $\\hat{\\mathbf{K}}$ has an overall scale such that $\\hat{\\mathbf{K}} = \\alpha \\boldsymbol{\\Sigma}$. In this case the gradient of the covariance matrix with respect to $\\alpha$ is simply $\\mathbf{C}$ and $\\hat{\\mathbf{K}}^{-1}\\mathbf{C} = \\alpha^{-1}$. This means we can write,\n", "$$\n", "\\frac{\\text{d}E(\\boldsymbol{\\theta})}{\\text{d}\\alpha} = \\frac{1}{2\\alpha}\\left(n - \\frac{\\mathbf{y}^\\top\\mathbf{C}^{-1}\\mathbf{y}}{\\alpha}\\right)\n", "$$\n", "which implies a fixed point updated is given by\n", "$$\n", "\\alpha = \\frac{\\mathbf{y}^\\top\\mathbf{C}^{-1}\\mathbf{y}}{n}.\n", "$$\n", "The availability of a fixed point update for the overall scale means that sometimes, if we have a process of the form,\n", "$$\n", "\\hat{\\mathbf{K}} = \\alpha \\mathbf{K} + \\sigma^2 \\mathbf{I}\n", "$$\n", "where $\\mathbf{K}$ might be an exponentiated quadratic, or another covariance, that represents the signal, and $\\sigma^2$ represents the contribution from the noise. Rather than representing directly in this form we might represent the model as\n", "$$\n", "\\hat{\\mathbf{K}} = \\sigma^2\\left(\\hat{\\alpha}\\mathbf{K} + \\mathbf{I})\\right)\n", "$$\n", "where $\\hat{\\alpha} = \\frac{\\alpha^2}{\\sigma^2}$ is a signal to noise ratio. Although some care will need to be taken if the actual noise is very low as numerical instabilities are likely to occur.\n", "\n", "## Capacity Control and Data Fit\n", "\n", "The objective function can be decomposed into two terms, a capacity control term, and a data fit term. \n", "$$\n", "E(\\boldsymbol{\\theta}) = \\frac{1}{2}\\log |\\hat{\\mathbf{K}}| + \\frac{1}{2} \\mathbf{y}^\\top \\hat{\\mathbf{K}}^{-1} \\mathbf{y}\n", "$$\n", "The capacity control term is the log determinant of the covariance, $\\log |\\hat{\\mathbf{K}}|$. The data fit term is the matrix inner product between the data and the inverse covariance, $\\frac{1}{2} \\mathbf{y}^\\top \\hat{\\mathbf{K}}^{-1} \\mathbf{y}$.\n", "\n", "The log determinant term has an interpretation as a log volume. The determinant of the covariance is related to Gaussian density's 'footprint'. Recall that the determinant is the product of the eigenvalues, and the eigenvalues represent a set of axes which describe the correlations of the Gaussian densities (the directions being given by the corresponding eigenvectors). The product of these eigenvalues gives the area (or volume) of the Gaussian density. Roughly speaking it represents the diversity of different functions the Gaussian process can represent. If the number is large, then the process can represent many functions. If it is small then the process can represent fewer functions. Although the terms 'many' and 'few' are badly defined here because the space is continuous and the Gaussian process represents a continuum of different functions. However, the intuition is maybe useful. \n", "\n", "The data fit term is a little like a quadratic well, trying to locate the data at the lowest point. The data fit term can be driven to zero by increasing the scale of the covariance. However, this causes the capacity term to also increase, so a penalty is paid for this. Ideally, the data fit term should be reduced by matching correlations seen in the data with a corresponding correlation in the covariance function." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 112 } ], "metadata": {} } ] }