{ "metadata": { "name": "", "signature": "sha256:d53c2e3857eecf1f98c4f9048d7c27918a97f32b015770d95d672b7e2446090e" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Inroduction to Gaussian Processes\n", "\n", "# Gaussian Process Summer School, Melbourne, Australia\n", "### 25th-27th February 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": [ { "output_type": "stream", "stream": "stderr", "text": [ "/Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/pytz/__init__.py:29: UserWarning: Module pods was already imported from /Users/neil/sods/ods/pods/__init__.pyc, but /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages is being added to sys.path\n", " from pkg_resources import resource_stream\n" ] } ], "prompt_number": 1 }, { "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": 2 }, { "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": 3 }, { "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": 4 }, { "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": 5 }, { "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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bCcnPJ/uEExh19NHuxeLYWK/4NISQWxw8b0g8RbysTC5F6EsTe4p4Rob3c7WM\nVMzmUmpqnqemZgWhoXmMHn05cXFnK/eMN3A6ZbSdpslwKy+ihF7Rb7pruql8uJLaF2uJPzeetAVp\nhGSFyI1m5eXUrF7N+n/9i+rVq8kyGJgQHk5cZycGq/Xn0ULp6e4oouTknsQqJlNv//ieniZN6+1S\n0d0qurAbVfGtPuN0dtPY+B41Nc/T0bGWpKTZJCdfSlhY3lB3bXjR0dE7gq6iwm168ERMDFx9NSxc\n6NVLK6FXeA1ro5Wqx6rY/cxuYk+LJW1hGmH5bp+H0+mksLCQFStW8P7773PaMcdw8Ukn8av0dLSK\najq2lNO9owIqygmpLye0q5EG/2QqnKmUiTQ6olKxJaZiSE8lNDeV6EmpjJoUR2aWRrQK6PA6JtMm\namtXUFf3KmFhE0hOvpi4uN/j56cqmP2Mri653qWbHgrtaVarew9Maqp7UqMfp6TIPFwDgBJ6hdex\nt9mpfqqaqmVVGI8xkr4onYjpEXR0uOPHt2yx8NlnZWzY0EF7ezyalkJcnJO8vMCeqJWcNCtjw6rJ\nNJQTbapEq6p0z4T0L1RXV+9dxZ67jEePlrfAycmH9mrpQWCzNVFf/wa1tS/R3V1DUtJfSEq6iNDQ\nMUPdtaHB4ZAr9frGRs8Njp5mNrt3tY8e7d7kmJLiFvOYmCELv1JCr/AqTU3uHZ47tjrZ/LGZHRud\nVItgLAZ/snIgO1vriSPPygJ//3I+//wl/v73F4mJieGiiy7iggsuOLCona4u967iysreXz5953Fd\nnfTd6KLvaXoOIj0fUUTEoREL6YHTaaW5+QNqa1fS0vIpsbGnkZj4F2JiTkLThtH224PBZOqdoqSu\nzp2ixDNdSUODTGKof3Y8JxC6qKekDKmIHwhK6BUHTUtL7/wr+rb9nTvlepK+szM727XDM81J5PoG\nup8pIzA+gLSFacSeHotm6P250107L730Eu+99x4FBQVcdNFF/c+o6XDIsBtd+PUvst565iNyOGTg\nu2diOj1ZXXy8O1upbpGRPv0F3xdCOGlr+5a6uldpaHibsLDxJCZeSELCufj7D7MFDSHk9uOmJvl/\n1hMNelp9vbutq5O/k5joHuATE3sP/nouqsREmep8mKOEXrFXLBYp3EVF7kRaulks7rwreh4WXdzj\n4vate8IhaFjVQMV9FQi7IHVBKgnnJWDw/3m+nvb2dt5++21WrFhBUVERs2bN4qKLLmLy5AHOj9LZ\n6RYDzyRrgJ5+AAAgAElEQVR1DQ1SRBoa3ILS2Chv2fW01Z5prHXTU1xHRbnTXhuN7nTYQUGDNlAI\nITCZ1lFf/zr19f/A399IQsIsEhPPJzg4fVD6sI+OyQ+VnjVWzxzb2upuddMzyHpmkm1ulr5tffCN\njXUPyvHxvQdrfQAPDx+WA3RfUUJ/CCOE1LPt292mp7utqZERK7m57oRaublS2Pub7lYIQfNHzVTc\nX0F3eTepN6SSNDcJv5C9uwl27tzZsyErLi6uZ0NWbGxs3zvhLazW3qmrPdNZewqUp3DpqbDb2uQt\nUESEFJ6ICHchHM+COKGhslCObnrxnKAgaXpRHb3ITkCANH9/hMGA2VZKU+sntLR9jBMnsXGnExN7\nBmHhue5/pF7PQa/poNd10Fu92I/NJs1qlW13tzzu7pa2Z70Hs1laZ2fv+g8mk7SODtn6+7vrQHjW\ngjAaew+Y0dG9B1N9gFVrML+IEvpDALtd5lzZvl0WofBs/f1lvvK8PGm6oGdmDs4da9vqNirur6D9\n+3ZSrk5h1OWjCIja+4UdDgeff/45K1as4D//+Q8nnngic+fO5eSTT8Z/IOvbDSRWqxQ7XfA8hdCz\nOI4umGazFFJdVD1F1iXCwmrFYWnB3t2Ew9IMDicBBiN+hggMIgDNU9Q90bTeVdoMBnfrGjjw9+89\nmHgONp4DkOfApFdy09vwcHerD3IjwD3iyyihH0F0d0tf+datbtu2TbpgkpLcRSj0ohR5efLu1hfo\n3NJJxdIKmv7TRPK8ZFKuSSFo1L4TrrW2tvKPf/yDFStWUFlZyYUXXsicOXMO2WIpTqedtrYvaWz8\nF42N72AwhBIffw7x8ecQHj5NpSM4xFFCPwzp7pa+cs/an1u2yM1D6enuUnF6ZaHc3OFT6dBSbqHy\nkUrqXqkj/px4Um9MJXTsL3d+69atrFixgldeeYWcnBzmzp3LueeeS8QI3+5qt7fR3PwhjY3v09z8\nASEhWcTFnU1c3O8IDc1X4q7oQQm9D2OzyRn65s29izrv2iVdK+PHS8vPl+3YscM/f7mOtdFK9ZPV\n7H5qN8bjjKQtSCPysMhf/B2bzcYHH3zAiy++yBdffMHvfvc75s2bx8yZM0eE6Akh6OraTlPTf2hu\n/g8dHT9hNP6KuLiziI09g6Cg0UPdRYWP4lNCr2naKcBjgB/wghBi6R7Pj0ihdzikeOuCvnmztJ07\n5T4LXdAnTBh5gr4/7CY7tctrqXy4kpAxIaQtSCP6pOj9CnddXR0rV65k+fLlCCGYN28es2fPJikp\naZB67h3s9nZaWj6juflDmps/BJzExJxGbOwZREcfj5/fMLlVUwwpPiP0mtyZUQScCFQDPwLnCyG2\neZwzrIVeCLmPRxdy3bZtk5FfupBPmCAtL0+uZSnAaXNS/3o9FQ9UYAg0kHpTKvF/iN9raKYnQghW\nr17N8uXLWbVqFQUFBVx88cWccsopPrmA63Ta6Oj4gZaWT2lu/gSTaT1G41HExJxCTMypyiWj6BO+\nJPRHAYuFEKe4Ht8MIIS43+OcYSP0DQ0/n6Fv3iyDEnRBnzhRHo8bJyPKFPtHOAVN/22i4v4KrDVW\nGZp50b5DMz3p6OjgjTfeYPny5VRVVTFnzhzmzp1LZubQlbYTwoHJtIHW1kJaWj6jre0rQkKyiIo6\ngZiYkzEaj8XPT432iv7hS0L/B+A3QohLXI//DBwhhLjK4xyfE/rmZrkY6ulH37xZ+tc9xVyfqftK\nlMtIoO0bV2jmj/sPzdyTTZs2sXz5cl599VWmTZvGJZdcwm9/+9sBL4nodNoxmdbR1vYlra1f0tb2\nFYGBSURFFbjseAID1YdE4V18SejPAU7xVaFvanJHt+ihi1u2yBDncePcYq4LenLyIbXxbkgxbTZR\n+WAlTe+7QjOv/eXQTE8sFgv//Oc/ef7559m6dSt/+ctfuOSSSxgzxjtJvOz2Ntrbv6et7Rva2r6m\no+NHgoMzMBp/RVTUrzAaf0VQ0PBaN1AMP3xJ6I8E7vBw3SwEnJ4LspqmicWLF/f8TkFBAQUFBX2+\n5p44nTIvluemIt1sNinonjZ+vMxnpATdN7BUuEIzV9YR9/s40m5MIzT3wBcri4uLeeGFF3jppZcY\nP348l112GWefffYBz/KFcNDZuZWOjh9ob/+OtrbVWCxlRERMw2g8BqNxJpGRRxEQENPXt6hQHBCF\nhYUUFhb2PF6yZInPCL0/cjH2BGA38AMDsBgrhHS3eCblKiqSVlwsd1Pru0Tz8tzx6P3d9q8YPGxN\nNqqfrKb6qWqMxxpJu3n/oZmeWK1W3nnnHZ599lk2b97MRRddxKWXXkp2dnbPOUI4MZt30tGxho6O\nn+jo+BGTaR2BgaOIjDyciIjDiYw8ivDwyar6kmLI8ZkZvaszp+IOr1wuhLhvj+cPSOj1mtd6FaKS\nEpkCoKREirkQ7mRcenIuPafLCN9jc0jh6HRQs7xGhmbmhJB2cxrRJ+4/NNOTHTt28MwzT7Ny5ctM\nmDCac89N5/DD27FYNhAQEEt4+DQiIg4jImIGERHTCQhQFVAUvodPCf1+L6Bpwm4XNDTIpFt6GvKq\nKnc1rooKmYk2IUEm5crKcuc9z86Wou6lUqWKYUJPaObSCgzBBtJuTiP+9/FofnumSbZjsZTS2bnF\nZRvp7NyExVKGwZDNt99G8dZb1VRXm7j44nn83/9dzSgv1/ZUKAaCYSf0/v6CqCi52OlZ0MWz1Ghq\nqkpmp/g5wiloer+J8ke2YQ0oJfriTgJn1GLuLqKraxtm804CA5MIDR1HWNh4wsImEh4+idDQXAwG\n9+Luxo0beeaZZ3jjjTc44YQTuPzyyykoKFDx7QqfZdgJfXe3UCKu+EWEENjtLZjNpVgsJZjNpZjN\nOzGbizGbi7Hb2wh0ZuHYNgpH0SjiJk9j1GlHEhE3Hj+/sP1fwEV7ezuvvvoqTz31FEIIrrjiCmbP\nnj3ic+wohh/DTuh9LY5eMfg4nXas1hq6uyvp7q7EYqmgu7sCi6Uci6UMi6UM0AgJySY4OIuQkCxC\nQnJcNoagoNFomtxR27G2g4r7K2j9vJVRl48i5eoUAmIPbvFUCMGXX37Jk08+yf/+9z9mzZrFFVdc\nQV5envffvGJEY3VY6bR2Eh3i3bUeJfQKn8HhMGO11mGz1WO11mG11mK11rhEfTdW6266u6ux2RoJ\nCIgnKCiFoKAUgoPTCQ5OJygojeDgTIKDMwgIiDqoa3cVd1GxtILGfzaSNCeJ1OtSCRp98AmFqqqq\neO6553juueeYNGkSV111Faeddhp+fiO09qqiTwghqDHVsKF2AxvrNrKxfiOb6jZR3FzMDUfdwF3H\n3+XV6ymhV3gVIQROpwW7vQ2How27vRW7vRWbrQW7vRmbrRm7vQmbrQmbrdFlDVitDQhhIzAwkcDA\nRAICEggMTCIwMJmgoGRXO5rAwNEEBiYOWMiipcpC1SNV1L5US/wf4km9KZXQnINPHNbd3c1bb73F\n448/TlNTE1deeSVz587FaBxm9VgV/cbutLO9cTvra9f3Mk3TmJw4mUmJk3osPy6fkADvp7xQQj+C\nkX8rJ0LYcTptCOE2p9OKEFacTitOZzdCdON0WnA69daC02nG4ejyaLtwODpdranH7PZ2HI4OV9sO\naPj7G10W7bIoAgJi8fePISAgmoCAOPz9YwkIiCUwMIGAgHj8/CJ8ZkHT2mil+vFqqp+uJubkGNIW\nphE+MbxPr/Xdd9/x+OOP8+GHHzJr1iyuvvpqr+28VfgWZpuZTfWbWFuzlnU161hXu44tDVtIiUxh\nStIUpiZNZXLiZCYnTSY5PHnQPu/DTuit1uZeIuMWHilITmeXh1DpwtXdI2qy1QXP7tE6elopjnrr\n3KMVgNt6P96Tvf89fv53EnscC4/z9Os493LsdPXbCchWfx/ysR0woGn+LgvAYAhwtUFoWmDPsdtC\nMBiCPSwUP78QVxvW0/r5hXtYBP7+Ea7W2CtCZbhjb7ez+2+7qXqsiojDI0i/JZ3Iw/uWga66upq/\n/e1vPPfccxxxxBFcc801HH/88T4zuCkOji5bF+tr17Nm9xrW1q5lze417GzeydjYsUxLnsbUpKlM\nTZbCHhE0tAv0w07ov/wyskdYDIYwl+iEugQo1CVUIR7CFYymBWEwBKJpga42wEP0/HsM/NA0aVIg\nDT3HoLkW8DTXsdZz3Pvxz3q9r3ez53vb4zmt57j3dT37YnC1fr36Kt+P53tR9BeH2bX56sFKQseG\nkn5rOsZfGfsk0mazmb///e889thjGAwGrrvuOs4//3yCDpUiA8OQTmunFPWaNdJ2r6G0pZRx8eOY\nnjydacnTmD5qOhMSJhDsHzzU3f0Zw07oletGMZQ4rU7qXq2j4r4KAhIDSL81nZjfxPRJ8IUQfPzx\nxzzyyCNs3LiRK664gvnz5xMbGzsAPVccKJ4z9Z9qfuoR9fEJ45mePF2aS9QD/YZHrLcSeoWiDwiH\noP6teiruqUAL0ki/NZ24s+LQDH37Lm3atIlHH32Uf/3rX8yaNYtrr722V24dxcDQZetiQ+0G1tSs\n4afdP7GmZg0lzSWMix/HjFEzmJ48nRmjZjA+YfywEfW9oYReoegHwilofK+R8rvKETZB+i3pxP/h\n5+kVDpSamhqefPJJnnvuOQoKCrjppps47LDDvNzrQxPPmbrugilpLiE/Pr9H0Kcny5l6kP/IcqMp\noVcovIAQguYPmim/qxx7q520RWkknJ+w31KH+8JkMrF8+XIeeeQRMjMzWbBgAaeccopauD1ATFYT\n62vXs7Zm7V596tNHTR+xor43hp3Qf93aylGRkRjUB17hgwghaPlfC+V3ldNd3U36Lekk/jkRQ0Df\nBN9ms/Hmm2+ydOlSNE3jpptu4k9/+pNP1rsdKlrMLayrXcfamrU9VtFWwYSECXKRdBj61L3NsBP6\ncd9/j8nh4E8JCfwpIYFp4eFqlqPwSVq/aKXsrjIsJRbSFqaRdFEShsC+Cb4Qgg8//JD777+fyspK\nbrzxRubMmUNwsO9FeAwUQgh2d+xmXe061tWsY32dnLE3djUyOXEy05Kn9YQ1josfR4CfqgOgM+yE\n3ul0srmzkzfq6/lHfT0C+GNCAufGxzNVib7CB2n7to2yO8vo2tZF2s1pJM9NxhDUN8EH+Pbbb7n/\n/vv58ccfufbaa5k/f/6IS6Rmd9opaixiQ92Gnp2k62rXAcj4dFeM+rTkaeTE5GDQ+v73PBQYdkLv\neQ0hBOtNJv5RX89bDQ0I4A/x8ZwTH89hERHKvaPwKdq+a6P8znI6N3WStjCN5Hn9E/xNmzZx3333\n8cknn3DFFVdw9dVXExMz/MoUNpub2Vi3kQ21G9hQJ21bwzZSIlOYnDSZKYlTmJIkbVTEKDWZ6wM+\nIfSapp0L3AHkAYcJIdbu47x9LsYKIdhgMvFWQwOrGhrodDo5Oy6Os+PiONZoxN+gRnyFb9D+Qztl\nS8ro3NhJ2s1pJM1Lwi+47xvbiouLuf/++3nnnXe49NJLue6664iPj/dij72D1WFle+N2NtVtYmPd\nRjbVy7a9u52JiRN78r5MTpzMxMSJhAf2LeWE4uf4itDnAU7gWeD6vgj9nmzr7OSfjY2809hIqdnM\n6bGx/DYujpOjo4lQC1kKH6D9x3bK7yynY10H6QvT+y345eXlLF26lDfeeIM5c+Zw4403kpSU5MUe\nHxgOp4PSllI212+W1iDb0pZSMqMymZg4kYkJ0iYnTSbdmK5m6QOMTwi9R2c+x0tC70mVxcK7TU28\n19jIt+3tHB0ZyZmxsZweG0tmiPczxSkUB0P7j+2U3eGa4S/qvw+/urqaBx54gFdeeYXZs2dz0003\nDUjJQ7vTTmlLKVsbtrK1YStbGrawtWErRY1FJIYnMiFhAuPjxzMhYQITEyaSG5frk+kBDgUOCaH3\npN1u5+PmZt5vauKD5mbiAwI4PTaWU2NimGk0EqhcPIohov0Hl+Bv7iT9lnSS5vQ9Sgfk5qsHHniA\nl19+mdmzZ3PzzTf3aYbfZetiR9MOtjVsY3vjdrY1bmNb4zZ2Nu8kOTyZcfHjyI/LZ3zCeMbHjyc/\nPl+5XXyMQRN6TdM+Afb2KVskhHjfdc6AC70nTiH4saOD/7pEf0dXF7+OjuaUmBh+Ex1NhprtK4aA\ntu/aKFtcRldRFxm3ZZA4u+9x+CAFf+nSpaxcuZI5c+awYMECEhISep2jhy4WNRVR1FhEUVMR2xu3\ns71xO3WddeTE5JAXl0debB758fnkx+WTG5dLaMDB5+pXDD7Dbka/ePHinscFBQUUFBT065qeNFit\nfNTczEctLXzc3IzR35+To6M5MTqaX0dHY1S+fcUg0vZNG7sW78Kyy0LG7RkkzOr7TluA3bt3s/iu\nxbz5xpsce/ax5J6ZS5Wjih1NOyhuKiY8MJzcuFzGxowlNy63R8wzojLwN6jP/nCisLCQwsLCnsdL\nlizxOaG/QQixZh/PD1oKBKcriueTlhY+bWlhdXs740NDOSE6mhOiozk6MpJgVR5OMQi0ftHKrtt2\nYa2zknFHBgl/Sthn8jQhBI1djZS0lLCzeWePFTcXU9xUjEM4yNAy6PpfF9XfV3PqBady1TVXMTV9\nKsZgVf1qqCnu6qLT4WCKl/dF+MSMXtO0s4HHgTigDVgnhDh1L+cNWa4bi8PBt+3tfNbSwv9aW9lk\nMjEjIoJfR0dznNHIEZGRhCjhVwwQemqFstvKsHfYCVsQRu3RtZS0llDaUkpJSwklzSWUtJTgp/mR\nHZPNmJgx5MTkkB2dzZjYMYyJGUNcaFxPhEtpaSlLlizhgw8+4IYbbuDKK68kNFS5YgabRquVfzQ0\n8EptLbssFu7KzORSLy+e+4TQH/AFfCipWYfdzjdtbRS2tvJ5aytbOjuZGhHBr4xGjjUaOcpoVK4e\nRZ8QQtBsbmZX6y52texiV+suSltKpTWXkrQmiXmfzyPIEMS22dsIPjGY7NhssqOzyYnJITok+qCu\nt3XrVm6//XZWr17Nbbfdxrx58wgIUOkDBpI2u513Ght5o76eb9vaOC02lgsTEzk5OnpA9vwoofcS\nJrud1e3tfNnWxjdtbfzY0UFWcDAzjUaOjozkaKORzOBgFS+sAKC9u52y1jJ2teySbasUdP1nBs1A\nVnQWmdGZZEZlkhWdJR9HZZIRlUGgXyCN7zRSdnsZfuF+ZN6dSfQJByfwe/LTTz+xaNEiSktLufvu\nu/njH/+IQUWheY0mm413GxtZ1dDAV21tHB8VxXkJCZwRG0v4AE8KldAPEDank7UmE6vb2vimvZ1v\n2tpwCMGRkZEcGRnJEZGRzIiIIFLN+kckbZY2ytvKKWst26tZHVbSo9LJjMrsEe+MqIweYT/QWblw\nCOrfrKdscRlBo4PIvCcT49H987V/9tln3HzzzdjtdpYuXcpJJ53Ur9c7lNnZ1cV7rj0860wmToqO\n5pz4eE6Pjd33d18I8PKEUAn9ICGEoKq7m+/a21nd3s4P7e2sN5lICw7m8IgIZkREMD0iginh4crX\n7+PorpWy1rIeMS9vLXcft5Vjc9h6xDvdmE5mdGav49iQWK/e3TntTuperqPszjLCJoSReXcmEVP7\nvqAnhGDVqlUsWrSI9PR0HnjgAaZOneq1/o5UzA4HX7a18UFTE/9tbsbkcHBmbCxnxcVxfFTUL3+3\nbTa47TYICIC77vJqv5TQDyE2VybOnzo6+Kmjgx87OtjW1UVOSAjTw8OZGhHB1PBwJoeHK3//IOIU\nTupMdT8Tcc/H/gZ/KdxR6WQYZZtuTO8R95iQvtWR7Xffu53sfm43FfdWYDzWSMadGYTlhfX59Ww2\nG8uXL2fJkiWceOKJ3H333aSnp3uxx8Mbhys679OWFj5paeG79nYmh4VxmmsT5uTw8ANLrlhZCeed\nB0YjrFwJcXFe7acSeh+j2yX+azs6WNPRwYbOTjaZTCQGBjIpPJzJYWFMCg9nYlgYWSEh+Cmf/0Hj\ncDqoMdX0uFHKW8t7zc4r2iqICIroNSPXRVwXdF8PRXR0Oqh6ooqqh6uIPSOW9MXphGT0fQNgR0cH\nDz/8ME888QQXX3wxCxcuJCoqyos9Hh7YnU7Wm0x85QrE+LKtjeTAQI6PiuKkmBgKoqIOflL2n//A\nvHlw7bVw442gFmNHvtDvDYcQFHd1sbGzk40mU4/4N9hs5IeGMiEsjPFhYYwLC2NcaCjpwcGHdIrm\nPYVcX/DUhbyyvZLYkNheM3JPEU8zphEW2PdZsC9ha7VR9XAV1U9XkzgrkbRFaQQl9b10Xk1NDbfd\ndhvvvfcet956K/Pnzx/RETqtNhvfd3Swuq2N1e3tfNfeTkpQEMcajRRERVEQFUVSUB//nhYL3HQT\nvPsu/P3vcMwx3u28B0rohzHtdjtbOjuldXWxpbOTrZ2dtNjtjA0NJT80lLzQUHJdbU5ICGEjwP8v\nhKCus65XxIpnW9lWSXRIdK9FTk9LM6Ydcsm1rPVWKu6roHZlLaMuG0XqTakERPVdoDdu3MgNN9xA\neXk5Dz30EGecccawjyjrcjjYYDKxpqODH1yu1EqLhRkRERxlNHJUZCQzjUZivTGwbdkC558PeXnw\n7LMQ3b+Iqf2hhH4E0m63s72ri+1dXRR5tCUWC7H+/owJDWVMSAg5LssOCSErONin0jd3Wjt74sd3\ntbjiyFtlLHlZaxkh/iFkRWfJSJWozJ5olczozENSyA8US4WFsiVlNL3XRMr1KaRcnYJfaN8GfyEE\nH3zwAddffz2jR4/m0UcfZeLEiV7usfcRQlBntbKxs5MNJhMbTCbWm0yUWiyMCw1lWkQEh0dEcHhk\nJONCQ70b1+50wuOPw913w9KlMHeu1yNs9oYS+kMIhyvyZ0dXFzvNZnaazRSbzZSazZRaLIT7+ZEZ\nHCwtJISM4GAygoNJDwoiLTiYUC/eDeizcn03p767s7SllJLmEtq628iIyiA7OrsnftwzrjwiaGSV\nzhtsOrd3UnZbGW3ftpF+azrJFyf3q4D5s88+y1133cU555zDnXfeSZyXFxP7ghCCGquV7V1dbOvq\nYqvr7ndzZycCmBgWxuTwcKa4Ah4mhIUNbPbaigqYMwfMZrngmpMzcNfaAyX0CkB+KWqtVnZZLNLM\nZsosFsq7uym3WKiwWIjw9yfNJfqpQUGkuGx0UBCjAwMZHRTUK3zMKZxUtVf1yrmi52ApaS4h2D+Y\n7Bi5ozM7OrvnOCs6i+SIZFUHdBDoWNNB6aJSzDvNZN6VScJ5+86jsz+am5tZsmQJr732GrfeeiuX\nX375gPvv9c9tqcVCiWvystNsZkdXFzvMZoINBvJCQxkXGkp+WBjjXWtaiYGBg+dqEgJWrIAFC+SC\n6003wSDfPSuhVxwQTiFosNmosFio6O6mymUVFgu7ujqotJhpdAj8hZ0AezvO7gbMXdWEOs0kBgaQ\nEhxGTngM44zJTI5JY1pcDtEhh17Uhq/S8nkLpQtLcVqcZN2bRcypfQ8P3bZtG9dccw1VVVU89thj\n/dpw1e10Uu36rFW6Pm/lrglImetYvxPN8XBHjg0NZWxICNFDvVBcVQWXXAK1tfDSSzB58pB0Qwm9\nYr8IIWjoamBH045eVtxcTElzCVHBUYyJHUNOzBhGR+cTbcwhOGw0fkHxNDugxmql1mqlzmqlzmaj\nzmqlw+Eg1t+f+MBA4gMCiAsI6GljPdoYf39iXK3R3/+QjiYaaIQQNL7byK5FuwiICyDr/qw+77IV\nQvDee+9x3XXXMWnSJB599FEyMjIAOWlottlocH0W6l1trctqrFZ2d3dTbbXSZrczKjCQlKAgUl13\nkukud6LuWhzo9AF9wumEF16AW26Bq66ChQvlRqghQgm9ogezzUxxc3FP4YkdTTt6Wg2N3LhcxsTI\nLIj6cU5MTp/85Tank0abjXqbjUabjQarlQabjSbX4ya7nWbX4ya7nRabDZPDQYS/P9H+/kR5mNHf\nH6Ofn2z9/Yn08yPS358IPz9pruNwl4UYDMM+QmQgEQ5B7Su1lC0uI3xKOJn3ZBI+oXfFKIcQmBwO\n2u122ux22h0OWl3HrXY7La7/X31nJz++8AI7X3mFqD/+Ec47j1bX/yU+IIDEwEASAwNJCAggOTCQ\nJJeNcrkE4wMCht/gXlQEl14qwydfeAF8YIFaCf0hhl5JaHvj9p4qQnpVobrOOjKjMsmNyyU31mVx\nuYyNHUtc6NAvrjmEoM0lIG0uYWlxHbe5RKZDFx+Hgw7X4w6HA5OHWZxOwvz8CDMYCPPzI9TPj1CD\noacNMRgIcQ0IwXtYkMFAkKbJ1mUBmkagphHgOu4xgwF/TcNf0/AD2brMABg8Wg2keRzrCN2E6Dl2\nCoHT9TdxCIHDdWx3Pbbp5nRiEwKr69gqBN1OpzQhsDidWJxOzA4HZqezx7ocDrrNDvJe7+LIFRa2\nHunHqosNlCcIOjz+hvrAGuHnR3RAgBx4XccxrkE5JiAAe00NL9x+O0UbNrBs2TJ+d9ZZg/rZGRQs\nFnjgARlVc/vtcMUV4CPhzEroRygWu4XipuKecnCeoh4aENpTFk4X9by4PNKj0g+JSkIOIeh0OHqs\nyyVsnR6C1+USQF0Eu10CadFF0iWU3bqQulqbh6DqwmvTxdj12IkUaoerFR6tp6h73nVoHqYPEJpr\nANEHD/89LMCjDTQYegYjz4Eq2DWwBbnaUNcApw98YX5+hJgEAU83Yn2+HuP58aTfko4xqW+b8j7+\n+GOuuuoqcnNzWbZsGZmZmf3+f/oEn3wihX38eFi2DNLShrpHvVBCP4zRqwnpYr69cTvbm2Rb3V5N\nZnRmL0HXS8NFBatFUMXBY62zUn5POXWv1ZFyVQop16XgH3HwE4Pu7m4eeeQRHn74Ya699lpuuOEG\ngvq6u3SoqayEG26AH36AJ56AM84Y6h7tFZ8Rek3THgTOAKxACTBHCNG2xzmHpNDbnXbKWst6Cfq2\nxtlSSOsAACAASURBVG1sb9yOUzjJjc0lPz6fvNg88uJk8ebMqEwC/Ebu1nTF0GEuNbPr9l20fNpC\n+i3pjLp0FIaggw+FLSsr4+qrr2bHjh08/fTTHH/88QPQ2wHCYoGHH4ZHH5Uz+QULwIerc/mS0J8E\n/E8I4dQ07X4AIcTNe5wzooW+xdzS4y8vairqcbeUtpSSGJbYI+b67DwvLo+EsAS1sKgYEkwbTJQu\nLKVrexcZd2aQeH4imt/Bfxbfffddrr76ao477jgefvhh4uPjB6C3XkIIWLVKxsJPngyPPALDwP3k\nM0Lf64VkDdlzhBB/3uPnw17orQ4rpS2lFDW6o1p0cTfbzT3+cn0hNC8ujzExYwgJ6HvmQYViIGn9\nopXSm0txdDlkDP5pBx+DbzKZuOOOO1i5ciX33nsvc+fO9b3qVj/9BNddB21tUuBPOGGoe3TA+KrQ\nvw+8LoR4bY+fDwuhtzlslLWWsbN5J8XNxRQ3Fcu2uZjq9mpSjamMjR1LbmxuT5sbl0tyeLKanSuG\nJd6KwV+/fj2XXnopISEhPPvss+Tl5Q1Abw+SkhIZD//ll3DnnTKNgY9E0xwo3hT6/a7KaJr2CZC0\nl6cWCSHed51zC2DdU+R17rjjjp7jgoICCgoK+tLXftNibukp3twrT0tLCdXt1YyOHE12dLaMO48d\nw29yfsPY2LHKd64YkWiaRvzv4ok9I5a6lXVsPW8r4VPDybo3i7DxB57yecqUKaxevZqnn36aY445\nhquuuoqbb755aBZr6+rgnntkCuFrroHlyyFseKSvLiwspLCwcEBeu98zek3TLgIuAU4QQlj28vyg\nzOj1fOZV7VVUtlVS0VZBRVtFrxqgTuHslWTLM1dLelQ6gX6BA95PhcJXcVgc7H56NxX3VxB7eiwZ\nd2QQnH5wGUUrKyu54oorKCkp4YUXXuCoo44aoN7uQXMzPPggPPcc/PnPcjafkDA41x4gfMZ1o2na\nKcDDwHFCiMZ9nNNnoRdC0N7dTkNXAw2dDTR0NVBrqu2xGlMNuzt2U91eTX1nPbGhsaREppBmTCMt\nMo00Y5osUuHKaR4dHK3cLArFfrC32al8qJLqp6tJ+ksSaYvSCIw78EmQEIK33nqLv/71r5x77rnc\ne++9hIeH7/8X+0JrKzz2GDz5JJx9tqzf6mPx8H3Fl4S+GAgEml0/Wi2EuHyPc8SrG17F7rRjd9rp\ndnTTbe/GYrfQaeuk09qJyWqi3dpOm6WNtu42Wi2tNJubaTY3E+wfTHxoPPFh8cSHxpMUnkRSeBKJ\nYYmMjhzNqIhRJIcnkxyRrGbkCoUX6a7tpvyucur/UU/KX1NIuTYF//ADj8Fvamri+uuvp7CwkOef\nf75fidJ+RnOz3OT01FNw5plyBj+IKYQHA58R+gO6gKaJC1ZdgJ/mh7/Bn2D/YIL8ggjyDyIsIIyw\nwDDCAsKIDIrEGGzEGGQkOiSamJAYYkJilHgrFEOMucQVg/+/vsXgf/TRR1x22WUcf/zxPPzww0T3\npzJTTY2Mg1++HH73O1i0CLKz+/56PsywE/rhEHWjUCh+GdMGE6WLSuna2kXGkgwSZx14DH5HRwcL\nFy7knXfe4W9/+xtnnnnmwV18xw4ZHvnmm9IHf8MNI8ZFsy+U0CsUiiGj9atWSheWYm+1k3l3JnG/\njTvgta8vvviCefPmcfTRR/PYY48RExOz75OFgG+/hYcegq+/hvnz4corh/0i64HiTaEfnN0NZWWD\nchmFQjHwRB0bxdSvppK9NJuyxWWsPWotLZ+1HNDvHnfccWzYsIGYmBgmTpzI+++///OTrFYZHnnE\nEfCX/2/vzMOjKtL9/6mEEJIQsnX2nSUQlhAIgjKCiOsMKCqO15HFq7iMzDgucwUZ547IqOPI6PXq\n82PujIAjLrgOgujlog5xQRQIYUmAEAhk3/el00mn6/dHnU4CRiWkk+4k9Xme9zmnzzl9+u3TXd+q\n89ZbdW5Xg5zOnFH58INE5B1N37Tog4MhOBh++lNll14K/XVCJI1G0460ScreLuP0f57GK96L+Kfi\nGTF9xHm994svvuCOO+5g1qxZvPDCC/g3NsLLL6sUycRElQf/s5/1u4FOjqL/hW7a2iAtDT7+GHbs\ngKNHYfZsuPpqZQkJffJUdY1G0zvYWm2UvFLCmTVnGHHRCOKfjD+vQVcNdXWsXLyYbZ9+yno3N65Z\nuhSWL4eJE/vAa9em/wn9uZ9RVaXmgt65U5mbG1x5JVx1Fcydq2/PNJp+Spu5jaK/FpH35zwCrgog\nbnUc3qO7mCGyuBhefVW14H19+XT2bO7csoV58+ezdu3a3su770f0f6HvjJRw/Dh8+qmyzz+H2Fgl\n+FdcoVr+I87vVlCj0bgG1norBS8UUPDfBQTfFEzs72MZFuYG27fDxo2wezfcfLN6fN+0aSAENTU1\nPPDAA+zevZtNmzYxc+ZMZ38NpzKwhP5cWltVmOdf/4LPPlMPB0hMhDlzlF16qRZ+jaaf0FrZQv5D\n31D0jplQ8RkxyUfxvO8WWLjwe+eg2bJlC8uXL2fZsmX84Q9/YOjQwTmWZmAL/bk0Nyux37ULUlNh\n3z4YN0619GfNUsLvynNhazSDDSnhyBF46y1lQ4fScuOd5JVdScmWJsKXhRO9Ipqhwd8v4KWlpdx1\n110UFRXx+uuvk5iY2IdfwDUYXEJ/LhaLmmP6889Vbu3XX0N4OPzkJx02Zozu3NVo+hIpISMD3nsP\n3n0XGhrg1lvhF7+A5OT28mgptJD7dC5lb5URcW8E0f8RjUdg1zPDSil5+eWXeeyxx1i9ejXLly8f\nVHNVDW6hP5e2NtV62L1b2VdfgdkMF18MM2eq5UUXge7c0Wgci80G334LH3ygzGJRcfeFC1UO/A88\nhKQ5t5ncJ3Mp/2c5kb+KJOqhKDwCuhb87OxsFi1ahMlkYuPGjYSFdTVr+sBDC/2PUVgIe/ao1v43\n38ChQ2rCoxkzYPp0ZePHw5DuPyRZoxnUNDSopInt2+Gjj8BkUnPOLFgAKSndvpM2nzaT+2QuFVsr\niPx1JFEPRuHh/13Bb21tZc2aNaxfv57169czb948R30jl0ULfXdpaVFiv3evaoHs3QsFBer5kRdd\npP6gKSkwduygHZyh0XSJPSvuf/9XjYH55hvVYJo/H+bNc9iMkeZTZnKfyqViW4Vq4T/YdQv/yy+/\nZPHixVx33XWsXbsWL6+B+7hOLfSOoLYWDhxQ8f79+1WmT2mpEv8pUzpswgQYpL3+mkFKaanKePv0\nUzXeRQg1ov3aa1XKcy9mvbUL/tYKIu6LIPqhaDyCzhb86upqfvnLX5KZmclbb73FxAE6uEoLfW9R\nUwPp6coOHFDLnBzVuZucrCqBpCRloaHO9lajcQylpfDllyqrbdcuKCqCyy5TAxivvNIpI9fNOWby\nnsmj/L1ywu8OJ/rhaIaGdjS4pJS88sorrFy5kj/+8Y/ce++9A66jVgt9X2I2Q2amCv0cPqzs0CEV\n3580SQ3VnjChw/z9ne2xRvP9SKmm/P36a2VffqmEfuZMNU7l8svVnayLhDCbc5vJW5tH2ZtlhC4J\nJfqRaIZFdTzeMCsri1tvvZX4+Hg2bNjQs7nuXQyXEHohxB+B6wEJVAL/LqXM7+K4/i30XSGlavVk\nZqqUsiNH1Pw9R4+Cr6/q6E1M7LBx4yAsTKd8avqeyko19uTbbzv6p4YPV8I+c6YaizJxossI+/dh\nKbaQ/5d8Sl4pwXSTiZiVMXiPUVMrWCwWVqxYwdatW3nzzTcHzIhaVxF6XyllvbF+PzBZSnlXF8cN\nPKH/Pmw2yM+HY8eUHT0KWVmqM8tiUbfAnW3MGGV+fs72XDMQKC6Ggwc7Qo9paWpeqalTVQeq3SIi\nnO3pBdNa2UrBiwUUrSvCf64/MY/G4DvFF4Bt27Zx99138+CDD7Jy5UrcfiC9sz/gEkJ/1kmEWAX4\nSSkf7WLf4BH6H6KqSt0ynzihxD87W9nJkzBsmHoc2ujRajlypLL4eFUo+/kfVuNgGhpUQyIzU91N\n2kOKVuvZiQTTpqn/1AD8/1jrrRT/vZj85/PxmehDzMoY/C/3p6CggNtuuw0fHx82bdpESD+eINFl\nhF4I8RSwBGgCLpZS1nRxjBb6H0JKFSM9dUrZyZNw+rTqBM7JgepqiI5Woh8b22ExMcoiI3VW0EDE\nZlPjQbKzVePg+PEOKytTqcATJqh+oqQktYyMHHThQZvFRukbpeQ9m4e7jzvRj0QTcEMAq9esZtOm\nTbzxxhtcdtllznbzgugzoRdCfAJ0NQztd1LKDzsd9ygwVkp5RxfnkI8//nj76zlz5jBnzpye+Dy4\nMJshN1eJf25uh+XnQ14elJRAUBBERSmLjDzbwsOV+fkNOhFweerq1G955owye+V+6pT6vf38OsJ7\n9r6esWNVpe/iMfW+RtoklR9Vkv+XfJpzm4l6IIrDsYdZtnwZ999/P6tWrXL5UE5qaiqpqantr594\n4gnXaNG3n0SIGOBjKeV3Elp1i76XsVrVHUFBQYcVFqrO4sJCFbctKlLHhYUpCw3tsJAQNSmcfRkc\nDIGBetRwT7BaobxcXXu7FRYqKyjoqKStVnV3Fh8PcXHKRo3qMD1txwVRt7eOgv8qoGpnFfImycr0\nlfgF+/Haa69hMpmc7d554xKhGyHEGClltrF+PzBdSrmki+O00LsCDQ2qQigpUVZaqkIApaVKlOxW\nUaHCRcOHq+HtgYEdFhCgzN+/w/z81AAau/n6qulnXbz1dF7YbOq61dYqq6lRVlWlslnsVlGhrLxc\nXc+aGnW97HdT4eFn32VFR6uwW0CAvsvqRZrzmil4sYCCjQVsMm1iZ+1O3tnyTr/JynEVoX8PGAu0\nAaeA+6SUZV0cp4W+v2GzKbGqqFCiZrfq6g6rqekQv/p6FYaorVXC2NQE3t6qsvDx6TBvb2VeXqoD\n2m6enqqfwdMTPDw6bMgQZe7uytzclDDazY6UymebTU1yZ7fWVtVqbm1V02BYLMqam5WZzcqamqCx\nUS0bGtT3aWhQ27y9VQXm76+E2c9PiXhQUEcFaL8TMpnUXZLJpEMrLoS1wUrpa6W89dRbPF36NA/c\n+ACrNqzCw7frSdRcBZcQ+vP+AC30g4+2NiWSdrG0i6hdVM3ms8W2swi3tnaY1arOZV/aBb3z/0lK\nVQHYKwF7peDufnalYa9I7Esvr44Kp3NF5OurKii7acEeMEgpSX89ncX3LybMHMZzdzxHwm8S8Bn/\n48+2dQZa6DUajeYCaW5u5ld3/orUnamsEWsYP348EfdGYLrJhPsw16nYtdBrNBpND3nllVdYsWIF\nTy1+ihkZM6hPryd0USjhy8IZnuT8jnAt9BqNRuMADhw4wM0338yCBQt4YvkTVL5eScnGEjxCPAi7\nPYyQ20IYanLOOBUt9BqNRuMgqqqqWLRoEWazmbfffpsQUwjVu6op+UcJlR9W4j/Hn9BFoQRdF4S7\n13dDO/86/S/MrWbmJTj2YSiOFPoBkAOn0Wg0F05gYCDbt29n9uzZXHTRRezdv5fAKwMZ//p4Lsm/\nBNONJopfLmZPxB6OLT1G5UeV2FpsHC0/yvw353P3h3c7+yv8KLpFr9FoNAbbtm3jrrvu4k9/+hPL\nli07a5+lyEL5e+VkfpDJusB1fDX+Kx6KeYiHb30Y7+HeDvdFh240Go2ml8jKymLBggXMnTuXF154\ngaHGXFI1zTU8u/tZ/pb2N24ffTt35N2B9QMrDQcbiHs8juiHoh3qhxZ6jUaj6UVqa2tZunQplZWV\nbNq8iXdy3+G5Pc9xfcL1rJ6zmmi/DlFvKWvBWmfFe7RjW/Va6DUajaaXaWppYuHTC9nZtJO5o+fy\n0k0vMc40rs8+X3fGajQaTS9hsVpYt28dY//fWNwT3Hk2+VkOPnaQ/Tv2O9u1C0ZPUajRaDRAs7WZ\nDQc28Ofdf2ZiyETev+V9pkdOB+CapGu44YYbOHToEM888wzu/WxqDB260Wg0g5rGlkb+nvZ3/rLn\nL0wNn8rvZ/2eGVEzvnNcZWUlt9xyC56enmzevBm/Xn4EqA7daDQaTQ+pNlfz5BdPMvLFkXyV/xXb\nf7GdD3/xYZciDxAUFMSOHTsYPXo0M2bMIDs7u489vnC00Gs0mkFFQV0Bj+x8hNEvjeZk1UlSb0/l\n/VveZ0r4lB99r4eHBy+++CIPP/wwl156KZ999lkfeNxztNBrNJpBweHSw9z+we0k/TUJq83KgXsO\n8I8b/kFicGK3z3XPPffw9ttvs2jRItatW9cL3joWHaPXaDQDFpu0sePkDp7f8zxHy4/y6+m/5r5p\n9xHgFeCQ8586dYrrrruOJUuWsGrVKoec045L5dELIX4LrAVMUsqqLvZroddoNH1KvaWeTYc28dLe\nl/Dy8OLhix/m3yb+G0PdHT8TZW1tLdXV1cTFxTn0vI4U+h6lVwohooGrgFxHOKPRaDQ94UTlCdbt\nW8drh1/j8rjL+dv8vzE7djaiF5/N6+fn1+sZOD2lp3n0zwMrgK0/dFDI2hCSQpNICk1icuhkJodN\nJtGUiOcQzx5+vEajGey0trWy/cR21u1fx+HSw9yZfCfp96YT4xfjbNdchp48HHwBMEdK+ZAQ4jSQ\n8n2hm4LaAg6XHuZQ6aH2ZU51DglBCSSHJTMlbArJYckkhyXjP8y/h19Jo9EMBs7UnGH9gfVsTN9I\nfEA8y6ct5+bxNw+YBmSfhW6EEJ8AYV3segxYBVzd+fDvO8/Lz7/cvn7PnHuYs3AO5lYzmeWZHCw5\nSHpxOu8efZfDpYcJ9g5mSvgUpoZNVcvwqYQN78oFjUYz2LBYLWzN2sqG9A3sL9rP4kmL2blkJxND\nJjrbtR6TmppKampqr5z7glr0QoiJwGdAk7EpCigEpkspy8459rw7Y9tsbZysOkl6SToHig+QVpxG\nenE6w4YMIyUihalhU0mJSCElPIUI34hejbtpNBrXQEpJekk6rx58lTcz3iQpNIk7k+/kpsSb8PLw\ncrZ7vYZLZd0A/FjopiefIaUktzZXCX9RGmnFytyFOykRKUwLn6aWEdOI8I3oydfQaDQuRGFdIZsz\nNrPp0CYaWhpYOnkpSycvZWTASGe71ie4otDnANP6Kr1SSkl+XX678O8v2k9acRpD3IaQEq5E3246\n7KPR9B9qmmvYcmwLbxx5gwPFB7gp8SaWJC1hVuws3MTgGt/pckL/gx/QR3n0UkryavPahd9u3h7e\nTIuYdlYFEOwT3Ov+aDSa86OhpYGPTnzE5ozN7Dqzi8vjLmd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"text": [ "" ] } ], "prompt_number": 6 }, { "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": 7 }, { "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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k4oRUsLgEOSGSs5RQ4oQkkvXN4qRI7JpPiGRusBpgsBtgcBhgdBhhsBe8FtsM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\nZ+P45fwv8crMK3hl9hVsJjbxyYFP4tM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"text": [ "" ] } ], "prompt_number": 8 }, { "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": 9, "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": 9 }, { "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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Qo+A6C/E3m8XQ55tvFi39f9d1ZGaSD//xMOdtnsfLn9pE92FpDF4ziMsOLGOV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"text": [ "" ] } ], "prompt_number": 10 }, { "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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xOp/vgnt8wNQVN2pKnbIS4FwscjhwmEx8sOjmS7w9FOLhupy+NBhb895xlSPv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"text": [ "" ] } ], "prompt_number": 11 }, { "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": 12 }, { "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": 13 }, { "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": 14 }, { "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": 15, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 15 }, { "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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D2tq6PAoWAJYsAdaurWgTW1yMQT4++DQgAOliZdMbwzAIGBuAODU+92Ix0KGD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\n1e/p2QxPz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"text": [ "" ] } ], "prompt_number": 16 }, { "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": 16 }, { "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": 17 }, { "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": 18 }, { "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": 19, "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": 19 }, { "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", " K_starstar = compute_kernel(X_test, 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": 20 }, { "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": 21 }, { "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": 22, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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CAKxSgqkJzTwAACiQYJoPAOhcgqlJT5p51NHGKbQVR+pzl3XLNt1oazupdMmVYnxsWV3E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\nfVHSs5ZYr8i8eHHZebbxTyx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"text": [ "" ] } ], "prompt_number": 22 }, { "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": 23, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 23 }, { "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": 24 }, { "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": 25, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 25 }, { "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": 26, "text": [ "" ] } ], "prompt_number": 26 }, { "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": 27 }, { "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": {} } ] }