{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Count Data Tutorial" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 28th May 2014" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### by Alan Saul, with edits by Neil Lawrence" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this tutorial we will look at using a Gaussian Process with a Poisson likelihood to model a response variable that is count data, i.e $ Y \\in \\mathbb{N}^{0} = \\{0, 1, 2, 3, ..\\}$.\n", "\n", "This allows us to incorporate an approximate Poisson Process regression model into our Gaussian Process framework, indeed as the Poisson rate parameter, $\\lambda$, becomes large the Poisson Process asymptotically approaches a Gaussian Process. This is important as if your counts are large, the rate, $\\lambda$ at which events is happening is likely to be large, and it may not be required to use the approximation routine used here. Instead it may be more appropriate simply use a Gaussian Process, this will drastically reduce the computation required without a significant loss in accuracy.\n", "\n", "The approximation we will use here is known as the Laplace approximation. The Laplace approximation was applied to Gaussian processes for classification by [Williams and Barber, 1999](http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=735807&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D735807). More details are given in [Rasmussen and Williams, 2006](http://www.gaussianprocess.org/gpml/), Section 3.4 and Section 5.5.1. It has also been suggested for Gaussian random fields by [Rue, Martino and Chopin, 2009](http://hachamam-inla.googlecode.com/hg/r-inla.org/papers/inla-rss.pdf). The Laplace approximation is a local approximation. It proceeds by finding the mode of the posterior distribution and computing the curvature to find the fit.\n", "\n", "First we perform some setup." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "%pylab inline\n", "import numpy as np\n", "import scipy as sp\n", "from scipy import stats\n", "import pylab as pb" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we show the Poisson distribution for several different rate parameters." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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3r2dowlBS+6S2u24o+zK4jE0dy/HG42yu3hzaDQshhI1iOjC010zV3YgRZryk\ngwdDt3+lFNePup7Xv349dBsVQgib2RYYlFL5SqlcpVRBG+ss8va8rTT+8Db/Qmvi4+HMM+Grr0Kx\n51OuHy2BQQgRXWwJDEqpTACtdYn1fLyXdWYC+R6LC6yRW4Oe/+HQiUOs37Oey4Zf5nOas84KfT3D\nlWlXUranjNqjIeokIYQQNrPrimEa4DoTVgKn/W3XWi+zXnNXoLUepbVeE2wG3t32LlkDs+jdrbfP\naexomdSza08uH3E5b1W8FdoNCyGETewKDElAjdvzFB/TJVvFT3OCzYCvzVTd2REYAK4fZVonCSFE\nNLCz8rntHmVeaK2XW8VPKUqp3GB2XlRZRF5GhASG0dfz5tY3aWhqCP3GhRAixOwKDE7MxD4ADqC6\nvQRKqQKllKvOoRpID3Tnew/tZVf9LrIGZvmVLiPDzP989Gige/ZucMJghicO58MdH4Z2w0IIYYMu\nNm33ZcxMbyVAGlAEoJRKamOSn0pgnfU4xZXG04IFC5of5+TkkJOTc9o6H+z4gIuHXkx8XLxfme7a\n1QSHr7+GsWP9StouV+ukS4dfGtoNCyGEh9LSUkpLSwNOr+zqlWs1Oa0E0rXWy61l67TW2dbjKcAy\nYK7W+mlrmeuKIU1r/Vsv29S+5PeXJb+ka3xXFuQs8DvfU6fCTTfB97/vd9I2rd21lttfvZ3Pf/J5\naDcshBDtUEqhtfa5eN+uKwZcwQBz1eBalu32uBAo9EizKhT7Xr9nPT+d8NOA0tpVz5A9KJvqI9VU\n1laS7gi4lEwIIWwXcz2ftdas272OrEH+1S+42NGXASBOxXHdqOuks5sQIuLFXGDYXredbvHdGNR3\nUEDp7bpigDD2gj5+HI4d8+3WIC2nhOjsYi4wBHO1ADB6NFRWwokTIcyUZVL6JD7c+SEHj4dwQKa2\nNDXB3LnQty8kJfl2698fVoWkRE8IEaViLjCs37Oe7IHZ7a/Yih49YNgw2Lo1hJmy9O3el4uHXkxR\npdcGV6F17BjcfDN89BHs3ev7FUNREfzrv5qbXD0I0SnFZGAI5ooB7KtngA7qBV1dDZMmmZEBV6+G\n5OT207hkZ8O6dfD555Cba4KKEKJTianA0Fzx7GfHNk921zP8Y8s/aNJN9uygogImToRLLoHnnzeX\nQP5KSYE33oArrzSB4p//DH0+hRARK6YCw7a6bfTo0oOBfQcGtR07A0OaI43+vfrzya5PQr/xjz4y\nAeH++2FWza6iAAAgAElEQVThQogL4uONi4OHH4ann4YpU+CJJ0BmohOiU4ipwBCKqwWwZ5pPd7a0\nTvrrX+E73zEn8lmzQrfdq6+Gjz82Vx/Tp4d2JiMhRESKqcCwfvd6sgcFXvHsMmaMGRajsTEEmfLi\n+tHX8/qWEAaG3/8e7r4b3nwTrrsudNt1GTEC3nsPHA644AL7LqeEEBEhpgLDuj2huWLo3RsGDICq\nqhBkyouLhlzE9rrt7KjbEdyGGhvhnntg+XL44APICv7YW9WjByxdCvPmweWXw8sv27cvIURYxUxg\n0FqzfnfwLZJc7Kxn6BLXhWtGXsMbW94IfCNHjpiy/w0b4P33Yfjw0GWwLbfdZpq0Pvgg/OIXpq+E\nECKmxExg+Mb5Db269uKMPmeEZHsRXc/w7bemxVDv3qb4KCkptJlrz7hxpknr+vXw//5fx+5bCGG7\nmAkMwfZ49nTWWfYWpV898mre3fYuR04e8S+h1nD77XDhhfDss9C9uz0ZbI/DYXpIP/ssFBa2v74Q\nImrETGAItsezJzuLkgCSeiSRNSiLNVV+Tm/9wguwfTs8/jgovyfJC60BA+CVV+Cuu2DjxvDmRQgR\nMjETGOy4YvjyS3uL0K8fdT2vbfajF/S+fXDfffCXv0C3bvZlzB9ZWaZV1I03Qk1N++sLISJeTAQG\nrTVle8pC0iLJJSkJEhPNVJ92cTVb9XmypLvvNsVI2aG7MgqJGTPge98zYzPJ+EpCRL2YCAxVzip6\nd+tNap/UkG7X7nqG0Smj6dW1F5/u/bT9lVetMsU1Dz9sX4aCsXChuX/ggfDmQwgRtJgIDKHq8ezJ\n7noGpRTfGf2d9lsnVVfDz34Gf/4z9OxpX4aC0aULvPSS6YH9/PPhzo0QIggxERhC1ePZk92BAXzs\nBX3vvWYy6n/5F3szE6zkZPjb30z/hrKycOdGCBEg2wKDUipfKZWrlCpoY51F/qbxJlQ9nj3Z3ZcB\n4JJhl/B19dfsO7TP+wr/+IcZjuKRR+zNSKicdx4sXmzqHPbvD3duhBABsCUwKKUyAbTWJdbz8V7W\nmQnk+5PGm+aK5xC2SHJx1THYOahot/hu5KXnee8FXVcHs2ebIS9697YvE6E2ZQr88IfmKufkyXDn\nRgjhJ7uuGKYBtdbjSmCS5wpa62XWaz6n8aaitoK+3foyoPeAwHPbiv79TdG53XPVtNoLeu5cM7pp\nbq69GbDDr35lphS9995w50QI4Se7AkMS4N6oPcWmNLbVL7h0RD3DNSOvoaSqhOMNx08tXLPGFCM9\n/ri9O7dLfLyphC4qMpXmQoioYWflcyDdcv1Os37PelvqF1w6op6hf+/+nDvgXEq/KTULDh+GggJY\nssR0pohWiYnw6qswf76ZREgIERXsCgxOwDXRsAOotilNyHs8e7K7L4NL/ln5vLDpBfPk3/4NLr7Y\nnrkVOtqYMeaKYcoU2LMn3LkRQvigi03bfRnIBkqANKAIQCmVpLV2+pPG04IFC5ofX3b5ZSHv8ezp\n7LPNcEB2u3XsrYz840jq336ThJdfjq2xh77zHTMa6y23wOrVpphJCGGb0tJSSktLA06vfB6Owd8N\nmyanlUC61nq5tWyd1jrbejwFWAbM1Vo/3Voaj21q9/xuqd7CpGcnse0X22w5BoDdu80o0x3R8vKH\nL0zljw++i+M/n4L8/PYTRJOGBlOJnpdnroiEEB1GKYXW2ueietsCgx08A8NLm15i5RcrWTVtlW37\n1NqMML11K/TrZ9tuAPjmJz/g64/eIG+9ExXukVPtsGuXGXRv5Uq49NJw50aITsPfwBDVPZ/tGgrD\nnVKnRlq1VVkZw1eV8NBNDtbuWmvzzsJk8GAzMuyMGWaYDyFERIrqwGB3iyQX25usNjXB7Nmoxx7j\nppy7WLZ+mY07C7Nrr4Vp08wUoVF0tSpEZ2JX5bPtmnSTbT2ePdneZPV//sf0pLv1Vm47vJ8xT47h\niaufIKF7go07DaNHHjFFSX/4gxlXSXQIrTVVx47xYX09H9bVUXnsmM9pe8XFkdm3Lxf07cuEhAQS\nu0TtqUP4IGo/3a01W3H0cNCvl80F/5jA8NZbNm28vh5++Uv4+99BKVL7pJKblssLG19gdvZsm3Ya\nZt26wYsvwkUXwSWXRN78EjHiSGMjnxw8yId1dXxUX8+H9fV0UYqJCQlMTEzk6uRk4nysy6pvaGDd\nwYP8+7ZtlB86xJDu3bmgb18uSEjggr59Ob9PH7rHRXUBhHATtZXPL258kVVfrqJwmv3zDX/zjTl/\n2TJpz5w5ZuYzt97BqytWM79kPutnrrdhhxFk5Uozf0NZWXR35IsQ+0+coKi2lg/r6viwvp6vjhzh\nvN69mZiYyEUJCUxMSGBo9+5BN2xoaGriiyNHWFtfz9qDB1lbX8+Wo0c5t3dvLrD2852UFPrKVUXE\n6DStku5/63769erH/Evn277fpiZISDCNakJ6/tq82UScTZsg9dQkQ026iYw/ZlA4tbBDisrCavZs\ncDrNFUQstsSymdaaj+vr+dOuXbxRU8OVSUlcnJjIxIQEMvv0oUcH9Rk53NhI2cGDrD14kFKnk/fq\n6vhev37cMXAgFyckxGYruyjSaQJDzjM5PHjpg0zOmNwh+87OhiefhAsvDOFGr73WtO2///7TXvrN\nu79hR/0Olly/JIQ7jEBHj5o39Wc/M8OACJ8cbWzk5f37+dOuXdQ2NPDTwYO5/YwzcHTtGu6sAbD3\n+HGe3bePP1u93X88cCC3pqZyRvfuYc5Z59QpAkOTbsKxyEHlzytJ6eXTWHtBu/VWuOIKM+VySLzx\nhgkIGzaYMncPuw/u5tynzmX7vdvp061PiHYaob76ylRGv/02nHtuuHMT0bYdO8biXbv4y969ZPft\ny92DB/tVV9DRtNZ8WF/PX/bsYdWBA1yWmMiPBw7k2uRkukqdRIfpFP0YtlRvIblncocFBQjxmEkn\nTpjhqJ94wmtQABjUdxCXDb+Mlze9HKKdRrAxY8wostOnw5Ej4c5NxNFaU1xTw40bN5K5bh0ntOb9\n8eP5x/nnc21KSsQGBTAnpIsTE3l6zBh2XHQRN/brx+PbtzP0ww+ZW1HBV4cPhzuLwouoDAzr99g7\n1LY3Ie3L8Ic/wOjRcM01ba5WkFnAsrIY7tPg7kc/gsxM+PnPw52TiNGoNU/v3s3Zn3zCvRUVXJuS\nwvaJE/ndyJGM6tUr3NnzW58uXbh94EDey8zknfHjiQOu+Owzbti4kbKDB8OdPeEmKgNDR/R49hSy\nvgx798KiReZqoR1Xj7ya3Qd389nez0Kw4winFDz1FLz7rqmI7uRKamsZv24dz+3bx5LRo9mQnc3M\nQYPoHSMDEJ7ZqxcLMzKouvBCJjkcfGfjRm7cuJFyCRARISoDQziuGNLSzDk96Cvf+fPhxz+GUaPa\nXTU+Lp47x9/J8rLTxhOMTX37wooV5qphy5Zw5yYsthw5wg0bNzJz82YWjBjB2+PGcXlSUsy26ukR\nH8/Phgyh4sILudLh4HorQHwqASKsoi4wNOkmyveUkzkws0P326ULjBxpWpgGbO1a01POj9FFfzz+\nx7y46UWOnOwkZe/jxsGvfw0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "K = np.arange(0, 30, 1)[:, None]\n", "rates = np.arange(0, 40, 1)[:, None]\n", "\n", "#Make a matrix with PMF for (rate x count) combinations\n", "rates_count = np.array([sp.stats.poisson(r).pmf(K) for r in rates]).reshape(rates.shape[0], K.shape[0])\n", "\n", "rates_plot = [1,3,8,20]\n", "\n", "#Plot each rate as a function of counts K\n", "for r in rates_plot:\n", " pb.plot(K, rates_count[r, :], label='rate = {}'.format(float(rates[r])))\n", "\n", "pb.title('Poisson probability mass distribution for different rates')\n", "pb.ylabel('PMF')\n", "pb.xlabel('Counts')\n", "pb.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As the rate increases, the Poisson distribution over discrete values begins to take on the form of a Gaussian density, over continuous values, as can be seen below." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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VS/5q9xBCnESW9vTSF7u/YFTCqJOSAhhjutavD4DON6+/TmTnGL47fxDvfPeO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\n5/bt48w1a1hy8CAuSRBeaTe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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Ks = [1,5,10,25]\n", "pb.figure()\n", "for k in Ks:\n", " pb.plot(rates, rates_count[:, k], label='K = {}'.format(int(K[k])))\n", "\n", "pb.xlabel('rate')\n", "pb.ylabel('PMF')\n", "pb.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The variance of the Poisson distribution is equal to its mean. The Poisson only has support for non-negative integers, whereas a Gaussian has support over all real numbers (including negatives). This can complicate matters because if we place a Gaussian process prior directly over the rate of the Poisson, then the implication is that the rate can go negative. In practice we normally consider a Gaussian process over the logarithm of the rate. This is known as a log Gaussian process. It is formed by combining a logarithmic link function with a Gaussian model. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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DawxEZJZA1zGsUEoVKaVW6Yuqg5xZdQO0mkUtgFxok+9BKWXxk1RaANTrzzOc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\nnQhQX69d+7BxIzBzptbUdNN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Hk/R0OF0zUa4YRMB0mBi01guAGqXUM+YjO0hxCeGz99+H886D8s9D91ae3khP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\nTspjwACroxHtuWPMHUQOfI+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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "small_rate = 2\n", "K = np.arange(0, 20, 1)[:, None]\n", "Kcont = np.linspace(0, 20, 100)[:, None]\n", "gauss20 = sp.stats.norm(loc=small_rate, scale=np.sqrt(small_rate)).pdf(Kcont)\n", "poisson20 = sp.stats.poisson(small_rate).pmf(K)\n", "pb.plot(K, poisson20, label='Poisson')\n", "pb.plot(Kcont, gauss20, label='Gaussian')\n", "pb.title('Gaussian and Poisson small rate')\n", "pb.ylabel('PDF/PMF')\n", "pb.xlabel('Counts')\n", "pb.legend()\n", "\n", "pb.figure()\n", "large_rate = 40\n", "K = np.arange(0, 80, 1)[:, None]\n", "Kcont = np.linspace(0, 80, 200)[:, None]\n", "gauss40 = sp.stats.norm(loc=large_rate, scale=np.sqrt(large_rate)).pdf(Kcont)\n", "poisson40 = sp.stats.poisson(large_rate).pmf(K)\n", "pb.plot(K, poisson40, label='Poisson')\n", "pb.plot(Kcont, gauss40, label='Gaussian')\n", "pb.title('Gaussian and Poisson large rate')\n", "pb.ylabel('PDF/PMF')\n", "pb.xlabel('Counts')\n", "pb.legend()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/maxz/anaconda/lib/python2.7/site-packages/scipy/stats/_distn_infrastructure.py:1590: RuntimeWarning: divide by zero encountered in true_divide\n", " x = asarray((x-loc)*1.0/scale)\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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kDs/leP0hEnrc4IMPzI5GCHPISDOiW9jsNhIvRU6CSOqZxMikkYybdUyqmUTM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\nb4deE4v50kTPHsuiM/5ySR/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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "rates = np.arange(0, 20, 0.01)[:, None]\n", "small_counts = 1\n", "gauss3 = sp.stats.norm(loc=rates, scale=np.sqrt(rates)).pdf(small_counts)\n", "#gauss3 = np.array([sp.stats.norm(loc=r, scale=np.sqrt(r)).pdf(small_counts) for r in rates])\n", "poisson3 = sp.stats.poisson(rates).pmf(small_counts)\n", "pb.plot(rates, poisson3, label='Poisson')\n", "pb.plot(rates, gauss3, label='Gaussian')\n", "pb.title('Gaussian and Poisson small count (K = {})'.format(small_counts))\n", "pb.ylabel('PDF/PMF')\n", "pb.xlabel('Rate')\n", "pb.legend()\n", "\n", "pb.figure()\n", "rates = np.arange(0, 100, 0.01)[:, None]\n", "large_counts = 40\n", "gauss40 = sp.stats.norm(loc=rates, scale=np.sqrt(rates)).pdf(large_counts)\n", "poisson40 = sp.stats.poisson(rates).pmf(large_counts)\n", "pb.plot(rates, poisson40, label='Poisson')\n", "pb.plot(rates, gauss40, label='Gaussian')\n", "pb.title('Gaussian and Poisson large count (K = {})'.format(large_counts))\n", "pb.ylabel('PDF/PMF')\n", "pb.xlabel('Rate')\n", "pb.legend()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the counts are believed to follow a Poisson distribution and are very large, then the underlying rate is also likely to be large and we might reasonably approximate the Poisson likelihood with a Gaussian likelihood by matching the mean and variance to the observed counts. \n", "\n", "If, however, counts are relatively small, and we wish to fit Poisson process regression into a Gaussian Process framework, we may need a more involved approximation to the posterior.\n", "\n", "The approximation used in this tutorial is called the Laplace approximation, it uses the location of the mode of the marginal distribution and the curvature of this mode (the hessian) to find a more appropriate Gaussian posterior approximation. As we shall see, in practice this is quite effective, and future tutorials will show us how it can be used to make Gaussian Process approximations to a range of different likelihoods, rather than just the Poisson, with relative ease.\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Approximate Poisson Process Regression\n", "======================================\n", "\n", "Poisson process regression, contrasting to the above example, allows the rate to change, for example over time. That is the intensity of the counts can change, initially perhaps the data suggests there were few counts due to a low number of counts, but later this intensity is assumed to rise as we start observing larger more frequent counts. \n", "\n", "An example of data that might follow this change in 'intensity' is the times at which buses come throughout the day, we will choose this as our toy modelling example.\n", "\n", "First we will generate some toy data, we will say that there are very few buses overnight, thus low counts, then during the morning rush-hour the number of observed counts rises, it then reduces and plateus until the evening rush-hour, and then finally as the evening draws on we observe again a reduced number " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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OhxEOh1N+v5AmijkIIUYBeBTAcCnlqqjnpJk2vMTbbwPTpwPTpqlW4k1WrQJG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\nAUd71gAwbx6tOr1iBXDOOcl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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "X = np.linspace(0,23.50,48)[:, None]\n", "intensities = 4 + 3*np.sin(10 + 0.6*X) + np.sin(0.1*X)\n", "pb.plot(X, intensities)\n", "pb.title('Real underlying intensities (rate at which buses are issued)')\n", "\n", "Y = np.array([[sp.random.poisson(intensity) for intensity in intensities]]).T\n", "pb.figure()\n", "pb.bar(X, Y, width=0.5)\n", "pb.xlabel('Time')\n", "pb.ylabel('Buses in half an hour')\n", "pb.title('Observed (noisey) counts of buses throughout day')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will now fit a Gaussian Process to this data using a Poisson likelihood and a Laplace approximation using GPy, as we shall see this is relatively simple, however to use Gaussian Processes we must make some assumptions about the intensity, we do this by defining a kernel." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import GPy\n", "kernel = GPy.kern.RBF(1, variance=1.0, lengthscale=1.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By defining a radial basis function (RBF) kernel (also known as a exponentiated quadratic, squared exponential or Gaussian kernel), we are making *a priori* a smoothness assumption about the intensity parameter throughout time in this case. This is a very general assumption that is required in many learning algorithms, that data 'close' to one another shares similar properties. In this case we are saying that times that are close to one another have a similar number of buses arriving. \n", "\n", "The measure of closeness is defined by the lengthscale (or timescale) parameter, $\\ell$, initially we set this to be 2.0, roughly speaking this says that as we have moved two time-steps (1 hour in this case) the intensity shares almost nothing with the intensity a hour previous.\n", "\n", "The variance parameter describes our prior assumptions about the scale of the function being learnt. Both of these parameters although initialised here are usually optimized to find the most appropriate parameters, as such we set them to something that seems sensible but do not need to worry about having them absolutely correct.\n", "\n", "There are a variety of likelihoods implemented in GPy, type 'GPy.likelihoods.' then press tab to list them.\n", "\n", "There is also a number of alternative inference methods implemented, a subject for another tutorial, however it is worth noting that not all likelihoods are yet compatible with all inference methods. \n", "\n", "We now choose the type of likelihood and inference method we would like to use, in this case the Poisson likelihood and Laplace inference method. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ "poisson_likelihood = GPy.likelihoods.Poisson()\n", "laplace_inf = GPy.inference.latent_function_inference.Laplace()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally we are ready to create our model Gaussian Process model" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Name : gp\n", "Objective : 121.214706928\n", "Number of Parameters : 2\n", "Number of Optimization Parameters : 2\n", "Updates : True\n", "Parameters:\n", " \u001b[1mgp. \u001b[0;0m | value | constraints | priors\n", " \u001b[1mrbf.variance \u001b[0;0m | 1.0 | +ve | \n", " \u001b[1mrbf.lengthscale\u001b[0;0m | 1.0 | +ve | \n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Iv+TC5fQNjHHoxCjnv/rN3HnnLUB14rFc9fJU4sZPJn6/g5sym7pmBJPPRkut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\nxiNxrDods+ngnFVz+ecPvpHz1vQwFjX41j07+cT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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "m = GPy.core.GP(X=X, Y=Y, likelihood=poisson_likelihood, inference_method=laplace_inf, kernel=kernel)\n", "print m\n", "\n", "#Predictions of counts of buses, including noise arising from Poisson noise\n", "m.plot()\n", "pb.xlabel('Time')\n", "pb.ylabel('Buses counts per 30 minutes')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we can see, the model as it stands is not all that great, the lengthscale parameter is soo small. We noted before that the kernels hyperparameters we chose were not the correct ones, only an initial guess. Next we try maximizing the likelihood of the model with respect to the parameters. Optimization within GPy is straightforward." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Name : gp\n", "Objective : 113.686968138\n", "Number of Parameters : 2\n", "Number of Optimization Parameters : 2\n", "Updates : True\n", "Parameters:\n", " \u001b[1mgp. \u001b[0;0m | value | constraints | priors\n", " \u001b[1mrbf.variance \u001b[0;0m | 1.5390223923 | +ve | \n", " \u001b[1mrbf.lengthscale\u001b[0;0m | 3.61055611736 | +ve | \n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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8GcR2UmkI2xmcGOYmahZ+NTWZSSU6aOOQ17fkT5zJg8S9iIU2uGyEZbt5Bl6P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\nkDGKAABk7P+LwdpIC+CKqwAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "m.optimize()\n", "print m\n", "\n", "#Predictions of counts of buses, including noise arising from Poisson noise\n", "m.plot()\n", "pb.xlabel('Time')\n", "pb.ylabel('Buses counts per 30 minutes')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we should have a clearly better fit to the data, we have made predictions of the number of counts we would see throughout the day with our model and it seems to fit the training data we have provided quite well.\n", "\n", "To get a better indication of the intensity we have infered we can plot the latent function $f$ of the model, this is the models prediction, before we have added Poisson noise to it. In fact the model is modelling the log intensity, this is what is displayed below" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(-2, 3)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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8vr+RGocfqzV5hKtwt6DfdSStwqsf7MEfjHDihNK4mAsAKSPYrZlb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\nRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#Posterior distribution of the intensity of buses, and true intensity in red\n", "m.plot_f()\n", "pb.plot(X, np.log(intensities), '--r', linewidth=2)\n", "pb.title('Real log intensity vs posterior belief')\n", "pb.xlabel('Time')\n", "pb.ylabel('Log intensity')\n", "plt.ylim(-2,3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get the predictions from the model is straightforward with the plot functionality." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Fklq/bGqm47XVJL5+fcE0vromsBxHaXKDhGHZiAIGftHZgb6xtwB+5IUPAbj8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\nY+xojNIU0QlHDcLRBDHdRDcsEoaFYdrOspLJ8UmRHLOUSA4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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pred_points = np.linspace(0,24,100)[:, None]\n", "#Predictive GP for log intensity mean and variance\n", "f_mean, f_var = m._raw_predict(pred_points)\n", "f_upper, f_lower = f_mean + 2*np.sqrt(f_var), f_mean - 2.*np.sqrt(f_var)\n", "pb.plot(X, intensities, '--r', linewidth=2, label='true intensity')\n", "#Plotting Y on an exponential scale as we are now looking at intensity rather than log intensity\n", "from GPy.plotting import Tango\n", "Tango.reset()\n", "b = Tango.nextMedium()\n", "pb.plot(pred_points, np.exp(f_mean), color=b, lw=2)\n", "pb.fill_between(pred_points[:,0], np.exp(f_lower[:,0]), np.exp(f_upper[:,0]), color=b, alpha=.1)\n", "pb.title('Real intensity vs posterior belief')\n", "pb.xlabel('Time')\n", "pb.ylabel('intensity')\n", "pb.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.11" } }, "nbformat": 4, "nbformat_minor": 0 }