Christian Forssen
87 lines
3.2 KiB
Plaintext
87 lines
3.2 KiB
Plaintext
{
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"<!-- dom:TITLE: Data Analysis and Machine Learning: Machine learning with Gaussian Processes -->\n",
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"# Data Analysis and Machine Learning: Machine learning with Gaussian Processes\n",
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"<!-- dom:AUTHOR: Christian Forssén at Department of Physics, Chalmers University of Technology, Sweden -->\n",
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"<!-- Author: --> \n",
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"**Christian Forssén**, Department of Physics, Chalmers University of Technology, Sweden \n",
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"<!-- dom:AUTHOR: Morten Hjorth-Jensen at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University -->\n",
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"<!-- Author: --> **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
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"\n",
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"Date: **Mar 19, 2018**\n",
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"\n",
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"Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
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"\n",
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"\n",
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"\n",
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"## What is a Gaussian Process?\n",
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"\n",
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"* We have considered splines and kernel regression methods. These\n",
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"\n",
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"require choice of somewhat arbitrary set of knots.\n",
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"\n",
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"* Antoher possibility is to setup a prior distribution for the\n",
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" regression function using a *Gaussian Process*.\n",
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"\n",
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"* This is a very flexible class of models that has distinct computational\n",
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" and theoretical advantages. It can be viewed as a potentially\n",
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" infinite-dimensional generalization of Gaussian distributions.\n",
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"\n",
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"* See the excellent (and free) book [Gaussian Processes for Machine\n",
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" Learning](http://www.gaussianprocess.org/gpml/) by Carl Edward\n",
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" Rasmussen and Christopher K. I. Williams. \n",
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"\n",
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"## Gaussian process regression\n",
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"\n",
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"* Realizations from a Gaussian process correspond to random functions\n",
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"\n",
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"* Let us first consider an unknown regression function $\\mu(x)$ that\n",
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" depends on a single, continuous variable $x$.\n",
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"\n",
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"* The Gaussian process is written as $\\mu \\sim \\mathrm{GP}(m,k)$, and\n",
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" is parametrized in terms of a mean function $m(x)$ and a covariance\n",
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" function $k(x,x')$.\n",
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"\n",
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"* The GP prior on $\\mu$ describes it as a random function for which\n",
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" the values at any set of $N$ prespecified points $\\{x_i\\}_{i=1}^N$\n",
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" are a draw from a $N$-dimensional normal distribution"
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]
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"$$\n",
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"$$\n",
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" \\mu(x_1), \\ldots \\mu(x_N) \\sim \\mathrm{N}\\left( \\left( m(x_1),\n",
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" \\ldots, m(x_N) \\right), K(x_1, \\ldots, x_N) \\right),\n",
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"$$\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"with mean $m$ and covariance $K$.\n",
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"\n",
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"## Topics\n",
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"* More matematical details\n",
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"\n",
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"* The role of the covariance function (different kernels)\n",
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"\n",
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"* multidimensional case\n",
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"\n",
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"* examples."
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]
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}
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