{ "metadata": { "name": "gauss_high_dim" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Behaviour of the Gaussian distribution with increasing dimension\n", "\n", "This notebook requires the brand new Meijer integrator, so make sure you have the latest release of SymPy." ] }, { "cell_type": "code", "collapsed": true, "input": [ "%pylab inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n", "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n", "For more information, type 'help(pylab)'.\n" ] } ], "prompt_number": 29 }, { "cell_type": "code", "collapsed": true, "input": [ "from sympy import *\n", "mint = integrals.meijerint_definite\n", "x, n, r = s.var('x,n,r')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 30 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'd like to investigate how the cumulative Gaussian distribution\n", "(the integral of the probability density function or PDF) changes over increasing dimension.\n", "The approach here is to, for each dimension, plot the integral of the multivariate PDF over\n", "spheres of different radii.\n", "\n", "The multivariate $n$-dimensional Gaussian PDF is\n", "\n", "$$\\mathcal{N}(\\mathbf{x} | \\mathbf{\\mu}, \\mathbf{\\Sigma}) = \\frac{1}{(2 \\pi)^{n/2}} \\frac{1}{|\\mathbf{\\Sigma}|^{1/2}}\n", "\\exp \\left(\n", "-\\frac{1}{2}(\\mathbf{x} - \\mathbf{\\mu})^\\mathrm{T} \\mathbf{\\Sigma}^{-1} (\\mathbf{x} - \\mathbf{\\mu})\n", "\\right).$$\n", "\n", "Let's work with a standard normal distribution, so that $\\mathbf{\\Sigma}=I$ and $\\mathbf{\\mu}=\\mathbf{0}$, then the expression above becomes\n", "\n", "$$\\mathcal{N}(\\mathbf{x} | \\mathbf{\\mu}, I) = \\frac{1}{(2 \\pi)^{n/2}}\n", "\\exp \\left(\n", "-\\frac{1}{2}\\mathbf{x}^\\mathrm{T} \\mathbf{x}\n", "\\right).$$\n", "\n", "This can also be expressed in spherical coordinates as\n", "\n", "$$p_n(r) = \\frac{1}{(2 \\pi)^{n/2}} e^{-1/2 r^2}.$$\n", "\n", "Our aim is to plot the integral of $p_n(r)$ over different values of $r$, and for different dimensions, i.e.\n", "\n", "$$P_n(x) = \\int_0^x p_n(r) dV_r.$$\n", "\n", "For an $n$-dimensional sphere (or *hypersphere*) the volume element is\n", "\n", "$$dV_r = 2 \\pi^{n/2} \\frac{1}{ \\Gamma\\left(\\frac{n}{2}\\right) } r^{n-1} dr,$$\n", "\n", "so that the full integral becomes\n", "\n", "$$\n", "P_n(x) = \\frac{1}{2^{n/2 - 1}} \\frac{1}{\\Gamma\\left(\\frac{n}{2}\\right)} \\int_0^x e^{-1/2 r^2} r^{n-1} dr.\n", "$$" ] }, { "cell_type": "code", "collapsed": true, "input": [ "f, cond = mint(1/(2**(n/2 - 1)*gamma(n/2))*exp(-S(1)/2*r**2)*r**(n-1), r, 0, x)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [ "R = np.linspace(0, 10)\n", "for k in range(1, 10+1, 2):\n", " plt.plot(R, [f.subs(n, k).subs(x, xx).evalf() for xx in R], label='d = %d' % k)\n", "\n", "plt.legend()\n", "t = plt.title(r'Cumulative Gaussian distribution (dimension n)', fontsize=16)\n", "t.set_y(1.1)\n", "plt.ylabel('$P_n(r)$', fontsize=16)\n", "plt.xlabel('$r$', fontsize=16);" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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lwGefyX7DL1wAWrbUbHliqRizTsxCQnYCoj+IhoOZg3IZvXgBjB8v+zsuTjZW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\nWE7G8QwOTjto+/Y8zc3wwgvWodjWrarKcIqisLC4mA9KStj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