{ "metadata": { "name": "WishartMatrix" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": "## Wishart Matrix and Marchenko Pastur Distribution\n\nA Wishart matrix is a $N \\times N$ random matrix of the form $W = XX^H$, where X is an $N \\times L$ random matrix with IID entries. A Wishart matrix is parameterized by $c = \\frac{N}{L}$. \nThe limiting eigenvalue denisty function of a Wishart matrix as $N, L \\rightarrow \\infty$ and $\\frac{N}{L} \\rightarrow c$ is given by the **Marchenko Pastur** density function.\n" }, { "cell_type": "markdown", "metadata": {}, "source": "Marchenko Pastur Distribution is given by\n\n
\n $f(x) = \\frac{1}{2\\pi c}\\sqrt{(a - x)(x - b)}$\n
\n\nwhere $a = (1 + \\sqrt{c})^2$ and $b = (1 - \\sqrt{c})^2$ are the upper and lower bounds of $f(x)$." }, { "cell_type": "code", "collapsed": false, "input": "def marcenkopasturpdf(x, c):\n # Marchenko Pastur Density Function for c > 1\n ub = (1 + sqrt(c))**2\n lb = (1 - sqrt(c))**2\n mp = sqrt((x - lb)*(ub - x))/(2*pi*c*x)\n return (mp)", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 74 }, { "cell_type": "markdown", "metadata": {}, "source": "Example below shows a case of Wishat matrix obtained from a random matrix of size $N \\times L$ with Complex Gaussian IID entries. The parameter $c$ is set to a fixed value and for a choice of value for $N$, $L = N/c$. \nA histogram of the eigenvalues of the computed Wishart matrix is compared with the actual Marchenko Pastur density for a given $c$." }, { "cell_type": "code", "collapsed": false, "input": "N = 256\nc = 0.8\nL = N/c\n\n# Random matrix realization\nX = sqrt(0.5)*(randn(N, L) + 1j*randn(N, L))\nWc = dot(X, conj(X.T))/L\nD, U = eig(Wc)\nhist(D, 40, normed=True)\n\nmp = marcenkopasturpdf(arange(lb, ub, 0.01), c)\nplot(arange(lb, ub, 0.01), mp, linewidth = 2, color='red')\nxlim((lb,ub))\nylabel('f(x)')\nxlabel('Eigenvalue')\n", "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 76, "text": "" }, { "output_type": "display_data", "png": 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QvTtx+fxzdLm56BIScEtIwBwYSN1vfkNdfDx1V10F3t6OrlwIYUfqDAHpCXSK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\n0dvV+fRh/6Cx7B80FgCt2cS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