{ "metadata": { "name": "", "signature": "sha256:79d0171f36c5c9bf7e262641fbb3eab58bc2c6f5e2f53b12c7e716681fadee83" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Sampling \u2014 exercises\n", "\n", "1. Consider the power-law distribution $p(S|\\alpha)\\propto S^{-\\alpha}$ from yesterday's problems. \n", "\n", " 1. Generate samples, $S_i$ from this distribution (for some fixed value of $\\alpha$), using rejection sampling or otherwise (but if your computer has a mechanism for directly generating power-law samples, please don't use that!). [In this case, do we need to know the normalization constant?] \n", " 1. In this case, do we need to know the normalization constant?\n", "\t 2. Do you need to make any additional assumptions?\n", " 1. Determine the mean and variance of the samples and check against an analytical calculation. \n", " 1. Plot the distribution and make a histogram of the samples." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division\n", "import math\n", "import numpy as np\n", "import scipy as sp\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this case, we'll also need to pick a maximum flux $S_\\textrm{max}$ in order to choose a well-defined uniform distribution as input to our rejection sampler.\n", "\n", "#### Notes:\n", "* If we really want to draw samples over the whole region out to arbitrarily large values of $S$, we should use a different technique -- see below.\n", "* Power-law distributions can have a lot of probability in the tails. This means that the variance will diverge if $\\alpha<2$, and that an $S_\\textrm{max}$ cutoff may miss important samples." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def powerlaw(s, alpha=1.5, s0=1):\n", " return (1/s0)*(alpha-1)*(s/s0)**-alpha\n", "\n", "### we know that it diverges a maximum as S->0; need to pick a minimum s0. \n", "### Units don't matter, so pick S0=1\n", "s0 = 1\n", "\n", "sarr = np.linspace(s0, 20*s0, 50)\n", "plt.plot(sarr, powerlaw(sarr))\n", "plt.xlabel(\"S\")\n", "plt.ylabel(\"p(S)\")\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 2, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAENCAYAAADpK9mHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nYoQAAESMEACAiBECABAxQgAAIvb/Ka5tpuKIRicAAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "def u_1(x, xmin=1, xmax=20):\n", " return 0 if (xxmax) else 1.0/(xmax-xmin)\n", "u = np.vectorize(u_1, otypes=[np.float], excluded={'xmin', 'xmax'})\n", "\n", "smax = 20\n", "pmax = powerlaw(s0)\n", "scale = pmax*smax\n", "\n", "sarr = np.linspace(s0,smax,100)\n", "plt.plot(sarr, powerlaw(sarr), label=\"power-law\")\n", "plt.plot(sarr, scale*u(sarr,xmin=s0,xmax=smax), label=\"%3.1f*uniform\"%scale)\n", "plt.fill_between(sarr, scale*u(sarr,xmin=s0,xmax=smax),powerlaw(sarr), \n", " hatch=\"/\", facecolor=\"w\")\n", "plt.ylim(0,pmax*1.1)\n", "plt.xlabel(\"$S$\")\n", "plt.ylabel(\"$p(S)$\")\n", "plt.legend()\n", "plt.figure()\n", "\n", "def rejection_sample(p, xlim=(-1,1), pmax=0.9):\n", " \"\"\" \n", " use rejection sampling to get samples from p(x), using uniform samples\n", " \"\"\"\n", "\n", " delx = xlim[1]-xlim[0] #### range of x\n", " scale = delx*pmax \n", "\n", " keep = True\n", " while keep: ###\u00a0loop until you're meant to keep a sample\n", " #### generate a sample from u: np.random.random generates from U(0,1)\n", " u_sample = delx*np.random.random()+xlim[0] \n", " \n", " fraction_to_keep = p(u_sample)/(scale*u(u_sample))\n", " keep = np.random.random()>fraction_to_keep \n", " \n", " return u_sample\n", "\n", "\n", "ssamples = np.array([rejection_sample(powerlaw, xlim=(s0,smax),pmax=pmax) \n", " for _ in range(10000)])\n", "plt.hist(ssamples, normed=True, bins=40)\n", "plt.plot(sarr, powerlaw(sarr))\n", "plt.ylabel(\"p(s)\")\n", "plt.xlabel(\"s\")" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 3, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAEJCAYAAAByupuRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIcQURiEghBBTGIWAEEJMYRQCQggxhfn/czZLyvaLph8AAAAASUVORK5CYII=\n", "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAENCAYAAADpK9mHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n", "text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "As noted above, this is probably not the best solution for this problem, as it requires specifiying a maximum flux, $S_\\textrm{max}<\\inf$.\n", "\n", "In this case, we can sample from the whole range by a change of variables. Consider the random variable $y=y(S)$ which we wish to have a uniform distribution $u(y)$ over $(0,1)$, so $u(y)=1$ over that range, $u(y)=0$ otherwise:\n", "$$\n", " p(S)\\; dS = u(y)\\; dy = dy\n", "$$\n", "We can integrate this from $S=S_0$ corresponding to $y=0$:\n", "$$\n", " \\int_{S_0}^S (\\alpha-1) \\left(\\frac{S'}{S_0}\\right)^{-\\alpha} \\; \\frac{dS'}{S_0} = \\int_0^y \\;dy'\n", "$$\n", "or\n", "$$\n", " 1-\\left(\\frac{S}{S_0}\\right)^{1-\\alpha} = y\n", "$$\n", "so we can solve for $S=S(y)$:\n", "$$\n", " S = S_0\\left(1-y\\right)^{\\frac{1}{1-\\alpha}} \n", "$$\n", "But this is just what we need: we can easily sample $y$ from $u(y)$ and then just calculate $S=S(y)$ from this formula. (As an aside, there's a slight simplification \u2014 if $y$ is from $u(0,1)$, then so is $1-y$, so we can just use $S = S_0y^{\\frac{1}{1-\\alpha}}$.\n", "\n", "Note that even in this case we'll need to pick and $S_\\textrm{max}$ for plotting purposes." ] }, { "cell_type": "code", "collapsed": false, "input": [ "alpha = 1.5\n", "S0 = 1\n", "expt = 1.0/(1.0-alpha)\n", "Smax=30\n", "nsamp = 10000\n", "ysamples = np.random.random(nsamp)\n", "ssamples = S0*ysamples**expt\n", "print \"Max and min: %f, %f\" % (ssamples.max(), ssamples.min())\n", "if Smax" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also change variables in another way Consider the quantity $y=\\ln(S/S_0)$, or $S=S_0e^y$. This quanitity has the distribution\n", "$$\n", "p(y|\\alpha)\\;dy = p(S|\\alpha)\\;dS = p(S=S_0e^y|\\alpha) \\frac{dS}{dy} dy = S_0^{1-\\alpha} e^{(1-\\alpha) y} \\propto e^{-(\\alpha-1) y}\n", "$$\n", "This is just an exponential distribution, which your package may be able to sample from directly." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2. Consider the bivariate distribution that is uniform between -1 and 1 for the quantity $x-y$ and a (univariate) normal with mean 0 and variance 1 for the quantity $x+y$.\n", " 1. How can we draw samples (pairs of random numbers) from this distribution using univariate uniform and normal random number generators?\n", " 1. Estimate the mean and covariance from the samples.\n", " 2. Plot the results in 2d, as well as the 1d marginals, with an appropriate color scheme.\n", " 3. Overlay the contours of the approximate gaussian with the estimated mean and covariance. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution\n", "\n", "We can define $v=x-y$ with a $U(-1,1)$ distribution and $w=x+y$ with an $N(0,1)$ distribution, and draw samples for each of these:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "nsamp = 10000\n", "\n", "def p_gaussian(x, mu=0, sigma=1): \n", " \"\"\" a univariate gaussian distribution \"\"\"\n", " return (2*math.pi*sigma)**(-0.5)*np.exp(-0.5*(x-mu)**2/sigma )\n", "\n", "def p_multigaussian(x, mu, covar):\n", " \"\"\" multivariate gaussian PDF \"\"\"\n", " \n", " detC = np.linalg.det(2*math.pi*covar)\n", " delx = x-mu\n", " Cinv_mu = np.linalg.solve(covar, delx)\n", " chi2 = np.dot(delx, Cinv_mu)\n", " \n", " return detC**(-0.5)*np.exp(-chi2/2.0)\n", "\n", "\n", "vsamples=np.random.uniform(-1,1,size=nsamp)\n", "wsamples=np.random.normal(loc=0, scale=1, size=nsamp)\n", "probabilities = p_gaussian(wsamples, mu=0, sigma=1)*0.5 #### 0.5 is the value of the U(0,1) distribution" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "But these are simply related to $x$ and $y$:\n", "$$\n", " x = (v+w)/2 \\qquad \\textrm{and} \\qquad y=(w-v)/2\n", "$$\n", "so we can just convert. (Note that we don't care about the Jacobian of the transformation since it is just a constant, irrelevant for us here.)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "xsamples = 0.5*(vsamples+wsamples)\n", "ysamples = 0.5*(wsamples-vsamples)\n", "\n", "nsamples = np.vstack((xsamples, ysamples))\n", "\n", "sorted_probabilities = np.sort(probabilities)\n", "\n", "n = len(probabilities)\n", "alphas = [0.683, 0.954, 0.9973] \n", "q_levels = [sorted_probabilities[np.round((1-a)*n)] for a in alphas]\n", "colors = np.zeros_like(probabilities)\n", "for col, lev in enumerate(q_levels):\n", " colors[probabilities" ] } ], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "with plt.rc_context(rc={'figure.figsize': (10.0, 10.0)}):\n", " plt.figure()\n", " plt.axes().set_aspect('equal');\n", " \n", " plt.subplot(2,2,3)\n", " plt.scatter(nsamples[0,:], nsamples[1,:], c=colors, marker='.', lw=0 ) \n", " plt.contour(xi, yi, zarr, levels=-0.5*(dchi2)*2)\n", " plt.xlabel(\"$x$\")\n", " plt.ylabel(\"$y$\")\n", "\n", " ### save the 2-d x and y limits for use in the histograms\n", " xlim=plt.xlim()\n", " ylim=plt.ylim()\n", " \n", " plt.subplot(2,2,4)\n", " plt.hist(nsamples[1,:],normed=True, bins=20, orientation='horizontal')\n", " plt.xlabel(\"$p(y)$\")\n", " plt.ylim(ylim)\n", " \n", " plt.subplot(2,2,1)\n", " plt.hist(nsamples[0,:],normed=True, bins=20)\n", " plt.ylabel(\"$p(x)$\")\n", " plt.xlim(xlim)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAmIAAAJdCAYAAACRehueAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n", "text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that in this case, there is a slightly simpler way to do this, since the $(v,w)$ plane is just a 45 degree rotation from $(x,y)$, but the somewhat more general problem where $(v,w)$ are related to $(x,y)$ by an arbitrary linear transformation couldn't be solved this way." ] } ], "metadata": {} } ] }