Spacerat/rand.ipynb

## rand.ipynb
{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Building increasing non-uniform random distributions using the uniform distribution ##\n",
      "\n",
      "Let U(x) is the uniform distribution - here we use 10000 samples."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def U(): return uniform(0, 1, size=10000)\n",
      "hist(U(), histtype='bar')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x109a17750>"
       ]
      }
     ],
     "prompt_number": 222
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### For a skewed distribution, you can simply multiply uniform distributions ####\n",
      "\n",
      "This is what people in this thread have suggested with $rand() * rand()$.\n",
      "\n",
      "$P(x) = 1 - (U(x) * U(x)) = -log(1 - x)$\n",
      "\n",
      "See http://mathworld.wolfram.com/UniformProductDistribution.html\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "squared = U() * U()\n",
      "arr = 1 - squared\n",
      "d = hist(arr, bins=50, histtype='bar')\n",
      "plot(linspace(0, 1, 50), -log(linspace(1, 0, 50)) * 250, 'g')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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N7zVN3o/C34p9qZ5V9RT+4nEMwyCzOJMNX2/g30f/zadZn9KxbUdib4pl9E2j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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109af7850>"
       ]
      }
     ],
     "prompt_number": 228
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### To get a ramp distribution, start with a triangular one ####\n",
      "\n",
      "Sum of two uniform distributions\n",
      "\n",
      "$T(x) = (U(x) + U(x)) / 2$\n",
      "\n",
      "Triangular distribution, mode = 0.5"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "triangular = ((U() + U()))/2\n",
      "hist(triangular, bins=50, histtype='bar')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10945c5d0>"
       ]
      }
     ],
     "prompt_number": 223
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### You can keep rolling the dice until you get a value less than the mode ####\n",
      "$ramp1(x): 2*T(x)$ with values of x>1 rejected."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ramp = array([v for v in triangular if v <= 0.5]) * 2\n",
      "hist(ramp, bins=50, histtype='bar')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1099f9890>"
       ]
      }
     ],
     "prompt_number": 224
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Or just subtract your number from the maximum if you get a value more than the mode ####\n",
      "\n",
      "$ramp2(x)$: Subtract 1 from $T(x)$ if $T(x) > 0.5$, then multiply by 2"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ramp2 = array([( 1 - v if v > 0.5 else v) for v in triangular]) * 2\n",
      "hist(ramp2, bins=50, histtype='bar')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10985de10>"
       ]
      }
     ],
     "prompt_number": 225
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}