week10/Lab10.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "#  Monte Carlo Errors and other estimation methods\n",
    "\n",
    "##### Keywords: monte-carlo, integration,  central limit theorem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The basic idea\n",
    "\n",
    "Let us formalize the basic idea behind Monte Carlo Integration in 1-D.\n",
    "\n",
    "Consider the definite integral:\n",
    "\n",
    "$$ I = \\int_{a}^{b} g(x) \\, dx $$\n",
    "\n",
    "Then it can be written as :\n",
    "\n",
    "$$ I =  \\int_{a}^{b}(b-a)  \\frac{g(x)}{(b-a)} = (b-a) \\int_{a}^{b}  \\frac{g(x)}{(b-a)} \\, dx $$\n",
    "\n",
    "Recall that uniform distribution between a and be has the probability density function:\n",
    "\n",
    "$$ f_{ab}(x) = \\frac{1}{b-a},x\\in [a,b]$$\n",
    "\n",
    "\n",
    "So the above integral $I$ can be written as:\n",
    "\n",
    "$$ I = (b-a) E_{U[a,b]}[g(x)]$$\n",
    "\n",
    "Note that the expected value of $f(x)$ where $x~U[a,b]$ is calculated as\n",
    "$$E_{U[a,b]}[g(x)] := \\int_{a}^{b} g(x)  f_{U[a,b]} dx$$\n",
    "\n",
    "And central limit says that:\n",
    "\n",
    " $$E_{U[a,b]}[g(x)]= \\lim_{n \\to \\infty} \\frac{1}{N}\\sum_{x_i \\sim U} g(x_i)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example. \n",
    "\n",
    "**Calculate the integral $ I= \\int_{2}^{3} [x^2 + 4 \\, x \\,\\sin(x)] \\, dx. $**\n",
    "\n",
    "We know from calculus that the anti-derivative is \n",
    "$$ x^3/3 + 4\\sin(x) -4x\\cos(x). $$ \n",
    "\n",
    "To solve this using MC, we simply draw $N$ random numbers from 2 to 3 and then take the average of all the values $f(x)=x^2 + 4 \\, x \\,\\sin(x)$ and normalized over the volume; this case the volume is 1 (3-2=1). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**2 + 4*x*np.sin(x) \n",
    "\n",
    "def intf(x): \n",
    "    return x**3/3.0+4.0*np.sin(x) - 4.0*x*np.cos(x) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Monte Carlo estimation= 11.8151950975 Exact number= 11.8113589251\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/ipykernel_launcher.py:29: RuntimeWarning: invalid value encountered in double_scalars\n"
     ]
    },
    {
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jH3Nu4fbtcwLJypVOHcm6dc79G284RV6uggJnaPdx45zb6NHObdQo575fv8ym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\nGim0+AJU18WWa7vHyPNohVO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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10628ce80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 2;    \n",
    "b = 3; \n",
    "\n",
    "# use N draws \n",
    "N= 10000\n",
    "\n",
    "X = np.random.uniform(low=a, high=b, size=N) # N values uniformly drawn from a to b \n",
    "Y =f(X)   # CALCULATE THE f(x) \n",
    "V = b-a\n",
    "Imc= V * np.sum(Y)/ N;\n",
    "\n",
    "exactval=intf(b)-intf(a)\n",
    "\n",
    "print(\"Monte Carlo estimation=\",Imc, \"Exact number=\", intf(b)-intf(a))\n",
    "# to store MC estimate for each N\n",
    "Imc=np.zeros(1000)\n",
    "\n",
    "# just to store N\n",
    "Na = np.linspace(0,1000,1000)\n",
    "\n",
    "# calculus result\n",
    "exactval= intf(b)-intf(a)\n",
    "\n",
    "# for each N run monte carlo to estimate integral\n",
    "for N in np.arange(0,1000):\n",
    "    X = np.random.uniform(low=a, high=b, size=N) # N values uniformly drawn from a to b \n",
    "    Y =f(X)   # CALCULATE THE f(x) \n",
    "\n",
    "    Imc[N]= (b-a) * np.sum(Y)/ N;\n",
    "    \n",
    "    \n",
    "plt.plot(np.arange(10,1000),abs(Imc[10:]-exactval), alpha=0.7)#np.sqrt((Imc[10:]-exactval)**2), alpha=0.7)\n",
    "plt.plot(np.arange(10,1000), 1/np.sqrt(Na[10:]), 'r')\n",
    "plt.xlabel(\"N\")\n",
    "plt.ylabel(\"|Imc-ExactValue|\")\n",
    "\n",
    "# \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obviously this depends on the number of $N$ as $1/\\sqrt{N}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Errors in MC\n",
    "\n",
    "Monte Carlo methods yield approximate\n",
    "answers whose accuracy depends on the number of draws.\n",
    "\n",
    "So far, we have used our knowledge of the exact value  to determine that the\n",
    "error in the Monte Carlo method approaches zero as approximately $1/\\sqrt{N}$ for large $N$, where $N$ is the number of trials. \n",
    "\n",
    "But in the usual case, the exact answer is unknown. Why do this otherwise?So, lets repeat the same evaluation $m$ times and check the variance of the estimate. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example. The same as before, but we fix the number of draws N = 10,000 and   repeat it 1000 times\n",
    "\n",
    "**Calculate the integral $ I= \\int_{2}^{3} [x^2 + 4 \\, x \\,\\sin(x)] \\, dx. $**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean: 11.8112880207 , std: 0.00421757811173\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x106ae0eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# multiple MC estimations\n",
    "m=1000\n",
    "N=10000\n",
    "Imc=np.zeros(m)\n",
    "\n",
    "\n",
    "for i in np.arange(m):\n",
    "    \n",
    "    X = np.random.uniform(low=a, high=b, size=N) # N values uniformly drawn from a to b \n",
    "    Y =f(X)   # CALCULATE THE f(x) \n",
    "    Imc[i]= (b-a) * np.sum(Y)/ N;\n",
    "    \n",
    "    \n",
    "plt.hist(Imc, bins=30)\n",
    "plt.xlabel(\"Imc\")\n",
    "print('mean:',np.mean(Imc), ', std:',np.std(Imc))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks like our telltale Normal distribution.\n",
    "\n",
    "This is not surprising\n",
    "\n",
    "### Estimating the error in MC integration using the CLT.\n",
    "\n",
    "We know from the CLT that if $x_1,x_2,...,x_n$ be a sequence of independent, identically-distributed (IID) random variables from a random variable $X$, and that if $X$ has the finite mean $\\mu$ AND finite variance $\\sigma^2$. \n",
    "\n",
    "Then, \n",
    "\n",
    "$$S_n = \\frac{1}{n} \\sum_{i=1}^{n} x_i ,$$\n",
    "\n",
    "**converges to a Gaussian Random Variable with mean $\\mu$ and variance $\\sigma^2/n$ as $n \\to \\infty$:**\n",
    "\n",
    "$$ S_n \\sim N(\\mu,\\frac{\\sigma^2}{n}) \\, as \\, n \\to \\infty. $$\n",
    "\n",
    "\n",
    "This is true *regardless* of the shape of $X$, which could be binomial, poisson, or any other distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sums\n",
    "\n",
    "$$S_n(f) = \\frac{1}{n} \\sum_{i=1}^{n} f(x_i) $$\n",
    "\n",
    "are exactly what we want to calculate for Monte-Carlo Integration(due to the MLE) and correspond to the random variable f(X) where X is uniformly distributed on the support.\n",
    "\n",
    "Whatever the original variance of f(X) might be, we can see that the variance of the sampling distribution of the mean goes down as $1/n$ and thus the standard error goes down as $1/\\sqrt{n}$ as we discovered when we compared it to the exact value as well.\n",
    "\n",
    "Why is this important?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## We need to fix the total samples (N = 10000) for each experiment. \n",
    "## Repeat doing the same experiment to get a more accurate answer \n",
    "## as approximation of the integral\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparing to standard integration techniques\n",
    "### Rectangle Rule\n",
    "\n",
    "If we divide the $a, b$-interval into $N$ \n",
    "steps and use some regular integration routine, what is the error? Consider the Rectangle Rule as illustrated in this diagram from Wikipedia:\n",
    "\n",
    "![](https://upload.wikimedia.org/wikipedia/commons/5/5b/Rectangle_rule.gif)\n",
    "The basic idea is that the function value at the left of the interval is used as the height of the approximating rectangle. In general, the differing methods consist of choosing different $x_i$ below. $\\Delta x = \\frac{b-a}{n}$\n",
    "$$I_{est} = \\sum_i f(x_i)\\Delta x = \\frac{b-a}{n} \\sum_i f(x_i)$$\n",
    "\n",
    "$x_i = a+i\\dot\\Delta x,$ where  $i = 0,1,2,...,n-1$. \n",
    "The error on the estimation of the integral can be shown to decrease as $\\frac{1}{n^2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "left_integral =  11.811387964 left_integral err=  2.90389439979e-05\n"
     ]
    }
   ],
   "source": [
    "def left_int(func,a,b,n):\n",
    "    delta_x = (b - a)/n\n",
    "    left_x  = np.arange(a,b,delta_x)\n",
    "    estimate= delta_x*np.sum(func(left_x))\n",
    "    return estimate\n",
    "\n",
    "a = 2.\n",
    "b = 3.\n",
    "N = 10000\n",
    "\n",
    "left_integral = left_int(f,a,b,N)\n",
    "left_err = np.abs(left_integral - exactval)\n",
    "\n",
    "print('left_integral = ',left_integral,\n",
    "      'left_integral err= ',left_err)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## what if I choose a different x_i ?\n",
    "If we divide the $a, b$-interval into $N$ \n",
    "steps and use some regular integration routine, what is the error? Consider choosing $x_i$ as the midpoint of each tiny interval\n",
    "\n",
    "Different from before (useing left point as height, we use the f(midpoint) as height).$\\Delta x = \\frac{b-a}{n}$\n",
    "\n",
    "$$I_{est} = \\sum_i f(x_i)\\Delta x = \\frac{b-a}{n} \\sum_i f(x_i)$$\n",
    "\n",
    "Now $x_i = a+ i\\Delta x + \\frac{\\Delta x}{2}$, where $i=0,1,2,... n-1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "midpt_integral =  11.8113589291 midpt_err     =  4.00740951534e-09\n"
     ]
    }
   ],
   "source": [
    "def midpt_int(func,a,b,n):\n",
    "    delta_x = (b - a)/n\n",
    "    mid_x   = np.arange(a,b,delta_x) +  delta_x/2\n",
    "    estimate= delta_x*np.sum(func(mid_x))\n",
    "    return estimate\n",
    "\n",
    "a = 2.\n",
    "b = 3.\n",
    "N = 10000\n",
    "\n",
    "midpt_integral = midpt_int(f,a,b,N)\n",
    "midpt_err = np.abs(midpt_integral - exactval)\n",
    "\n",
    "print('midpt_integral = ',midpt_integral,\n",
    "      'midpt_err     = ',midpt_err)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Trapezoid Rule\n",
    "We calculat individual area us a trapezoid\n",
    "\n",
    "![](https://upload.wikimedia.org/wikipedia/commons/4/40/Trapezoidal_rule_illustration.svg)\n",
    "The basic idea is that the little areas is\n",
    "$$\\frac{(f(x_i)+f(x_{i+1}))\\cdot \\Delta x }{2}$$\n",
    "So the area of estimation is: $\\Delta x = \\frac{b-a}{n}$\n",
    "$$I_{est} = \\sum_{i=0}^{n-1} \\frac{(f(x_i)+f(x_{i+1}))\\cdot \\Delta x }{2} = \\frac{\\Delta x }{2}\\sum_i (f(x_i)+f(x_{i+1}))$$\n",
    "\n",
    "$x_i = a+i\\dot\\Delta x,$ where  $i = 0,1,2,...,n-1$. \n",
    "The error on the estimation of the integral can be shown to decrease as $\\frac{1}{n^3}$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "trapz_integral =  11.8113589171 trapz_err     =  8.02189781268e-09\n"
     ]
    }
   ],
   "source": [
    "def trapz_int(func,a,b,n):\n",
    "    delta_x = (b - a)/n\n",
    "    left   = np.arange(a,b,delta_x) \n",
    "    right  = np.arange(a+delta_x,b+delta_x,delta_x) \n",
    "    estimate= delta_x*(np.sum(f(left))+np.sum(f(right))) / 2\n",
    "    return estimate\n",
    "\n",
    "a = 2.\n",
    "b = 3.\n",
    "N = 10000\n",
    "\n",
    "trapz_integral = trapz_int(f,a,b,N)\n",
    "trapz_err = np.abs(trapz_integral - exactval)\n",
    "\n",
    "print('trapz_integral = ',trapz_integral,\n",
    "      'trapz_err     = ',trapz_err)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The basic reason for this can be understood on a taylor series expansion of the function to second order. When you integrate on the sub-interval, the linear term vanishes while the quadratic term becomes cubic in $\\Delta x$. So the local error goes as $\\frac{1}{n^3}$ and thus the global as  $\\frac{1}{n^2}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example. The same as before, but we fix the number of draws N = 10,000 and   repeat it 1000 times\n",
    "\n",
    "**Calculate the integral $ I= \\int_{2}^{3} [x^2 + 4 \\, x \\,\\sin(x)] \\, dx. $**\n",
    "\n",
    "We know from calculus that the anti-derivative is \n",
    "$$ x^3/3 + 4\\sin(x) -4x\\cos(x). $$ \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Monte Carlo estimation= 11.8149037481 Exact number= 11.8113589251\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return x**2 + 4*x*np.sin(x) \n",
    "\n",
    "def intf(x): \n",
    "    return x**3/3.0+4.0*np.sin(x) - 4.0*x*np.cos(x) \n",
    "\n",
    "# copy the past monte carlo method:\n",
    "a = 2.\n",
    "b = 3.\n",
    "exactval=intf(b)-intf(a)\n",
    "\n",
    "\n",
    "N = 10000\n",
    "exactval=intf(b)-intf(a)\n",
    "Y =f(np.random.uniform(low=a, high=b, size=N))   # CALCULATE THE f(x) \n",
    "Imc= (b-a) * np.sum(Y)/ N;\n",
    "mc_err = np.abs(Imc-exactval)\n",
    "\n",
    "print(\"Monte Carlo estimation=\",Imc, \"Exact number=\",exactval)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def left_int(func,a,b,n):\n",
    "    delta_x = (b - a)/n\n",
    "    left_x  = np.arange(a,b,delta_x)\n",
    "    estimate= delta_x*np.sum(func(left_x))\n",
    "    return estimate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Monte Carlo estimation= 11.8149037481 left_integral =  11.811387964 Exact number= 11.8113589251\n",
      "Monte Carlo err  = 0.00354482298142 left_integral err=  2.90389439979e-05\n"
     ]
    }
   ],
   "source": [
    "\n",
    "left_integral = left_int(f,a,b,N)\n",
    "left_err = np.abs(left_integral - exactval)\n",
    "\n",
    "print(\"Monte Carlo estimation=\",Imc,\n",
    "      'left_integral = ',left_integral,\n",
    "      \"Exact number=\", exactval)\n",
    "\n",
    "print(\"Monte Carlo err  =\" ,mc_err,\n",
    "      'left_integral err= ',left_err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error comparison\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# to store MC estimate for each sampling, until we reach sampling 10000 times \n",
    "# to store left\n",
    "N     = 10000\n",
    "Imc   = np.zeros(N)\n",
    "Ileft = np.zeros(N)\n",
    "\n",
    "# just to store each time number of samples\n",
    "# an array of 1,2,3... 10000\n",
    "Na = np.linspace(1,N,N)\n",
    "\n",
    "\n",
    "# keep track of computational time\n",
    "Imc_time   = np.zeros(N)\n",
    "Ileft_time = np.zeros(N)\n",
    "\n",
    "# for each N run monte carlo to estimate integral\n",
    "# and run left integral\n",
    "\n",
    "for K in np.arange(1,N+1):\n",
    "    t0        = time.time()\n",
    "    Imc[K-1]  = (b-a) * np.mean(f(np.random.uniform(low=a, high=b, size=K)));\n",
    "    t1        = time.time()\n",
    "    Ileft[K-1]= left_int(f,a,b,K)\n",
    "    t2        = time.time()\n",
    "    Imc_time[K-1]  = t1-t0\n",
    "    Ileft_time[K-1]= t2-t1\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EqsLdd9618AJt/RMRERGJJI6FOwBr7Sd4u1RrH7uv1usS4MpGrn0N73IojuuQ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104b8e550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (10,8))\n",
    "plt.plot(np.arange(10,N),abs(Imc[10:]-exactval), alpha=0.7, label = 'monte error')\n",
    "plt.plot(np.arange(10,N), 1/np.sqrt(Na[10:])   , 'b', label = '1/sqrt(N)')\n",
    "plt.plot(np.arange(10,N),abs(Ileft[10:]-exactval), alpha=0.5, label = 'Left error')\n",
    "plt.plot(np.arange(10,N), 1/(Na[10:])            , 'orange', label = '1/(N*N)')\n",
    "\n",
    "plt.ylim([0,0.1])\n",
    "plt.xlim([10,N+1])\n",
    "\n",
    "plt.xlabel(\"N\")\n",
    "plt.ylabel(\"|Imc-ExactValue|\")\n",
    "plt.legend()\n",
    "# \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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6SOXl5erXr58uv/xy9evXz+1iuSLa+mCMybPWdo83LTVjAAAgJcOHD1fXrl11/PHHq127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\nxkREREQypGBMREREJEMKxkREREQy9D+3wHTs/bDhMgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106b716a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (10,8))\n",
    "plt.plot(np.arange(0,N),Imc_time,color = 'b', alpha=0.7, label = 'monte carlo time')\n",
    "plt.plot(np.arange(0,N), Ileft_time, color = 'r', label = 'rectangle rule time')\n",
    "\n",
    "plt.legend()\n",
    "# \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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