10_01_montecarlo_integration.ipynb

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    "Monte Carlo integration\n",
    "=======================\n",
    "\n",
    "Imagine that we want to measure the area of a pond with arbitrary shape.\n",
    "Suppose that this pond is in the middle of a field with known area $A$.\n",
    "If we throw $N$ stones randomly, such that they land within the\n",
    "boundaries of the field, and we count the number of stones that fall in\n",
    "the pond $N_{in}$, the area of the pond will be approximately\n",
    "proportional to the fraction of stones that make a splash, multiplied by\n",
    "$A$: $$A_{pond}=\\frac{N_{in}}{N}A.$$ This simple procedure is an example\n",
    "of the “Monte Carlo” method.\n",
    "\n",
    "Simple Monte Carlo integration\n",
    "------------------------------\n",
    "\n",
    "More generaly, imagine a rectangle of height $H$ in the integration\n",
    "interval $[a,b]$, such that the function $f(x)$ is within its\n",
    "boundaries. Compute $n$ pairs of random numbers $(x_i,y_i)$ such that\n",
    "they are uniformly distributed inside this rectangle. The fraction of\n",
    "points that fall within the area contained below $f(x)$, <span>*i.\n",
    "e.*</span>, that satisfy $y_i \\leq f(x_i)$ is an estimate of the ratio o\n",
    "fthe integral of $f(x)$ and the area of the rectangle. Hence, the\n",
    "estimate of the integral will be given by:\n",
    "$$\\int _a^b{f(x)dx} \\simeq I(N) = \\frac{N_{in}}{N}H(b-a).\n",
    "$$\n",
    "\n",
    "Another Monte Carlo procedure is based on the definition:\n",
    "$$\\langle g \\rangle=\\frac{1}{(b-a)} \\int _a^b{f(x)dx}.\n",
    "$$ In order to determine this average, we sample the\n",
    "value of $f(x)$:\n",
    "$$\\langle f \\rangle \\simeq \\frac{1}{N}\\sum_{i=1}^{N}f(x_i),$$ where the\n",
    "$N$ values $x_i$ are distributed unformly in the interval $[a,b]$. The\n",
    "integral will be given by $$I(N)=(b-a) \\langle f \\rangle .$$\n",
    "Monte Carlo error analysis\n",
    "--------------------------\n",
    "\n",
    "The Monte Carlo method clearly yields approximate results. The accuracy\n",
    "deppends on the number of values $N$ that we use for the average. A\n",
    "possible measure of the error is the “variance” $\\sigma^2$ defined by:\n",
    "$$\\sigma ^2=\\langle f^2 \\rangle - \\langle f \\rangle ^2,\n",
    "$$ where\n",
    "$$\\langle f \\rangle = \\frac{1}{N} \\sum_{i=1}^N f(x_i)$$ and\n",
    "$$\\langle f^2 \\rangle = \\frac{1}{N} \\sum_{i=1}^{N} f(x_i)^2.$$ The\n",
    "“standard deviation” is $\\sigma$. However, we should expect that the\n",
    "error decreases with the number of points $N$, and the quantity $\\sigma$\n",
    "defines by (\\[mc\\_sigma\\]) does not. Hence, this cannot be a good\n",
    "measure of the error.\n",
    "\n",
    "Imagine that we perform several measurements of the integral, each of\n",
    "them yielding a result $I_n$. These values have been obtained with\n",
    "different sequences of $N$ random numbers. According to the central\n",
    "limit theorem, these values whould be normally dstributed around a mean\n",
    "$\\langle I\n",
    "\\rangle$. Suppouse that we have a set of $M$ of such measurements\n",
    "${I_n}$. A convenient measure of the differences of these measurements\n",
    "is the “standard deviation of the means” $\\sigma_M$:\n",
    "$$\\sigma_M ^2=\\langle I^2 \\rangle - \\langle I \\rangle ^2,\n",
    "$$ where\n",
    "$$\\langle I \\rangle = \\frac{1}{M} \\sum_{n=1}^M I_n$$ and\n",
    "$$\\langle I^2 \\rangle = \\frac{1}{M} \\sum_{n=1}^{M} I_n^2.$$ \n",
    "It can be proven that\n",
    "$$\\sigma_M \\approx \\sigma/\\sqrt{N}.\n",
    "$$ This relation becomes exact in the limit of a very\n",
    "large number of measurements. Note that this expression implies that the\n",
    "error decreases with the square root of the number of trials, meaning\n",
    "that if we want to reduce the error by a factor 10, we need 100 times\n",
    "more points for the average.\n",
    "\n",
    "### Exercise 10.1: One dimensional integration \n",
    "\n",
    "1.  Write a program that implements the “hit and miss” Monte Carlo\n",
    "    integration algorithm. Find the estimate $I(N)$ for the integral of\n",
    "    $$f(x)=4\\sqrt{1-x^2}$$ as a function of $N$, in the interval\n",
    "    $(0,1)$. Choose $H=1$, and sample only the $x$-dependent part\n",
    "    $\\sqrt{1-x^2}$, and multiply the result by 4. Calculate the\n",
    "    difference between $I(N)$ and the exact result $\\pi$. This\n",
    "    difference is a measure of the error associated with the Monte\n",
    "    Carlo estimate. Make a log-log plot of the error as a function of\n",
    "    $N$. What is the approximate functional deppendece of the error on\n",
    "    $N$ for large $N$?\n",
    "\n",
    "2.  Estimate the integral of $f(x)$ using the simple Monte Carlo\n",
    "    integration by averaging over $N$ points, using (\\[mc\\_integral2\\]),\n",
    "    and compute the error as a function of $N$, for $N$ upt to 10,000.\n",
    "    Determine the approximate functional deppendence of the error on $N$\n",
    "    for large $N$. How many trials are necessary to determine $I_N$ to\n",
    "    two decimal places?\n",
    "\n",
    "3.  Perform 10 measurements $I_n(N)$, with $N=10,000$ using different\n",
    "    random sequences. Show in a table the values of $I_n$ and $\\sigma$\n",
    "    according to (\\[mc\\_integral2\\]) and (\\[mc\\_sigma\\]).\n",
    "    Use (\\[mc\\_sigmam\\]) to estimate the standard deviation of the\n",
    "    means, and compare to the values obtained from (\\[mc\\_sigma2\\])\n",
    "    using the 100,000 values.\n",
    "\n",
    "4.  To verify that your result for the error is independent of the\n",
    "    number of sets you used to divide your data, repeat the previous\n",
    "    item grouping your results in 20 groups of 5,000 points each.\n",
    "\n",
    "### Exercise 10.2: Importance of randomness \n",
    "\n",
    "To examine the effects of a poor random number generator, modify your\n",
    "program to use the linear congruential random number generator using the\n",
    "perameters $a=5$, $c=0$ and the seed $x_1=1$. Repeat the integral of the\n",
    "previous exercise and compare your results.\n",
    "\n",
    "#### Challenge 10.1: \n",
    "\n",
    "Exercise 10.2\n",
    "\n",
    "\n"
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from matplotlib import pyplot\n",
    "\n",
    "x = np.arange(0,1,0.02)\n",
    "pyplot.plot(x, 4*np.sqrt(1-x**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 82 3.28 0.1384073464102067\n",
      "200 163 3.26 0.11840734641020667\n",
      "400 306 3.06 0.08159265358979306\n",
      "800 621 3.105 0.036592653589793134\n",
      "1600 1265 3.1625 0.020907346410206973\n",
      "3200 2527 3.15875 0.01715734641020683\n",
      "6400 5046 3.15375 0.012157346410206937\n",
      "12800 10077 3.1490625 0.00746984641020676\n",
      "25600 20061 3.13453125 0.007061403589792903\n",
      "51200 40245 3.144140625 0.002547971410206795\n",
      "102400 80438 3.142109375 0.0005167214102068662\n",
      "204800 160702 3.1387109375 0.0028817160897931515\n",
      "409600 321473 3.139384765625 0.0022078879647930982\n",
      "819200 644142 3.145224609375 0.0036319557852069195\n",
      "1638400 1287372 3.142998046875 0.0014053932852067241\n",
      "3276800 2573360 3.14130859375 0.00028405983979329363\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "# Hit and miss Monte Carlo integration\n",
    "ngroups = 16\n",
    "\n",
    "I = np.zeros(ngroups)\n",
    "N = np.zeros(ngroups)\n",
    "E = np.zeros(ngroups)\n",
    "\n",
    "n0 = 100\n",
    "for i in range(ngroups):\n",
    "\n",
    "    N[i] = n0\n",
    "    x = np.random.random(n0)\n",
    "    y = np.random.random(n0)\n",
    "    I[i] = 0.\n",
    "    Nin = 0\n",
    "    for j in range(n0):\n",
    "        if(y[j] < np.sqrt(1-x[j]**2)):\n",
    "            Nin += 1\n",
    "            \n",
    "    I[i] = 4.*float(Nin)/float(n0)\n",
    "    E[i] = abs(I[i]-np.pi)\n",
    "    print (n0,Nin,I[i],E[i])\n",
    "    n0 *= 2\n",
    "    \n",
    "            \n",
    "pyplot.plot(N,E,ls='-',c='red',lw=3);\n",
    "pyplot.plot(N,0.8/np.sqrt(N),ls='-',c='blue',lw=3);\n",
    "\n",
    "pyplot.xscale('log')\n",
    "pyplot.yscale('log')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 3.083548695053129 0.05804395853666433\n",
      "200 3.0169076292704244 0.12468502431936868\n",
      "400 3.1806738953569527 0.03908124176715955\n",
      "800 3.116184732780529 0.025407920809263906\n",
      "1600 3.174486501377891 0.03289384778809801\n",
      "3200 3.1495504408897914 0.007957787299998298\n",
      "6400 3.14574823312641 0.004155579536616827\n",
      "12800 3.1349325642964856 0.006660089293307525\n",
      "25600 3.1435910835455867 0.001998429955793579\n",
      "51200 3.1387551229509194 0.0028375306388737087\n",
      "102400 3.1403323363592297 0.0012603172305634125\n",
      "204800 3.139379645209978 0.0022130083798153066\n",
      "409600 3.1408746244781374 0.0007180291116557491\n",
      "819200 3.142975298734189 0.0013826451443956778\n",
      "1638400 3.1414994374643403 9.321612545276636e-05\n",
      "3276800 3.14129904834528 0.0002936052445132731\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Simple Monte Carlo Integration\n",
    "ngroups = 16\n",
    "\n",
    "I = np.zeros(ngroups)\n",
    "N = np.zeros(ngroups)\n",
    "E = np.zeros(ngroups)\n",
    "\n",
    "n0 = 100\n",
    "for i in range(ngroups):\n",
    "\n",
    "    N[i] = n0\n",
    "    r = np.random.random(n0)\n",
    "    I[i] = 0.\n",
    "    for j in range(n0):\n",
    "        x = r[j]\n",
    "        I[i] += np.sqrt(1-x**2)\n",
    "        \n",
    "    I[i] *= 4./float(n0)\n",
    "    E[i] = abs(I[i]-np.pi)\n",
    "    print (n0,I[i],E[i])\n",
    "    n0 *= 2\n",
    "    \n",
    "            \n",
    "pyplot.plot(N,E,ls='-',c='red',lw=3);\n",
    "pyplot.plot(N,0.8/np.sqrt(N),ls='-',c='blue',lw=3);\n",
    "pyplot.xscale('log')\n",
    "pyplot.yscale('log')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.1457636696861324 10.69446669333447 0.7986376278173068\n",
      "3.1372366968536767 10.639779583215944 0.7975254911305765\n",
      "3.150995532231518 10.719719006029514 0.7909461618865272\n",
      "3.1430803624629067 10.687552183237157 0.8085980183372001\n",
      "3.1549093390648597 10.725146283018782 0.7716933453001129\n",
      "3.1524565362061563 10.719672155349185 0.7816899426802681\n",
      "3.128145606617917 10.599685249016789 0.8143903128138117\n",
      "3.1384054020541434 10.645817378212246 0.7962289105696172\n",
      "3.1445580189862543 10.683852412854495 0.7956072780837395\n",
      "3.1332986511743366 10.615776688479952 0.7982162510290358\n",
      "=============================\n",
      "3.136743104879589 10.660760589836224 0.8216032838265797\n",
      "3.154784234492714 10.728172796832657 0.7755092306288791\n",
      "3.1387311928691286 10.653576800128173 0.8019432990385091\n",
      "3.135742200838215 10.625982366303749 0.7931032161860561\n",
      "3.143268496642032 10.687526668004205 0.8073898260219465\n",
      "3.158722567820987 10.751911344054774 0.7743830835931629\n",
      "3.1415836052163484 10.683297105508757 0.8137495569446074\n",
      "3.1445771197094867 10.691807260965554 0.8034419991651429\n",
      "3.1845050824958947 10.884185738328162 0.7431131178859758\n",
      "3.125313595633803 10.566106827709524 0.798521756656033\n",
      "3.1575195303371184 10.73188976937331 0.761960184912974\n",
      "3.1473935420752115 10.70745454132511 0.8013684326283634\n",
      "3.124137415684765 10.567418034780752 0.8071834426992712\n",
      "3.1321537975510703 10.631952463252777 0.821565051739185\n",
      "3.1296417982029836 10.57924507556743 0.7845872905082256\n",
      "3.1471690059053077 10.712389680857035 0.8077169291260322\n",
      "3.143595399324597 10.681346784020045 0.7991547493652735\n",
      "3.145520638647886 10.68635804168898 0.7920579535291754\n",
      "3.1259537277454843 10.587599370989583 0.8160126629836935\n",
      "3.1406435746031978 10.643954005970329 0.7803119432739773\n",
      "=============================\n",
      "3.1428849815338245 10.673146763274831 0.7954207561239635\n"
     ]
    }
   ],
   "source": [
    "n0 = 100000\n",
    "I = np.zeros(n0)\n",
    "r = np.random.random(n0)\n",
    "for j in range(n0):\n",
    "    x = r[j]\n",
    "    I[j] = 4*np.sqrt(1-x**2)\n",
    "\n",
    "def group_measurements(ngroups):\n",
    "    global I,n0\n",
    "    \n",
    "    nmeasurements = n0//ngroups\n",
    "    for n in range(ngroups):\n",
    "        Ig = 0.\n",
    "        Ig2 = 0.\n",
    "        for i in range(n*nmeasurements,(n+1)*nmeasurements):\n",
    "            Ig += I[i]\n",
    "            Ig2 += I[i]**2\n",
    "        Ig /= nmeasurements\n",
    "        Ig2 /= nmeasurements\n",
    "        sigma = Ig2-Ig**2\n",
    "        print(Ig,Ig2,sigma)\n",
    "        \n",
    "group_measurements(10)\n",
    "print(\"=============================\")\n",
    "group_measurements(20)\n",
    "print(\"=============================\")\n",
    "group_measurements(1)\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variance reduction\n",
    "------------------\n",
    "\n",
    "If the function being integrated does not fluctuate too much in the\n",
    "interval of integration, and does not differ much from the average\n",
    "value, then the standard Monte Carlo mean-value method should work well\n",
    "with a reasonable number of points. Otherwise, we will find that the\n",
    "variance is very large, meaning that some points will make small\n",
    "contributions, while others will make large contributions to the\n",
    "integral. If this is the case, the algorithm will be very inefficient.\n",
    "The method can be improved by splitting the function $f(x)$ in two\n",
    "$f(x)=f_1(x)+f_2(x)$, such that the integral of $f_1(x)$ is known, and\n",
    "$f_2(x)$ as a small variance. The “variance reduction” technique,\n",
    "consists then in evaluating the integral of $f_2(x)$ to obtain:\n",
    "$$\\int _a^b{f(x)dx}=\\int _a^b {f_1(x)dx} + \\int _a^b{f_2(x)dx} = \\int\n",
    "_a^b{f_1(x)dx}+J.$$\n",
    "\n",
    "Importance Sampling\n",
    "-------------------\n",
    "\n",
    "Imagine that we want to sample the function $f(x)=e^{-x^2}$ in the\n",
    "interval $[0,1]$. It is evident that most of our points will fall in the\n",
    "region where the value of $f(x)$ is very small, and therefore we will\n",
    "need a large number of values to achieve a decent accuracy. A way to\n",
    "improve the measurement by reducing the variance is obtained by\n",
    "“importance sampling”. As the name says, the idea is to sample the\n",
    "regions with larger contributions to the integral. For this goal, we\n",
    "introduce a probability distribution $P(x)$ normalized in the interval\n",
    "of integration $$\\int _a^b{P(x)dx} = 1.$$ Then, we can rewrite the\n",
    "integral of $f(x)$ as $$I=\\int _a^b{\\frac{f(x)}{P(x)}P(x)dx}\n",
    "$$ We can evaluate this integral, by sampling\n",
    "according to the probability distribution $P(x)$ and evaluating the sum\n",
    "$$I(N)=\\frac{1}{N} \\sum_{i=1}^N \\frac{f(x_i)}{P(x_i)}.\n",
    "$$ Note that for the uniform case $P(x)=1/(b-a)$, the\n",
    "expression reduces to the simple Monte Carlo integral.\n",
    "\n",
    "We are free to choose $P(x)$ now. We wish to do it in a way to reduce\n",
    "and minimize the variance of the integrand $f(x)/P(x)$. The way to to\n",
    "this is picking a $P(x)$ that mimics $f(x)$ where $f(x)$ is large. if we\n",
    "are able to determine an apropiate $P(x)$, the integrand will be slowly\n",
    "varying, and hence the variance will be reduced. Another consideration\n",
    "is that the generation of points according to the distribution $P(x)$\n",
    "should be a simple task. As an example, let us consider again the\n",
    "integral $$I=\\int _0^1 {e^{-x^2}dx}.$$ A reasonable choice for a weigh\n",
    "function is $P(x)=Ae^{-x}$, where $A$ is a normalization constant.\n",
    "\n",
    "Notice that for $P(x)=f(x)$ the variance is zero! This is known as the\n",
    "zero variance property. There is a catch, though: The probability function\n",
    "$P(x)$ needs to be normalized, implying that in reality, $P(x)=f(x)/\\int f(x)dx$, which\n",
    "assumes that we know in advance precisely the integral that we are trying to calculate!\n",
    "\n",
    "### Exercise 10.3: Importance sampling \n",
    "\n",
    "1.  Choose the weight function $P(x)=e^{-x}$ and evaluate the integral:\n",
    "    $$\\int _0^{\\infty} {x^{3/2}e^{-x}dx}.$$\n",
    "\n",
    "2.  Choose $P(x)=e^{-ax}$ and estimate the integral\n",
    "    $$\\int _0^{\\pi} \\frac{dx}{x^2+\\cos ^2{x}}.$$ Determine the value of\n",
    "    $a$ that minimizes the variance of the integral.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VIYuPVn9Uo/e3aqU/xuOP66DLmDd0KCxbpgu/9uypnfttmgFjgi48k/qkSdqN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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x102408cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.xlim(0,10)\n",
    "x = np.arange(0,10,0.1)\n",
    "pyplot.plot(x,np.exp(-x));\n",
    "pyplot.plot(x,np.exp(-x**2));\n",
    "pyplot.plot(x,x**1.5*np.exp(-x));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Trapezoids:  1.3293401896452883\n",
      "Simple Monte Carlo:  0.20055243365045128\n",
      "Importance Sampling:  1.3288525836135752\n"
     ]
    }
   ],
   "source": [
    "\n",
    "# Trapezoidal integration\n",
    "def trapezoids(func, xmin, xmax, nmax):\n",
    "    Isim = func(xmin)+func(xmax)\n",
    "    h = (xmax-xmin)/nmax\n",
    "    for i in range(1,nmax):\n",
    "        x = xmin+i*h\n",
    "        Isim += 2*func(x)\n",
    "\n",
    "    Isim *= h/2\n",
    "    return Isim\n",
    "\n",
    "def f(x):\n",
    "    return x**1.5*np.exp(-x)\n",
    "    \n",
    "    \n",
    "print(\"Trapezoids: \", trapezoids(f, 0., 20., 100000))\n",
    "\n",
    "# Simple Monte Carlo integration\n",
    "n0 = 1000000\n",
    "r = np.random.random(n0)\n",
    "\n",
    "Itot = np.sum(r**1.5*np.exp(-r))\n",
    "print(\"Simple Monte Carlo: \", Itot/n0)\n",
    "\n",
    "x = -np.log(r)\n",
    "Itot = np.sum(x**1.5)\n",
    "print(\"Importance Sampling: \", Itot/n0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.xlim(0,np.pi)\n",
    "x = np.arange(0,np.pi,0.05)\n",
    "pyplot.plot(x,1./(x**2+np.cos(x)**2));\n",
    "pyplot.plot(x,np.exp(-x));\n",
    "pyplot.plot(x,np.exp(-2*x));\n",
    "pyplot.plot(x,np.exp(-0.2*x));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Trapezoids:  1.5811879708476726\n",
      "Simple Monte Carlo:  1.5815947622343893\n",
      "Importance Sampling:\n",
      "0.1 1.5244330723519488 0.00321381588763609\n",
      "0.2 1.50095680220008 0.0020648357111915415\n",
      "0.30000000000000004 1.4904411006780327 0.0015512787071996397\n",
      "0.4 1.4968606674298741 0.0012547420671273522\n",
      "0.5 1.5146208065214364 0.0010525050870957287\n",
      "0.6 1.5384283496617999 0.0008903917445926586\n",
      "0.7000000000000001 1.5621715230895488 0.0007325196183547532\n",
      "0.8 1.5795007309626732 0.0005398433954381976\n",
      "0.9 1.5848844693695763 0.00018141907194581285\n",
      "1.0 1.5741096356771225 0.0004972401813311438\n",
      "1.1 1.5451370063730057 0.0007367892503514896\n",
      "1.2000000000000002 1.4986266667374715 0.0009147552770853103\n",
      "1.3000000000000003 1.4371939198631032 0.0010517960421495326\n",
      "1.4000000000000001 1.364831049309352 0.0011532665188242917\n",
      "1.5000000000000002 1.2862725322212676 0.0012231156728855844\n",
      "1.6 1.2052048099895123 0.0012636640779222586\n",
      "1.7000000000000002 1.1250482091736054 0.0012800101146486459\n",
      "1.8000000000000003 1.0480214022503385 0.0012761065837385973\n",
      "1.9000000000000001 0.9756899351567899 0.0012583224714410996\n",
      "2.0 0.908556238655731 0.0012275630544004439\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Trapezoidal integration\n",
    "\n",
    "def g(x):\n",
    "    return 1./(x**2+np.cos(x)**2)\n",
    "\n",
    "print(\"Trapezoids: \", trapezoids(g, 0., np.pi, 1000000))\n",
    "\n",
    "# Simple Monte Carlo integration\n",
    "n0 = 1000000\n",
    "a = np.arange(0.1,2.1,0.1)\n",
    "I = np.arange(0.1,2.1,0.1)\n",
    "\n",
    "r = np.random.random(n0)\n",
    "\n",
    "I0 = np.sum(1./((r*np.pi)**2+np.cos(r*np.pi)**2))\n",
    "print(\"Simple Monte Carlo: \", I0/n0*np.pi)\n",
    "\n",
    "# Importance Sampling\n",
    "print(\"Importance Sampling:\")\n",
    "x = -np.log(r)\n",
    "              \n",
    "i = 0\n",
    "for ai in a:\n",
    "    norm = (1.-np.exp(-ai*np.pi))/ai\n",
    "    x1 = norm*x/ai\n",
    "    Itot = 0.\n",
    "    Nin = 0\n",
    "    I2 = 0.\n",
    "    for xi in x1:\n",
    "        if(xi <= np.pi):\n",
    "            Nin += 1\n",
    "            Itot += g(xi)*np.exp(xi*ai)\n",
    "            I2 += (g(xi)*np.exp(xi*ai))**2\n",
    "\n",
    "    Itot *= norm\n",
    "    I2 *= norm\n",
    "\n",
    "    I[i] = Itot/Nin\n",
    "    i += 1\n",
    "    print(ai,Itot/Nin,np.sqrt(abs(Itot**2/Nin**2-I2/Nin))/np.sqrt(Nin))\n",
    "    \n",
    "pyplot.plot(a,I,ls='-',marker='o',c='red',lw=3);\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 10.4: The Metropolis algorithm \n",
    "\n",
    "Use the Metropolis algorithm to sample points according to a ditribution\n",
    "and estimate the integral $$\\int _0^4 {x^2e^{-x}dx},$$ with\n",
    "$P(x)=e^{-x}$ for $0 \\leq x \\leq 4$. Plot the number of times the\n",
    "walker is at points $x_0$, $x_1$, $x_2$, ... Is the integrand sampled\n",
    "uniformly? If not, what is the approximate region of $x$ where the\n",
    "integrand is sampled more often?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Trapezoids: ', 1.5237933888733828)\n",
      "('Metropolis: ', 1.5251458730929552)\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b782b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "delta = 2\n",
    "xmin = 0.\n",
    "xmax = 4.\n",
    "\n",
    "def f(x):\n",
    "    return x**2*np.exp(-x)\n",
    "\n",
    "def P(x):\n",
    "    global xmin, xmax\n",
    "    if(x < xmin or x > xmax):\n",
    "        return 0.\n",
    "    return np.exp(-x)\n",
    "\n",
    "\n",
    "def metropolis(xold):\n",
    "    global delta\n",
    "    xtrial = np.random.random()    \n",
    "    xtrial = xold+(2*xtrial-1)*delta\n",
    "    weight = P(xtrial)/P(xold)\n",
    "    xnew = xold\n",
    "    if(weight >= 1): #Accept\n",
    "        xnew = xtrial\n",
    "    elif(weight != 0):\n",
    "        r = np.random.random()\n",
    "        if(r <= weight): #Accept\n",
    "            xnew = xtrial\n",
    "    return xnew\n",
    "\n",
    "\n",
    "xwalker = (xmax+xmin)/2.\n",
    "for i in range(100000):\n",
    "    xwalker = metropolis(xwalker)\n",
    "\n",
    "I0 = 0.\n",
    "N = 300000\n",
    "\n",
    "x = np.zeros(N)\n",
    "\n",
    "x[0] = xwalker\n",
    "for i in range(1,N):\n",
    "    for j in range(20):\n",
    "        xwalker = metropolis(xwalker)\n",
    "    x[i] = xwalker\n",
    "    I0 += x[i]**2\n",
    "\n",
    "binwidth=0.1\n",
    "pyplot.hist(x,bins=np.arange(xmin-1, xmax+1, 0.1),normed=True);\n",
    "\n",
    "print(\"Trapezoids: \", trapezoids(f,xmin,xmax,100000))\n",
    "print(\"Metropolis: \", I0*(1.-np.exp(-4.))/N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x106da8710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.hist(x**2,bins=np.arange(xmin**2-1, xmax**2+1, 0.1),normed=True);\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Challenge 10.1\n",
    "\n",
    "- Calculate the integral $\\int_0^1 x^2 dx=1/3$ using simple MC integration and importance sampling with $P(x)=x$.\n",
    "\n",
    "- Calculate the integral $\\int_0^1 \\sqrt{x}dx=2/3$ using simple MC integration and $P(x)=1-e^{-ax}$. Find the values of $a$ that minimizes the variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}