Lecture4_VarianceReductionInMonteCarloPO.ipynb

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      "# Lecture 4: Variance Reduction\n",
      "\n",
      "AM207: Pavlos Protopapas, Harvard University\n",
      "\n",
      "Feb 5, 2014\n",
      "\n",
      "------\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will discuss methods that may be used to reduce  the standard error of \n",
      "the sample mean approximation (or the integral). \n",
      "We will look into four different methods and we compare them\n",
      "to mean Monte Carlo methods with examples. \n",
      "\n",
      "$$\n",
      "\\newcommand{\\var}{{\\rm var}}\n",
      "\\newcommand{\\cov}{{\\rm cov}}\n",
      "\\newcommand{\\corr}{{\\rm corr}}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
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     "source": [
      "## 1. Control Variates\n",
      "\n",
      "The basic idea is to transform the function we would like to integrate into \n",
      "a different function where the variance of this new functions is much smaller\n",
      "(or flatter).\n",
      "\n",
      "Suppose again that we wish to estimate  $E[Y]$, where $Y = f(X)$ is the output of a simulation experiment.\n",
      "Suppose the mean of a function $Z=g(X)$, $E[Z]=\\tau$ is known or easily\n",
      "estimated with a simulation. The transformation $f(X) \\rightarrow f^{\\star}(X) = f(X) - [g(X) - \\tau]$ will\n",
      " yield the same value, $E[ f^{\\star}] = E[f]$. \n",
      " \n",
      " The control variate method finds the best way to do so for \n",
      " any arbitrary function $g(X)$. To do so we introduce a new function $f^{\\star}$:\n",
      " \n",
      "$$ f^{\\star} (X) = f(X) + b[g(X)-\\tau] $$\n",
      "\n",
      "where $b$ is a constant. \n",
      "\n",
      "The expectation of this new function is identical to the original function, \n",
      "\n",
      "$$ E[ f^{\\star}] =  E[ f] + b \\, E[ g] - b \\, \\tau=  E[ f].  $$\n",
      "\n",
      "\n",
      "Also the variance of the new function is \n",
      "\n",
      "\\begin{equation}\n",
      " \\var(f^{\\star}) = \\var(f)+b^2 \\,\\var( g) +2 \\,b\\, \\cov(f,g)  \n",
      " \\end{equation}\n",
      "\n",
      "\n",
      "Since we are free to choose $b$, we should choose it to minimize $\\var(f^{\\star}) $. Simple calculus then implies that the\n",
      "optimal value of $b$ is given by\n",
      "\n",
      "$$ b_c = -\\frac{\\cov(f,g)}{\\var(g)} $$\n",
      "\n",
      "Substituting for $b_c$ above we see that\n",
      "\n",
      "$$ \\var(f^{\\star}) = \\var(f) - \\frac{[\\cov(f,g)]^2}{\\var(g)}  = \\var(f) (1-\\rho^2(f,g)) $$\n",
      "\n",
      "where $\\rho(f,g)$ is the correlation coefficient of $f$ and $g$. \n",
      "This means that if $\\rho^2$ is close to one, the variance of $f^{\\star}$ becomes close\n",
      "to zero. \n",
      "The function $g$ is called the control variate for $f$.\n",
      "\n",
      "Now back to our integral. To use the control variate $g$ in our simulation we would like to modify our\n",
      "algorithm so that after generating n samples of $f(X)$ and $g(X)$ we would simply set\n",
      "\n",
      " $$ I = \\int_{V} f(x) dx \\approx \\frac{V}{N} \\sum_i f^{\\star}(x_i) =  \\frac{V}{N} \\sum_i \\left[ f(x_i) +b_c (g(x_i) -\\tau) \\right] $$\n",
      " \n",
      " There is of course a problem with this.  We usually do not know $\\cov(f,g)$. We overcome this problem by\n",
      "approximating it from the samples. "
     ]
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     "source": [
      "###Example\n",
      "\n",
      "In this example we would like to estimate the integral of the function $f(x) = \\cos(x) \\exp(-x)+0.2  $ from 0 to 10. \n",
      "\n",
      "$$ I = \\int_{0}^{10} dx [\\cos(x) \\exp(-x)+0.2] $$\n",
      "\n",
      "and we introduce a control variate $g(x) =exp(-x)+0.2 $. This function is chosen such \n",
      "as the correlation of $f,g$ is close to one. [HOW DO I KNOW THIS?]"
     ]
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     "input": [
      "plt.figure(figsize=[14,8])\n",
      "\n",
      "xmax = 10\n",
      "xmin = 0\n",
      "N=10000\n",
      "\n",
      "w = 0.5\n",
      "\n",
      "# ==================================\n",
      "# derivative of the function\n",
      "# =================================\n",
      "f = lambda x: np.cos(w*x)*np.exp(-x)+0.2;\n",
      "g = lambda x: np.exp(-x)+0.2;\n",
      "fint= lambda z: z/5.0- (4.0*np.cos(z/2.0))/(5.0*np.exp(z)) + (2.0*np.sin(z/2.0))/(5.0*np.exp(z))\n",
      "fint_vr = lambda z: z/5  -1/np.exp(z)  \n",
      "\n",
      "# ============================\n",
      "# plot it\n",
      "# ============================\n",
      "x = np.linspace(0,10, 100)\n",
      "\n",
      "plt.subplot(1,2,1)\n",
      "plt.plot(x, f(x), 'b', label=u'$cos(x)\\, \\exp(-x)+0.2$') \n",
      "plt.plot( x, g(x), 'r', label = u'$\\exp(-x)+0.2$')\n",
      "plt.legend()\n",
      "\n",
      "Ivmc = np.zeros(1000)\n",
      "Imc_vr = np.zeros(1000)\n",
      "\n",
      "\n",
      "#exact value \n",
      "true_val = fint(xmax) - fint(xmin)\n",
      "\n",
      "tau =(1.0/(xmax-xmin))*(fint_vr(xmax)-fint_vr(xmin));\n",
      "\n",
      "# DO THIS FEW TIMES\n",
      "for k in np.arange(0,1000):\n",
      "\n",
      "    # ----------------------\n",
      "    # lets do the normal MC\n",
      "    # ----------------------\n",
      "\n",
      "   \n",
      "    X = np.random.uniform( low=xmin, high=xmax, size=N)\n",
      "    Ivmc[k] = (xmax-xmin)*np.mean(f(X));\n",
      "\n",
      "    # ----------------------\n",
      "    # CONTROL VARIATE\n",
      "    # ----------------------\n",
      "    \n",
      "    q = np.cov(g(X),f(X));\n",
      "    \n",
      "    b =- q[0,1]/np.var(g(X));\n",
      "       \n",
      " \n",
      "    \n",
      "    \n",
      "    Imc_vr[k] = (xmax-xmin)*(np.mean(f(X)) + b *(np.mean(g(X)) -tau));\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "yreal = true_val*np.ones(30);\n",
      "\n",
      "plt.subplot(1,2,2)\n",
      "plt.hist(Ivmc,30, histtype='step', label=u'Basic MC');\n",
      "# bar(b1,h1, 1, 'FaceColor', 'none', 'EdgeColor', 'red')\n",
      "plt.hist(Imc_vr,30, histtype='step', label = u'Control Variate');\n",
      "# bar(b2,h2, 1, 'FaceColor', 'none', 'EdgeColor', 'blue')\n",
      "plt.plot([true_val, true_val], [0 ,100], 'y.-') \n",
      "plt.legend()\n",
      "#legend('VMC', 'VR', 'True Value')\n"
     ],
     "language": "python",
     "metadata": {},
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LL+mNN97Qpk2b5OHhoVWrVql58+Z65ZVXJEkbN27U/v37tWnTJv3zn/+0Xmvg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\nCQ4AAACAoEGCAwAAACBokOAAAAAACBokOAAAAACCBgkOAAA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       "text": [
        "<matplotlib.figure.Figure at 0x104bea990>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 1.1 Multiple Control Variates\n",
      "\n",
      "Up until now we have only considered the possibility of using one control variate. However, there is no reason\n",
      "why we should not use more than one and I will now describe how to do this. \n",
      "Suppose again that we wish to estimate    $E[Y]$ where $Y = f(X)$ is the output of a simulation experiment.\n",
      "We also suppose that the means of the functions $E[Z_i]=\\tau_i$ where $Z_i=g_i(X)$ for $i= 1\\ldots m$ \n",
      "is known for each $i$.\n",
      "\n",
      "We can then construct unbiased estimators of $E[Y]$ by setting \n",
      "\n",
      "$$  f^{\\star}(X) = f(X)+b_1 \\left[ g_1(X) - \\tau_1\\right] +b_2 \\left[ g_2(X) - \\tau_2\\right] +\\ldots +b_m \\left[ g_m(X) - \\tau_m\\right] $$\n",
      "\n",
      "To choose a good set of $b_i$'s, we compute $\\var(f^{\\star})$ and obtain\n",
      "\n",
      "$$ \\var(f^{\\star}) = \\var(f) +2 \\, \\sum_i^{m} b_i\\, \\cov(f,g_i) + \\sum_{i}^{m} \\sum_{j}^{m} b_i \\, b_j \\cov(g_i, g_j) $$\n",
      "\n",
      "As before, we could use use calculus to find the optimal set of $b_i$'s \n",
      "by taking partial derivatives with respect to each $b_i$, \n",
      "set them equal to 0 and solve the resulting system of $m$ linear equations and $m$ unknowns. \n",
      "This is easy to do but as before, however, the optimal solutions $b_{i,c}$'s\n",
      "will involve unknown quantities so that they will need to be\n",
      "estimated using the samples. A convenient way of estimating the $b_{i,c}$'s\n",
      "is to observe that $b_{i,c}=-c_i$'s, where $c_i$'s are the solution to the linear regression \n",
      "\n",
      "$$ f = a + c_1 g_1(X) + \\ldots + c_m g_m(X) + \\epsilon $$ \n",
      "\n",
      "where  $\\epsilon $ is an error term. \n",
      "\n",
      "To get an intuition for why remember that we want to minimize $\\var(f^{\\star})$ which means minimizing the value of $\\var(f(X)+b_1 \\left[ g_1(X) - \\tau_1\\right] +b_2 \\left[ g_2(X) - \\tau_2\\right] +\\ldots +b_m \\left[ g_m(X) - \\tau_m\\right])$.  Since the variance of constants is 0, that means minimizing $\\var(f(X)+b_1 \\left[g_1(X)\\right] +b_2 \\left[g_2(X)\\right] +\\ldots +b_m \\left[g_m(X)\\right])$  Now solving the multiple linear regression equation above gives the set of coefficients $c_i$ that best represent f as a linear combination of the $g_i(X)$'s.  Taking $b_i = -c_i$ gives the best possible (in a least squares sense) variance reduction as a linear model of the $g_i(X)$'s -- reducing the variance to the residuals of the linear regression which by definition are minimized."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## 2. Antithetic Variates\n",
      "\n",
      "Suppose as usual that we would like to estimate $E[f(X)] = E[Y ]$, \n",
      "and that we have generated two samples,\n",
      "$Y_1=f(X_1)$ and $Y_2=f(X_2)$. Then the expectation of\n",
      "\n",
      "$$ Z = \\frac{1}{2} \\left[  Y_1 + Y_2 \\right] $$\n",
      "\n",
      "$E[Z] = E[Y]$\n",
      "\n",
      "and \n",
      "\n",
      "$$ \\var(Z) = \\frac{ \\var(Y_1)+\\var(Y_2) + 2 \\cov(Y_1,Y_2)}{4}. $$\n",
      "\n",
      "Therefore, the more negatively correlated $Y_1=f(X_1)$ is with $Y_2=f(X_2)$, the more the variance will be reduced.\n",
      "If $Y_1$ and $Y_2$ are iid, then the $\\var(Z) = \\var(Y_1)/2=\\var(Y_2)/2$.\n",
      "On the other hand, we could reduce $\\var(Z)$ if we could make it so that\n",
      "$\\cov(Y_1, Y_2) < 0$. \n",
      "\n",
      "Lets try it out. Generate two samples that are correlated "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Y1= np.random.uniform(low=0, high=1, size=10000)\n",
      "Y2a= np.random.uniform(low=0, high=1, size=10000)\n",
      "\n",
      "rho =-0.9\n",
      "\n",
      "Y2 = rho + np.sqrt(1-rho**2) * Y2a\n",
      "\n",
      "Z= 0.5*(Y1+Y2)\n",
      "print np.var(Z), np.var(Y1), np.var(Y2), np.cov(Y1,Y2)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.0246650436434 0.0836271679952 0.0155481163169 [[ 0.08363553 -0.00025758]\n",
        " [-0.00025758  0.01554967]]\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The method of antithetic variates describes how to do this idea in reducing the estimator error of an integral. \n",
      "Suppose $X \\sim U(0, 1)$ iid random variables and\n",
      "\n",
      "$$ \\theta=E[f(X)]=E[f(U)] $$\n",
      "\n",
      "If we construct two samples $U$ and $1-U$, each of length $n$ and now define\n",
      "$$ Y_1 = f(U)  \\hspace{2cm} Y_2 = f(1-U) $$\n",
      "\n",
      "then the expectation of \n",
      "$$Z = \\frac{1}{2} \\left[  Y_1 + Y_2 \\right] $$\n",
      "$E[Z] = E[Y]$ as before. \n",
      "\n",
      "If we to compare this with the mean MC with  $2n$ sample, the question\n",
      "remains as to which algorithm is better. \n",
      "Note that both algorithms require approximately the same amount of time. \n",
      "That is, comparing the two algorithms amounts to computing which estimator has a smaller variance.\n",
      "\n",
      "Say we want to calculate \n",
      "\n",
      "$$ I = \\int f(X) dX $$\n",
      "\n",
      "and let $X\\sim U(0,1)$ for $2n$ samples and $Y=f(X)$, \n",
      "\n",
      "$$ \\var(\\hat{I}_{2n}) = \\frac{\\var(Y)}{2n} $$ \n",
      "\n",
      "on the other hand \n",
      "\n",
      "$$ \\var(\\hat{I}_{n+n})  = \\frac{\\var(Z)}{n}  = \\frac{\\var(Y_1+Y_2)}{4n} = \\frac{\\var(Y_1)}{2n}+\\frac{\\cov(Y_1,Y_2)}{2n} $$\n",
      "\n",
      "So again to reduce the $\\hat{I}_{n+n}$, we would like to make the $\\cov(Y_1,Y_2)$ as negative \n",
      "as possible. \n",
      "\n",
      "If the $Y$'s are monotonic functions of $X$ then this guarantees if $Y_1 = f(X)$ and $Y_2=f(1-X)$ then\n",
      "$Y_1,Y_2$  are negatively correlated.\n",
      "\n",
      "To get an intuition why, consider the region where $X_1=U$ has a high value (i.e. is close to 1).  Then by construction $X_2=1-U$ has a low value (i.e close to 0).  Since $f$ is monotonic then the value of $Y1 = f(X)$ will also have a somewhat high value and $Y2 = f(X)$ will have a somewhat low value.  Thus $Y1$ and $Y2$ are anticorrelated."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Example\n",
      "\n",
      "In this example we are trying to calculate  the integral of $f(x) = 1/(x+1)$ from $[0,1]$. To do so \n",
      " we generate $X_1=U$ from a uniform distribution and $X_2=1-U$. Because $X_1,X_2$ are correlated,\n",
      " one can show that $Y_1=f(X_1),Y_2=f(X_2)$ are also strongly correlated if the function is monotonic.\n",
      " "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N =10000\n",
      "xmin = 0 \n",
      "xmax = 1\n",
      "\n",
      "\n",
      "# ==============================\n",
      "# DEFINE FUNTION \n",
      "# ==============================\n",
      "Y = lambda x:  1/(x+1)\n",
      "intY = lambda x: np.log(x+1) \n",
      "\n",
      "Iact = intY(xmax)-intY(xmin)\n",
      "\n",
      "Ivmc=np.zeros(1000)\n",
      "Iant=np.zeros(1000)\n",
      "# ==============================\n",
      "# GENERATING THE SAMPLE X1, X2 iid \n",
      "# and X1 amd Y1 negatively correlated \n",
      "# ==============================\n",
      "for k in np.arange(0,1000):\n",
      "    U = np.random.uniform(low=0, high=1, size=2*N)\n",
      "    X=U\n",
      "\n",
      "\n",
      "\n",
      "    # ==============================\n",
      "    # VANILLA MC\n",
      "    # ==============================\n",
      "    Ivmc[k] =  np.mean(Y(X))\n",
      "\n",
      "    # ==============================\n",
      "    # ANTITHETIC MC\n",
      "    # ==============================\n",
      "    U = np.random.uniform(low=0, high=1, size=N) \n",
      "    X1= U\n",
      "    X2= 1-U\n",
      "    Iant[k] = 0.5*( np.mean(Y(X1))+np.mean(Y(X2)))\n",
      "\n",
      "\n",
      "plt.hist(Ivmc,30, histtype='step', label=u'Normal MC')\n",
      "plt.hist(Iant,30,  histtype='step', label=u'Control Variates')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "<matplotlib.legend.Legend at 0x10519ab10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x105167890>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## 3. Stratified Method\n",
      "\n",
      "The key idea is to achieve a more regular sampling of the\n",
      "most `important' dimension in the uncertainty.\n",
      "Start by considering a one-dimensional problem:\n",
      "\n",
      "$$ I= \\int_{0}^{1} f(x) \\, dx $$\n",
      "\n",
      "and lets assume $x$ is drawn from a uniform distribution. \n",
      "\n",
      "$$ I= \\int_{0}^{1} f(U) \\, dU $$\n",
      "\n",
      "Instead of taking $N$ samples, drawn from uniform\n",
      "distribution, we break the interval into $M$ _strata_ of equal width and take $L$ samples for each\n",
      "strate, $N=LM$.\n",
      "\n",
      "Let $U_{ij}$ be the value of the $i$-th sample and strata $j$. The average from strata $j$ is\n",
      "\n",
      "\\begin{equation} \n",
      "I_j = E[f(U_j)]\\approx  \\frac{1}{L} \\sum_i f(U_{ij}) \n",
      "\\end{equation}\n",
      "\n",
      "and the overall integral without stratification is:\n",
      "\n",
      "\\begin{equation}  \n",
      "I =E[f(U)] \\approx \\frac{1}{N}\\sum_{i,j}  f(U_{ij}) \n",
      "\\end{equation}\n",
      "\n",
      "With stratification we would average the estimations in each strata\n",
      "\\begin{equation}\n",
      " I_s \\approx \\frac{1}{M} \\sum_j I_j \n",
      " \\end{equation}\n",
      "and that is obviously the same as the estimation without stratification.\n",
      "When we take strata we average the estimate for each strata as\n",
      "\n",
      "\\begin{equation} I_s = \\frac{1}{M} \\sum_j I_j =  \\frac{1}{M} \\sum_j \\frac{1}{L} \\sum_{i}  f(U_{ij})=  \\frac{1}{N} \\sum_{i,j}  f(U_{ij})= I\n",
      "\\end{equation}\n",
      "\n",
      "To do the analysis of the variance, lets remind ourselves that the \n",
      "MC estimated variance is  $\\sigma_I=\\frac{\\var(f)}{N}$. The variance without stratification is then \n",
      "\n",
      "\\begin{equation} \\var(I) = \\frac{\\var(f)}{N} = \\frac{E[f^2] - E[f]^2}{N}  = \\frac{E[f^2] - I^2}{N}\n",
      "\\end{equation}\n",
      "\n",
      "The expectation of $f^2$ can be expanded as \n",
      "\n",
      "\\begin{equation}\n",
      "E[f^2] \\approx \\frac{1}{N} \\sum_{ij} f^2(U_{ij}) =\\frac{1}{N} \\sum_j L \\left[ \\frac{1}{L} \\sum_i f^2(U_{ij}) \\right]=\\frac{L}{N}  \\sum_j E[f^2(U_j)] = \\frac{1}{M}  \\sum_j E[f^2(U_j)] \n",
      "\\end{equation}\n",
      "\n",
      "however from the definition of $\\sigma_j^2$ \n",
      "\n",
      "\\begin{eqnarray} \n",
      "\\sigma_j^2 = E[ f^2(U_j)] - E[f(U_j)]^2 &=& E[ f^2(U_j)] - I_j^2  \\nonumber \\\\\n",
      " &\\Rightarrow& E[ f^2(U_j)] =\\sigma_j^2 + I_j^2   \n",
      "\\end{eqnarray}\n",
      "\n",
      "substituting this, we get\n",
      "\n",
      "\\begin{equation}\n",
      "\\var(I) =\\frac{ \\frac{1}{M}  \\sum_j  (\\sigma_j^2 + I_j^2)- I^2}{N} = \\frac{1}{NM}\\sum_j  \\sigma_j^2 + \\frac{1}{N} \\left( \\frac{\\sum_j  I_j^2}{M} -  I^2 \\right)\n",
      "\\end{equation}\n",
      "\n",
      "the variance with stratification is \n",
      "\n",
      "\\begin{equation}  \\var(I_s)= \\var\\left(  \\frac{1}{M} \\sum_j I_j  \\right)  = \\frac{1}{M^2}  \\var\\left(\\sum_j I_j  \\right)  = \\frac{1}{M^2}  \\sum_j \\var(I_j)\n",
      " = \\frac{1}{M^2}   \\sum_j  \\frac{\\sigma_j^2}{L} =  \\frac{1}{M^2 \\,L}   \\sum_j  \\sigma_j^2\\end{equation}\n",
      " \n",
      "where we ignore all covariances since the deviations about $I_j$ are all iid and uncorrelated. Finally, \n",
      "\n",
      "\\begin{equation}\n",
      "  \\var(I_s)= \\frac{1}{M^2} \\frac{ 1}{L}   \\sum_j \\sigma_j^2  =  \\frac{1}{MN}  \\sum_j \\sigma_j^2 \\le \\var(I)\n",
      "\\end{equation}\n",
      "\n",
      "where the last step is based on the fact that \n",
      "$$  \\frac{\\sum_j  I_j^2}{M} -  I^2  > 0 $$\n",
      "by the Cauchy-Schwarz Inequality.\n",
      "\n",
      "Thus stratified sampling reduces the variance.\n",
      "\n",
      "\n",
      "Sub-dividing into strata always reduces the variance, so the optimum choice is to use 1 sample per\n",
      "stratum. \n",
      "However, we need multiple samples in each stratum to\n",
      "estimate the variance and obtain a confidence interval.\n",
      "Despite this, interesting to analyze what happens with\n",
      "1 sample per stratum."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "###Example\n",
      "\n",
      "For a  one-dimensional application we take $ x/(x^2+1)$ and integrate from $[0,1]$.\n",
      "We break $[0,10]$ into $M$ strata and for each stratum, take $N/M$ samples\n",
      "with uniform probability distribution. Compute the average within each \n",
      "stratum, and then calculate the overall average. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=[14,8])\n",
      "Y = lambda x: x/(x**2+1.0);\n",
      "intY = lambda x: np.log(x**2 + 1.0)/2.0;\n",
      "\n",
      "## Ploting the original functions \n",
      "plt.subplot(1,2,1)\n",
      "x = np.linspace(0,10,100)\n",
      "plt.plot(x, Y(x), label=u'$x/(x**2+1)$')\n",
      "\n",
      "N= 10000\n",
      "M = 10   # number of strate \n",
      "\n",
      "xmin=0\n",
      "xmax =10\n",
      "\n",
      "# analytic solution \n",
      "Ic = intY(xmax)-intY(xmin)\n",
      "\n",
      "Imc = np.zeros(N)\n",
      "Is = np.zeros(N)\n",
      "for k in np.arange(0,N):\n",
      " \n",
      "    Umin=0\n",
      "    Umax =10\n",
      "    # First lets do it with mean MC method \n",
      "    U = np.random.uniform(low=Umin, high=Umax, size=N)\n",
      "    Imc[k] = (Umax-Umin)* np.mean(Y(U))\n",
      "\n",
      "    #stratisfied it in Ns regions\n",
      "    step = (Umax-Umin )/M\n",
      "    Umin = 0 \n",
      "    Umax = step\n",
      "    Ii = 0\n",
      "    \n",
      "    \n",
      "    for reg in np.arange(0,M):\n",
      "        x = np.random.uniform(low=Umin, high=Umax, size=1.0*N/Ns);\n",
      "        Ii = Ii+(Umax-Umin)* np.mean(Y(x))\n",
      "        Umin = Umin + step\n",
      "        Umax = Umin + step\n",
      "\n",
      "\n",
      "    Is[k] = Ii\n",
      "\n",
      "plt.subplot(1,2,2)\n",
      "plt.hist(Imc,30, histtype='step', label=u'Normal MC')\n",
      "plt.hist(Is, 30, histtype='step', label=u'Stratified')\n",
      "plt.legend()\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "<matplotlib.legend.Legend at 0x108fa9dd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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QnaZmwZEkSZIOXMwUnHifogZw+umwdCnsdO1USZIkSfshpgpOPE9Rg+g1fs4/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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108f9b5d0>"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}