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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Introduction\n",
      "-------------------\n",
      "\n",
      "Brzezniak (2000) is a great book because it approaches conditional expectation through a sequence of exercises, which is what we are trying to do here. The main difference is that Brzezniak takes a more abstract measure-theoretic approach to the same problems. Note that you *do* need to grasp the measure-theoretic to move into more advanced areas in stochastic processes, but for what we have covered so far, working the same problems in his text using our methods is illuminating. It always helps to have more than one way to solve *any* problem.  I urge you to get a copy of his book or at least look at some pages on Google Books. I have numbered the examples corresponding to the book. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Examples\n",
      "-------------\n",
      "\n",
      "This is Example 2.1 from Brzezniak:\n",
      "\n",
      "> Three coins, 10p, 20p and 50p are tossed. The values of those coins that land heads up are added to work out the total amount. What is the expected total amount given that two coins have landed heads up?\n",
      "\n",
      "In this case we have we want to compute $\\mathbb{E}(\\xi|\\eta)$ where\n",
      "\n",
      "$$ \\xi = 10 X_{10} + 20 X_{20} +50 X_{50} $$\n",
      "\n",
      "where $X_i \\in \\{ 0,1\\} $. This represents the sum total value of the heads-up coins. The $\\eta$ represents the fact that only two of the three coins are heads-up. Note \n",
      "\n",
      "$$\\eta = X_{10} X_{20} (1-X_{50})+ (1-X_{10}) X_{20} X_{50}+ X_{10} (1-X_{20}) X_{50} $$\n",
      "\n",
      "is a function that is only non-zero when two of the three coins is heads. Each triple term catches each of these three possibilities. For example, the first term is when the 10p and 20p are heads up  and the 50p is heads down.\n",
      "\n",
      "To compute the conditional expectation, we want to find a function $h$ of $\\eta$ that minimizes the MSE\n",
      "\n",
      "$$ \\sum_{X\\in\\{0,1\\}^3} \\frac{1}{8} (\\xi - h( \\eta ))^2 $$\n",
      "\n",
      "where the sum is taken over all possible triples of outcomes for $ \\{X_{10} , X_{20} ,X_{50}\\}$ and the $\\frac{1}{8} = \\frac{1}{2^3} $ since each coin has a $\\frac{1}{2}$ chance of coming up heads.\n",
      "\n",
      "Now, the question boils down to what function $h(\\eta)$ should we try? Note that $\\eta \\in \\{0,1\\}$ so $h$ takes on only two values. Thus, we only have to try $h(\\eta)=\\alpha \\eta$  and find $\\alpha$. Writing this out gives,\n",
      "\n",
      "$$ \\sum_{X\\in\\{0,1\\}^3} \\frac{1}{8} (\\xi - \\alpha( \\eta ))^2 $$\n",
      "\n",
      "\n",
      "which boils down to solving for $\\alpha$,\n",
      "\n",
      "$$\\langle \\xi , \\eta \\rangle = \\alpha \\langle \\eta,\\eta \\rangle$$\n",
      "\n",
      "where \n",
      "\n",
      "$$ \\langle \\xi , \\eta \\rangle =\\sum_{X\\in\\{0,1\\}^3} \\frac{1}{8} (\\xi  \\eta ) $$\n",
      "\n",
      "This is tedious and a perfect job for `sympy`.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy as S\n",
      "\n",
      "eta = S.Symbol('eta')\n",
      "xi = S.Symbol('xi')\n",
      "X10 = S.Symbol('X10')\n",
      "X20 = S.Symbol('X20')\n",
      "X50 = S.Symbol('X50')\n",
      "\n",
      "eta = X10 * X20 *(1-X50 )+ X10 * (1-X20) *(X50 )+ (1-X10) * X20 *(X50 )\n",
      "xi = 10*X10 +20* X20+ 50*X50 \n",
      "\n",
      "num=S.summation(xi*eta,(X10,0,1),(X20,0,1),(X50,0,1))\n",
      "den=S.summation(eta*eta,(X10,0,1),(X20,0,1),(X50,0,1))\n",
      "alpha=num/den\n",
      "print alpha"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "160/3\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This means that\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta) = \\frac{160}{3} \\eta $$\n",
      "\n",
      "which we can check with a quick simulation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "from numpy import array\n",
      "\n",
      "x=np.random.randint(0,2,(3,5000))\n",
      "print (160/3.,np.dot(x[:,x.sum(axis=0)==2].T,array([10,20,50])).mean())\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(53.333333333333336, 53.243528790279981)\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example\n",
      "--------\n",
      "\n",
      "This is example 2.2:\n",
      "\n",
      "> Three coins, 10p, 20p and 50p are tossed as before. What is the conditional expectation of the total amount shown by the three coins given the total amount shown by the 10p and 20p coins only?\n",
      "\n",
      "For this problem,\n",
      "\n",
      "$$\\eta = 30 X_{10} X_{20} + 20 (1-X_{10}) X_{20}  + 10 X_{10} (1-X_{20})  $$\n",
      "\n",
      "which takes on three values (10,20,30) and only considers the 10p and 20p coins. Here, we'll look for affine functions, $h(\\eta) = a \\eta + b $."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.abc import a,b\n",
      "\n",
      "eta = X10 * X20 * 30 + X10 * (1-X20) *(10 )+ (1-X10) * X20 *(20 )\n",
      "h = a*eta + b\n",
      "\n",
      "J=S.summation((xi - h)**2 * S.Rational(1,8),(X10,0,1),(X20,0,1),(X50,0,1))\n",
      "\n",
      "sol=S.solve( [S.diff(J,a), S.diff(J,b)],(a,b) )\n",
      "print sol"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "{b: 25, a: 1}\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This means that\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta) = 25+ \\eta $$\n",
      "\n",
      "since $\\eta$ takes on only four values, $\\{0,10,20,30\\}$, we can write this out as\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta=0) = 25  $$\n",
      "$$ \\mathbb{E}(\\xi|\\eta=10) = 35  $$\n",
      "$$ \\mathbb{E}(\\xi|\\eta=20) = 45  $$\n",
      "$$ \\mathbb{E}(\\xi|\\eta=30) = 55  $$\n",
      "\n",
      "The following is  a quick simulation to demonstrate this."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x=np.random.randint(0,2,(3,5000))  # random samples for 3 coins tossed\n",
      "eta=np.dot(x[:2,:].T,array([10,20])) # sum of 10p and 20p\n",
      "\n",
      "print np.dot(x[:,eta==0].T,array([10,20,50])).mean() # E(xi|eta=0)\n",
      "print np.dot(x[:,eta==10].T,array([10,20,50])).mean()# E(xi|eta=10)\n",
      "print np.dot(x[:,eta==20].T,array([10,20,50])).mean()# E(xi|eta=20)\n",
      "print np.dot(x[:,eta==30].T,array([10,20,50])).mean()# E(xi|eta=30)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "25.1587301587\n",
        "34.3410852713\n",
        "43.7323358271\n",
        "56.6238973536\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example\n",
      "-----------\n",
      "\n",
      "This is Example 2.3\n",
      "\n",
      "![alt text](files/ex23.jpg)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that \"Lebesgue measure\" on $[0,1]$ just means uniformly distributed on that interval. Also, note the the `Piecewise` object in `sympy` is not complete at this point in its development, so we'll have to work around that in the following."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x=S.Symbol('x')\n",
      "c=S.Symbol('c')\n",
      "\n",
      "xi = 2*x**2\n",
      "\n",
      "eta=S.Piecewise((1,S.And(S.Gt(x,0),S.Lt(x,S.Rational(1,3)))),  #  0 < x < 1/3\n",
      "                (2,S.And(S.Gt(x,S.Rational(1,3)),S.Lt(x,S.Rational(2,3)))), # 1/3 < x < 2/3,\n",
      "                (0,S.And(S.Gt(x,S.Rational(2,3)),S.Lt(x,1))), \n",
      "                )\n",
      "\n",
      "\n",
      "h = a + b*eta + c*eta**2 \n",
      "J=S.integrate((xi - h)**2 ,(x,0,1))\n",
      "\n",
      "sol=S.solve( [S.diff(J,a), \n",
      "              S.diff(J,b),\n",
      "              S.diff(J,c),\n",
      "              ],\n",
      "              (a,b,c) )\n",
      "print sol\n",
      "\n",
      "print S.piecewise_fold(h.subs(sol))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "{c: 8/9, b: -20/9, a: 38/27}\n",
        "Piecewise((2/27, And(x < 1/3, x > 0)), (14/27, And(x < 2/3, x > 1/3)), (38/27, And(x < 1, x > 2/3)))\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Thus, collecting this result gives:\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta) = \\frac{38}{27} - \\frac{20}{9}\\eta + \\frac{8}{9} \\eta^2$$\n",
      "\n",
      "which can be re-written as a piecewise function as\n",
      "\n",
      "$$\\mathbb{E}(\\xi|\\eta) =\\begin{cases} \\frac{2}{27} & \\text{for}\\: 0 < x < \\frac{1}{3} \\\\\\frac{14}{27} & \\text{for}\\: \\frac{1}{3} < x < \\frac{2}{3} \\\\\\frac{38}{27} & \\text{for}\\: \\frac{2}{3}<x < 1 \\end{cases}\n",
      "$$\n",
      "\n",
      "The following is a quick simulation to demonstrate this."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.random.rand(1000)\n",
      "f,ax= subplots()\n",
      "ax.hist(2*x**2,bins=array([0,1/3.,2/3.,1])**2*2,normed=True,alpha=.5)\n",
      "ax.vlines([2/27.,14/27.,38/27.],0,ax.get_ylim()[1],linestyles='--')\n",
      "ax.set_xlabel(r'$2 x^2$',fontsize=18);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f75050>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This plot shows the intervals that correspond to the respective domains of $\\eta$ with the vertical dotted lines showing the $\\mathbb{E}(\\xi|\\eta) $ for that piece."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example\n",
      "-----------\n",
      "\n",
      "This is Example 2.4\n",
      "\n",
      "![alt text](files/ex24.jpg)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x,a=S.symbols('x,a')\n",
      "\n",
      "xi = 2*x**2\n",
      "\n",
      "half = S.Rational(1,2)\n",
      "\n",
      "eta_0=S.Piecewise((2, S.And(S.Ge(x,0), S.Lt(x,half))), \n",
      "                  (0, S.And(S.Ge(x,half), S.Le(x,1))))\n",
      "\n",
      "eta_1=S.Piecewise((0, S.Lt(x,half)),\n",
      "                  (x, S.And(S.Ge(x,half),S.Le(x,1))))\n",
      "\n",
      "\n",
      "\n",
      "v=S.var('b:3') # coefficients for quadratic function of eta\n",
      "h = a*eta_0 + (eta_1**np.arange(len(v))*v).sum()\n",
      "J=S.integrate((xi - h)**2 ,(x,0,1))\n",
      "sol=S.solve([J.diff(i) for i in v+(a,)],v+(a,))\n",
      "hsol = h.subs(sol)\n",
      "f=S.lambdify(x,hsol,'numpy')\n",
      "print S.piecewise_fold(h.subs(sol))\n",
      "t = np.linspace(0,1,51,endpoint=False)\n",
      "\n",
      "fig,ax = subplots()\n",
      "ax.plot(t, 2*t**2,label=r'$\\xi=2 x^2$')\n",
      "ax.plot(t,[f(i) for i in t],'-x',label=r'$\\mathbb{E}(\\xi|\\eta)$')\n",
      "ax.plot(t,map(S.lambdify(x,eta_0+eta_1),t),label=r'$\\eta(x)$')\n",
      "ax.set_ylim(ymax = 2.3)\n",
      "ax.grid()\n",
      "ax.legend(loc=0);\n",
      "#ax.plot(t,map(S.lambdify(x,eta),t))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Piecewise((0, x < 1/2), (2*x**2, And(x <= 1, x >= 1/2)))\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "<matplotlib.legend.Legend at 0x840a310>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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cOMaMGUP58uWpVatWxjalSpXiypUrnv4rWJ5leqkWOGN3ZS5sfjaeuGU0Nd9s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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x830c350>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The figure shows the $\\mathbb{E}(\\xi|\\eta)$ against $\\xi$ and $\\eta$. Note that $\\xi= \\mathbb{E}(\\xi|\\eta)= 2 x^2$ when $x\\in[0,\\frac{1}{2}]$ . Assembling the solution gives,\n",
      "\n",
      "$$\\mathbb{E}(\\xi|\\eta) =\\begin{cases} \\frac{1}{6} & \\text{for}\\: 0 \\le x < \\frac{1}{2} \\\\ 2 x^2 & \\text{for}\\: \\frac{1}{2} < x \\le 1 \\end{cases}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This example warrants more a more detailed explanation since $\\eta$ is more complicated. The first question is why did we choose $h(\\eta)$ as a quadratic function? Since $\\xi$ is a squared function of $x$ and since $x$ is part of $\\eta$, we chose a quadratic function so that $h(\\eta)$ would contain a $x^2$ in the domain where $\\eta=x$. The motivation is that we are asking for a function $h(x)$ that most closely approximates $2x^2$. Well, obviously, the exact function is $h(x)=2 x^2$! Thus, we want $h(x)=2 x^2$ over the domain where $\\eta=x$, which is $x\\in[\\frac{1}{2},1]$ and that is exactly what we have.\n",
      "\n",
      "We could have used our inner product by considering two separate functions,\n",
      "\n",
      "$\\eta_1 (x) = 2$ \n",
      "\n",
      "where $x\\in [0,\\frac{1}{2}]$ and\n",
      "\n",
      "$$\\eta_2 (x) = x$$ \n",
      "\n",
      "where $x\\in [\\frac{1}{2},1]$. Thus, at the point of projection, we have\n",
      "\n",
      "$$ \\mathbb{E}((2 x^2 - 2 c) \\cdot 2) = 0$$\n",
      "\n",
      "which leads to\n",
      "\n",
      "$$\\int_0^{\\frac{1}{2}} 2 x^2 \\cdot 2 dx = \\int_0^{\\frac{1}{2}} c 2 \\cdot 2   dx $$\n",
      "\n",
      "and a solution for $c$,\n",
      "\n",
      "$$ c = \\frac{1}{12} $$\n",
      "\n",
      "Assembling the solution for $x\\in[0,\\frac{1}{2}]$ gives\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta) =  \\frac{2}{12}$$\n",
      "\n",
      "We can do the same thing for the other piece, $\\eta_2$,\n",
      "\n",
      "$$ \\mathbb{E}((2 x^2 - c x^2) \\cdot x) = 0$$\n",
      "\n",
      "which, by inspection, gives $c=2$. Thus, for $x\\in[\\frac{1}{2},1]$ , we have\n",
      "\n",
      "$$ \\mathbb{E}(\\xi|\\eta)= 2 x^2$$  \n",
      "\n",
      "which is what we had before."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example\n",
      "-----------\n",
      "\n",
      "This is Exercise 2.6\n",
      "\n",
      "![alt text](files/ex26.jpg)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x,a=S.symbols('x,a')\n",
      "\n",
      "xi = 2*x**2\n",
      "eta = 1 - abs(2*x-1)\n",
      "\n",
      "half = S.Rational(1,2)\n",
      "\n",
      "eta=S.Piecewise((1+(2*x-1), S.And(S.Ge(x,0),S.Lt(x,half))),\n",
      "                (1-(2*x-1), S.And(S.Ge(x,half),S.Lt(x,1))))\n",
      "\n",
      "v=S.var('b:3') # assume h is quadratic in eta\n",
      "\n",
      "h = (eta**np.arange(len(v))*v).sum()\n",
      "\n",
      "J=S.integrate((xi - h)**2 ,(x,0,1))\n",
      "sol=S.solve([J.diff(i) for i in v],v)\n",
      "hsol = h.subs(sol)\n",
      "\n",
      "print S.piecewise_fold(h.subs(sol))\n",
      "\n",
      "t = np.linspace(0,1,51,endpoint=False)\n",
      "\n",
      "fig,ax = subplots()\n",
      "fig.set_size_inches(5,5)\n",
      "\n",
      "ax.plot(t, 2*t**2,label=r'$\\xi=2 x^2$')\n",
      "ax.plot(t,[hsol.subs(x,i) for i in t],'-x',label=r'$\\mathbb{E}(\\xi|\\eta)$')\n",
      "ax.plot(t,map(S.lambdify(x,eta),t),label=r'$\\eta(x)$')\n",
      "ax.legend(loc=0)\n",
      "ax.grid()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Piecewise((2*x**2 - 2*x + 1, And(x < 1/2, x >= 0)), (2*x + (-2*x + 2)**2/2 - 1, And(x < 1, x >= 1/2)))\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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rKF68OLy9vfOsxwsAt27dylmykmH0RkKCFPIeP662JfLhsvgR0Z1c//9NCPGjEKI0EaXk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\n5IoVgfr1pWFm3t7SULObN6Uyo0ZJIzA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       "text": [
        "<matplotlib.figure.Figure at 0x8414510>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The figure shows that the $\\mathbb{E}(\\xi|\\eta)$ is continuous over the entire domain. The code above solves for the conditional expectation using optimization assuming that $h$ is a quadratic function of $\\eta$, but we can also do it by using the inner product. Thus,\n",
      "\n",
      "$$ \\mathbb{E}\\left((2 x^2 - h(\\eta_1(x)) )\\eta_1(x)\\right)= \\int_0^{\\frac{1}{2}} (2 x^2 - h(\\eta_1(x)) )\\eta_1(x)  dx = 0$$\n",
      "\n",
      "where $\\eta_1 = 2x $ for $x\\in [0,\\frac{1}{2}]$. We can re-write this in terms of $\\eta_1$ as\n",
      "\n",
      "$$ \\int_0^1 \\left(\\frac{\\eta_1^2}{2}-h(\\eta_1)\\right)\\eta_1 d\\eta_1$$\n",
      "\n",
      "and the solution jumps right out as $h(\\eta_1)=\\frac{\\eta_1^2}{2}$. Note that $\\eta_1\\in[0,1]$. Doing the same thing for the other piece,\n",
      "\n",
      "$$ \\eta_2  = 2 - 2 x, \\hspace{1em} \\forall x\\in[\\frac{1}{2},1]$$ \n",
      "\n",
      "gives,\n",
      "\n",
      "$$ \\int_0^1 \\left(\\frac{(2-\\eta_2)^2}{2}-h(\\eta_2)\\right)\\eta_2 d\\eta_2$$\n",
      "\n",
      "and again, the optimal $h(\\eta_2)$ jumps right out as\n",
      "\n",
      "$$  h(\\eta_2) = \\frac{(2-\\eta_2)^2}{2} , \\hspace{1em} \\forall \\eta_2\\in[0,1]$$ \n",
      "\n",
      "and since $\\eta_2$ and $\\eta_2$ represent the same variable over the same domain we can just add these up to get the full solution:\n",
      "\n",
      "$$ h(\\eta) = \\frac{1}{2} \\left( 2 - 2 \\eta + \\eta^2\\right) $$\n",
      "\n",
      "and then back-substituting each piece for $x$ produces the same solution as `sympy`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example\n",
      "-----------\n",
      "\n",
      "This is Exercise 2.14\n",
      "\n",
      "![alt text](files/ex214.jpg)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x,a=S.symbols('x,a')\n",
      "half = S.Rational(1,2)\n",
      "\n",
      "xi = 2*x**2\n",
      "eta=S.Piecewise((2*x,    S.And(S.Ge(x,0),S.Lt(x,half))), \n",
      "                ((2*x-1),S.Ge(x,half)),\n",
      "               )\n",
      "\n",
      "v=S.var('b:3')\n",
      "\n",
      "h = (eta**np.arange(len(v))*v).sum()\n",
      "\n",
      "J=S.integrate((xi - h)**2 ,(x,0,1))\n",
      "sol=S.solve([J.diff(i) for i in v],v)\n",
      "hsol = h.subs(sol)\n",
      "\n",
      "print S.piecewise_fold(h.subs(sol))\n",
      "\n",
      "t = np.linspace(0,1,51,endpoint=False)\n",
      "\n",
      "fig,ax = subplots()\n",
      "fig.set_size_inches(5,5)\n",
      "\n",
      "ax.plot(t, 2*t**2,label=r'$\\xi=2 x^2$')\n",
      "ax.plot(t,[hsol.subs(x,i) for i in t],'-x',label=r'$\\mathbb{E}(\\xi|\\eta)$')\n",
      "ax.plot(t,map(S.lambdify(x,eta),t),label=r'$\\eta(x)$')\n",
      "ax.legend(loc=0)\n",
      "ax.grid()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Piecewise((2*x**2 + x + 1/4, And(x < 1/2, x >= 0)), (x + (2*x - 1)**2/2 - 1/4, x >= 1/2))\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9KN+1/s5renYzI0fPT0S6OnBOX3PCUVhN+JFwGlVolPajXRe5jkUHF/FNr28o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\nT0SlvZZjZP7r5RuX+T3ydy7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bAewBUoApInLAuqgdx8Hv7wtghqZpu9H9/ndE\nJNqyoJ1A07SfgCZAsKZpZ4Bh6DaFYV1Rg5wVCkWuxPrd2xQKhcIClPgpFIpciRI/hUKRK1Hip1Ao\nciVK/BQKRa5EiZ9CociVKPFTKBS5EiV+CoUiV/L/msea8Ip5xD4AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x81753b0>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As before, using the inner product for this problem, gives:\n",
      "\n",
      "$$ \\int_0^1 \\left(\\frac{\\eta_1^2}{2}-h(\\eta_1)\\right)\\eta_1 d\\eta_1=0$$\n",
      "\n",
      "and the solution jumps right out as \n",
      "$$h(\\eta_1)=\\frac{\\eta_1^2}{2}  , \\hspace{1em} \\forall \\eta_1\\in[0,1$$\n",
      "\n",
      "where $\\eta_1(x)=2x$. Doing the same thing for $\\eta_2=2x-1$ gives,\n",
      "\n",
      "$$ \\int_0^1 \\left(\\frac{(1+\\eta_2)^2}{2}-h(\\eta_2)\\right)\\eta_1 d\\eta_2=0$$\n",
      "\n",
      "with  \n",
      "\n",
      "$$h(\\eta_2)=\\frac{(1+\\eta_2)^2}{2} , \\hspace{1em} \\forall \\eta_2\\in[0,1]$$ \n",
      "\n",
      "and then adding these up as before gives the full solution:\n",
      "\n",
      "$$ h(\\eta)= \\frac{1}{2} +\\eta + \\eta^2$$ \n",
      "\n",
      "Back-substituting each piece for $x$ produces the same solution as `sympy`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xs = np.random.rand(100)\n",
      "print np.mean([(2*i**2-hsol.subs(x,i))**2 for i in xs])\n",
      "print S.integrate((2*x**2-hsol)**2,(x,0,1)).evalf()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.282679822464177\n",
        "0.270833333333333\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Summary"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We worked out some of the great examples in Brzezniak's book using our methods as a way to show multiple ways to solve the same problem. In particular, comparing Brzezniak's more measure-theoretic methods to our less abstract techniques is a great way to get a handle on those concepts which you will need for more advanced study in stochastic process. \n",
      "            \n",
      "As usual, the corresponding [IPython Notebook](www.ipython.org) notebook for this post  is available for download [here](https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Conditional_expectation_MSE_Ex.ipynb)  \n",
      "\n",
      "Comments and corrections welcome!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "References\n",
      "---------------\n",
      "\n",
      "* Brzezniak, Zdzislaw, and Tomasz Zastawniak. Basic stochastic processes: a course through exercises. Springer, 2000."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}