Lecture2.ipynb

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      "#Lecture 2: Linear Algebra\n",
      "\n",
      "## Topics\n",
      "\n",
      "* Review of Linear Algebra\n",
      "* Gaussian Elimination and LU Factorization\n",
      "* Cholesky decomposition\n",
      "* Matrix calculus and Mean/Var optimization\n",
      "* Norm and condition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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     },
     "source": [
      "# Review of Linear Algebra"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "subslide"
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     "source": [
      "## Vector\n",
      "* Is a set of elements, which can be real or complex:\n",
      "\n",
      "$$\n",
      "\\renewcommand{bs}{\\boldsymbol}\n",
      "\\begin{matrix}\n",
      "\\bs u = \\left(\n",
      "\\begin{matrix}\n",
      "u_1 \\\\\n",
      "u_2 \\\\\n",
      "\\vdots \\\\\n",
      "u_n\n",
      "\\end{matrix}\n",
      "\\right) \n",
      "\\hspace{2cm}\n",
      "\\bs v = \\left(\n",
      "\\begin{matrix}\n",
      "v_1 \\\\\n",
      "v_2 \\\\\n",
      "\\vdots \\\\\n",
      "v_n\n",
      "\\end{matrix}\n",
      "\\right) \n",
      "\\end{matrix}\n",
      "$$\n",
      "\n",
      "## Basic vector operations\n",
      "* Vector addition: $\\bs{w = u + v}$\n",
      "* Scalar multiplication: $\\bs w = a \\bs u$, where $a$ is a scalar\n",
      "\n",
      "## Notation\n",
      "* column vector: $\\bs {u, v, x, \\beta}$ \n",
      "* row vector: $\\bs u^T, \\bs v^T, \\bs x^T, \\bs \\beta^T$\n",
      "* scalar: $a, b, \\alpha, \\beta$ \n",
      "* matrix: $A, B, P$"
     ]
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     "source": [
      "## Vector addition\n",
      "* Associativity: $\\bs{u + (v + w) = (u + v) + w}$\n",
      "* Cumutativity: $\\bs{u + v = v + u}$\n",
      "* Identity: $\\bs{v + 0 = v}$ for $\\forall \\bs{v}$\n",
      "* Inverse: for $\\forall \\bs{v}$, exists an $-\\bs{v}$, so that $\\bs{v + (-v) = 0}$ \n",
      "* Distributivity: $a \\bs{(u + v)} = a \\bs{u} + a \\bs v, (a+b) \\bs v = a \\bs v + b \\bs v$\n",
      "\n",
      "## Scalar multiplication\n",
      "* Associativity: $a(b \\bs v) = (ab)\\bs v$\n",
      "* Identity: $1 \\bs v = \\bs v$, for $\\forall \\bs v$"
     ]
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     "source": [
      "##Vector space $\\Omega$\n",
      "* a collection of vectors that can be added and multiplied by scalars.\n",
      "\n",
      "##Vector subspace\n",
      "* a subset of the vector space $\\Omega' \\subset \\Omega$  that is closed under vector addition and scalar multiplication"
     ]
    },
    {
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     "source": [
      "##Linear combination\n",
      "* $\\bs v = a_1 \\bs v_1 + a_2 \\bs v_2 + ... + a_n \\bs v_n$\n",
      "* Linear combinations of vectors form a subspace $\\Omega_v = \\text{span}(\\bs{v_1, v_2, ... v_n}) \\subset \\Omega$\n",
      "* Linear independence: $\\bs {v = 0} \\iff a_k = 0$ for $\\forall k$ \n",
      "* Basis: any set of linearly independent $\\bs v_i$ that spans $\\Omega_v$\n",
      "* Dimension of $\\Omega_v$ is the number of vectors in (any of) its basis"
     ]
    },
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     "source": [
      "## Inner product\n",
      "* $\\langle \\bs u, a \\bs v_1 + b \\bs v_2 \\rangle = a \\langle \\bs{u, v_1} \\rangle + b \\langle \\bs{u, v_2} \\rangle$ \n",
      "* $\\langle \\bs{u, v} \\rangle = \\langle \\bs{v, u} \\rangle^c$ \n",
      "* $\\langle \\bs{u, u} \\rangle \\ge 0$ \n",
      "* $\\langle \\bs{u, u} \\rangle = 0 \\iff u = 0$ \n",
      "* $\\bs {u, v}$ othogonal if $\\langle \\bs{u, v} \\rangle = 0$ "
     ]
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     "source": [
      "## Dot product\n",
      "* the standard inner product: $\\bs{u \\cdot v} = \\sum_{k=1}^n u_k^c v_k$  \n",
      "* the magnitude of a vector $\\bs b$ is $\\vert b \\vert = \\sqrt{\\bs b \\cdot \\bs b}$\n",
      "* the projection of vector $\\bs a$ to the direction of vector $\\bs b$ is: $ a_1 = \\frac{\\bs a \\cdot \\bs b}{\\vert b \\vert} = \\vert \\bs a \\vert \\cos(\\theta) $\n",
      "\n",
      "<center>\n",
      "<img src=\"img/projection.png\">\n",
      "</center>"
     ]
    },
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     "source": [
      "## Matrix\n",
      "* Represents a linear response to multiple input factors:\n",
      "\n",
      "$$ \\overset{\\text{Outputs}}{\\longleftarrow}\\overset{\\downarrow \\text{Inputs}}\n",
      "{\\begin{pmatrix}\n",
      "a_{11} & a_{12} & . & a_{1n} \\\\\n",
      "a_{21} & a_{22} & . & a_{2n} \\\\\n",
      ".      & .      & . & \\\\\n",
      "a_{m1} & a_{m2} & . & a_{mn}\n",
      "\\end{pmatrix}} $$\n",
      "\n",
      "* Matrix addition and scalar multiplication are element wise, similar to those for vectors"
     ]
    },
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     "source": [
      "## Matrix multiplication\n",
      "\n",
      "<center><img src=\"img/matrixprod.png\"></center>\n",
      "\n",
      "$$\\begin{array}\n",
      "\\\\ \\bs u = A \\bs v &\\iff u_i = \\sum_{j=1}^{n} a_{ij}v_j \\\\\n",
      "C = AB &\\iff c_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} = a_i \\cdot b_j\n",
      "\\end{array}$$"
     ]
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     "source": [
      "## Matrix represents linear transformation\n",
      "Linear function on vectors:\n",
      "* $L(\\bs{u + v}) = L(\\bs u) + L(\\bs v)$ \n",
      "* $L(a \\bs v) = a L(\\bs v)$ \n",
      "\n",
      "Any linear transformation bewteen finite dimensional vector space can be represented by a matrix multiplication, therefore we can write $L \\bs u$ instead of $L(\\bs u)$."
     ]
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     "source": [
      "##Properties of linear transformation\n",
      "* Associativity: $A(BC) = (AB)C$\n",
      "* Distributivity: \n",
      " * $A(B+C) = AB + AC$\n",
      " * $(B+C)A = BA + CA$ \n",
      " * $\\alpha (A+B) = \\alpha A + \\alpha B$\n",
      "* But **not** commutative: $AB \\ne BA$"
     ]
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     "source": [
      "##Matrix definitions\n",
      "* Identity matrix $I$: $I A = A I = A$\n",
      "* $A^T$ is the transpose of $A$: $a^T_{ij} = a_{ji}$\n",
      "* Symmetric matrix: $A = A^T$\n",
      "* $A^*$ is the adjoint of $A$: $a^*_{ij} = a_{ji}^c$\n",
      "  * real matrix: $A^T = A^*$\n",
      "  * self-adjoint (Hermitian) matrix: $A = A^*$\n",
      "* Inverse matrix: $AA^{-1} = A^{-1}A = I$\n",
      "* Orthogonal matrix: $A^T = A^{-1} \\iff AA^T = I$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "# Gaussian Elimination and LU Factorization"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Linear system\n",
      "* In matrix form, a linear system is $A \\bs {x = y}$\n",
      "* It has a unique solution if $A$ is a full rank square matrix"
     ]
    },
    {
     "cell_type": "code",
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     "input": [
      "%pylab inline\n",
      "lecture = 2\n",
      "\n",
      "import fmt\n",
      "import sympy as sp\n",
      "from IPython.display import display\n",
      "\n",
      "sp.init_printing(use_latex = True)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
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     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
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     "prompt_number": 2
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     "input": [
      "a = sp.Matrix([[2, 1, -1], [-6, -2, 4], [-2, 1, 2]])\n",
      "y = sp.Matrix([8, -22, -3])\n",
      "X = sp.MatrixSymbol('x', 3, 1)\n",
      "x1, x2, x3 = sp.symbols('x_1, x_2, x_3')\n",
      "x = sp.Matrix([x1, x2, x3])\n",
      "fmt.displayMath(a, fmt.joinMath('=', x, y), sep=\"\", pre=\"\\\\small \")"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
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     "outputs": [
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       "latex": [
        "$$\\small  \\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\-22\\\\-3\\end{matrix}\\right)$$"
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     "source": [
      "##Gaussian elimination\n",
      "\n",
      "Eliminate the $x_1$ terms using the first row, this operation is a linear transformation:\n"
     ]
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     "input": [
      "A = sp.MatrixSymbol('A', 3, 3)\n",
      "L1 = sp.MatrixSymbol('L_1', 3, 3)\n",
      "L2 = sp.MatrixSymbol('L_2', 3, 3)\n",
      "l1 = sp.eye(3)\n",
      "l1[1, 0] = -a[1, 0]/a[0, 0]\n",
      "l1[2, 0] = -a[2, 0]/a[0, 0]\n",
      "\n",
      "fmt.displayMath(L1*a, fmt.joinMath('=', x, L1*y), \"\\;,\\;\\;\", l1*a, fmt.joinMath('=', x, l1*y), sep=\"\", pre=\"\\\\small \")"
     ],
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       "latex": [
        "$$\\small  L_{1} \\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=L_{1} \\left(\\begin{matrix}8\\\\-22\\\\-3\\end{matrix}\\right)\\;,\\;\\;\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 2 & 1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\2\\\\5\\end{matrix}\\right)$$"
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       "text": [
        "<IPython.core.display.Math at 0x9f01f28>"
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     "source": [
      "Use the 2nd equation (row) to eliminate the $x_2$ terms:"
     ]
    },
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "l2 = sp.eye(3)\n",
      "a2 = l1*a\n",
      "y2 = l1*y\n",
      "l2[2, 1] = -a2[2, 1]/a2[1, 1]\n",
      "u = l2*a2\n",
      "fmt.displayMath(L2*a2, fmt.joinMath('=', x, L2*y2), \"\\;,\\;\", u, fmt.joinMath('=', x, l2*y2), sep=\"\", pre=\"\\\\small \")"
     ],
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       "latex": [
        "$$\\small  L_{2} \\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 2 & 1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=L_{2} \\left(\\begin{matrix}8\\\\2\\\\5\\end{matrix}\\right)\\;,\\;\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\2\\\\1\\end{matrix}\\right)$$"
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        "<IPython.core.display.Math at 0x9f01e80>"
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     "source": [
      "the $L_1$ and $L_2$ are both lower triangular matrix"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Ui = sp.MatrixSymbol('U^{-1}', 3, 3)\n",
      "U = sp.MatrixSymbol('U', 3, 3)\n",
      "L = sp.MatrixSymbol('L', 3, 3)\n",
      "fmt.displayMath(fmt.joinMath('=', L1, l1), fmt.joinMath('=', L2, l2), fmt.joinMath('=', U, u), pre=\"\\\\small \")"
     ],
     "language": "python",
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      {
       "latex": [
        "$$\\small  L_{1}=\\left(\\begin{matrix}1 & 0 & 0\\\\3 & 1 & 0\\\\1 & 0 & 1\\end{matrix}\\right)\\;,\\;\\;\\;L_{2}=\\left(\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & -2 & 1\\end{matrix}\\right)\\;,\\;\\;\\;U=\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)$$"
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     "source": [
      "The resulting matrix $U = L_2L_1A$ is upper trianglar"
     ]
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     "source": [
      "## LU factorization\n",
      "\n",
      "The triangular matrix is easy to invert by variable replacement"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y3 = l2*y2\n",
      "a3 = l2*a2\n",
      "ui = a3.inv()\n",
      "fmt.displayMath(fmt.joinMath('=', Ui, ui), \"\\;,\\;\", fmt.joinMath('=', x, Ui), fmt.joinMath('=', l2*y2, ui*y3), sep=\"\\;\", pre=\"\\\\small \")"
     ],
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       "slide_type": "-"
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     },
     "outputs": [
      {
       "latex": [
        "$$\\small  U^{{-1}}=\\left(\\begin{matrix}\\frac{1}{2} & - \\frac{1}{2} & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\;\\;,\\;\\;\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=U^{{-1}}\\;\\left(\\begin{matrix}8\\\\2\\\\1\\end{matrix}\\right)=\\left(\\begin{matrix}2\\\\3\\\\-1\\end{matrix}\\right)$$"
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        "<IPython.core.display.Math at 0x9f7e208>"
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     "prompt_number": 7
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     "source": [
      "Now we can group $L = L_1^{-1}L_2^{-1}$ and obtain the LU factorization\n",
      "\n",
      "$$L_2 L_1 A = U \\iff A = L_1^{-1} L_2^{-1} U \\iff A = LU $$\n",
      "\n",
      "* $U$ is a upper triangular matrix.\n",
      "* There can be infinite numbers of LU pairs, the convention is to keep the diagonal elements of $L$ matrix 1. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "l =  l1.inv()*l2.inv()\n",
      "fmt.displayMath(fmt.joinMath('=', L, l), fmt.joinMath('=', U, a3), fmt.joinMath('=', L*U, l*a3), pre=\"\\\\small \")"
     ],
     "language": "python",
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     "outputs": [
      {
       "latex": [
        "$$\\small  L=\\left(\\begin{matrix}1 & 0 & 0\\\\-3 & 1 & 0\\\\-1 & 2 & 1\\end{matrix}\\right)\\;,\\;\\;\\;U=\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\;,\\;\\;\\;L U=\\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)$$"
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        "<IPython.core.display.Math at 0x9f7e588>"
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     "source": [
      "* The LU factorization is the matrix representation of Gaussian elimination"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
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     "source": [
      "## Pivoting\n",
      "\n",
      "* The Gaussian elimination does not work if there are 0s in the diagonal of the matrix.\n",
      "* The rows of the matrix can be permuted first, so that the diagonal elements have the greatest magnitude. \n",
      "\n",
      "$$ A = P \\cdot L \\cdot U $$\n",
      "\n",
      "where the $P$ matrix represent the row permutation.\n",
      "* The permutation (pivoting) also improve the numerical stability"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.linalg import lu\n",
      "\n",
      "def displayMultiple(fs) :\n",
      "    tl=map(lambda tc: '$' + sp.latex(tc) + '$',fs)\n",
      "    r = '''\n",
      "  <table border=\"0\"><tr>'''\n",
      "    for v in tl :\n",
      "        r += \"<td>\" + v + \"</td>\"\n",
      "    r += \"</tr></table>\"\n",
      "    return r\n",
      "\n",
      "a = sp.Matrix([[0, 3, 1, 2], [4, 0, -3, 1], [-3, 1, 0, 2], [9, 2, 5, 0]])\n",
      "p, l, u = map(lambda x: sp.Matrix(x), lu(a))\n",
      "Pi = sp.MatrixSymbol('P^{-1}', 4, 4)\n",
      "A = sp.MatrixSymbol('A', 4, 4)\n",
      "P = sp.MatrixSymbol('P', 4, 4)\n",
      "fmt.displayMath(fmt.joinMath('=', A, a), fmt.joinMath('=', P, p), pre=\"\\\\small \")"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
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     "outputs": [
      {
       "latex": [
        "$$\\small  A=\\left(\\begin{matrix}0 & 3 & 1 & 2\\\\4 & 0 & -3 & 1\\\\-3 & 1 & 0 & 2\\\\9 & 2 & 5 & 0\\end{matrix}\\right)\\;,\\;\\;\\;P=\\left(\\begin{matrix}0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0\\\\1.0 & 0.0 & 0.0 & 0.0\\end{matrix}\\right)$$"
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        "<IPython.core.display.Math at 0x9f7e128>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayMath(sp.Eq (Pi*A, p.inv()*a), pre=\"\\\\small \")"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
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      {
       "latex": [
        "$$\\small  P^{{-1}} A = \\left(\\begin{matrix}9.0 & 2.0 & 5.0 & 0\\\\0 & 3.0 & 1.0 & 2.0\\\\4.0 & 0 & -3.0 & 1.0\\\\-3.0 & 1.0 & 0 & 2.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x9ed3470>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Complexity of numerical algorithm\n",
      "\n",
      "Complexity of a numerical algorithm is stated in the order of magnitude, often in the big-O notation:\n",
      "\n",
      "* binary search $O(\\log(n))$\n",
      "* best sorting algorithm: $O(n \\log(n))$\n",
      "\n",
      "Most numerical linear algebra algorithm is of complexity of $O(n^3)$\n",
      "* e.g.: the LU decomposition (why?)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "#Cholesky Decomposition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Symmetric positive definite Matrix\n",
      "* $V$ is positive definite if $\\bs x^T V \\bs x > 0$ for $\\forall \\bs{x \\ne 0}$\n",
      "* $V$ is semi positive definite if $\\bs x^T V \\bs {x \\ge 0}$ for $\\forall \\bs{x \\ne 0}$\n",
      "* Semi-positive definite does not mean every element in the matrix is positive"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Covariance matrix\n",
      "\n",
      "The most important and ubiquitous matrices in quant Finance, \n",
      "\n",
      "* capturing the linear dependencies of random factors $\\bs r = [r_1, ..., r_n]^T$ \n",
      "* we use $\\bar{\\bs r} = \\mathbb{E}[\\bs r]$ to denote their expectations\n",
      "\n",
      "\n",
      "$$\\begin{array}{l}V &= \\mathbb{E}[(\\bs r - \\bar{\\bs r})(\\bs r - \\bar{\\bs r})^T] = \\mathbb{E}[\\bs r \\bs r^T] - \\bar{\\bs r}\\bar{\\bs r}^T \\\\\n",
      "&= (\\text{cov}(r_i, r_j)) = (\\rho_{ij} \\sigma_i \\sigma_j)\\end{array}$$\n",
      "\n",
      "Covariance of linear combinations: \n",
      " \n",
      "$$\\begin{array}{l}\n",
      "\\text{cov}(\\bs x^T \\bs r, \\bs y^T \\bs r) &= \\mathbb{E}[(\\bs x^T \\bs r)(\\bs r^T \\bs y)] - \\mathbb{E}[\\bs x^T \\bs r]\\mathbb{E}[\\bs r^T \\bs y]\\\\\n",
      "&= \\bs x^T \\mathbb{E}[\\bs r \\bs r^T] \\bs y - \\bs x^T \\bar{\\bs r}\\bar{\\bs r}^T \\bs y = \\bs x^T V \\bs y\n",
      "\\end{array}$$\n",
      " "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Correlation matrix\n",
      "\n",
      "* $C = (\\rho_{ij})$ is the co-variance matrix of the normalized factors $\\bs s = [\\frac{r_1}{\\sigma_1}, ..., \\frac{r_n}{\\sigma_n}]^T$\n",
      "* $V = \\bs \\Sigma C \\bs \\Sigma $, where $\\Sigma$ is a diagonal matrix of $\\sigma_i$\n",
      "\n",
      "Both covariance and correlation matrices are symmetric positive semi-definate (SPD):\n",
      "\n",
      "* $\\bs x^T V \\bs x = \\text{cov}[\\bs x^T \\bs r, \\bs x^T \\bs r] = \\text{var}[\\bs x^T \\bs r] \\ge 0$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Example: weekly rprice and returns"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pandas as pd\n",
      "f3 = pd.read_csv('data/f3.csv', parse_dates=[0]).set_index('Date').sort()\n",
      "\n",
      "fig = figure(figsize=[12, 4])\n",
      "ax1 = fig.add_subplot(121)\n",
      "f3.plot(title='Historical Prices', ax=ax1);\n",
      "\n",
      "ax2 = fig.add_subplot(122)\n",
      "r = np.log(f3).diff()\n",
      "r.plot(title='Historical Returns', ax=ax2);"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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ZpOVvvgknnqgnvAiwfj106AAbN8KPP5rcvtWJUioZeBm4BDgBuF4pdXyZzXKA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\n8aEYuXOnDmnIyYEePWDKFH3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kSj2lTFk9RccB+30MWw833vlGYM4LlM+L8nqDr9lDIliL\nh9NadAWvF/F5cPmcQU9wmltv14BQTLDD4yAlkEotJYUSv+UJtnL9BjzBBapFhAgWhyN4/dJRD9dx\nNm0GTielntKg9zPIY4/B8OEM+fbbUFlFnmAoL2QCnmCAHTuCxcofIxyiip7gqDHB6enlhXKrVhWf\nE2iRvYWGWaGUftml2XRp1gUvToqL9WE9Hn384tSWtBBdD40KSqC0FH+pk2T8OAo9iMeLlOqY4P1F\n+znljVN0I8jL04Kiqp7gcFHd+ZAyyepry9YZfOKlsCgb34H9oYKA/bm5UFyMq2Uz0kpD38mugl1s\nyY390BFBGREc/p36xc8z85+JslMM5s7VM6ZVM7UmgnfsCInWo47Sr2zPOUd/kQcPQp8+NWtPmzba\nG/zCC/Hv88QTWjzs2aPfZkyadPjsMxw+VuxfQd92fWmQ3KC2TQniduuY3Guu0aE9InqilxhvhSN4\n5RX9W3X55TrPb3i++1699O/gaafBP/6hJ5IJOFUyM/X2PXvGd57U5FTaNG7DzV/eTJGriA9XfVj1\nizXUDbKz9euBsp7g8PAC0D98ZV/RV+YJXrSIYcOf5M4XFurj5OaGRHCzZlBYSPLBLJoXu/XNcPnl\nOB1F7CrYxZ0z78Tl0+dum7OBNLePOwPDsX/8kdINu5j9USZ5rbpTvC8sHGLXrqAnLGuX9RqwTB7Y\nQDq0QlchPvFx6wo4etXO0OX5vCifN+QJTtYiz+kMeYLF6yPZD95mm0mpwBPs8/vw+D2kWueT1FRe\nmBQ566PbrUVwnr9FKBwCeOCbe/h03afg9+Nr1RKA4sbpeB0lePweMkvCvH0ejx4Us2lTZP0XFOgR\nsPGI4EBM8DFWbnJLgDd05pcbhfbuu9Zb3rIiOHxgnLVOKotvLusJBmjVitlbZ3Pm5DOj7nbJtlc5\na9vTlLj1ebNKsujarGtQBAdClXG5KEgpI6gdDoqLdN25Cxx4nR4O7tR1kVmSqcNMSkr0IMamTeP2\nBA8bZj1PhnvBu3SJa99wwp4/AuZWPrfF+vXBsKSvV39OclFx6PsN3L+WJ7i0bXPSHJ7gA1ahq5Cs\n0qz4DSwzMC6cnNIcHprzEE5vFSY/KSqKOUA2UWpNBIcTeAiaN0/HZZfN6FATKKXjyV5+OQO/X78y\njuXYWLlSjzzt3RtattQC+L77yjkSqhW7xtSAPW3z+X1sy9sW1bapq6fyytJXWLBrAWO+H8OATgPK\nH6AGqKjevv9eC9X8fPj0U1iwQL85eTmOcQqzZ+tBv2+/rQeXlxnIDeg3rC++qPXFF1/oqcUbWM8A\n990HI0fG/512Tu/M0r1LeeLcJ/hs3Wdx7XOo2LG9Hen4C/KhY8dynuDtG1z4k5JDHeaoUboBh1OZ\nJ9iKQdzdKhlvm9a64QdEcHo6FBbiLy0myefXAurbb5k041Gu/exa2K5zv5LkoXXWBrb36cQtK8Hf\ntQvMn0+TE47mUcazM+kYKCjA57J+eH2+oAi5+L/6/s9ZPj/CcxoIgch35uP1e+lqXUJQBHt9KL8v\n6AkWyxP87SY9YGzxwV0on1d7gNuuI9kPviRFQ2+kCHZ4HST7IJBW35eUSqHX8gRbwjTgCc71t8CX\nH/IEZ+XuoWRhBiQn4/Z5KE6F7Gat8JToa9uTF+YJXrcuKEAzwus/P1/nqy07MA4q9gRbscoBYdLY\nlc/+2x6L2PTtt/VDNkVF+ofdajvi1Mf2OT14C/L0ZZYW64clETIef1zHcQWIFg4B0LIln6z9hEV7\nFhGNZHcpyY0nceEHFwIhT7BPlfcE56uWkTuXluIu1jHCnkIHqXgpys7hmQn3kl2ajdPrxJeTrQVA\nFcIhliyxHgw8ntDDSFgozt7CvUHRHouBA3XIfACHA3buzIi90/Tp8KF2VDQIOO4DIRHhIri4mKI2\nzUh3QYlH21LoLiSrJLoIfnHRi6zPWh9ZWMYTHN5n7y/WHug8R15se8NJUARXNv7FFiI4JUU/Hb32\nmvZ81Rbdu2sv9P79OtZ927aKt737bh2D/Oab+hPIuFPpvO6Gw8qO/B08/NPDAHyz6RuGfTys3DZ+\n8fNYxmM8+vOjXDH1CkafOjoiE0RtU1iowxSeeUYL2r/+VQvg4mKYNSv2+KP8fD3J06efVjK1qsWp\np8L77+uHvgADB8LVsUMvI+jSrAsNUxpyec/L2Z6/Pf4dDXWKgpx9Orl/GRF8cLeL0pRmoR/RvLzy\ngjfgCc7Ohv/+t/zBi4v59cxjePCaFvzaMIf8AzsiPcFFRUhJKcorLJqljz34P59w4RTt8n3z17fg\nkjG0zNzA91f3p5EXSvr3CYrcPqxjb4NjaOgqYPP+tRHnBWht/VYv/uFt3vw1NNgrMEVygbMAn99H\nlzIiWFme4FJPKcrdFJL18QIeX/ELWJ5gWm8iWcCVlkKat5RSGpGW5OHgQR0K0dALzhTBk6RFcDAd\nmrashF0AACAASURBVCVM3S4hBR8lqS0ozQx5gp0l+aQv17HUHpeLS2+ErLQmuEv1tS9dGyaCV67U\nWR3KUlAAbdviKXXzTOAttSWW9x6M9I4XFOdEhkSEzZYmBZHfe1GR5fUsKopoO84Cva+ryE1ptrYv\n92AhDBtG/k/f6h/S5ctDBwoXwSIwdiwAO325FLgKqIgUt4OjcxuyaM8iRISs0iy6NOvC218WMmB1\nT/aWbguK4EJ/Ot5A9g0AhwNPqZUqLV//lV8+o+lzE8ku1dfsyNqnRXBqalzhED9v/5ni1vN1CIzH\no8VG4PoApk5l4eg/MmX1lAqP8dO2n5i0fBIFBZET1TkcccxtUlAQ/N4ai3WtgcFxZcIhCls1Jt2t\nPcCg/5YLrbG474f7uGXGLZGFMWKCA/HUuY5cRCQ+50mCIjivEp1tCxEM+gHlb3/T8Yi1RcuWoNSQ\n4DiPzZvLb7NpE/zzn3oA0bBh+pXy1VfrMQbDhsFXX8UWKYeCneMga9M2j8/DygMrAd3JPLfwObJK\nsli4eyHrs9cz6A+Rsb4zNswgvUE6e+7bw4H7DzD6tNHBAW9bt+pB5DVFeL35fLpvP/ZY/SMbyJF9\nww26c/vyS91vRpspNS9Pt7tJk3Rc77nnxnf+0aP1eINwERzNtlh0Tu/M8W2Op32T9pS4S4JescOJ\nne+FI5XCnL3Qrh0Hs3dGvMb0O8uIYIdDN9QfftBPWKAbcEEB/PijTglVlqIiilL97MjfQV4aZO7Z\npAVCp05s9+Xw26Z57Nq3DuX18/UULe76/b6ffpnAMbAhaws0Pkh61lZWnNiKFR0gs1dn3M6QKtgn\nHfGTRCOnN+K8AK2sJtty697gjz5EeoJ94qO1A7wpSaFsYD7LE+x1MGZhKlfu0wI7IIJPa9YJ5fPR\ngFRomkl6cmOKm+pXnSU0oQFunE44kOMgzQeuZIU3CfK8xThSrB+SgAh2+vGjcKU10yLY8jy6iwtp\nslOP4ve4XXiToAjBY4ngmVmv89yC56wvsVBnVQCGQPBVN8XFZHpbk7PPzUsvWddmCbM/vReZgeFg\nvnY/SkkJ0qJFhBJr4osUpEVF1uqiIh1KYz10lOTqCncWu3Fk66DorP1FyJ7d3DX5zwxu2xYcDkQs\nM5xOHXLg9eoDPvEEANn5eyvMfgGQ7HXStVB/WcXu4qAn+MbVPkav2EyWe1cwHKLY14hSFZaTtbQU\nn1WH7jx9jc3zizg7iaBH1Jm1H2nVivUFW/G7K09J9/Wmr3F2/iEkgjMzdTxxQCzecAPXfryanQU7\ntai47LJyx1hzcA3zdy7E7Y7M+a6jkIbENiBsauxGfmt2u2ie4KIidh/Tjr4HYdS0G/h1368xwyGa\npDZhyd4lRGTEjZYnOC8Pxo8n9atvaOHQIvhA8QGu+fSaUFy7lNHOgSwoRUX48/OYveWH2NdYhsqS\nFthGBNsBpfRg5EC2klmzdPqUn34KbTN3rvYAt24dehukFNx/v8468dJLuu831BxfbviS898/H7fP\nzcoDK0lSSXy46kMW7l6IiLA+O/Saxi9+HprzEE+f/zSNUxuTlhI5E8S99+oBY7XBjBla6C5dqtOq\nBsZ8dOum43XPPBOefVZ7hcNxu6FHD+jbV79NefTR+M958cW6zZ5ySuJ2d2nWhb7t+qKUolN6p7hT\npu0u2B3MvWqwPyV5WUi7djjysyMyDojDRUlSeuhH1OnUjTIjQ3eYEPIEL1+uXYPTpunXbhbL5xaT\nl+LD7XOT3xBy92+FggLm7c7hsumjaOaCovyDpPiErK1apKb44YRCHcOzt2A/jZMK8SenUqQ8nD0K\nxjdcytq9K4PnOOBuRXFyc5rkh4SxWL+Q3aUlBWnQNDOPIrcu++EHWLJcC818Z36wrea1aRr0BCeJ\nF+XXA+N65KRwlEMLssAAOL/Xj/L6aOBPhSaZNE9Ox5GmPXC+JD/J4qF3b1i70UlDL7iSkvAmwQHv\nPn7sk8Oa/jcG69Xt8OFLElzNfbR/6Fb8E17U5aWFJK/RasgXEMF+P55Sy+tKPp+ss/L5ejzQsCFf\nPHgfjhRCDzNFRWzKaU3KwX3ceeCRoHcUIM2tL3ZV5irm7ZyH163Lc3YWk53aMUIEp0uhDg2YOjVw\nWDzb94REsOWq3LhHPywUFxThzsujJBVyDxbjz86mUbELX6mOVV24UPdRuFw6ZVxKCs6tof7FW1oS\n0xOc6nXQtdhDkl+nR8suzaZ9k/bkNNLfUbGnUDuXHU6KGxRS2siNT1k7Oxz4nboteAu1+GtaVESa\nD7Zkak+mOzsTT7MmHPTk43FUHsJQ4CzAp0pDIrikBNq1w1Gcz451OqRjea9mbM3byr6XntYDQcrg\n9DrJt8R5uAi+7ufRjNsXZXKECAMKwkRwEnkNCYngcE9waSkrpSkrOkKLWXPZkb+DQlchxe5i3WbK\neGQ7NTk6aFvIUMsT/I9/hDyDv/wCr7xC35c/5eydwMKFwd8Lt0+fPyMjLCJARMeq33cfFBeT5Bd+\nWV+BwHK5tFcnjDd+fYPvtnwTs0qMCC5DkyYZLFig37y8+qpuH+PGhdavWaO/l5NOKr/vBRfocLiF\nCw+PbXaOg6wN29w+NyXuEjJ2ZJDryOXHbT+yMnMlz13wHE/Of5IN2Ru4tMelTPtmWnCfH1avYOdO\nYUDLi8sdLzdX/2YvXaq/8w0bIsf3zJoFV16p81dPnlw91xCot3HjdBzuo49qT3BFXHedboOBOKff\nftOivWdPHS+8di0cf3z851dK5+pu2LBi2yrjtlNv48nzdL7BLs26sHDXQj5a9REP//Qwb/waPWfP\nvqJ99PhfD7q+2JXP130edZtY2PleOFJx5mfjbNWcJm7IytoRzCkpLhdFSnuC58yBvP1WoOWePVBQ\nwE3Tb8JTlK87znnzIDWV/8/eecfbUZXr/zt193Z6klNCeicBQhICEiB08EqXpsAVUa/lih3wIpcr\nXgvqvXasICJFUIkUISSBUBJISAjpOemn192nz/z+WPuck4TQbD/18nw+fMjZe83sNTNrZp71rud9\nXu/S97Luf4UEyXXht3cW6JfFSzAbht79+3D7czzdvxm3593UeDpSqYzqB2T3CRLsSTAtpzOtOJWu\nYhdxqYCtxyk7ZRKZel7u34h2wByrw6wiRwq9f4Q0SZUbPFm02VktER8oDpPgX/8anlg6EgkeStzK\np8LDzwUVF7mSGKfaIVTEMQxFgl8cbBORYF+DWA8JJY5XeeuGlCJSEDB9isfWVoOYDUVdRIIdRWb/\ncQ+ytuq0YXLimB6eBIWqyvL1K4LgO+UizRUC5tk2rgx5PNxycfg4t/RuEV7Gtk2gaqyeOonnPIaP\ntaNzB1vSz1HjdHI5d4tV+srxNuk1ANy07CYu/P5N7GoVn8cpsiG8F6trhJTKVKzEKhm8hQJ89MF5\n3HfvF/Fr63GL4tpt7HwegFKhSJDP0xeFXF8BeWCQKgOW794NhkFHh5g3PbX5UXaW20HTsHa1s52J\nPDp3NK5ZJme+PgnWXQPNhw/vrqd02WfpLfdSG6ulMy4uQtEVJNjMWVjREmXdo38oGGzbSBU5hJcX\nRD9qGqw34O7HRWDF6ummHA/hyBy06vB6yJo5ArVCgodIZ309djHHpmX3gq4j2zaP7XgMO3/4dXzT\nNckaeTjxFrr6KhPPffs4ufUOTGsFwd2/ek3hkqefrsw5K5HgI48E3ZPZnwS3rUvotg+MBFsW+wsu\nT46Due1wxPfvQe4boLYI+UceEsvmB2BfhwjdDumHRUdNMUG6/Xbo6BDP7A0boK+PaF+Of9kGcz71\nDfblhD+u8/uH4PHHmfKps2nfV+n/CxWtt67j5QTxLve+jndxb6/w3D0Aq9tW83znU4dvX8E7JPgQ\n1NUJ4frJJ4t//+53IpdgqLDKxo1w/PEiKnc4zJv39j3h38Gfho888hE+8+RneHrv01x39HXc+syt\nrO9azxWzruCJK55g44c3cs6kc3hg8wPDAvxP3bEEtp3LrbdKw0TyF78QTiU33CBI7iOPiOfAokUH\nT4Cuv15I6r76VVi+/C93HJ4nZAz33w8XXPDGbVVVaNd37BCBtbPPFn368IeFBCIe/8v1662iJlpD\nc6oZECT4qt9fxffXfJ/bnr2NX2745WG3+fm6n/P+I9/PN077Bj9++ceHbfOnYEf/jrdvvfMO3hIK\n/V081dlP1IGdn712eLYlWRb5QJDgRx8FY8AQL/j2dsjleHLnk3jFyprkyy9jn3EqSgDq2nV8+Zkv\n09sLcQr0KuIlnA3Drlc7MLqybHC3U+w7lojhIhkGih9g9hXwdZ29DRFkPUSdrdNT7iQuFSjrGmWn\nTGOyEVs5IPkH2FeqIuunCGULFA9Jvk4UHbpGJagvBBRMQdT274f2rpFIsFKurNEG/jAJVhByiLJT\nRnFCKIj276m7rtI0QPZ8lEDIIeJyfDjSGPFcXEVm+iSHbbsNEjYUQj6uDLpSA7WbaB9QR+QQhicI\nckREJfsiTWJHpomKIOuBa+MGGsXAwy6Vof0YQs//JzPrZ7KmYw04Dkv+qLPiJQM1gGyF5PV07qa7\nRtgbZhikvX3ErWG0mqF57qus3LuSQf0V+nOCEIax6MiUybW9Vj+2Id+P27qHRmM7SWOAyECB32zf\nj50b5NXuVykWRfSxVCwQlEsMRMDr6kLyPDIm+JZYSq8EJclmu1ndux40DWdPG+uZzb3T4yglg60f\n20b8MEoE3wfdN/CB237Tz6w/PsSr3a8yuXoy3TERjS87ORwHin0mbsykpPn0H2DKoRbEe8M7gJDq\nHgxqIpL95AP95KMqtgKOUeJDHzpYpzvkJw3iOLqzOUi28UvrQgoDlS/q6tBsj1DrHpg3j4jhkrNy\nxA7Ms3v2WfGiQpDgvFWA47/K3v6KpvinP2VLej4KLrz/fSKh6QBpwi23wGNLXMjl8E2LDRtAtWFf\nCpbe08mCj97BHS9Usq4HBsA0aS+6FHWI2XDU9x7ig4/18J9r4kgPCP3usGwGsH0TCXm4aAwg5CwV\nzXPHS8/QXewWJNjziBdMTtitoA/m2Z8TFnbKvffDPfcwat2j1HZWTOyHyviWSgxWrO7M/kMKfAz1\noT8vTrZtC9lFKkXRKbKn+MaJWu+Q4ENw0kmLmD1bJCa98ALEYqJ07AsviCjcsmWiMMfnP3/47efN\nE5HE1yvi8+fg71kH+bfuW2ehk1+9+ise2vIQXcUu/vfM/yWiRrjh+BuojdVy9OijGZUYxbVHXcsp\nJ53C9U9cz0+XL2VL/Pv85PqLWbJEEM5iUViQNTbCL38pNLnz5o3IWu68c8RHf/duIUXQ9YMLTfw5\nWLhwEbNmicTsM84Qkdk3w5QpQqZz880ib+ill+DKK/8y/TkQf8o1bUw2MiYxhpVXr2TJpUte13bu\n0dZHuWTGJcxpmMOe7J6/WN8e3vYw33nxO297f+/gzWHkevnewLcJu7CrKF5IO7YHYNtk/QRLNz7F\nc/tXItkjkeAgl6O31IdUNiASwWlo5FMdY7h7JtS8upMfv/xjurshfvTtdEsFVCdN1qvFz/YQtgd5\nzlpJoW0+sh8QzotIcHLWd1g3+wi+98mFMH48C2rS9FvdxOUBOrR22vuzNKWacGTQDngO7y9W0e+l\nCJVMivrBSx+pkotdV0MAvLIqy5YtggR3dgtymbNySBUyKnn+QZFgP/MyZaeM6mrDZLQa4dd7VLQe\n2QtQPRViPcSlkUhw2PdxZZmTT3B4cplB3IZihQQrQTUqUfb3BcMk2CiV8WQYXRRSFMkUpLyKKLIv\ndho4Dm55DAXJwS2VUK06/OVfZHJmGjv6d4Dj0F/QaOsrcbwE3b0V94OBAv2yKMqRIk/bPg+zJAhy\n2JLZf+Jp/Mex/0Mydxxd5RF5WWccCvtHfHiHsHagG+end/HvfJuoZ1JtwIvRpwi78OPVP4OKpMIs\nFsE0yIZB7xXkJmNAfUeU1o0mAwOA7BCYZVYPvEKg6wR799NNPf1+kXhBsN/zK11yfZf+stAImCZE\n5RLbqiWShrgu46vGUx+vx5MECQ4V2gUJ7rewoyZlPRiOBJc00IuC/AaFgeFjOxkIMq1IZhX+QD/7\nXXAUUSXwrrtGDB/ueuUuPn69ScW6md/8Bja25qB6O23K0xhDEey6OnTHQ93SBnPnEq/0NWaJPgYB\nZJ94UejpESR4wOgDzaAnWxANfvEL7mz+D65mj1jdWLx4WPsN0LvP4IJPjIHdu3FL4pyZOZd9KSju\naYNYD3u6t4klwVyOwDTpKjmUtRG9fHOPRZMXJ7ZCRPFv/g9BsgcHIVAMQl41ZafM6tUV1YtpIlUm\nUr+5/39YwQrhOKAKLfKkrIduOXR1i8RLees2YeEGTB9YKTh8W5sod5rLYQ6KyZ/TX9El33473Hkn\nQRBgmD6XnFHR8heL4gWdz1OwCuw3DrEDPATvkOBDcP31gvDW1IwsSx93nLBDe+opobk8sMjGoaiv\nF1rhQ20YPU+M4bdTzv0dvD7u3XgvF027iJJT4tzJ56IrOsvev4zPHf+5g9pJksTtp9/OM3uf4d+e\nvpDL9fu47MS5bNkidLaf/7xIbvz618UEeMKEkW1nzhSToFWrhCwmHhfSgZUrRST27XqjHw7Llol7\n9uGH3/o2U6eKFYnvfe/tSR/+Fpg3Zh4fO/ZjyJLM2PTYkapKByAIAjb3bmZm3Uxa0i3sze09KKGi\np9RzUIb+28Fz+5+js9hJR+HPKPf5Dg6LuA0FHUxVRIcALpm3B7tg0a85mKUsq1ouQPXzBJZN0NaG\nNziIj4ebMyklG+hpPJrvHrmZ6y8ZRVVvgWx/O+2dDgnXoaAHKOVGDG8iNUY/nuRhqAGlPbMwozqp\nnInqQ7LpSV7ytuHOmAaZDFW2QkBAXO+mqENrVxeNiddGgvvlODkE0SsrB3vwpso+4UwNXXGIDA7Q\n89CzfGnnFXT1jESCN75UIcH+wZFgGgQpUF0ZNRBfaK7INfB8Fz2QUDwZIoPEpCieBGbl911F5qiZ\nDtOOrJBgHVw7jeKPJqxE6cwG+Jb43UI5jyfBT44S0baIK6LrCS+K7FcIk+ei2o0UZQe3bKD4Uerq\nIEINDy/txzUc8mWN/kIBW4GePrHEHOTLZL2RF1t/a5ZirpdcCPrayxDr4ZS6y5HtNI4zkmnUmQC7\n+7WRuQ63A2ewyDhJRIlrslUkN3+HkhRh5fqVyEPOFuUSqmkyGIZ4xcqtyoDe/QYD7SISPHpKGzEn\nDOEwg24B2tvopp6cXCBtiufGeVsASeLrnzue99wnHIHKZUGCl7aMJLudMf4MAMKOiDYc0b0Tx4Hy\ngIkZKVLSoD+k4sjQH4FwUZwfqTiigQ25IO84lfOemYBeHOT3L/nYCmQHDAxjxGzhhqdu4P4Vm+jv\nF30sFMAkx6oHW6kplpCDCrusqkIKILmtC/vo2SQsqNGaiFYkDVu3wl3fKww7rkS7+hmzq417HwC3\ne68glrrOs9HT8IbcLfbtGy7nHAQwcd9TxEs90N09TIJl22N/ChqkDlAserMdgrwYBnbBwgo5OEqS\nmkpwd/xAQKOhEerqxUGluhJ+37kT0Axks5qSXeLyy2HJEg6qNCjt2siafatE4wPIOcCeV1qRfdBa\ndw1LrOb7z4nDbWvDmDCD9k1ZOrbnKMY0/MEBXtrUR+fap+Gll3hy15O86zvvpdRVIcGFwnAJ4qJV\noNd540z3d0jwIVi5csVronGnny4Sla68UjhYvFm07nCSiGuvFZrPY499bbnut4q/Zx3k37pv9226\nj8tmXsa5k87l8pmXv2Hbz1+/hq9OXU7tqh/z8XNPBkQ094orxKTmDPFcJHRwjhySJApHLFkiVpeO\nOELkZsydKwq8HDrRORzerGrkt761gn//9xHf+beCBQuEDMJx/rok+E+5phdMu2B4IjI6Mfo1ZLSz\n0ElXsQtVVqmN1RLVoiRDSXYM7OD5/YJMLNm2hE8/+enhzPy32jfP93hu/3NMrZnK2o61b7nPg8Yg\nRbv45g3/ASFJ0hmSJG2VJGmHJEmfe502/1v5/hVJkuYcrg3AWH8cRV0QtSMqnOCI4ga8skVfootQ\nrgFW3khYKrPxxc1I5TLFDtFQs8rsI8nGzDioaqUhdzG5qEK85LKpYw8JG5xwgsBIEU1MpcUokgtJ\n3Fy3gcAJYUZ0Mjkb1YeEBfmQKNlNOEzH7gIRG8blPEEiQ91UaY04Cgdpggcu/uAwCS7KB0eCawyJ\ncKaWrjjUGjl6XumkgS4Gsi66C1JbO3u3DUWCg4MiwUOlkGUf1ED8YGAJ4vNibjdaIA9HaqPE8aQA\nUx4hwTgOTeMMEhYUQuCiYroaUTVGuMrHqtiJFc0cniyzckKCL+hfII7QX4atKGolvCy7DnG1iYJs\n45fLVFsq9fUQlKp5eGkfvR0OOUPDCoosk6C3EgnWygZetGX4fPRsz5LL9pNVQ+DkmLcnTOTB+wic\nCJ47suTdkQD6evEOoRJt5TWsaX2CyaqwWEqbNrs6YmSlFHZ3DtUZkniUUG2LbBjShT5sRSFjwpag\nh7hqsCe/k9TlH0Y3wrx38hcwJQ+lqx0zVUtJK1FV6UrCEr8/dsV61nWuw/EcQYKlMk+1iAz2tgSc\nOfFMcc4cuG86XPrKKuaXl1EetCiFCpQ16IhHuG86lHWIlASxUowR3fEqF6Y+/p/8/MV1RMwc7YqN\nI0Nfl+jMUJ5Z3srTO/a7cK5IVjMMcOQc8zp9TthvoFYGkRNLYKhwRFcvpz6+hIQFn48Ji6qyCp2d\nAUo5P1za+l9+spIXf+RyySZYsPF3gpycfTYlU+FB0hyKgQE43VmCjyAuviHGk+K67E/C5HQPsm4x\nMNhBkEoRGAZm1iQ22gY9M0yCx2ZhTHeZYjxCSUni5Yr0tVv0PfkyqBZOoYr9XWV27oS1axl++flI\njOv12P/cZpxElOf8PXRVNNk+sHPdLlqyIJsWFIv0RVUa6OIPfwB/XxtPdc1gYHeWsFukUFODP5Bl\n/m3/yrKNS2D3bnYP7mZb3zYWzz0gElwpp22XCwS88bL8OyT4LeDoo8V5PbDU+hvhUBJcKgnf1s2b\nhZ74re7nHRwem3o2sTe3l1OOOIV7LriHxeMWv27bjg4hpfrI5c2UX7roIBeEc88VspVTTnn93zrj\nDKER3rnz4IS1ceNGSn2/Hnp6RCGY1zPr3rBhRGbzdnDWWeK5N336376y4ttBJpzBdM1hy7S8lafl\n2y3ct+k+ptaMsPex6bGc9svTOPNXZ1J2yqzct5K8lRcaxreI9nw7Ny67kSk1Uzhvynm81PHSW9ou\nCALO/fW5fO7Jw/LDf2hIkqQA3wXOAKYBl0qSNPWQNmcBE4IgmAh8EDhMaRWBcEmiKIfoCYeY1gum\nKpE58z0oNS+Sb9pIqFBNXaiJsOeyY+s9dOhjCJt5FA+UwONns15hSXUz1fZRWLkUg2FIm7By81bi\nNthaEn//fJqaZtOSlcmFodBdDYAZCVGTd0UkuEIWrd5GiESI+DJXbIDvPSIi1SguVk/TcCS4rErc\nMrcas37nMAnOEsFQRjxhawyJSHUD+UyEWqvI3m0GiZBDVa3D8z+Fb137GxRHOAdIvs//vHoTxLpR\nsVE9CfNGE8WXhiPBpYotlxuU0QIJpSIEjgYRXBlMOSa+VyWwbZJV5kgkGIWyqxHVoqRGe5gFQRjL\nZh5PkqnXxxMaNTJp04phQhUSrDkuMakZI+rg5cqseuQPvJuHaVmyDyL99HU65MQ6P54Eg4NikhIy\nLaLVE4f3+dITgwz0ZCnpKSJygVV3lRn/X1cROBF8b+SB1i2PQRscHI5AlirexqrnY+a6aPTEUneV\na1AMoljRFFoxj+LaFFUZxyyj2w6FUJg6o5+dsQQZA5IhG8k02OU+xxbnjyR8jXxuDKbkoXW34U/r\nxgqiRCvzZKVy3ucORmlONrGpdxPlMoQx2VhTyw1HXY0vScxvnC/Gsgt3HQmL9u7jd4VTKGXzlDWT\nsgZWMsqVF4Alh4hWXBg0Y8Q2TwaunG+RdB1qvUEGEwaOAnO+93Xezy+44w74yEc9SmaB6/JLONNf\nQxAE7Ci/iK8JItuYB61CgluzCqYKGcvhpaYHMFSJsU8I3W1Rl9jbk0W3CgSVSHBnnYhsb62Gk/Ys\nF4xz4UIMAwbCUTaLPEZRwAahKBgv72RzrZiYlfoFCVZdj30pyJT7iCQsfNtiXY9FT99e/LKJVmMR\nS46mtiTGru5BpjPLQFTD0uIcPa2fq27KMPM7/4oShAisOI8vK5NOw4vrDPyKXKdXT5IuRhjnNrFX\nKbFLLbC2QexzUzpGnb6buQPj6InWAbAz45IgxxVXBBS3tvFsdgYTqrMkA4MdRjPhgoE/5nniNgS7\ndtFV7KEotXHhaeL8uNkiL94tIlRuqUBVMDKuD4d3SPAheD2toa6L0vJvBaefLordDJU1XLZMRA+T\nSUF4nnnmT6ss944mWOC7L36X646+Dk1589KCX/86vP/9i1ixQuh8D3jvMWfOm0sKFiwQ1+3DHz6Y\nBI8aNeJz/npYs0bopW65ZUTfP6QvXrtWJN3deuuiYc3Y28UTT1Tsg/5K+HOvqSRJNMQbhqsDrdy7\nEsd3uPWZW5leO8Lebz7xZr606EvMGzOP3275Lc/ue5bTxp/GEztf32vwwL693Pky078/ne392/nJ\nuT9hYfNCnt33LKvbVnPz8ptfdx9P7XqKi39zMb3lXh7Y/ACO51CyS2+vlOffN44FWoMg2BMEgQPc\nC/zLIW3eDdwJEATBaiAtSVL94XYmlUuUyi10qGkm98GemhAZE0K+R/6lL1Cta3z7P1NoQUDa8mn3\nMyi+R9qEsgbfWAjLFJmxmRaK2RgDIfHd2N/+hHltElXSOQSPf4MjZ8+jseCR18Js2yZelmY4TEMh\nQKuQ4HwI7rhdRIKPaayjtgzJCokE6G0d0QSrfsCXFxcg2cZVn0gAkAsi9IRGskirjIAJLbMZ9ubX\n+AAAIABJREFUM2Euo8wybdsN4rrN1LoBju6E7S1xFMfB1BQkz2Np+4OQ3oOiWKhOiJAaQvWCYRKc\n6xcP+OnRCFogIVVIcDiIHRQJrsuVYepU4ulKYpwuIsFlWyOmR0k0+FhFg288/w3KRg5fljhm/Dgy\nxz073HfX7SHsC4ITsj1iUgvlZA4nn6fKMjm392fMfH4jUryPnjYb0xckeK4mkatk3Edtm9Fjpw3v\nszaxA8fMkWqqo8YQ+y42TCCwogQVEuwDauEMZMPAReg8OzRBZM7gSxyjzR/WZGuBh6NGCZJV1Epl\nNM+hqOq4ZpmQ42GFqhnt5NiWEGPqtISCYhtk+lcRkiNkXJlcuQFLAbV/P6un3owVHZkIDLlxTNg5\nyD2/KPDsvmf53PPXEA5MPCXFw+b7iJoRVFn0M+JCaxXcfOxUlsqnUiz0oGhVlDWQ9QzK7+6mrEnE\ny0VsGTSrMJzQuAg490ix3D7e7sVsKGIrECtkuZ1Ps35XG8t2PMf8Nvjh8n6u2rGL3dnd/Nw/GTTx\nbBlzAAlevq2EqYqor6FDXle54NEPAFBSVNZ1rSVR9SJWb57BQZAq8pgfzIV0eRBnIM8l16Xp7YXB\naY3cPUv0M6sq7NghtO21qX1sSNaKsdkr+qD7HmayiiAIaJKyhFzoiQ0QdkH1LKyQxYVHvY+a0gjx\nUVyPLh2cUIJMy1KiRQNlsIuIGqG+Ksovf13m+k94XLruvcimhaVI9CVldEeietUXGYyEeSVtsfSI\ngF1HHMm65hJjcjVEBqNslYUcZ3caErHdsPBrqN1t/LGQIWRmibsOa/MzacyqEOsjZsm4rXvYvKeb\nIDJAbUK4lDz7WB59ryDBXqlIrXM0b4R3SPBfAdOmwUc+IqyrxowR/77iCvHd3LkikSmReHMS9Q5e\ni6yZ5d5N9/LBo9/YD7FUEhOOe+8VJHTaNLj8ENWEJIlr80byFlUV9jJPPSVKEA9h9OgRx5DXw5o1\nwuv3a1+DO+4QE58zzxS682OOEfu85po3OeA3wOjRB5P6v0ccKIlYvmc51x19HZfNuIyb3nXTcJuz\nJp7FVbOv4r0z3stP1v2EQXOQG0+4kQc2P/CWfuOZvc9wxawreOiSh5hYPZHjmo7jpY6XuPyhy/nG\nC9/gtpW38cfWPx60TRAEXPbQZcwfM581165hQdMCZv9oNu+57z3M/uHs162M9A+GMcCBWUttlc/e\nrE3j4Xam20VKpUl0uONRA9iR0agqQ6h3PGW/jqhsMbspCUDCBlNRyCkRRhUFCQZoC9Ywe2wLhf4o\n/bpP2oTv7H6YtBUQavsQ6TQcdelk1ADKemy4amchFifmCLKTsAUJbts8BkcJE/Pk4eSdIRLc0zoS\nCdZ9cHQbFAczJW6Yn2ZO4mcNI8tCmXJAdd1Yxo1ZQNyzwSyTitocXyOWe8pKgOzaGLqG5PkYXhlU\nC1U2URwd3wclkFAr2vYhEuwFZTR/JBIc9qO4MhjKAdqrUomMNkDckijq4EkaRVsjEY4hJ11Mq48b\nl91IycrhSwoTa8axvrxueHNtsA7N93EliZDrUyvPwVVrKZVaiXku47IvExowyYzpp7vNwUYHvYgj\nqRTzIjIZtV1mzhIT04IOzSd9jbTqU9PSzNiK9CVwPXw7Ar4gUbYikfKPAsceJsG9cWFurpkhKI5o\nhwEi1VGkdBVxu0TIg5IWojBYJuz4EBVDrrXBo6oskVBMYl6epx7+AbeOu41qN8bGvUl8RSVilMiH\noJI3hqVGUXyJnpQYZFN2DPLUY9/j4f0/J+xbKHqGtm3HEDkghyPkBpiEWFWjofguhtGPHq7BDCs4\noRDerhMpqwEJQzhXROwieX3kRZEpChLcZA1SbhggqMhrqumH65vYPvdUTtoNu9KgByZ5K48jjdiH\nNeYr41KG34ZvwNCgryJTz6kjycRlOcTPB68mUfUKZneOBx4As93i2nPhziMhjcHO9QX2FBUG5nyB\nb160nT2LBekrhm3um/cZHvm9Q0zqYqu0EEcGvdIP3feJRxt45YzZfHT7y+ge5FJ5Ig7ovklRMxk7\noQaNg+3WOnUHNxLHij9GlQEZq59YKETL6ChKpMRFCx/hQwMiycXQoCcOsSBA25dA5RS+e3yI/z06\nw7rvLqc9CV95poeZ+8rskTK4EuyOR0j4JWom/A7fNdlx9SWQGyRh+6yxTmBizoG+ycQtCc01KewS\n16KtTSTV/fb+bkZ7+3H0KK5RJFV+YxP8d0jwIfhLaVuvu074yp5wgiBiQ2RnyI/VNA9fPfRv0be/\nBv5WffvVhl9x+vjThyu8HQ733CO010EgKvtt3Pjn9U2WhcSlrm7ks7dCgteuFdKXr39d/PuRR8Sq\nwI9/LJJ3ly6FNWv+vL79NfGXuKZNqSb25fbxcufL3PnKnXzs2I/xnbO+I/Sch+CE5hNYsWcFC5sW\ncnzz8eTMHFf97qrDFtQ4sG/ru9ZzZP2IcXc6nGZqzVTOm3Ien5z/SZbvWc6HHvkQi+9azJwfzWFb\n3zY29mwkrsf51HGfIhFK8LtLfseJLSeytW8rc8fM5fbnb2fnwM7X/O4/GN5q7cpDp4GH3e7fKBJ5\nbgq/HIBvA49IHhlTIoRNZ3Q/L3sDVEniTb6lCJtjRfK6zjXr4AUX2A2F0b/nuOnNlPbv4wVXyCEA\nVgDP7dhEJgOR+iR/SEdZ5enDdlNPBgorEJHdlAm7skB2J32lMNv29tPTI/YxRIJbX96Bs1+0d2Rg\nj/j9XFQkuj3vSDzhiMZlFZ4JYEVrK7anozkqPfIrrPcHmB3aT2taZ23BotDfjanoSJ5PYUcW7BdR\nZQvV1XnyyRVstovDEck1PbtYAawv9aAGEmssE3ZDyI/gSgHPBAYrgFN/ciLMmUPu+Rfo6VIphMBD\nZY3Vj73LQI65OM4gdqvNK61PEsgysxtmky0lWAGYukb45ct53rN5XIGwB2l9LJHOd7O7Ql7Thf3s\nsXvRS/twDAclrEJ/G3c4AaWyWEJeZ/mUHBFJa0tJ7Nyxm/1BCr26jpYs/F5TWFku4NsRFN9nBbDM\nCxHW6pFch6dlcTyDVSK6vr57G8/3jTwgVwCF6Ba0miqufwGaNwQ8p0rkOvuxFdhbkFgBbGrMEXPh\nh4NZNiEmogsHPV4y+vnFwxswPXHNavZeCfvGAuBGEqw34I++x5WLHuA39edT9eBWpF2g+h5aJE3e\nW81qysO2YWttl8LeNGXVQMNmw+Au2ksq31uc5Ml5k8B8mdWuQ8IwGIzALruXJ+SRY9mwfRUrADUI\nGEwY9EgpVlSOVfLh5js9Fi+HxydA2LdZvnw57AZscX/0d8NTiMlhPgQrffijAoqkUNBC/JjxPBWJ\nodg1yGY1O3PwkpvlxdUBctZiRw4KHRB1LeRinq74U9D438y15zP+qMVcNx02GHBD9nYeu3cd640C\nzlHv4r6xowhJBrCCtY5DJjGa5SePI7lrN2ZnPQVdTDRfCCz6dg8QTmaGj3kFQt/bnTB4VirT07mJ\nKgNCgUNse0DayPLFL+RY8ouPD7cvqwHdCZsNtk2fei+t2RqiUgZtbwg79wq/Sd7FQPUk7PZB1gRF\n+iMSu7UG1tkezZ2rebU6jRO3eMayecGD7cosZrZPZMaz32JL2cOUdEKte2hZJ7P0VeGbXerezl30\nczFluleU6V7xKm+Ed0jwXwkNDXDbbfCVr8DChQd/19MjEjd//eu/XonlfzSU7BL/s+p/3pR4LNm+\nhIumXXTQZ47DcKnrvj4Rsb3pJuGb/adKDd4MbyaHeOUVofc95xz4138VXr47dwr9cakE73qXWBX4\nZ8fk6sls69vGvRvv5aNzP8r0utcXMU+omkB9rJ7jm49HlmQevfxRlu1exta+ra+7TRAErO9az+yG\ng+s+P33V03zt1K/xXyf/F09e+STPXPUMl828jOm101myfQlLdy1l8REjWnJJkvjW6d/iuWue4/r5\n1/O157/GCT8/4aASuv+AaAeaDvi7CRHpfaM2jZXPXoMfEmH82K8zQ3ov/w44R0BVSUbHpqVhASdp\nKlUVknJKAOdOT1CIqHxyFbzHAtQFEBlkcn0LqfqjGFUDmWwGQxVLzF19pw5X4Tx97gKOjZ04TIJr\nmiayqNKPlAXhVfeBfzpd2Qi6J7FQVVkEFDUJxQ+Tz57KHad34cgVEnwE0CLTFxIhYyU8h2h0EiCI\nyCJg0eLFGK6G7unMiWU4PRliorWfV1NJ5ss+oxIJDFVH8n28FhcSU1AlG9XVmT9/EUcrsWESPKY0\nmkWAj4Xmw7FSBI4QJNjwbVoatrEIKAQmTJ3KAh1m65HhSPAEGmmZ00QQd8ArImXGo9QX8WWFC6dd\nyHeu+DKLAC8WJU6REyWFuSHxMonEIhw75ULOPSDyedUoj+pMEa1qM4WzroFp/UzRQhjlHLl+h8UB\nXHLJ+QD0VIdpVAucrOlIY8cyqwcmZ+KcIGl4ZhTVF+drgRQhHKpBdT1O0FwWAY+822HF1COYGR7D\nu/yRudUi4GtfOYHR09Is3BuwCJidSpFSuzFUmclNLeJ4mkZhRjWOMUxOq9jNTdnfxWLfoq90BrYv\nIugnj7+d779HuB94sSTzNYUFasCqlzM83n4W57oSkUawNZlUJE3AKSxExy2Lk7I4gETzOJxQGQ2H\nKTocPXEShZokdjIB/mmMT0HEcRiISJwgw/R4JbFMkjhd8YbH47+d/VlmNaSG/z5yX5zZMQ+pBR6Z\nBGFHJjYhKcZgUbyQLjDgOBUMFSKJsZxcW8vYDFwy4xJKfgMup7LwhbVEfI26h9bzLj/EqXg88ViW\nlGMTagbGqziaTHO4jfHzmsF8D58+7lPUpkfzs/PhHAfkIOCz/3U/5wQqX77jAzzzuVMJYZFMLuJE\nCWrTY9gVNjjTc5nqH4elCveXBehI4yBUIcGLgGN0nSxp+gbmU5Pew7g6iapBobGfMjrBxKMncsov\n7+cz9+wV9xMiEtweL7NIUTllXJI9hWpkO01m4iguvXQRax+5EmqmcaYHYzO19MUC9tLE6R5cpKdZ\nO76X0yadIsabAt4RE5jgtnH5RadzEhJdSZfFra9y9ytRxktCsjIz+ioXqVXcWA2T5oaY0PLGvvHv\nkOBD8JfUtn72s4fP+q+tFfZbMOwI8pbwz6AJ7u8XjgyH4sEtD3LDshu4YdkNh9+u3M8Hl3yQ5/Y/\n95pEuD/8QRS2sCx4/HFBMC+9FKLRt9e3t4PDRYJ37BBJCCCKX1x/vbjWmYz4//Llop+p1IhLzD/D\nNX0jTKmZwtb+rWzr38aMuhlv2FaSJD678LO8Z4qwOJpRN4MTWk5gdftrq88sWrSIzy/9PBc+cCGt\nA62v2XdEiyAdoHNpSjVxzZxrOH/q+SzbvYxfvforzpt63kHbhNQQzalm5oyaw+5P7Ob0Cafz7VXf\n/lMP/e8Ba4CJkiSNlSRJBy4BDjXjexh4H4AkSfOBbBAE3YfbWagqxs9/DpMWjcbVw+xPm1SVZUJY\nyOkkVdndaHPnAcLBwVTBSso8OiHCjA8D668CoCXdQk0qRjYMmVxGOAwAJTU1XIhKmzUbYvUMVmoU\n7AuPyJRjNphWLVVV0FsIMzGoodoQa+NFP4UuRenrg2xvVFTyEiv1SP1T6JCFjtR2Exi++OEhqQaj\nR2M4KiFf46rL8siOTVVXB50tNWiOTwibshRG9n1MrwyKVYkEq1iWKMM7RIKlikvE9KiQSQSVwge6\nF8EMXH5yFNw/rVJha+pU0ns6Ga2pwyTYRSMRilIK96C7PsHOk+hxNg0nO42qES8WJZ2iPpzHU/Rh\n71s9qlI9Jk3NAc40ej7Lxi8PMsnrFJOCRBfzInFMa5Btr3ZR1EHXdLjpJraMT3LLUz76nv1w5JHE\nbRhIJsBz8czIsM7XkXRikWpUz0P14Sdz4JmmIs6Us8j32Ri9RQJJEgk1wPS5UbTaNHIl8BO3HVLy\nAGVZQ4mIh3Vm0gzK8TAnHxAcymzYiOw5GESwfbGUqqbjnHuhIMR6dRIlACUI+ONjSdqkiUwrJogV\nqrF0iaqokOgYRDD6hZ457HmkUo044RIaDmqQI5OqJ67HiWghcEOUKweaj+gkfJOCLjp1XCYF+/bh\nqwolXeKDCz8ImjjGh4+O84k9MaIO/PfcWvLhMGErxObuVo7bBx9bLfaZssTk7M45EtdfdhstDZPp\ni8KFUy/kmGOnceV/TiKc1AlJNq2tkDTFs8z6lymk/BIFNUpaamJQ9QiXS3RPuwepXM/ixYtoiDfg\nKmDqFQeG3DZ020NOxJh2xERC2DQ3Q8gLaKgdwy41T8q0cTtTqB3nYGgSJmEM1yCcFImpZRWyWoxB\nMvS1n0VWswg7Zary4rqd/qpB2lXxbItt1dBaJ246Q4XuWEBc8vjw3CR2sga3mCEVGnGxkNIpxhgG\njXM8vn1cmDWhSZR0eF9+LC81+Jw67lQA+hMa806JQzrNpFg7MVuiM+EzJg8TsmAMdNMVh+nKDjoT\nIQwNQqUIke43zh16hwT/f4IkwWmnDftf/5/Bz34mvHYHB+GR7Y/gVxJJHtj8AJ9e8GlWta16zTYl\nu8SpvzwVTdb41fm/IhVOHfT9qlXCBubmmwUJPv74v/5xHI4Ef/WrQvsLIup7YGntWbOEFKKpCX70\nI0GG/y9gSs0UtvZtZVvfNqbUTHnT9tcvuJ5J1ZOG/z529LHc/sLt3Lvx3uHPgiDg5uU3c+/Ge1my\nbQkXTLuAiBY53O5eg0VjF/FY62P0lHqGH66Hw9j0WC6dcSnLdi97S/v9e0QQBC7wUeCPwGbgviAI\ntkiSdJ0kSddV2jwK7JIkqRX4EfCR19ufkogxeTL825dH42RqGIj6ZEwISxaxUYJoDLHWuA2G4rNp\nvM3DJ85iUz2k7emc0vUIYxJjqE5GyYWhzkoSdeBLv7qWD3xEp6qq8mOf+ARrF3wUELr8XVLDcD/i\njoRNmPHjIWuFWfsMZCqGBUWnHl2OMGoUbNqgiUpelbecmp1GlyKYYSDHMfyDyys+vyXDj+/UCHs6\nvpkH2yba1k1+YhUhVyIkFyhLEfCG/NEsFMlB8zRME9TAHybBQ0UzfMmouGNUPndDeDI8OB0uuRhR\nYWvCBFLt/VQHMgUdfEkj0DRieoz9ymZ0RyXV00hRaiWoJHYpUeEuEcpU0xDL42shvMp3oZhKoiF2\n0LFJJaEDrUuUOW6UmPxpehTL6ePeFfcNy0i49VbWHNvIT4dyiWaLFZZcOo3kuXhWZPgYPVkjHqtB\n80TC4kfPgktmXk1tYjx9nTZJuYA0evTIclw0ylCo//vn1LN29lxSfglDCuEEgkw1Tp2HmTjYvk7a\nupUgGgckPKMRD5lQKowUEp0O1yVRfQnVh/raBHuCFloGVUbljqGk+tQmxKAyiGBlDQLHRQ6gqW6E\nBGsUqaoaTUyPCRKMRLmSeF2MhEl6JoYW0B0DtzYDe/fijxuHWl1LWA2DLtrePtvg7E0FUiYMZKex\nKHETEUfh/qWtzOqGd7UL/Y/mi8nZ766ax5yJxyNFouTiGqMSo0jc+AniV18Euk5YdpjBq6ScgJwS\nIqn1kHJc8n4VNVoz+Yq0fGdsNaolJooNcXGvWDFxHv09u/BiUZBlLpt7NWHfpWnOFnTfp7mhmS6z\nj1JIpaqQpyE9CUMNsOUwJadENCVIcFsS+oOUIMH+KPIhj5BpUl3ScBSJj/+2g2OXb8cZ7Oe2E2BT\nlRj/hgbdcZGEV6f0Q00N5YEM6cgICVarktTaJqWYxW+OqmXAaaYQglGb9/P5D93NuIzISM9lwiKA\nNnkyJ4/eSsIJ6IzDmALU5UqkCg7daZUJRjs9dQ6GKqHkwoSzb5zf8Q4JPgR/S93thAkic/Ot4h9d\nE/zFLwp5SGMjfO+hdZzz63N4sf1FvrLyK6ztWMv1C66naBcPKrAQBAHXPHwNs+pn8d2zvsu7J7/7\nNftdtUoUvXj2WVHU5NCS1n+N89bQIIplbDugImNr68j13LlTlDcewrRpwiqtsREuuWSkvPE/+jV9\nM0yqnsTm3s1s79/OhKoJb77BITh1/Kmkw2k+t/RzfOaJz+D6Ljcuu5EHHn2AVR9YxS2LbuGG4w+/\nenA4VEWquOf8e3jkskdQ5DfOKjx2zLGs7Vw77FfsB/4bOlb8PSIIgseCIJgcBMGEIAi+UvnsR0EQ\n/OiANh+tfH9kEAQvv+7OYhVideSRdHz8ZgYiUGUGhLC47bvJg5rKwM5yO7efPwprrvAAbKhO8K5R\nZyFJEolwlGwYaq0wUQdCySq+9CX45jcrO2hqwm4SN1BtLWzzhYbclSDmgI3OEUdAdzbMfqOPjBHg\nyCrF3vnoUpSWFtiwThOVvCqXOVScRKcs2HIiGcf0RJ+HdMlbtkrYvkrUj2HmB8C2UUwLpyYqSLCa\np+hHhn2BUSxUXFRfwbJAC4JhgrjwWDFmNpbKKF6AXgl/qm4YTwK9MJEfnvAYZaeMEQ9j9feQdkQk\n2Jd00IRF2k5vA6MMh3Vrfyh+t0J0h5JLpGSKhkiOQA8RyIIRhaIK8XpxrTxJEcSzWpAZ3Sgzadwo\nvP/wWIuE4w2wrOszWOER0tw7aQxfvLBaPNzGjqU3qVKsqkbyXNxy9CASHI5F0D0RAXcUWDzxBNSo\njuTYRL2CMFQfIsGRCEOh/odOb8aqqSdlQTmIM6YSZT1q3ELMRJSlkojQl0bVwP79SEkRtU8lqigS\nJxaXRCRJ1yGZRA0kFB8ikRjhxhowTcb7EUqKz6QaMY4MoliDZZyCiaGoTBxTh6366FiEfItkqoqY\nFiNSiVyvGCOIeSkWI+mZOAo0fxKeQYJCAfWoownVCOIpaeLcr6n3yAyaZGyFnFvLgubziXgynWYr\nicEGEsU4lgK+BK4i8cK/vkBTqgnCYfyqNC2pFhHFaWwETSMRtnmVWVRbFm0p4YySMmFgx8XMTp5K\nIQRmSMFSLEJOPStWrKA+JvrkJcRLJryvEyrnry41GgmYvGA9uh/Q3DCWnlIPubTGKeMGOHpeBEMD\nRwmhyRpaJEYgyywdBx9KfoYObSx7vEnkNQ/VMJiWktC8ytjWwwTZAbJh6K/MY7pjsDMDiuWwYts2\ntFE1OPk0NbHM8HjTakVQKx82iKspKNdghFSkvj6mzl5MdUSM3UK6EuiYPp30vg1oXkA5Vs3o3hRy\nEDCzB+y6asYOZOmqLWM4GSJWCLn3sAtbw3iHBP9/RF3diLH2PzvWrxcJYffeC5/5DPxy23fQZI0r\nf3slT+x6gqXvW0oqnOLUcacy/6fz2dYn2OXHH/s4bfk2fnjODw9a3gZBOM84Q1RP+/znRRT46qtf\nU5DmrwJNE+4f3/wm3H23iHDv2CEK1biukEWMHTvSflrFfajxsHn3/7yI63GOHXMsAQEhNfTmGxyC\nabXTeO6a55hWO40frv0hj7c+zg/W/IAvn/xlGuINfOGELzC5ZvLb2uelMy9lZv3MN22XDqdpTjWz\nfPdyAO5YewcXP3Dx2z6GfxoMkeBoFPeSKxmMQMb0CWEJH8FD0G73c8n0S0hGxcurJpEYthlMVkhw\njS0RcyCSqiYSObgaZ6Iik6irg/ZciosvhFwYoo6Ejc64cdA+EEE3E2TKMgPJcRTtWsJKlOZm2LhR\nPigSHClNpkMWEdFEKoHhiZdvsqKd7egABw3NSFAuDIDjIFsOZCKEXIhpRfJudDiqi2qh4omonn2w\nHOKoWRUPTMlF9SR0OUAJQkiOjCeDUmzm6NoFlJ0yTw2uJW1Bk5KmEBIkWKqQ4L2uqHY12h1ACcAb\n0hIMZVgnk8xqyRFO6gSV+yscU0hUvGTbMrOETc4QCTYdotEUsiSjhUMEUieXvqoQO+3a4fOeDCWp\nidbAJLEis78hilVTh+R74IpIsIeEp+hiyb4S6b7pxC8yt2k2alQjQQFfUkRp27PPHh43pNMUoypS\nKoUWiZEyYeLUaibXiotwzOhjsFMxYYUWSxCbOA08DymZIJOB6gaNIvHhIAKhUIUEi3MvazpNzRJt\n6ljmlD0MFWaMEpNvR41Q7je49QYDM4jxkXddimPUoysGYd8jkooT1+NEK1WTHm4WRM2KJki4Dm5F\nWiNVpA8ceeQwqZdCOgVdFNjwVIXReZ9cUEWmIUzEBapaiXZMJJ6txVYkDF3GUw6gX+EwHznzPxiV\nOCCJRddR3BFhd2/CYoIznowJg69+mFMmzycfAj8u7suwK8hvU6qJi6ZdhJ+M40uQ6c6hpKsr41HC\n0RQ0SRRJGZ1uoq/cx0BcYZzcT+O4CIYKrqqSDCXFRCMaZTACvU2n8u3jf8PzHI8R1giVbGpdl10V\nPpvKOyi5ItkwDFb46sXnj+O5ZpBdD/J5oqPTYGSoT45EgsO14tnRHyqR0JNQrsGKhIT1UW0tNdEa\nPnQ2PPi+Y8QGM2bA6tVYYRViVTRUKhjWl8CvjNnWWJlkVS0xK4T+TiT47eFvqdGsr397JPgfVT/a\n2yuq7X35y0IC0nyEzS79d1wz5xpaB1q56YSbmFYrWOLd59/N+498Pzcuu5FXul7hwS0P8tjlj4kl\npwNQKomiEfPni6IVqZSIrv7sZ8MStLfUtz8H550nos+f+xzccIN4ie7bJ/4bNergfgyR4ENLbv+j\nXtO3g6VXLuXVD79xhu6b4bHLH+Pao67lg0s+yAVTL+C8M897843+Avja4q9x0QMXceNTN3LL07fw\n9FVP/01+9+8SsZFoYTqhkQtB3PEJB+bI0sYBsBQ4atRRpCok+L9vSfAeIfcmGYlVIsFFfAniB0SG\nhjC0y6Ym6O0M8cAMQWijTjBMgjsHwyxAPBv2jj+Z3YwjpERobgbHlkQkWIXjE1cSc1rYJ+fpjcs0\njo5jH1Jdq71dePQqRgK7MCgiwZaNn4wS8kCXizihyAgJViyUwEMNRD7CgXKIISP4mVFQfBldgrBf\nAy6i4IYbIx6OUnbK7A2yVNsqWtmiqEMg6Ui6IMHDUWxsUj2NSEPWageQYDmXRQqFCKmpdorZAAAg\nAElEQVQZXBQSSYlMtUyJKAOpsSIJo6IziZk+0agg/4vrRiFLZS7bAMnrRywnU6EU1dHq4b9/8P5p\n7DluIbLvghNB86Akh/EVjVBCkGBHhi+ddAuqrKLHdKrpxw4lhNF6XZ3QtGgapNP010SJaTHckCuS\nHJPVbDpnHsdfDalwCicZZ65eOca6OhFBTiR4+mmob9YpERsZiqEQJBLokoSKBIpCUxO0m9WM83wM\nDWaPHYcsg6NFGWgr84sfmVhBjEl1LRw5YQK6YhLyfSKpODE9NkyCXSlG88cybBs/DpmRydS7RtWL\nB/zo0cPnVdJD5CqXxkrHidkBfqSGuuYwYd+HUJG4DWmliC+HcCL6wST4/PNFMsuB0HUxu6rAksfw\nyQmn4chgehlmTExQDMuoleS1aFDPokWLiGpR7r/ofoJkkn21OhNyCnJqRELoaSohtx9bkYhpMaJa\nlK6oR6irF0IhDFXC0RVBggEiEYJImGOmjGLiRLj4YuitCXNEXiaSNzj5ffDr984gmTOIlWzMRJif\nz4Zlp4znA+86ByQINJVFrku6MQ6dR3Fsy0gic2y06FuvXBBSx/0LSVc3i+VWWaY6Ws2P5kJ+QrPY\nYPp0WL0aO6yjJmpQKhXh+jIh6o8TErfOMSkaxqSIWBrjYu9Egv9uUVcnnCL+0TBgDLzltvfdJ8bs\nu989YhPXE12Bmp3MGRPOQFd0jmsa0S+ossqnj/s0RbvIgp8u4OrZV4/cjAfg+98XgYqbbz7s+/dv\ngsmThQSiUBCJcDNniuj0tm0HF9YAUZDjuutG3l3/lxDRIm+aFPdWcM6kc9AVndtOue0v0Ku3hrMn\nnc2qD6zim6u+SXWkmiMbjnzzjf5ZcQAJTsRlgkClrFVYmqqOtKsMclOF2Q2zmTxekOBjZiaG79VM\nXESC64wBrANfuAdgqG1LC9hGhZgoELUDLEKMGwflIMxoOhgIqnj2sh+wefDdnFR1BS2VCsCOIuEq\nEtcfcRcxuZpue5Cjbqzh53dECJSDcws6OuA9F2kcOy2KVcgS2Day42LHI4R8Hy0w0P8fe+cdHlWV\n/vHPmXunTwpJCBA6SC8CgogUAQWUYkewIKCi/tS1rb2tru6Krqu46Lo2VHR1xS5WUIkNGwJioygW\nei9JSJlyfn+cmUzJJJkZUgY8n+fJk7llzn3vnTt3vve93/OeJg6MUNUDsxyTAKYUQTtEtAiWQmCR\nYPjBZpFMLfkOKsBvASrcOKzKR7ravxX3Pi/mXpVFC1hsCLsVi7BUCi+AZq/eTX5mQdQxJisLdu8G\nux2b3Y0Pk5NPVg6IEtz4QgOCBDPBdj+4nGq/bW4b1n0empYGcHZrV7mdLHuWygSHPvZ+A2nRuieG\n9FVmgktNJwHDiitDSRCvhcondVa3jRx24nNmhGMN9VIePJjHLxuG2+ZmWLdBNCk3EU4npw6YyuN3\nq17ivuwMykzA6VLnXF4eZGTQqxc4MqyUGZ5oEZyZiVWCKZUIzstT+96GANJhp1VzOzNnqkxwyfZS\nnJSyT6pzskf7zpiiHIffj93tVqLQblNOC+FmXc5uLO7c4LmkNmlxutVJeeKJMHOmmmezsyf4kXib\nZOK3CC6+PI+OPRw4gh5yj1eSaSnCIjLwOR34zYgPd9Ik1XEkEptN1VIN4hEjOaI0l932HCjLpn1O\na7LyWmJmN0EgeOCu6DFuiocO5I12Xtrvkuo8CeK3mYxeuEDZcgwbGbYMNti92DZvA5uNctOC12ap\n/E4Kl4urRv2FKZM8jB6tftM3tvDQY7vAtreEzR7Y3SwL964SssugoFV3vm8G8pGHuXDABQBIux12\n7iSnrQfzu+mcf0S4aL89mAleL3eT486EogLyW3SstNFkO7KxiHA89OwJv/+O6fbQsrl6old894Pk\nbdxNh9bqGBZ1akOTnAKyS1ppEZwsDenRTDYTnA7+0Z2lO2lzXxvmfjO3ct7ChTB5ciGBmCG6v/5a\ndYJbsEBlgUNjeL+x6RF8y86kjRzK7SNur9KpKdOeydtnvs3zpz7PlYOurBLDscfCjTeqEddqGugi\nRH0dN4dDVf/o2VNdD5cuVT/cjz6qMt6ROJ1KKDdUbHVBusU2sv1Ifrr0J/Ld+Q0aW9e8rpzS7RRO\n7X5qg20zLYkQwU4n4LexzRNjcenQodIaYTpdFGQUkOl0YLVYo+ww2W4X212QV7wF4XZH1XkOESmC\n8an3BixCVQJwqExwGQ5+Yy07aaL6XBW15OTWF9MmmDTyBUWw0wkeSy47SndQRDlOqxNpBmugthU8\n0K4769ZBl+4mzVxOyov3IHw+zHIv0u0CgraNPEeUJ9giA5gBqewQhD3B+P1Ii8F3+8DwS6wWP5m2\nbGSFxGcBKjwYBrisLr4p+xV7USnmjl3syjCRFiuG3coP235ARvxCP3DLVjKzg0osVgTbbOTkmzjc\nJnZ7WAT7Q17fa69lY/u84MeoMuCf7NuLuzhLxRxxd55pz6z0YQLMOnYWR7Y/Solgr/IES7eTjFwb\nbjdUGBZ8EcOp2jxKBKvObMGTJSSCbTZ2dm2L2+qmdZssTL8PXC6shrXS1uTPVjV3LW63OgmCIhhA\nWK1kFHjoFBoJNyiCjUDQimKaTJoEnfu66WAGMN0ZGIay4HmtTkq3FjGXsykLPj3okN8Fm6zA7g9g\ndWcwoGAAfVp2p2tXsOICIbF6VIF4aVGP9j4uKlZet8xMlQkBLHYHpS613J+TgzfDxchuA8DhwB7s\nU+DxBpRNxLDic9oJmLWMdBQaCcnjYe9bn9BrRB6sXInRpCMjRxi0yGjBsF7jsWRmccfIOxgzsG3U\ndbHoiotY0EGSXeyLsitJw8LQ1z/F45XYDBsem4eNDi/C74fcXMqtBl67CHdAd7lwZOYwbhyccoqa\ntbkgkz4b/NCxI+VW2OTy49xVRHYZnHTE9MrzKMuu2hB2O4U7d9K0rZv8/Jjf7WBs6wJ7yPWo9Y2s\njMrHphZhIceZExbBQQuKZ9MOunVQPTidnVqBw4EIihBru440b9qetrtG0cqu7RBpS36+sgrEisd0\n5rGlj9EzvyfXv389UkrKy+Gxx+C119QgFZE8+SRccUVlB2POeOkMbv/wdj5Z/yHZv05jQM9cJhZc\nE3c7QggmdJlATsTFGFRJueXLYe3acJm5xqRHDyWCQSXDevZUx+KMMxo3roOV0LCnDc3jxz/OzcNu\nbpRtpw0RIlgIIGBjS0bEDewdd6hRYII/ahN6T0QIgdPqJMOeEdVUE48SwWbAh6dJs7i+7pAnuG1b\nIFh/OBB8hPzZ1zaaN1c9/nPZwU5yKpNdTieVmeCQCB44EG68MpedpTvZ592Hw3Rgmln4BZw7til/\nPvRofvgBspta8VgcZMtg+a195Ui7nQrDILMcXE2d0Z5gGcCkGjuEaWKtyMQS8GOIAFYrSJ9Uw+96\n3ZgmuK1uvt27BiEE0uXCcLqQholhN9WoiRPDoybmereEhVGkCC4uVmLQaq3MyDscsE+48TuDQnTw\nYIpaKTEXsp4I06TpPhclVjNKlYzvPJ5pfaZFfxbZBiZ+8DqU9zbDSX6BFZcLvMKCX0SL4Fx2IDPi\nZIKBDFsGbqtbxRv6wCKQ2dlUGMEKGBGZYACsVtr39IT7WwTtEIaUGAEJhsFhh0HHXm4KSg36HjK0\nsl2fzUVg42aO4As1shvQIqcNpvTh8EusTg8X9L+AiYeO44cfwCrUMW7aRP3QSIsTj82DYXNEd/gA\nLFY75R47WfYsRF4ejpx8hrQZAnY7Vr+XQzdBhs+LK1BMwLDid9mRZi3XslDHP4+HzOMGk9ledVZs\n0TOP998PHcwMyMjghqE3YDOifYBZ9ix2hA57RCbYWRTOLlsNqxosqCvsmH03nH46fqeBK0+GRafL\nFfX5gcp273FZEEELx7dyC+7te3D6oG9nNS/DnkG2Q91wCacTpKT7ADe33Razn8HYim2Qn5XFn/+M\n6ggZUeQ/z5UX/bSofXvw+8lqpc4LIzu4rHVrvIagZXZrcDrJdZXSHJ0JToqG9Gja7erc2r07sfXT\nwT/6/PfPc9cxd+H1e1m9eQMFBfDqm6VcdHUPPoipJvXeezBmjHpd7itn/ur53FJ4CzP6zaB9Sw+B\ngMqegrIUvPlm9Pt37YJOnagsmF9ermovX3BBch3M6vO4TZqk7Fwh3ngDPv6YykxUbaTDZ1odOrYw\ndtNeazWJg54IEQxgCdjYlhEx78Yb1RczKFi6tFSPJp2mkwxbtAjOzrBRbBh4TXuVH9gQUZlgvxKl\n/mD2LL+VDYcD/KaDUfjZSn7lQBtOp/r+Wa3gMy34TAtNmsCEsTYMYeANeLEbdlw2N2UmZGVmI/wO\nKiogK8+KxeenszN8gbHYHXgNk8xywO1UYgvAqMAMqExwqERapUD2+cBicETxECwBHyZ+bDaQ3gDS\nEMhyd2UmeGfZLsjOJtA0D5dViWCLw0qn3E7RTx+2bo0vgkH9mJhmpQgWAsoMT1gEA6ZDHecMj0os\nDC8oIL/MQYk12qPVI78Hw9pG+1MzswQ+DLLsNqwBCLiUiHW7oUKY+CK+G3aPlSbswpKVEf5AIj7j\nvi360rdF33CniZjPX2Zn08djIBzOuCI4yv8WzARbAmAJED4+bjdi82aMrLDv229zEtihSvgVoCoQ\n2R0erAGJ3S+xuqPPUZ9D1cBskq1qMkuLwdpL13L06DGqM0oEgdwm7M5108zTDGt+8/DnYrGA1cq7\nz0D/0rXY/GUYNhO/y0XAmsANvc0WPj55ebB6tSqXEiIzM3xsiL4uZjuy2RG6v4iwWljLveHmg3aI\nZQXgm3Y2WCx47QYVjkCNItht8/BbcycMHYrDdLDOUY7rl/WY2TlkBcufZdozcVld3Dv6XoTDyXDA\nne/mvPNi9jF401xkhxxnNvfcg3qUEfEjn+vMjRbBwRHIrDnBfQ8dg/79ueGtP9Mltws4neR5Ssnz\naxGc1iRriWhMNhVt4pddv2DfOphd3w9g4Mlf0aULDL3yAR60tmHBKtVpSEp1Xq+Ur/G1fByAxesW\n0y2vG+M7j+dPh/+JMWPgqKNg2TLV9oMPquypPzhC7s8/w+WXq/+ffaay5QMGqGv8jTc2xt7H57TT\nlD0jhMdTtUSbRnNQECOChbSxIytaODB2rLozBCVQUD+GOc6cqNU8HmDJxZR78qq0G7UOys7qCFop\nZKgzkU35Nq3BmrJbaBYlgt1u5c33GwJfRAekcn85LTwtEELgsrrY4oYWbfIoL3Fwwglg2E0I2iAA\nfFYD07Dis4ZEsAtTogaWNsoxpMQmJMXFVHbQUW/0IQ0DAgEMAliCIjjg9SMtFqjwYJpUlnwUWdmY\nLVry3pT3VCbYEV3gX7rd6ocilD0MHttKsWWzRWWCASqsbqQ7QgQHX9tDwthmo3mZnXJb7TW2PR7w\nY9DEaVfZbrcbbDaVCcbAHymCM2xYkJhZwe2EbA1BTutxGmcferZ6jAZRnb8AfO3bsCnfqT7IGDtE\nKDNaSbDihBEIZoJD++92qyE9I2wAfpsLsXtX1LZsTg82fwCHD0xn9HlY7FAVimyZKoMeME2aupuq\nUZBOOCFq3a0jDuexi47gk+mfkN3qkKjMKw4HzUqgpV9lnzNzrZgZmRj2BDqIRIrg3FyVCYoUwbm5\nlX7vWDLsGWwNHaqzz467jtVixWMLnhfB75jXZlJi8ZFpi/Dghh6tBHHb3My8sCdMnMjqS1bz8sUf\nBW/8LJXZ3wxbBkIIrhh0hTpfHY7wTUokEZngST2C147rroNLL61cpVd+L9pltwu/56GHYMWK8HkR\n8TnfNeouLux/ITidjDyilHx/TEH/GLQIjqGhfZA5OWqgh0RoDI9mebmybAD8/eXXsf4+ilNPNhnf\n93BaHfcsf7tnD65uH9Nxbwe2FTzDpk1qyGA59A6ypk5n5qd/R0rJA189wOSek5l/+nxaZLTg9tuV\nVWLePJU9nT1bXb9WrFDbmjVLlRybNAk+/VTZIIqL4eWXq1Z/qI1087ZGomNLjXSO7aAlRqwa0sbO\n7JgObYcdpsYKh8psZc/8nrx71rtRq3k8wDv3U5HZtNZMsNMJeU2CIjjkowxeBKwZDgqhSiYY1BNT\nn2mJ7oAETOg8AYDRR9vpcTF48ltx8zUZvPQSSkh6vVjKygHwWU2shhWbx0lmORguBwFQo56Z5ZhS\nYiLZuxfM4Chx6o0+MAy+8Sk/osVXQZ9lTyB9SgSHMsF3Hn0nb5/5NiI7G5GfT4/8HkjDxHRGi2DR\nsWN0JthiUccgNhMcITK8djfSFf7MrK6grzZUB3f7dg4p6Ylhr/2xlcWiKmfkuqzKDuFWmeDcXKgI\nuJGEt2M4Vfu23KBAGThQ9aaKJS/Y+e6HH6JmV/TqzjUjWoXvZo4+OjwCUmwm+KmnYMgQDL9Un0nI\nm+x2q+MVkSUNOJwYe5UI/ieqr4nDruJ2eYOP7COwyUxYNwi3Jwe/AItNCex4155RHUdx7eBrlUjO\nzY0SwSJ0wxLCaqWgRWc65idQ2tFqDZ/QIbEbKYKnT4e/hzsKR8ZmERYym7dle9GWqOMQiWExKq1K\n9mDlEZ/NSrHwhjOv//63+l5H4La6KWrVFKxWWme1pk1eR+jaFbZvx2PzMKrDKNy2iOuFw0FhdT/c\nmZlIIbjm6FvomBMsrt+kSZSwfWj8QxzV7qjwezwe5YfMiMkEB/dbCAFOJ9byYkRIwFRD4xjsNJW4\n3WoQhXTlyiuVUP3oI5j77WOMzb2NP82Ddj3P55r3rmbK5z0orijmln63cLPjXp5+WrJ6+8/4B9zP\ne1Pe5vSXTmfoE0PZVLyJZ056Jqrt4cPV9e3MM5X9Z9w4ZSXYsQPeegueew727lWd67KzK5+AaDSa\nxiDWDoGNL3t2BaNp9HqhH7vgj78QQomDCEI6xpddfSY40lKal2NjPSBNE58hMINix5alBMJW8qM8\nwSF8pgW/Efa7Fk4t5LAC9YN+912CB//u4rbht9E2q63Sj6bKBIvSUjUal82CaTGxuz1klW7D73Lg\nF4IMPOwxyjECYCUQFMERmWC/HwwD6VUZZVFczLDX/8wbfW5GGhYCQRFsN+0ce8ixkPVP1UkE+Cl/\nEGbLiNEVL7xQlYt64oloAeRwhIVCSGhFZIL3ufMxc8MVA1zBkdMqvbiGQb6xF1GNQIrFL0wy7SbW\nfWBkesCqytdt8mcQJW9CNyihR9UWS/X+sN9/V48OIxjYciDn9zsfKn5Vj8RDdYZDsUeKykMOgZIS\nTF8Av0VghLzNbrf6DCL2TTpcWEu2sJS+XMU/+TPgMB14DfB4iW4XuNbzA7c97SBjqqDYpipTVEdz\nT/PKkdoYODAsxqFqSSDTVPFZq2+vklg7BESfA3Z7lbgj+enSn+L2o/i1uYNnu3q5AfBY1Zcx5Cn2\n2a3s9XvjVmwJ4bbG+c7Onw8LF2IRFhZMiRlUyG6v4v2uxONBfPopfxk0qNrtVUvoQhKnTjlOp+o8\nlJenngpUgxbBMTS019DtVjVvE6GhYtu4UQ320LKlGtzixBPhrw+soci1kUevHYOqzd2CZ05+hrnf\nzOXez+7lislXcN89/+bvz3xMebvXOXvKdAa2GsiDYx+kqKKI8Z3HV6kCkZWlbjK/+AKGDlW2pcJC\nuOwytbxfP3UtOf54uP569QQkFbS3NTV0bJooYsSqiY29TZvCzDnR64V+3Gv4cQ79dgVy8sAVv8SL\n06m+/w4H/PlSO1O+AQwTv2lU/nAde6KD4d/Cv41mlb+xkb+1/hgRHJVNAno07UFBRkH42hTMBFNa\nSrHLgtUUWC1WTKebzN2wyVaOXwgyLdnsMcsxA2DFz549YBCRCfZ6EVaTwyMzpP4KlQl2WAiUuKOq\nypGdrbxxwEddzqd394hlDz0Ec+eq7EDk42SHIzoT7PdHieDnRz7M+BFhMZad1Sy8j8Dw9u3x71mB\niCcg4uAXJm6bH9NvwdKtMzjtuN1qgBFvpJAN3QQlUruydesqszLsGVxx+hVwepz1hw6NLscHYBhY\nfQF81ohjEzpXI0Sw4XFiLdlJOeHz0mE68FpU6bjY83XU4DxuqwCPPUCJlcrt1nrt6d9f/VVuJEYE\nh7LZyYrgUCY4L6/a1WNjq64j8crmJneNsXED6ngbwqjs8+C3W9lTWhKuDhEHt81dZfAqDjlE/cXD\n4WB4NbYNQNWTToWMDPWdiCewnU71OLlVKy2C05lkRHBDceutquzZP/6hzs1x42DS7V/TYsQReNzR\nnp6zDz2bKb2nIITgvnEzuVyei/Tu5tqjvwDguE7H1bgtIeDpp9XTj99/h5tuUteauXPD17q77lIx\n9OlTY1MajaY+ibEtmMKGyxrH1xj6ca+hKHZIH8m8puAujbuOEPD+++oaOXaMHb4BTBN/hNi54DIH\n3A57HfmV2itKBBsGfrP6OopfzvgyeoZpQmkp+P2UOkz8SEyLibDbaS5d2Jp1wS8sZIhshFmMBeUF\njmeHsJgGWY4Kysts2KnA8JUT8Pkp9xn4SjzR9siCgsqKAzZbnOS406l+KCIFYPfuKkMcqiLg9UYt\nf+gRI1pnhT6P0EyrFWP3TmgVM4JPNQQsJg7TR4+crmSPO0516kANYV3uj9j3UPsJZpiTIjIrHCJ4\nIAOWiM85dIJFxGBmOHFU7KKI8DyH6aDCgLIAOGJEXcgB4K2wUGIXYE3Sh1e5kTgiOJlMcE12iBTY\nful53Bd4AZuhtp9hy4gqX+h12ineXUpeLZlgQQL1SUM4HNU+8dkvcnPh//4vfq3Udu2UqOjXD776\nqtomavUECyHmCCG2CCG+jZiXI4RYKIRYLYRYIITIjlh2vRBijRBipRBidPxW05eG9homI4IbIrai\nInjhBXUNueMONQRx//7gy1tOvxbxxyMWQlBYWMip3U9lZKdBDOtwBB2adIi7bjx69FC/Ad26webN\nyiJxekwWYNiw+E88EiGd/aM6ttRI59gOWmJ+xFq3tNGtcxyhG2OHiEdIo1jyq7dDgLJMQdivKKxm\nZYUIQHkNgWJXPqYJEydGaw6/1RK9fm1YrarzgcNBwGpSakishnoEn10uaJZfgB+TjrY+GA7VmcOU\nPvbuBSNSBAftEJub78IarKVs8fuQXh9lXgt43dEi+N571bCawFlnwahRMXE5HMo2EPmmRYtUVtDp\nrFIdIvSWqG2EDkzIE7xxoyrBk6BYDVhMLlh7Dc1WrEJECDgvVkr9EYIumUxwNST1/Q7upMWMEKmh\ncyriR8Oa5aIJu3BmO7j4YjUvZIcoj3Oj5HAoJ0rXrlDqMCpFa9LXntBxF0LFZZqJZ4Kt1vDNp92u\n3leDCE4ktpLbbuK9pkVYLWr7Hpun8vsFUHh0B2YOqKjRDtEqsxWts6pm8avFbqfQ6619vWSx2VSH\nonj076+elNRSSiqRTPATwGxgbsS864CFUsq7hRDXBqevE0J0ByYB3YGWwHtCiM5SygOoEm7Dki6Z\n4A0b1PC/ZWUwYoQSoaedJjnikoeZ+sHzMKSQCQPm19rOEyc8Qbm/PKUYsrKUBUNnfDWaNCRGrOZl\n28nL2r9McPmZ50B+WbXrhQhlqppnt8KxPUJsBrdR5G6GEKr/QiQBwyBg1prrCWOaKhPgcCBtfvZV\n7FOPlO12KCnBnmGnyO9ggOt4Ftr+AoCBr6odIlgn2OL1YnHYobhIzd5VTHkLAyrcUbbRSLV69NHx\nDoC9ynqVhERwTHWIKsTJBLNzZ8JiVRomx2wIyoDIKhTYkL4IgRMSwfWRCY6HxQJCYFrjiOCIGOzZ\nTpqwi4rmDh54QM1zmA52WUAY8c+RkAXvF7uJJRHRGo/Q8M+lpeHPKVhdo1Yi7RCgRlyKqVGcLB6b\nh4AMVHqAM+zRmeDynEw2ZUK+O7/aNi4+/OLkNupwNPxwqaapxEwtIrjWq4OU8mNgV8zs44Gngq+f\nAoIjwnMC8JyU0iul/BX4CTg8ibAbnYb2Grpc6eEJvu021UHvtddgxgw46SQYcOJXzN/9dy7sfyEt\nPC04cUD1H2UoNsNi4LLG7+2dCL17V+mIut+ks39Ux5Ya6RzbQUuMCLYZtio+f7Wg9kyw3R608nVq\nVb2PMAJDGFiEBZczC6sj4vpiGGRe8DABd/ysld9qELCmlgnGaqXY4g2LYMB02/FjYKmwYtpVJtgi\n/cFMsC/cTrA6xHCrNSrjV7xpL6X+LCiJHuK2VkICIp4IdrnUMY/JBFfbRsgT3KmT+vFJVKxGbjti\nO81bWWnSNGK7dSCCk/5+x1TGiCeCHTkustkNtvB5aTfteA2oqOVGqdxhRQRFdtKxORwqnezxVJ5X\n9OiR2A9dpB0CVC/yGj7jRGILlUQLieDYTHDodWVHv7rAbmd4TJm1BuFf/6JqYeJoUvUEN5NShqrb\nbgFC3+gC4POI9dajMsKaamjsTLDPp2wIGzfCL78oH3CHDurJzdnXf8k3m8cwqeckJvWc1CDx/O9/\n9WMd0mg0+0kcEewwa8gE1yCChYCbb64cAbVWhBAUTi3EWPKPKhm0rSeej2tZ/PcFTKP24WkjCYng\nrCywQalX1VKt3BeXi4Aw8JeZGHY1ypFRaYeoKoIpLY0SLZnspWzpZUCCo+mEqEkER2aC4y0PYber\nAx9aJ/Q5Jegzk0aEXIgQ9h262qA8xY5xdYVhRIvDOCLYmePEgozKSJoWkwoDApaaRbDXaY2ygCSF\nw6HKec2apQrLm6byUx91VO3vjc0E1wEh8Rvq2JZhy4gabS70upk7yRu1mnA4oitmNBRxOl7Gst8d\n46SUUggha1ol3sxp06bRLpjWz87Opk+fPpV3MSFfS2NMR3pqGmJ7bjcsXVpIYWHt68fGWBfbf+cd\nMIxC7r8f8vOHk58PC99fiGkx+WrPVwxpPSSh9pYvX87ll1++3/FkZtb98Z41a1banF+x0w19viUz\nHRtjY8dTH+dbItOzZs1i+fLllderPywxIthhOuI/9QmJkVoef/7lL8ltfmjboS/8ZScAACAASURB\nVGDeV0Vc795dyPHHD4/7Hr/VTN4OUV4ODgcWm6S8jKhMMDk5BISBLDUxLWWUmgIz4GfPbqmGFQ4R\ntEMUlpQwPDs8almBaw9lRSk8Fg4dy3hZQJcrric4bhsRQq7wp58YDqr8TiJEtm3GZH7ruGNcYWFh\n5fcvIQyj1kywK1dlVIUz+vh7LRYkNd8oeZ0OLMEMctKxORzqbq9v3yqfQa1EeoITIJHYhBA09zRn\nxz41eEesHaK4ohggat5+43BQuGULNUfWSEgpa/0D2gHfRkyvBJoHX7cAVgZfXwdcF7HeO8DAOO3J\ndGXRokVx5x//3PFy+qvT63x7jzwi5bnnJrZubGy/7vpVlvvKU972r79K2aGDlP/9b3jeB2s/kI47\nHPLUeafKDvd3kMs2LUsptnRCx5YaOrb4BK9fCV07D5Y/QMr166OOw++7f5f7KvbFP0hWq5Rr1yZ2\nQJNh4kQpBwyImlXTufDKce3lSxMOSbz9n3+WEqTs21euHdxDvtAN+cL3L0g5ZYqaX1QkNxit5E2X\nL5T5VyF3203pM22yWZ5PLQ/9deki5aGHykUZGVJ26lQ5/7umR8mpPCGT/glcuVK1MWNG1WWDBkk5\nc6aUl18u5dCh1bfx0ktSejyVk4tOOUUmE8jmZr3C+7d6dXjBCSdIOWpUeHrfPrXON98k3HYsSX+/\ns7KkbN06PL1uXeXnVRnWux9JCXLlsZdFvXVZM0N+U5BRY/NzT+8hX/jr5NRimzJFytmz1evDDpNy\n0qTE3ztunJT33JPw6onG5vP75LaSbVJKKdfsWCNPef6UymWjnx4tubWONdp118lFZ5xRt20mQU3X\n7FTz068DU4OvpwKvRsyfLISwCSHaA52AL+O8P22JdxcVkAFeX/U6K7asqPPtRdoh5s9XfTISjW3S\ni5N4fdXrKW977lw15O8ZZ6hpKSVnv3o2L058ka82fMUhOYdwaLNDE2orqTvjBkbHlho6Nk0U+dEd\nZVpntY7vCQaVHayPjjCmWcUOUdO5ELCaBKxJPPCMyGJbHS7KzZhaq243Uhj4Sg3MAFQYFiwBH/v2\n+qLbCWaCh0sZlTXNMfbSrM1+ZIJrskMkmQke/vTTiQ9XCjjc8e0QlUM2R07Dftkhkv5+x7NDhKox\nBHE0UeeqxRmd4fQZFrw1HTdg4cR+/Hr0YanFdt55ajhxUJ9BLduKwpacHSLR2AyLQZ5L1Rs+JOcQ\nXjztxcplu0pju4DVAWPGMPyCC+q+3TogkRJpzwGLgS5CiHVCiOnATGCUEGI1MDI4jZTyB2Ae8APw\nNnBRUIUfsJR6S2l1r+pdmOPMSeq9JRUl1Lb7kSL4+OPhb39LrG2v38vyzctZuX1lreu++9O7HP/c\n8cRWKPn4YyWCQ6zasQpDGIzrPI4FUxbw3CnPVS2IrdFo/pgk8xi3U6f6qQ4QRwTXxJ48D3tzkxBj\nEZUtbE4PZWbQExy6SAtBwGLg22diBkBaHYhAAH9FHBFsGFBRESVcm7n2MvWCOhbBITtEbdUhQuuE\ncDoTN2UDWbnV2CFit2sYyv/ZUNUhQtuMPDZZWcqDG/H7JdxKTFrc0cc/YDPAVbMwzXPlke3IrnGd\nahk2THW0geTtEDZb9SOt1RO7yupBBA8fro5DGpJIdYjTpZQFUkqblLK1lPIJKeVOKeUxUsrOUsrR\nUsrdEev/XUp5iJSyq5Ty3ZraTkci/ZAAn677lLbZbfn83M/ZULQhqbaOePwIrlpwVY3rxHaMW7w4\nsdi+3/Y95f7yWkVwQAa49r1reWv122S2Ws/y5Wq+zweffw5HHhled9EvixjRfgQAnXM7JyX6Y49b\nOqFjSw0dmyZlli2rn45RcURwTefCe+O68eHY7tUuj9s+gMOBPSiCTYsZdZGWFgNviYERgMxMN9Ji\nwU5MWcigCC6sqIgSiJaivXTr56ypdn98avIEO52JV4eI9AQn+x2qyRMcu90jjgiPZpcCKcUWKYIt\nFrj00uh1gmLSiBHBWA1ELSL4zqPvZFqfaanFFkmyIviKK2DMmIRXr4vrYr8W/RJ+ApwM6XrNboTu\negcW7699n1EdRnFIziFsLNqY8Pt+2vkT20q28ex3z/LDth+qXc/tViP7PfOMmv76awjEVFWWsqo4\nXrJxCZ1zO7NqxyqklEx6cRJrd62NWmflSnh12Uf4Aj7sq8+i5fg5vPKKWrZ6NWT2+pAz3h5Ds3ua\nMfrp0Vy98Gomdp+Y8D5qNBpNg5JkJthqsWKKJB4/R1S2cDgzKTdQg2VEiOCAxcRfYscMgN3hRhpm\nfBEcEoaRAnHPHoTTETWqbkLUlAmeMEGV20qkTnCqFQ5itx0rgmPb/fTTpD6n/SbWDhGPoAjOLYjJ\nBJsGflvNx8Vu2qsdgjgpQraVRBk4EFq02P/tJsFzpzzH1+d/3aDbbEy0CI4h1lPz3i/vcUyHY8hx\n5lDqLeWxpY/hD/hrtDkEZIDZX8xmfOfxnNPnHB75+pFq13W7Yd06uOwyaNtWTW/eHL3Or7+qIdP7\n9VOxSSlZvG4x0/tMZ2PRRo568ijmfT+Ped9HV4q/6Sa447X/Msh1Nh3X3cK2Dvdzz6+TOfnRP7Hi\nxxJ2HjWN4zsfz/zT5zOs7TA2/nkjYzuNTeZwVZLOHk0dW2ro2DRpR5KeYKvFqkRsMu1DlB3CF/BF\nP66zGPiLrZy1/QeEobKv1WWCh0e2CcoekYpXOrTP8UTwlClqhKHYbGgsIdtEkKS/Q7GWhxC1ie8U\nSMkTXNO+Q6W3NqNpjAi2mgRqEcH7FVsk+3sjUgt1cV20CAuGJYmyggmSrtdsLYLjsGPfDr5Y/wUz\nXp/Bqu2rOKLVEQghKPeXM2P+DHr8uweD5wxmZ6nqVLCtZBv9Hu7HPu8+AJ799lkKfyvklqNuYcZh\nM3hmxTOUekvjbivk29+5U93wtW+v6vVGZoN/+EFNf/45bCzaSId/deD1Va8zuedk1l66lmaeZkzs\nPpHXVr0W1faSNev4zv8KvmVncvpx7Sk8531O7DOCN7/+hqu/G0UzowsXH34xh7c8nJuG3VTjMIka\njUbT6JhmjfWHq6xuMSuHh02ICE+wsNspN1HX+X37KleRhkFFqR8Tle0VpkEbfo9uJzhscmXMkaQi\ngoUIjzBSU+w1idGuXal8FJgKobY3boz2EsezQzQ0td0AQNhbG3P+SKtBwF5/wjSKehbBmuTRIjiG\nwsJCJr4wkcFzBjN3xVwGtxlcWTzaIiyc0esMjmx9JPnufF74/gUAHv76YZZtXsZba96ipKKEJ5c/\nyU1Db6JNVhvaZbfj8JaH8+IPL8bdXmTpzebNwyLYMODZZ9XQ7j/+qKafeaaQt9e8zfq962mV2Yp2\n2e2wm3ZemPgCz5z8DCu3r2Rz8WaKyouY9vIM1rX7G5alF7LotZZMmAB9W/Thv1dcwKBtj7He/xVj\n8s6v0+OWrujYUkPHpkk7rNakPMFWI/VMMJmZ7HZA68zWahz5iUGrWKQINgxEUREfETPwQahOcHD9\nKFKtmuFw1Cz0avMEC6FGRgqSsic4pkoIMaPi1QVJx5aIHSJ0kxBz/KVpEkjixqpBPcFJks7XxXSN\nrZFv39KLVdtXceMHN7LKs4oHxz5Im6w2WET4PqH8pvJKX9AzK57h5R9fZnLPycz+cjZn9jqTSS9O\nYmL3iXy96WsmdJlQ+b5xncbx8e8fM+XQKVW2Geo70r+/EsFNmigRDGqExJwcNX/CBPj2Wyj7+V1m\nHzeb0R1HR7VjM2wce8ixzF81n445HXnq28cQhzpo9fZ3HNJVjdIY4s6rOnPkSV8z8bkeaDQazQFD\nKp7gZLychhHOut50E/dbblGi5S+Do9apKPVjCuILL6s1XB2COOvsjwiuSejVgy0hCtNUHc5ihXg6\nZIITsUOAygbHimCrFZKwQ+wXyXqCNfWOzgRH8NKPL7HcsZwze53JBf0v4LhOxzHmkHDPzMiL6eiO\no/nglw+4Z/E9jO00luuGXEe+O5/nv3+eUR1G4Sgug0cfhb176d2sd9waw5uKNuEzd7NpE/TuHc4E\nrw32b7viCnj4YWWHOOcc+Omn4Xy1cQldHSP44dMOVdqb2H0is7+cze1zP8ISsJNn6ciaLzvyzjtR\nlWIYNAj+eU1vBg6oO99Puvp9QMeWKjq2AxshRI4QYqEQYrUQYoEQIm6NJyHEHCHEFiHEtw0dY9Ik\n6wlONhMMSkw6HFVLioUIimBD+uILGtOs3hMMqZe82t9McAwpeYLjZUzTwROciB0ClC84jgiWjsQz\nwft17XG766d+dpB0vi6ma2xaBEewcO1Cnj/1ee4dc2+t6+a78+nVrBf/WPwPzu93Pj3ze7L+ivW0\nzGjJ+M7j4ZJL4Pzz4fPP6dWsF99t/Y6ADBt9V25fSdcHu3Lnx3fSvDnccIOqqd2+vaoQ0aQJ3Hsv\nnHoqzHznSW7e0Ic23bayuWgLn795CDfdVDWmk7qehH1PDwoDf2NQ8d28PHUO1ZX5vfLKhh3aXaPR\nNDjXAQullJ2B94PT8XgCOLaaZelFkplgj81Dhi3JerVxHplHYRj4ysN2iHjLq60OAamLoNo8wUOG\nwCmnpNZ2IhhG/GMfrzpEQ5OIHQLCA4tEYrWCvf6EaRRXXgkXX9ww29IkhBbBQbaWbGXppqVYfk28\nZ+TkHpPJc+UxsNVAQI3CsmDKAia3HANvvKFGidm8mWxHNnmuPH7e+XPle5dsXILTdPLJuk8A6NgR\nWrZUIvi778K2q3JfOXcuuxzTYlJ6yAyaeHvwxecG330X3WEZYMsWwS/3P8bRrcfx/F9OYUj7w/f/\nwCRIuvp9QMeWKjq2A57jgaeCr58CToy3kpTyY6AeKuTXA3FGoqvpXLj6yKu5/IjLk9tGKBNcDdJi\nEPD6MfBXnwkGVSc4cjrUAaS+PME9esD48Qk3l5InOF4meOzYOhffKXmCU7RDWGw2RBKfyX5de5o0\ngewUB91IgHS+LqZrbFoEB3l4ycNM7D4Rly3xIQrP7Xcu75z1Ttg3vGUL3WUetjffgZEj1UUpWO+s\nW9NurNqxqvK9W4q3MKHzBJZvXs4Ty56onN+mjbIuNGumpj/45QN6NevFrcNv5Rfb6+xY05FXX1XW\niaVLo+N55BE47SQ37533Ki0zW6Z2IDQazcFCMynlluDrLUCzxgymTjj//KqDINSA3bRjNxN/1A3U\nXoHCUCK42kxwhAiu/P/OOzB5srq4p5o1TXbI3bqmuix8794weHDV+Q1JonaI/v1VtimCQ4eeSr8j\nT62nwDTpzh/OoR2QAf764V+5ceiNlV4xKSVPffMUz5/6PIcVHJZwWw7TQc/8nuEZp56qRrWYPBmO\nOw6KilQRYKBjk478vPNn1u1Zh9PqZGvJVjo06cDQNkM55/VzOLX7qWTYM7BaoVWrcCb4qW+e4pRu\npzC+83hOG38a2VvGsaNCieAvvlD1g9U+qGoSTz5ZF0cpedLV7wM6tlTRsaU/QoiFQPM4i26MnJBS\nSiHEfg1hP23aNNq1awdAdnY2ffr0qfwcQlmexpgePnx43bZvtVL4++9QWBh/ucVge8VS1myhUpQW\nBo/RcFDvB9ixQ01nZVFot8Pu3Qx3OECI1OIrK2N4UOjV1f6GSGj97dsZHhTB9f35huYl/P7iYpCS\n4RHvjbv+3LlVlmdcc7OaTnB7dX6+/YGmQ9T39mbNmsXy5csrr1c1IqVs8D+12cbh3Z/eldyKvHD+\nhfLuT+6WUkr5zeZvZLtZ7WQgENi/xo89VkqQMiNDyl9+kfK//5Vy8mQppZT3Lr5XDntimHT9zSWn\nvjJVTn1lqnx86eNSSil7P9RbLtmwpLKZ4cOlnHFxkfxx248y/x/5cm/Z3iqbmjtXyokTw9M//yxl\nQYGU+7sLGo2mZoLXr0a5dibzB6wEmgdftwBW1rBuO+DbGpbX4RFMc1q1knLOnGoX/9xpjDyWt+RT\nk9+U8rjj1DU/8q9VK/X/7LPV/6uuUm+8+mopmzRJPa7Ro6X8179Sf//+cu65Unbr1njbr4mhQ6Uc\nP76xo9CkKTVds/9wdoiHv36YTHsmjyx9hFdXvQrAY0sfY2L3iYiIO/Sk8Xph2zZ4/HHlS2jXTqVr\ng3aIDk068NFvHzG5x2ReXfkqa3auoZlbPZ3smtc1yirRvONWHm2awcxPZjKu0zgy7KpjR2RsAweq\nTHCIZcvUyJnVdYSrb1I+bg2Aji01dGwHPK8DU4OvpwKvNmIs9Uadnwu1eYINAwM/dl9JrXaIQoDc\nXDVtq+pnToraOsYlSdLHLclOiftDSrHV4bGpiXS+9ujYkucPJYJ/2fULH/76ITcOvRGBYPnm5Yx7\ndhz/++5/XH3k1ak3vGMHFBTAypWqk8DkyWp+s2aVIrhjTkcAzux9Jke1O4rF6xbTzBMUwbldWbl9\nZWVzzqP/CcD81fPpktsl7iY7dVJui9AQy8uXq5EzNRqNJshMYJQQYjUwMjiNEKJACPFmaCUhxHPA\nYqCzEGKdEGJ6o0SbLiTgCfZQzIQ3/091CAsEoGfP6PdH/g+JYLs99fJokB6e4CQGlWhQEq0OodHE\n8IcSwbM+n8W5fc/lhC4ncHqv0+ndrDcFngK+OO8LmrqbAil6Dd99F7ZvV2VxIkfTad4cNm0ClCe4\ne9PuDG49mBHtRgCqzBpAl7wuUSJ4vW8ZR7U9ip2lO+mc27lyfmRsQsCAAfDll2p66VLo2zf50OuK\nlI5bA6FjSw0d24GNlHKnlPIYKWVnKeVoKeXu4PyNUspxEeudLqUskFLapZStpZRPVN9q+lHn50It\nmWAMgwI2UmFzw7Rp6mIcKW4jMsHDoe4ywbVVh0iSpI9bdSXS6oGUYmugTHA6X3t0bMnzh7l12rFv\nB0+veJoV/7eCVpmtePqkp9m+bzvZjuzkRhSKZds2eOwxuOgiWLFCjagTIidHXbh+/RVnu3Z8f9H3\nAAxurXrShkRwz/ye3P7R7ZVv+37b91w/5Ho+/O1DOuV2qnbTIUvEhg1qNLkhQ1LfDY1Go9FQuwg2\nTfLZSpkrJzzP5YpaDoRFWV2J4GuuqTpkcUOSzpngJAcK0WhC/CEywf6An7NeOYvpfabTKrNV5fw8\nV14VAZy0b+XOO9UFcOZM+Pjj6GVCqFJps2bB6tWVs/s078MlAy7BYaoLYte8rvy2+zf2efexu2w3\ne8v3MqbjGASCjk06VhvbwIEqE/zuu2pgjaZNkwu9LklXvw/o2FJFx6Y5EKjzc6EWsScMg6Zsq10E\nm6byBOflqen9FcE9e9apCD6oPMENmAlO52uPji15Dvpbp/mr5nPPZ/dgCIOZx8ys+w38+KMaASaj\nmlGJRo6EGTOUXeKZZwA1lOfssbMrV7EZNrrmdeW7rd+xp2wP3fK60TGnI8+f+jxOa/UessMPh6++\nUrWF27ev073SaDSaPyYzZ0K/ftUvNwzy2B4tgquxQwB1lwlubNI5E9yAIlhzcHFQZ4LX7lrLOa+f\nw1m9zuKl015KaAz5pH0rK1dC167VLz/rLLj1VthV84BMhzY/lDnL5jDllSlcOehKLMLCxB4Ta4yt\naVM1CNH33ze+CE5Xvw/o2FJFx6Y5EKjzc+GYY2ruwGaqTHB5pAju1Sv8OiTGLBblCc4JrpdmIjjp\n49aAmeCUYmsgO0Q6X3t0bMlzUIvgh5c8zLRDpzHjsBk0cTap28a9XrjrLli/XpVDqw6HQ1WLWLmy\n+nWAqwZdxfzV8/nbyL8xuefkhMPo0wc8HjUao0aj0Wjql5AdotwVcdG9+WZYs0a9DomxQED9DwlH\nhyPaNnGgoTPBmoOQg1YEL9u0jDnL5zDjsBlJvS9h38rMmXDddaoiRG13oB06qN5rZWXVrtIjvwfr\nrljHuf3OTSq2Qw9VzTdWfeAQ6er3AR1bqujYNAcCDX0uiFAm2B2RCTYMyMpSr0PDIrdqRWFwhDIA\nxo+Hu+9uuEBrQXuCUyOdrz06tuRJG0+wlBIRoeTKfeVc8tYlDGkzhLbZbRnebnhS7V36zqX8beTf\nokqM1SmPPQaFhUoE14bVqvwKzz4Lb7wBEybA9KqlOC0i+XuSIUNg69ak36bRaDSaFBCmQTZ7qPDk\nRC8IVQaKsEPQunV4udvd+L61/aEBS6Qlja4OoUkRoUaUa+CNCiFD2w3IAJNenMSIdiO4aMBFgKrm\nMPGFiRRXFLN43WIAvrnwm8oBJyLZXbabE/93InNPmovX76VDkw6s3bWWQY8PYv2V67EZ9fCl/e03\n1Stt8+bEU7Dz58Mtt6hBNVwueOGFuo9Lo9HUO0IIpJSN/OylYYm8Zv/R+W38RbR98yHeOvclxj52\ncnjBrl3K/ztmjCrZc//9cOmljRdoXTNnDvz+u+rjkm6ce67KxN97b2NHoklDarpmN+qt08aijZzw\nvxP4aedPAFw04CIq/BVc+val7Cnfw9tnvk1ABrj3s3uZ/tp0/jrir/zp7T+xZMYS7KbyJl333nUs\n2biE3g/1xhvwMu3QaSxYu4AZ/WbUjwAGVQpt2LDkPAgTJqi/VavURVKj0Wg0BxzCVJneKpngUAY4\nlJE8kLO+8TjnnMaOoHq0J1iTIo3mCd5Ttocb3r+BwwsO5+PpH7Nk4xJe/OFF+vynD7/v+Z2XT3sZ\nm2HDYTq4bsh19MrvxYinRrC1ZCsnzzuZs14+i8vevowPf/uQD6Z+wHGdjuPL877k8WWPc+PQG7lj\n5B0pxVWtb2XNmnCt3/nzYdSo1Ha8UyfYvVsNslFXsaUBOrbU0LGlRjrHpmlYGvpcsHpLAdjbImZI\n+5Adwu9X/4cNS+vz9KCKrQHtEAfVcWtA0jW2RssEHznnSPLd+cw7dR65rlx2le7insX3MPOYmUzo\nPCHKH2wRFh4c9yDXDbmOb7Z8w9tr3qZb024s3bSURVMX0dzTnOdOeQ6ArVdvJdOeWfcBX3ihKsh7\n//3wzjvw73+n1o7FAocdBkuWqCGVmzZVGWKNRqPRpD2/XXQXPd66m5lNqvEE//ab+h/qKKepf3Qm\nWJMijeYJPvuVs3nyhCcrxe6mok009zSPEr9pw8qV0LcvtGyp6kcOGwYPPph6e9ddp9p5+mklgO+7\nr+5i1Wg09Yr2BP+x+fxzGDQIHn0UzjsvYkF5uSqD9vDDajjPQw9ttBj/cFx+ubrpuO22xo5Ek4bU\ndM1uNDvEEyc8ESV4W2S0SE8BDGqktwsvVJnbgoL9E8AAAwao6hI//6w6Gmg0Go3mgCBUAa1K4jGU\nCW7SRAvghkZXh9CkSKOJ4FTKgTUEVXwrgYAqbTZliqrzeNVV+7+RAQPUIBvXX5+UCE5XTw3o2FJF\nx5Ya6RybpmFp6HMhpLUssT9hoRkRyZx0Pk8Pqtis1vDdST1zUB23BiRdY9O3TrXxwgvKt9u3Lzz/\nfN202aYNLF+u2p0zp27a1Gg0Gk29E9Ja1YrgKgs09c5ll2lPsCYlGr1OcKOxb5/K8no8ajoQgPfe\nU1UfQnfyfj/07Kk6w40eXfcxBALKG7xnT1qNKa/RaKpHe4L/2KxeDV26KJfcmWfGLLRY4OWX4cQT\nGyU2jUZTlbT0BDc6V18NN94Ynr72WlW/99tvw/PmzYPc3NTLodWGxaI6261fXz/tazQajaZOqdYO\nEZqpM8EazQHDfn1bhRC/CiFWCCGWCSG+DM7LEUIsFEKsFkIsEEJk102odcSf/6w6tz35pLIkgBrA\n4sknYdo0CiNHnHngAeUBrs8Oe23aJOwLTldPDejYUkXHlhrpHJumYWnwOsHV2SFCM7UneL/RsaWG\nji159veWVQLDpZR9pZSHB+ddByyUUnYG3g9OpwdbtsDjj6vSZFdcobK+xcVwxhkqK3z++Wq4y5Ur\n4bPPYO1aGDeufmNKQgRrNBqNpnGpMRNsGDoTrNEcQOyXJ1gI8QvQX0q5I2LeSuAoKeUWIURzoFBK\n2TXmfY3jL7vvPpX9feopNd2sGUyaBBs2wIsvqjv4yZPhlVeUH3jWLLjkkvqN6aabwGaDW26p3+1o\nNJo6QXuC/9hs3676NL/4IpxySsxCt1stOO64RolNo9FUpaZr9v5Wh5DAe0IIP/CwlPJRoJmUcktw\n+Rag2X5uo+54+WVVlizE0KGq5u+yZeFHWHPnQkUF/O9/MH16/cfUpg189VX9b0ej0Wg0+02tnuB0\nrXev0WiqsL/PbQZLKfsCxwEXCyGGRi4Mpg7SI32wZYuyPxx9dHjec88py0Pv3pWzChcvVhUjzjuv\nYUqutG6tPcH1jI4tNXRsmgOBtKkTHJoZsSCdz1MdW2ro2FIjXWPbr0ywlHJT8P82IcQrwOHAFiFE\ncynlZiFEC2BrvPdOmzaNdu3aAZCdnU2fPn0YPnw4ED5YdTo9bx7Djz8e7Pbo5W3b1s/2Ep1u04bC\nH3+EwsJa1w/RqPFWM718+fK0iidyenmwA2S6xHOgTIdIl3ga63ybNWsWy5cvr7xeaf7Y1NgxTnuC\nNZoDipQ9wUIIF2BIKYuEEG5gAXAbcAywQ0p5lxDiOiBbSnldzHsb1l/m86l6v48+qiwQ6URRETRv\nDnv36mLfGs0BgPYE/7Hx+1U2+M03YezYmIV5eaq05siRjRKbRqOpSn3VCW4GfCyEWA58AbwhpVwA\nzARGCSFWAyOD043Djh0wdSqMGKHq8Q4Z0mihVEtGhhLBa9Y0diQajeYgIpFylUKI1kKIRUKI74UQ\n3wkhLm2MWA8kQoneuAlfi/YEazQHEimLYCnlL1LKPsG/nlLKO4Pzd0opj5FSdpZSjpZS7q67cJPk\nzjuhvFxVfHjyyYQuTrGPghuEfv3g669rXa1RYksQHVtq6NhSI51jSyMSKVfpBa6QUvYAjkD17ejW\ngDHuNw19LgihLBGJ2CHS+TzVsaWGji010jW2/a0Okb5s2QJz5qjOcC1bNnY0NXPYYbB0aZwxODUa\njSZljgeOCr5+CigkRghLKTcDm4Ovi4UQPwIFwI8NF+aBh2nqEeM0moOBjZ/ZUQAAGfpJREFU/aoT\nnPJG69pf9tZb6u+ee8DhUPMuuUTdss+eXXfbqS8WLFBZ60WLGjsSjUZTCweKJ1gIsUtK2ST4WgA7\nQ9PVrN8O+BDoIaUsjlmmPcERZGWpcvJVrL+tW6vymoMHN0pcGo2mKvVZJ7jxefZZuPpqVebs8svh\nP/+B996D116Db75p7OgSo29fVas4ENBZBI1GkzBCiIVA8ziLboyckFJKIUS1KlYI4QFeBC6LFcAh\nGryiTxpPS1nIihUwcmTMcsMAIRo9Pj2tp//I00lV9JFSNvif2mwd8MADUrZpI+WKFVJu2yZldraU\nf/+7lJ06STl/fkpNLlq0qG5iS5bWraU85hgp16ypdpVGiy0BdGypoWNLjcaMLXj9apRrZzJ/wEqg\nefB1C2BlNetZgXeBy2toqy4PYZ3SGOdCfr6UH34YZ0H79lJ+9lnlpP4OpYaOLTV0bPGp6ZqdfmlH\nrxfGjYMlS6pfJxCA//4XbrsNCguhVy9Vmuaee2DzZpg4EcaPb7CQ64R+/VQGe8GCxo5Eo9EcHLwO\nTA2+ngq8GrtC0CbxOPCDlHJWA8Z2QFNtxziL9gRrNAcSje8JlhJWroRHHlE+3q++ghtuUAO0T5qk\nbAKvvgqtWoUbOOEE+O03VfGhT58Gj79eWLhQeckqKuDppxs7Go1GUw0HkCc4B5gHtAF+BU6TUu4W\nQhQAj0opxwkhhgAfASsIj+55vZTynZi2ZGP8VqQr7dopJ96RR8Ys6NxZJWgGDGiMsDQaTRzS0xMs\nJZx2murQ5nLBSSfB8OGqbu7s2aqm75FHwtat8PDD8MUXUFICnTrBihWwahXYbI0Wfp0zapQS+uPG\nNXYkGo3mIEBKuRM1eFHs/I3AuODrT9i/evF/SBItkabRaNKbxvu2fvqpErMrV8Ly5SoTfPLJqnft\n2LGq++2yZfD666pyQseO8Je/KNvAvHn1JoBDButGoUsXlQleuTLu4kaNrRZ0bKmhY0uNdI5N07A0\nxrmQaIm0dD5PdWypoWNLjXSNrfEywf/8J1x4oRK9IWbNUkMchwa1sNlg0CCVAbbb1bzRoxs+1obC\nYlEZ8ZdfVpYQjUaj0aQdplnNKPcWPWKcRnMg0Xie4C5dVAY4VNdXo1i4EG69VWXKNRpN2nGgeILr\nEu0Jjubcc9VlOjKHA8D778PQoQeXVU+jOcCp6ZrdeCK4tFQL4HgUF0OzZrBjhz4+Gk0aokWwRqPR\nHDjUdM1uPE9wmgq8RveteDzQvbsq8bZ5c9SiRo+tBg742ObMge+/h4cegs8/r/eYQhzwx62RSOfY\nNA1LOp8LOrbU0LGlho4teXQ31nTk5JPVY7XFixs7koOXyKzW22/Dn/+s6k3/859q1EGNRqPRaDQH\nNY1fJ1gTn2uuAacTzjsvjvGsgdmwQQ1GEuqcGEt5OZSVqYoe1bFxoxKbRx4Jv/6q9qlnz3oJNyFu\nuAF+/x0GD1ZVR156ScW0ezecfjr8+GPjxPXBB/D3v8P06ZCdDUcdpZ4OaNIGbYfQaDSaA4f09ATr\nC2rNzJmjBg/p06fuMsIffABHHKHqMifKU0+pKh6HHw5HH63KuA0YAB06qOU+H5x4ohrc5LPPwj2j\nt26FRYvg669VR5H771fzV6yA3FxV8u6NN+pmv6pjzx5luxEiuqPK7t0q/n79VPz/+Q907Rren+xs\nuO8+2LsXiopURZLIqvg+n+oanmwvcL8fdu6Epk3V9Pr1qiRe6Fju2QOHHqqOV1mZ8oWvXw933aVu\nNLZsgQkTwusfDHz/vboZOfbYhu1VX1KibC8dO6qRD5JAi2CNRqM5cEhPT3Cakja+la5dobRUjaC3\naxfccguFixal3l5hoRJzY8eqNkP4fGqEo5dfhjPOgLVr1byiImUZ+Oc/1Yh9xxyjRNkzz8BhhynL\nRkkJjBkDUlK4a5cawQ/Uej17qmGsnU4lgMvKVCZ461b4+GP15/XuzxEKU1GhKo0ArF6tsue33qoG\nH+nQgcIWLdS8V19VAvixx+C449Qw1YWFYQEMqvbRySfDu++qDHhFBRx/vMoMb98Of/0rFBQoz7bP\nl1h8Tz8N116rYigogOuvV6K6f38K+/eHs85Sx/7EE9VgKU8/DS+8oG5aLrpIHddXX1XTZ58dbeWo\nR+r0u/DzzzBiBFx+OZx/vhoYZ8gQNe+qq8LZ923b6i42KeGjj+Cnn9R0UZHK+v/732r7N9ygbu4u\nvRReeUU9DdBC74Ajba7ZcdCxpYaOLTV0bMnTeHWCNTXTrRu0bKlE00MPwe23p54B/OYbmDhRDTyy\nZo0SwgsWqOzitGkq47l5M5x6KlxwgRKCZWWqRrPVqsTzmDHh9vbuVSJiwACVTX3qKfV3zTUqQ/r2\n22o0wAceiB9PXp7KwH34oRLXIbZvV5nb0OP/ffuUOD/99KpFORcvVvt1zDFq/NK//lUJ9vnzISdH\nWTe++kqJ4s8/V23/+99w9dVKDL31VvXHa+7c6OncXCU+N2xQx+7991V2/H//UwK2OrxeJcYff1y9\nf8cOFfeQIXDxxfDgg7BkicoQl5crAXbRRdFtXH65+gO13vDhcOaZ6maktpGpPvoINm1SGeZ589Rx\nPPfcmt8Tj/LyGobISoDvvlNi98or4YcfoHdvNZ2RAcOGqbavukp9lvv2Qd++8Oij6hyJJBBQTxsi\n54eE7rx58M476jy5+WZ47TX12Xu96rMfNUqVHTzqKJV5HjtWfTZ79qhz/l//UsfqjTfUd2XQIGXh\n6dEjtX3WaDQaTfojpWzwP7VZTa0EAlLeequU+flSgpQzZ6bWxrBhUv7nP+F5994r5UknSdmunZTP\nPitlWZmUCxeq/4cdJuWVV0rZsqWUnTpJuX17/Hbff1/KG2+UsqIiPO+TT6QcM0bKSZOk3Lu35rie\nf15tf8MGKbdskfKOO6S021VcL78s5QknqH3OzpbykUfC+7Jzp5Rffy1lXp6Up5yi2sjLU/H36yfl\n5MnRMcXyyitSvvlmYscuhM+njtm8eeF5L70k5RFHqG0FAlKuXi3lM89IuWaNWr51q5TXXCPl4MFS\nrlwZ3d7XX0tZXp5cDCFKS6UcMkTKs86S8rffqi7/9FMpX31VyosvlrJNG3U8zzlHxZKfL+UTT0i5\neLGUv/xS83ZWrZLyvvtU/NnZUnbuLOUZZ0i5aZPa399/l/Kii6R87TW1vt+vtnv//VK+8476/P/0\nJ3XOHnOMlP/6V2L7t3evlLNmSdmihZQPPSTlf/+rzrXjjpOyY0cp27aVsm9fKfftU/syaJCUXbpI\nef31Uv74o/qMjjlGtfH++yqun36Scu5cKT/7rOZtb9wo5W23qfPIapWyaVMpx42Tcvx49R0qLpZS\nShm8fjXKtbOx/vQ1W6PRHKjUdM3WnuB0Z+lSZT848UT1WPe009Tj+kQyVG+9pWwHb76phqAOZVO3\nbFFZ5unTVcYtHgsXKjtBt251ty+xTJmistNCKJ/su++q7HJWlrIH3Hyz8oyOHq0yu1KqbGpurhpK\ne+pU1Y6UDT9Kk9erRvfbuFFlzjduhDZtVJZ77lzlvW7dWmXxDzmkbre9eTPceKPKTM+dqzLfFRXq\nsz7rLFVir1cv9fSgSZPw++bPh9mzVSZ02zZlsxg5UtkQPv0U8vOVjWX2bGW/GDsWJk9W59qGDSpL\nOmeOanPrVpXF/fhjOOcc9YTht9/Ufr//vrKdDBmiPr9OnZS9I5kBBN58U9kTyspUfH37qiz2EUeo\nc+Pll9Xrq69WTymq67SZCqHzzONRGXebTe3Trl3w0ksIm017gjUajeYAocZ+HNWp4/r8I42zCosW\nLWrsEKIJBFT269NP5aJ27aQ0TSmnTau63kMPqezgtGlSfvWVylC2aiVlbq6U771Xdf2XXqo9W5sE\nKR23339XGcr33lNZPClVFnHt2uj1NmyQ8rvv1N+aNVJ+8UX9x5YIgYCUDzwg5aOPqoxjSYnKHvbv\nL+Ull9RvbOXlKlOfkSHloYdKmZOjsveFhYm9f/FilW11OqXs3VvK889X7bRpo54+bN0aP7YlS1RG\nNRBQ05s2SXnssVKOGBHOwFdUSPnii+p4+Hyp7V8tLHrrrXppt1rKy9XTiR49dCY4zUi7a3YEOrbU\n0LGlho4tPjVds7UnON0RQmWDmzRRHap69FCZ4KeeCmdCf/pJZU3/8Q+VgRs7VmXOOndWGd14Xs6T\nT27Y/YhH69aq8kEkkd7jEAUF6i/dEEJ5e0O4XOp4v/GG8kfXJzZbeNjxa69VVSVq8ifHMmiQyu7u\n2wdud+LvO+yw6OnmzdUTBynD55nVCqecknibqeB01m/7sdhsqvPcokWqSopGo9FoDni0HeJA5Ntv\nVSein3+G//s/ePFFJYRuvVUtX71aVT2YNKnm2r0ajSZpdIk0jUajOXDQdYIPRo75//buPkiuqk7j\n+PdhAgGVt7iRRF4Mu0MQyqAW8qYICRALtxZdynUpXCUiEuRFU2W5SJW7tXFhBZUqUSmhSncl7OLL\norCiLEJQASk1KjARBCqya9xFSQwQrCgvksnPP86Z2E7NJDM33fee2/18qm6l+/bt6Sc9t39z7r2n\nzzkp9fHdtCn1nZwzp/5+sWYDyI1gM7P28DjB01DqWHYwLtvVV6cG8PLlMHdu4w3g1rxvhXG2akrO\nZvUqeV9wtmqcrRpnmz73CW6r4eHuzSRnZmZmNmDcHcLMbBrcHcLMrD3cHcLMzMzMrIMbweOU2m8F\nnK0qZ6vG2awNSt4XnK0aZ6vG2abPjWAzMzMzGzjuE2xmNg3uE2xm1h7uE2xmZmZm1sGN4HFK7bcC\nzlaVs1XjbNYGJe8LzlaNs1XjbNPXk0awpJMlPSzpZ5I+2IvX6JWRkZGmI0zK2apxtmqcrd0kzZK0\nUtIaSbdJ2muCbXaVtErSiKQHJV3aRNYdUfK+4GzVOFs1zjZ9XW8ESxoCrgROBg4FTpd0SLdfp1ee\neuqppiNMytmqcbZqnK31LgJWRsR84Fv5/p+IiGeBRRHxKuAwYJGkY+uNuWNK3hecrRpnq8bZpq8X\nZ4KPBB6JiLUR8TzwJeDNPXgdMzOb3JuAFfn2CuCvJ9ooIp7ON3cBhoAnex/NzKx5vWgE7wv8f8f9\nR/O6Vli7dm3TESblbNU4WzXO1nr7RMT6fHs9sM9EG0naSdJI3uY7EfFgXQG7oeR9wdmqcbZqnG36\nuj5EmqS3ACdHxNn5/tuBoyLivR3beKwdM2utUoZIk7QSmDPBQx8CVkTE3h3bPhkRs7bxs/YEbgUu\niog7xj3mmm1mrTVZzZ7Rg9f6JbB/x/39SWeDtxvGzMymLiIWT/aYpPWS5kTEOklzgV9v52f9RtLN\nwGuAO8Y95pptZn2nF90hfgwcJGmepF2A04CbevA6ZmY2uZuAJfn2EuC/xm8g6c/GRo2QtBuwGLiv\ntoRmZg3qeiM4IjYDF5Auqz0IfDkiHur265iZ2TZdBiyWtAY4Id9H0kvzGV+AlwLfzn2CVwFfj4hv\nNZLWzKxmjUybbGZmZmbWpIGcMW6iQeNLkbuQFKnkbLB1jOriSOpF3/uukLRr0xkmI2l2/rfY98/q\n4ZpdTcnZoNyaDWXXnVLrdhtr9kA1giUdJelrwGclnVXSjiTpGEnXAcslzS+pOORs1wOXSzq0sGyv\nlXQxQESMNp2nU97f/gO4VNICScV8uUjSEZJuAK6QdGIpv1MlL5T0JeBrkLpYlfTeWX1cs6txza7O\ndXvamVpdswemESzpcOAq4Ct5WQQMNxoqk7QA+BTwDdI3uM8Gzmg0VCbpJaQZAP8beAJYBryr0VCZ\npCWkSQA+JOm0vK7xI9BcFJYDnwNuIY3Ccj7w6iZzwdZslwFXkwrW/wHvBGY3mWtMJL/Ld18s6bx8\ne2BqlSWu2dW4Zlfjul1N22t2K0J2ydHA/0TEvwO3AbuRdqQSvA54OCK+SPoAPgO8XdKBzcYCYAGw\nJiI+D1wO3AC8WdL8ZmMBaVKWE0hTdF8OZRyBRupo/wtgSURcB1wCvIw0G1ejcrY7gcURsQK4hjRT\n2G+azAVbC72UhvNaD7wbOFfS3hExWsJZD6uVa3Y1rtkVuG5PXz/U7L5tBEt6m6R/ljQ2ZfNXgRPz\nZZifkmax+6SkiwrItgo4QNJwRPwWGCXt3Gc3kG2hpKM7Vq0GXiPpL/LR3o+Be4D3FJDtDmBdRNwG\n/GLsEhu9Gf96utm+CKyWNDMingA2AXPrzjVRtoi4JSKelPR64PvAgcBVkk5vOJvyWYXHgHnAz0m/\n44vyZ6O4S6fWPa7ZlbO5Zlfkur3DuVpfs/uuEZyPSs4F/h5YC3xc0tKIWAccCuwKXBgRR5OOpl4n\n6ZgGs70TeAz4LnCNUv+3I4DrgSGlsTvryLa7Ul+jG4FzJM0CiIjHgf8E3pc33QjcDrwgH/01li0b\n+5C9B1gmaZ+IeL6OXNvJ9vuIGI2I5yTtTJo05uG6cm0rm6Sxz/1G4MyIOJJ0luHEus4WTZQtIrbk\nx+YD/xsRjwIrgXOB6yXNzO+l9RHX7MrZXLO7n891exq5+qFm910jOF82OBr4aET8G3AesFDSX+ai\nehLweN78XlJ/rt83lO180uD0r4qIfwDOAa6JiL8CfgYcFhHP1JGN9B58B/g74FfAWyH9ESAV95dL\nOinv9E+QzsrUdSlmwmwRsSUitkgaiogHcs6xsVDf2HS2jm0OAdZHxBpJe0g6sslsQOSMD0TEt/O6\n7wKzSGc+msxGvj8s6SbSJdM7gbUR8Vzdfyyt91yzK3PN7kG+jm1ct6eWC1pcs/uiESzpDEnHdxzN\nPQTsK2lGRNwO/IRUVOcAnwUuzEdVpwGvIBWIJrKtBO4HFknaPyJ+GhE35u1OAFZ1HP31KttCpf47\nz5Hem9uBNcDhkl6e/wjcT7pMdIWk4ZxNpD5JTWWbn7fb+v5ExFnAEkkbgVfmPwZNZhs7An4x8LSk\nM4Hvkfrs9cRUskVETPDenAhsAX5Hj0wh28F5092BdaRLa4dHxCmkS8+H9yqb1cs1e4eyuWb3Np/r\n9tRztb5mt3ayjLwzzAW+QNoJHgFeRDoNfwqpUF4TEQ8rfVnhE8DyiBiRdC2wF+mLFssi4sEGs80D\nrgA+HBH35aPNj+bnnRMRj9SUbVlEbMjbzCdNs/psRFzc8dwLgYPzsrTG922b2fLzDiC9j7OA8/NZ\nhsaz5fWXAh8kfSv6iohYXUI2peGmjgU+BjxKuuTc1Ut/O5Btr4h4quPn/Ml9ax/X7K5nc83uUb68\nfiDr9sDV7Iho3QLMyP8eDFw3tg74DHAt6Wj3X0lD1uyZH18BXJJv7wzMLizbxfn2bGBhzdmuBG4Y\nt+2pOfMw6QMwlNfPLCzbbqRv7+4JHFVYthfmda8FTiss2675c7AAOKWgbAfl3+lM0pmrnXqRzUu9\ni2t217O5Zvcu38DW7UGs2UWMzzdVSsNtXALsJOkW0in4zbB1mJX3kr6wcCjpUtCpwH7AR0id8X+Q\nt30e2FBYtlV52w2kb1fWmW0Z8CtJx0fEnXn9jZIOAW4lFdSFwEORLomUlm1RpDMcq0rLJmlRRHyv\nm7m6lY30vt1PunRaSrZv5mwLI+Ihcj84ayfX7J5lc83uYb5Bq9sDXbObboVPdQGOB0ZIg6efTeoQ\nfjJp3MgjO7Y7H7g13z4MuJn0QbsReJGzTZrtXOCOjvt/S+pn9DngJc7mbG3P5qXepU/qYsnZ/Plu\nUb5Ss5Waq66l8QDT+EUdB7yj4/5V+ZdyJnBPXjcEzCHNLnRgXrc3sK+zTSnb9R3ZjgOOczZn65ds\nXupd+qgulpzNn++W5Cs1W6m56lraNDrEj0jjzo3NQHI3cECkWXGGJL0v0sDM+wHPR8TPASJiY0T8\n0tmmlG1zR7a7IuIuZ3O2Pspm9eqXulhyNn++25Ov1Gyl5qpFaxrBEfFMRDwbf5yBZDF/HDvyXcAh\nkm4m9d2619mczdmczZpT8r7gbP2XrfR8pWYrNVdtdvRUct0L6ZuKQ8AtwHBeN0y6THUssJ+zOZuz\nOZuXMpaS9wVn679specrNVupuXq9tOZM8JiI2EwaIuRx4LB8hPKPwGhE3B1p2j5nczZnczYrQMn7\ngrP1X7bS85WardRcPdd0K7zKAhxDGsT5buCspvM4m7M5W7uyefG+4Gz9m630fKVmKzVXL5dWzhgn\naT/SwOWXR0Qtc8hPlbNV42zVOJu1Qcn7grNVU3I2KDtfqdlKzdVLrWwEm5mZmZntiNb1CTYzMzMz\n21FuBJuZmZnZwHEj2MzMzMwGjhvBZmZmZjZw3Ag2MzMzs4HjRrCZmZmZDRw3gq0VJI1Kuk/SA5JG\nJL1fkrbznJdJOr2ujGZmlrhmWxu4EWxt8XREvDoiXgEsBt4I/NN2nnMg8LaeJzMzs/Fcs614bgRb\n60TEBmApcAGApHmS7pJ0T16OyZteBrw+n41YJmknSR+X9ENJqyUtber/YGY2KFyzrVSeMc5aQdKm\niNh93LqNwHzgt8CWiHhO0kHAFyLiCEnHAx+IiFPy9kuB2RHxL5JmkuZHf2tErK31P2Nm1udcs60N\nZjQdwKwLdgGulPRKYBQ4KK8f3//sDcACSX+T7+8BDANr6whpZmaAa7YVwo1gayVJfw6MRsQGScuB\nxyLiHZKGgGe38dQLImJlLSHNzAxwzbYyuU+wtY6k2cDVwKfzqj2Adfn2GcBQvr0J6LwcdytwnqQZ\n+efMl/SC3ic2MxtcrtlWKp8JtrbYTdJ9wM7AZuBa4BP5sc8AX5V0BvBNUn8zgNXAqKQR4PPAp4B5\nwL15qJ5fA6fW9j8wMxscrtlWPH8xzszMzMwGjrtDmJmZmdnAcSPYzMzMzAaOG8FmZmZmNnDcCDYz\nMzOzgeNGsJmZmZkNHDeCzczMzGzguBFsZmZmZgPnD7HTH/7w3WUcAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9f96908>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "Correlation and covariance matrix of weekly returns:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cm = r.corr()\n",
      "cv = r.cov()\n",
      "    \n",
      "fmt.displayDFs(cm, cv*1e4, headers=['Correlation', 'Covariance'], fmt=\"4f\")"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "html": [
        "<center><table style=\"border-style:hidden; border-collapse:collapse; text-align:center;\"><tr style=\"border:none\"><th style=\"border:none; text-align:center\">Correlation</th><th style=\"border:none; text-align:center\">Covariance</th></tr><tr style=\"border:none\"><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td>1.0000</td>\n",
        "      <td>0.0321</td>\n",
        "      <td>0.4191</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>0.0321</td>\n",
        "      <td>1.0000</td>\n",
        "      <td>0.3011</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td>0.4191</td>\n",
        "      <td>0.3011</td>\n",
        "      <td>1.0000</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td>8.0432</td>\n",
        "      <td>0.2461</td>\n",
        "      <td> 5.6109</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>0.2461</td>\n",
        "      <td>7.2876</td>\n",
        "      <td> 3.8375</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td>5.6109</td>\n",
        "      <td>3.8375</td>\n",
        "      <td>22.2823</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td></tr></table></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0x39ec2e8>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "* Why don't we compute the correlation and covariance of the price levels?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Cholesky decomposition\n",
      "\n",
      "If $A$ is symmetric semi positive definate (SPD) matrix \n",
      "\n",
      "* $A$ can be decomposed as $A = LL^T$, where $L$ is lower triangle\n",
      "* In another word, the $U = L^T$ in $A$'s LU decomposition\n",
      "* $L$ can be viewed as the \"square root\" of $A$\n",
      "\n",
      "Cholesky decomposition is unique:\n",
      "\n",
      "* The rank is same between $L$ and $A$\n",
      "\n",
      "The Cholesky decomposition of previous correlation matrix:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lcm = np.linalg.cholesky(cm)\n",
      "lcv = np.linalg.cholesky(cv)\n",
      "fmt.displayMath(\"\\\\small \", sp.Matrix(lcm).evalf(4), fmt.joinMath('=', sp.Matrix(lcm.T).evalf(4), sp.Matrix(lcm.dot(lcm.T)).evalf(4)), \n",
      "                sep=\"\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ \\small \\left(\\begin{matrix}1.0 & 0 & 0\\\\0.03215 & 0.9995 & 0\\\\0.4191 & 0.2878 & 0.8611\\end{matrix}\\right)\\left(\\begin{matrix}1.0 & 0.03215 & 0.4191\\\\0 & 0.9995 & 0.2878\\\\0 & 0 & 0.8611\\end{matrix}\\right)=\\left(\\begin{matrix}1.0 & 0.03215 & 0.4191\\\\0.03215 & 1.0 & 0.3011\\\\0.4191 & 0.3011 & 1.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa6842e8>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Recursive algorithm\n",
      "\n",
      "A SPD matrix $A$ and its Choleski decomposition $L$ can be partitioned as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s_A, a11, A12, A21, A22 = sp.symbols(\"A, a_{11} A_{21}^T A_{21} A_{22}\")\n",
      "s_L, s_LT, l11, L12, L12T, L22, L22T = sp.symbols(\"L L^T l_{11} L_{21} L_{21}^T L_{22} L_{22}^T\")\n",
      "\n",
      "A = sp.Matrix([[a11, A12], [A21, A22]])\n",
      "L = sp.Matrix([[l11, 0], [L12, L22]])\n",
      "LT = sp.Matrix([[l11, L12T], [0, L22T]])\n",
      "\n",
      "fmt.displayMath(fmt.joinMath('=', s_A, A), fmt.joinMath('=', s_L, L), fmt.joinMath('=', s_LT, LT))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ A=\\left(\\begin{matrix}a_{{11}} & A^{T}_{{21}}\\\\A_{{21}} & A_{{22}}\\end{matrix}\\right)\\;,\\;\\;\\;L=\\left(\\begin{matrix}l_{{11}} & 0\\\\L_{{21}} & L_{{22}}\\end{matrix}\\right)\\;,\\;\\;\\;L^{T}=\\left(\\begin{matrix}l_{{11}} & L^{T}_{{21}}\\\\0 & L^{T}_{{22}}\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa2ed860>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "From $A = LL^T$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayMath(fmt.joinMath('=', A,  L*LT))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ \\left(\\begin{matrix}a_{{11}} & A^{T}_{{21}}\\\\A_{{21}} & A_{{22}}\\end{matrix}\\right)=\\left(\\begin{matrix}l_{{11}}^{2} & L^{T}_{{21}} l_{{11}}\\\\L_{{21}} l_{{11}} & L_{{21}} L^{T}_{{21}} + L_{{22}} L^{T}_{{22}}\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x39ec2e8>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We immediately have:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayMath(fmt.joinMath('=', l11, sp.sqrt(a11)), fmt.joinMath('=', L12,  A21/l11), \n",
      "                fmt.joinMath('=', A22 - L12*L12T, L22*L22T))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ l_{{11}}=\\sqrt{a_{{11}}}\\;,\\;\\;\\;L_{{21}}=\\frac{A_{{21}}}{l_{{11}}}\\;,\\;\\;\\;A_{{22}} - L_{{21}} L^{T}_{{21}}=L_{{22}} L^{T}_{{22}}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x9fa1b38>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that $L_{22}$ is the Cholesky decomposition of the smaller matrix of $A_{22} - \\frac{1}{a_{11}}A_{21}A_{21}^T$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Correlated Brownian motion\n",
      "\n",
      "Ubiquitous in quant Finance\n",
      "\n",
      "* The common underlying process for asset prices and other risk factors\n",
      "* The vast matjority of quant finance models are based on Brownian motions.\n",
      "\n",
      "Steps in drawing correlated n-dimensional Geometric Brownian motion paths:\n",
      "\n",
      "$$\n",
      "\\frac{dx^k_t}{x^k_t} = u^k dt + \\sigma^k dw^k_t, \n",
      "\\;\\;\\;\\;dw_t^j\\cdot dw_t^k = \\rho_{jk} dt\n",
      "$$\n",
      "\n",
      "1. compute the Cholesky decomposition of the correlation matrix: $C = LL^T$\n",
      "2. draw a n independent normal random number vector $\\bs \\epsilon = (\\epsilon_1, \\epsilon_2, ..., \\epsilon_n)^T$, these are the normalized risk factors\n",
      "3. $L\\bs \\epsilon$ is the correlated normal random vector, and the correlated Brownian increments is $dw_t = L\\bs \\epsilon\\sqrt{dt}$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Simulated paths"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = np.random.normal(size=[3, 20000])\n",
      "dw = (lcm.dot(e)).T\n",
      "\n",
      "dws = dw*np.diag(cv)\n",
      "wcm = np.exp(np.cumsum(dws, 0))\n",
      "\n",
      "figure(figsize=[12, 4])\n",
      "subplot(1, 2, 1)\n",
      "\n",
      "dw = (lcm.dot(e)).T\n",
      "dws = dw*np.diag(cv)\n",
      "wcm = np.exp(np.cumsum(dws, 0))\n",
      "plot(wcm)\n",
      "legend(cm.index, loc='best');\n",
      "title('$L \\epsilon$')\n",
      "\n",
      "subplot(1, 2, 2)\n",
      "dw2 = (lcm.T.dot(e)).T\n",
      "dws2 = dw2*np.diag(cv)\n",
      "wcm = np.exp(np.cumsum(dws2, 0))\n",
      "plot(wcm)\n",
      "legend(cm.index, loc='best');\n",
      "title(('$L^T \\epsilon$'));"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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oNUREmFdWp07OvbHK3uwLF+TlW7fU+e7ckcvWK+f+fefrpjyO2E+s371r3jUw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\nS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GgmCIIgCIIgCAeQ0UwQBEEQBEEQDvg/s19JzzxXPssAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9f9a0f0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* The identical results can be obtained by using cholesky decomposition of the covariance matrix.\n",
      "* $L \\bs \\epsilon$ and $L^T \\bs \\epsilon$ are different, which one gives back the intended correlation?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "df1 = pd.DataFrame(np.corrcoef(dws.T), columns=cm.index, index=cm.index)\n",
      "df2 = pd.DataFrame(np.corrcoef(dws2.T), columns=cm.index, index=cm.index)\n",
      "\n",
      "fmt.displayDFs(df1, df2, headers=['$L \\epsilon$', '$L^T \\epsilon$'])"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "outputs": [
      {
       "html": [
        "<center><table style=\"border-style:hidden; border-collapse:collapse; text-align:center;\"><tr style=\"border:none\"><th style=\"border:none; text-align:center\">$L \\epsilon$</th><th style=\"border:none; text-align:center\">$L^T \\epsilon$</th></tr><tr style=\"border:none\"><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td>      1</td>\n",
        "      <td>0.03515</td>\n",
        "      <td>0.4232</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>0.03515</td>\n",
        "      <td>      1</td>\n",
        "      <td>0.2931</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td> 0.4232</td>\n",
        "      <td> 0.2931</td>\n",
        "      <td>     1</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td>     1</td>\n",
        "      <td>0.1354</td>\n",
        "      <td>0.3878</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>0.1354</td>\n",
        "      <td>     1</td>\n",
        "      <td>0.2696</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td>0.3878</td>\n",
        "      <td>0.2696</td>\n",
        "      <td>     1</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td></tr></table></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa2d3e10>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "The resulting covariance (correlation) is: \n",
      "$$\\mathbb{E}[L\\bs \\epsilon  (L\\bs \\epsilon)^T] = \\mathbb{E}[L\\bs{\\epsilon  \\epsilon}^T L^T] = LL^T = C$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Cautions on correlation/covariance matrices\n",
      "\n",
      "The covariance/correlation matrix are extremely useful, but they are difficult to deal with when their size grows large:\n",
      "\n",
      "* In practice, we often deal with thousands or tens of thousands risk factors\n",
      "* Correlation matrix is often easier to work with than the covariance matrix, because it is \"normalized\"\n",
      "* It is very difficult to keep a large correlation matrix semi positive definite (SPD)\n",
      " * Correlation matrix size of a few thousands is the practical limit\n",
      " * Small changes in isolated values can invalidate the whole correlation matrix\n",
      "* Cholesky decomposition often fails for large matrices due to non-SPD or numerical noise.\n",
      "* Adding new entries to a large correlation matrix can be extremely hard"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "# Matrix Calculus and Mean/Var Optimization"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Scalar function \n",
      "\n",
      "$$f(\\bs x) = f(x_1, ..., x_n)$$\n",
      "\n",
      "* Derivative to vector (Gradient): $\\frac{\\partial f}{\\partial \\bs x} = \\nabla f = [\\frac{\\partial f}{\\partial x_1}, ..., \\frac{\\partial f}{\\partial x_n}]$\n",
      "* Note that $\\frac{\\partial f}{\\partial \\bs x}$ is a **row vector**, whenever the vector or matrix appears in the denominator of differentiation, the result is transposed."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Vector function \n",
      "\n",
      "$$\\bs{ y(x) } = [y_1(\\bs x), ..., y_n(\\bs x)]^T$$\n",
      "\n",
      "* Deriative to vector (Jacobian matrix): $\\frac{\\partial \\bs y}{\\partial \\bs x} = \\left(\\frac{\\partial{y_i}}{\\partial x_j}\\right)$\n",
      " *  $\\frac{\\partial \\bs y}{\\partial \\bs x}$ is in the transpose of $\\bs y$, ie, with $\\bs y$ rows and $\\bs x$ columns\n",
      " * $\\frac{\\partial \\bs y}{\\partial \\bs x}\\frac{\\partial \\bs x}{\\partial \\bs y} = I$, even when $\\bs x$ and $\\bs y$ have different dimension. \n",
      "* Derivative to scalar: $\\frac{\\partial \\bs y}{\\partial z} = [\\frac{\\partial y_1}{\\partial z}, ..., \\frac{\\partial y_n}{\\partial z}]^T$\n",
      " * remains a column vector"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Vector differentiation cheatsheet\n",
      "\n",
      "* $A, a, b, \\bs c$ are constants (ie, not functions of $\\bs x$)\n",
      "* $\\bs {u=u(x), v=v(x)}, y=y(\\bs x)$ are functions of $\\bs x$\n",
      "\n",
      "| Expression | Results | Special Cases |\n",
      "| :------: | :-----: | :-----: |\n",
      "| $\\frac{\\partial(a \\bs u + b \\bs v)}{\\partial \\bs x}$ |  a $\\frac{\\partial{\\bs u}}{\\partial \\bs x} + b\\frac{\\partial{\\bs v}}{\\partial \\bs x}$ |$\\frac{\\partial{\\bs c}}{\\partial \\bs x} = 0I, \\frac{\\partial{\\bs x}}{\\partial \\bs x} = I$ |\n",
      "| $\\frac{\\partial{A \\bs u}}{\\partial \\bs x}$ |  $A\\frac{\\partial{\\bs u}}{\\partial \\bs x}$ | $\\frac{\\partial{\\bs A x}}{\\partial \\bs x} = A, \\frac{\\partial{\\bs x^T A}}{\\partial \\bs x} = A^T$ |\n",
      "| $\\frac{\\partial{y \\bs u}}{\\partial \\bs x}$ |  $y \\frac{\\partial{\\bs u}}{\\partial \\bs x} + \\bs u \\frac{\\partial{y}}{\\partial \\bs x}$ | |\n",
      "| $\\frac{\\partial \\bs{u}^T A \\bs v}{\\partial \\bs x} $ |  $\\bs u^T A \\frac{\\partial{\\bs v}}{\\partial \\bs x}  +  \\bs v^T A^T \\frac{\\partial{\\bs u}}{\\partial \\bs x} $ |$\\frac{\\partial \\bs{x}^T A \\bs x}{\\partial \\bs x} = (\\bs{A + A}^T)x, \n",
      "\\frac{\\partial \\bs{u^Tv}}{\\partial x} =\\bs{u}^T\\frac{\\partial \\bs{v}}{\\partial x} + \\bs{v}^T\\frac{\\partial \\bs{u}}{\\partial x}  $ |\n",
      "| $\\frac{\\partial{\\bs g(\\bs u})}{\\partial \\bs x}$ |  $\\frac{\\partial{\\bs g}}{\\partial \\bs u} \\frac{\\partial{\\bs u}}{\\partial \\bs x} $ | chain rules of multiple steps,  $\\frac{\\partial \\bs y}{\\partial \\bs x}\\frac{\\partial \\bs x}{\\partial \\bs y} = I$  |\n",
      "\n",
      "* Very similar expressions to  univariate calculus.\n",
      "* Replace $A^T$ by $A^*$ for complex matrix"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Portfolio optimization\n",
      "\n",
      "* Powerful mean/variance portfolio theory can be expressed succinctly using linear algebra and matrix calculus.\n",
      "\n",
      "Suppose there are $n$ risky assets on the market, with random return vector $\\tilde{\\bs r}$ whose covariance matrix is $V$,\n",
      "\n",
      "* $\\bs w$: a portfolio, its elements are dollar amounts invested in each asset\n",
      "  * $\\bs w^T \\tilde{\\bs r}$ is the portfolio's P&L in dollar amount\n",
      "  * $\\sigma^2 = \\bs w^TV\\bs w$: the variance of the portfolio P&L\n",
      "* $\\bs w_b$: a benchmark portfolio, e.g. a sector index\n",
      "  * $\\bs \\beta_b = \\frac{V\\bs w_b}{\\sigma^2_b} $: the beta vector of individual assets to the benchmark portfolio\n",
      "  * $\\bs w_b^T \\bs \\beta_b = \\frac{\\bs w_b^T V\\bs w_b}{\\sigma^2_b} = 1$, the benchmark portfolio itself has a beta of 1\n",
      "* $\\bs w_m$: the market portfolio, as defined in CAPM\n",
      "  * $\\bs \\beta_m = \\frac{V\\bs w_m}{\\sigma^2_m}$: the betas vector to the market portfolio $\\bs w_m$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Excess return forecast\n",
      "* $\\bs f = \\mathbb{E}[\\tilde{\\bs r}] - r_0$ is a vector of excess return forecast of all assets\n",
      "  * $r_0$ is the risk free rate\n",
      "* $\\bs f$ is a view, which can be from:\n",
      "  * Fundamental research: earning forcasts, revenue growth etc\n",
      "  * Technical and quantitative analysis\n",
      "  * Your secret trading signal"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Optimal portfolio and Sharp Ratio\n",
      "Given the view $\\bs f$, the optimal portfolio $\\bs w$ to express the view is:\n",
      " * minimize the variance of portfolio P&L (risk): $\\bs w^TV\\bs w$\n",
      " * while preserving a unit dollar of excess P&L: $\\bs w^T \\bs f = 1$\n",
      " \n",
      "which solves a portfolio with the largest Sharp ratio under the view $\\bs f$.\n",
      "* the optimal portfolio can then be scaled to match the dollar amount to be invested\n",
      "* the scaling does not change the portfolio's Sharp ratio and optimality "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Why we take the covariance matrix $V$ as a constant, but treat expected return $\\bs f$ as a variable?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Characteristic portfolio\n",
      "\n",
      "The optimal portfolio can be solved analytically using Lagrange multiplier and matrix calculus:\n",
      "\n",
      "$$\\begin{array}\n",
      "\\\\  & l &= \\bs w^TV \\bs w - 2 \\lambda (\\bs f^T \\bs w - 1) \\\\\n",
      "& \\frac{\\partial l}{\\partial \\bs w} &= 2  \\bs w^T V - 2 \\lambda \\bs f^T = \\bs 0^T \\\\\n",
      "\\end{array}$$\n",
      "$$ V \\bs w - \\lambda \\bs f = 0 \\iff \\bs w = \\lambda V^{-1} \\bs f $$\n",
      "\n",
      "plug it into $\\bs f^T \\bs w = 1$:\n",
      "\n",
      "$$ \\bs f^T(\\lambda V^{-1} \\bs f) = 1 \\iff \\lambda = \\frac{1}{\\bs f^T V^{-1} \\bs f}$$\n",
      "$$\\bs w(\\bs f) = \\frac{V^{-1}\\bs f}{\\bs f^T V^{-1} \\bs f}$$\n",
      "\n",
      "$\\bs w (\\bs f)$ is also known as the characteristic portfolio for the forecast $\\bs f$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Relativity of return forecast\n",
      "\n",
      "The optimal portfolio is invariant if $\\bs f$ is scaled by a scalar $a$:\n",
      "\n",
      "$$\\bs w(a\\bs f) = \\frac{1}{a} \\bs w(\\bs f)$$ \n",
      "\n",
      "* the portfolio $\\bs w$ and $\\frac{1}{a} \\bs w$ have no material difference \n",
      "* identical after the scaling to match the dollar amount to be invested\n",
      "\n",
      "Therefore, the return forecast $\\bs f$ is only meaningful in a relative sense:\n",
      "* it is only important to forecast the relative size of excess returns\n",
      "* e.g. excess return forecast of [1%, 2%] and [10%, 20%] of two assets would result in identical optimal portfolio\n",
      "\n",
      "The return forecast $\\bs f$ should only be interpreted in a relative sense, we use $\\propto$ to make it explicit."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Important characteristic portfolios\n",
      "\n",
      "Identical returns: $\\bs f \\propto \\bs e = [1, 1, ...., 1]^T$: \n",
      "\n",
      "* A naive view that all assets have the same excess returns (zero information)\n",
      "* $\\bs e^T\\bs w = 1$ means it is a fully invested portfolio of \\$1\n",
      "* $\\bs w_e = \\frac{V^{-1}\\bs e}{\\bs e^TV^{-1}\\bs e}$ has the minimum variance amongst all fully invested portfolio\n",
      "\n",
      "Beta to a benchmark portfolio: $\\bs f \\propto \\bs \\beta_b = \\frac{V\\bs w_b}{\\sigma_b^2}$:\n",
      "\n",
      "* $\\bs{\\beta_b w} = 1$ means the portfolio has a beta of 1 to the benchmark portfolio $\\bs w_b$\n",
      "* $\\bs w_{\\beta} = \\frac{V^{-1}\\bs \\beta_b}{\\bs \\beta_b^T V^{-1}\\bs \\beta_b} = \\frac{\\bs w_b}{\\bs{\\beta_b^T w_b}} = \\bs w_b$, which is the benchmark portfolio itself\n",
      "* the benchmark portfolio itself is the optimal portfolio amongst those with unit beta."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Implied view of a portfolio\n",
      "\n",
      "From any portfolio, we can backout its implied excess return forecast $\\bs f$:\n",
      "* It is nothing but the betas to the portfolio, following previous slides\n",
      "* Only meaningful in a relative sense as well\n",
      "* Normalization required to produce absolute excess return forecast\n",
      "\n",
      "Consider the market portfolio: the market collectively believe that $\\bs f \\propto \\bs \\beta_m$: \n",
      "* the absolute implied excess return by normalizing to the market excess return:\n",
      "\n",
      "$$\\mathbb{E}[\\tilde{\\bs r}] - r_0 = \\bs \\beta_m \\left(\\mathbb{E}[\\tilde r_m] - r_0\\right)$$\n",
      "\n",
      "* where $\\tilde r_m$ is the market portfolio's return, this is exactly the CAPM.\n",
      "\n",
      "The implied excess return is a much better return estimate than historical data.\n",
      "* How many years of historical data do we need to reliably estimate expected returns?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "~500 years of data"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Portfolio optimization example\n",
      "\n",
      "The covariance matrix estimated from historical weekly returns are:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayDF(cv*1e4, \"4g\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td> 8.043</td>\n",
        "      <td>0.2461</td>\n",
        "      <td>5.611</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>0.2461</td>\n",
        "      <td> 7.288</td>\n",
        "      <td>3.838</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td> 5.611</td>\n",
        "      <td> 3.838</td>\n",
        "      <td>22.28</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa0fd1d0>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "We consider three risky assets, with the following excess return forecast:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "er = np.array([.05, .02, .01]).T\n",
      "df_er = pd.DataFrame(np.array([er]).T*100, columns=[\"Excess Return Forcast (%)\"], index = f3.columns).T\n",
      "fmt.displayDF(df_er, \"2f\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Excess Return Forcast (%)</th>\n",
        "      <td>5.00</td>\n",
        "      <td>2.00</td>\n",
        "      <td>1.00</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa0ffb70>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The optimal portfolio for the given forecast is:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cvi = np.linalg.inv(cv)\n",
      "w = cvi.dot(er.T)/er.T.dot(cvi).dot(er)\n",
      "df_er.ix['Optimial Portfolio', :] = w\n",
      "\n",
      "fmt.displayDF(df_er[-1:], \"3g\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Optimial Portfolio</th>\n",
        "      <td>17.6</td>\n",
        "      <td>8.35</td>\n",
        "      <td>-4.83</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa685c88>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Implied views"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Suppose we are given the following portfolio:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "w = np.array([10, 5, 5])\n",
      "vb = w.dot(cv).dot(w)\n",
      "ir = cv.dot(w)/vb\n",
      "\n",
      "df = pd.DataFrame(np.array([w, ir])*100, index=[\"$ Position\", \"Implied Return %\"], \n",
      "                  columns = [\"SPY\", \"GLD\", \"OIL\"])\n",
      "fmt.displayDF(df[:1], \"4g\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>$ Position</th>\n",
        "      <td>1,000</td>\n",
        "      <td>500</td>\n",
        "      <td>500</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa8b59b0>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "We can compute its implied return forecast as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayDF(df[1:], \"2f\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Implied Return %</th>\n",
        "      <td>4.73</td>\n",
        "      <td>2.50</td>\n",
        "      <td>8.04</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa2d64a8>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "* note that these forecast are only meaningful in a relative sense"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Does the investor really have so much confidence in OIL?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "#Norm and Condition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Ill-conditioned covariance matrix\n",
      "\n",
      "The Mean-variance optimization is very powerful, but there are potential pitfalls in practice:\n",
      "\n",
      "* Suppose we have the following covariance matrix and excess return forecast, then we can compute the optimal portfolio."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "nt = 1000\n",
      "es = np.random.normal(size=[2, nt])\n",
      "\n",
      "rho = .999999\n",
      "e4 = rho/np.sqrt(2)*es[0, :] + rho/np.sqrt(2)*es[1,:] + np.sqrt(1-rho*rho)*np.random.normal(size=[1, nt])\n",
      "es = np.vstack([es, e4])\n",
      "\n",
      "cor = np.corrcoef(es)\n",
      "cor1 = np.copy(cor)\n",
      "cor1[0, 1] = cor1[1, 0] = cor[0, 1] + .00002\n",
      " \n",
      "sd = np.eye(3)\n",
      "np.fill_diagonal(sd, np.std(r))\n",
      "cov = sd.dot(cor).dot(sd.T)\n",
      "cov1 = sd.dot(cor1).dot(sd.T)\n",
      "\n",
      "e, v = np.linalg.eig(np.linalg.inv(cov))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "er = v[:, 2]/10\n",
      "\n",
      "df_cov = pd.DataFrame(cov*1e4, index=f3.columns, columns=f3.columns)\n",
      "df_cov1 = pd.DataFrame(cov1*1e4, index=f3.columns, columns=f3.columns)\n",
      "\n",
      "pf = pd.DataFrame(np.array([er]), columns=f3.columns, index=['Expected Return'])\n",
      "covi = np.linalg.inv(cov)\n",
      "pf.ix['Optimal Portfolio', :] = covi.dot(er.T)/er.T.dot(covi).dot(er)\n",
      "\n",
      "fmt.displayDFs(df_cov, pf, headers=[\"Covariance\", \"Optimized Portfolio\"], fmt=\"4f\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><table style=\"border-style:hidden; border-collapse:collapse; text-align:center;\"><tr style=\"border:none\"><th style=\"border:none; text-align:center\">Covariance</th><th style=\"border:none; text-align:center\">Optimized Portfolio</th></tr><tr style=\"border:none\"><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td> 8.0234</td>\n",
        "      <td>-0.1578</td>\n",
        "      <td> 9.7193</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>-0.1578</td>\n",
        "      <td> 7.2697</td>\n",
        "      <td> 8.5246</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td> 9.7193</td>\n",
        "      <td> 8.5246</td>\n",
        "      <td>22.2275</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Expected Return</th>\n",
        "      <td>0.0384</td>\n",
        "      <td>0.0325</td>\n",
        "      <td>0.0864</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Optimal Portfolio</th>\n",
        "      <td>3.8417</td>\n",
        "      <td>3.2496</td>\n",
        "      <td>8.6419</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td></tr></table></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0x390c5f8>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "* A few days later, there is a tiny change in covariance matrix, but ..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pf1 = pd.DataFrame(np.array([er]), columns=f3.columns, index=['Expected Return'])\n",
      "covi1 = np.linalg.inv(cov1)\n",
      "pf1.ix['Optimal Portfolio', :] = covi1.dot(er.T)/er.T.dot(covi1).dot(er)\n",
      "\n",
      "fmt.displayDFs(df_cov1, pf1, headers=[\"Covariance\", \"Optimized Portfolio\"], fmt=\"4f\")"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "html": [
        "<center><table style=\"border-style:hidden; border-collapse:collapse; text-align:center;\"><tr style=\"border:none\"><th style=\"border:none; text-align:center\">Covariance</th><th style=\"border:none; text-align:center\">Optimized Portfolio</th></tr><tr style=\"border:none\"><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>SPY</th>\n",
        "      <td> 8.0234</td>\n",
        "      <td>-0.1576</td>\n",
        "      <td> 9.7193</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>GLD</th>\n",
        "      <td>-0.1576</td>\n",
        "      <td> 7.2697</td>\n",
        "      <td> 8.5246</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>OIL</th>\n",
        "      <td> 9.7193</td>\n",
        "      <td> 8.5246</td>\n",
        "      <td>22.2275</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td><td style=\"border:none; padding:20px; text-align:center\"><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>SPY</th>\n",
        "      <th>GLD</th>\n",
        "      <th>OIL</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Expected Return</th>\n",
        "      <td>0.0384</td>\n",
        "      <td>0.0325</td>\n",
        "      <td> 0.0864</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Optimal Portfolio</th>\n",
        "      <td>0.5848</td>\n",
        "      <td>0.0864</td>\n",
        "      <td>11.2791</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></td></tr></table></center>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0xa2d7ef0>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Ill-conditioned linear system\n",
      "\n",
      "Consider the following linear system $A\\bs x = \\bs y$, and its solution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = np.array([[1, 2], [2, 3.999]])\n",
      "x = sp.MatrixSymbol('x', 2, 1)\n",
      "y = np.array([4, 7.999])\n",
      "fmt.displayMath(fmt.joinMath('=', sp.Matrix(a)*x, sp.Matrix(y)), \n",
      "                fmt.joinMath('=', x, sp.Matrix(np.round(np.linalg.solve(a, y), 4))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\7.999\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}2.0\\\\1.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x9f01ef0>"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A small perturbation on vector $\\bs y$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "z = np.copy(y)\n",
      "z[1] += .002\n",
      "\n",
      "fmt.displayMath(fmt.joinMath('=', sp.Matrix(a)*x, sp.Matrix(z)), \n",
      "                fmt.joinMath('=', x, sp.Matrix(np.round(np.linalg.solve(a, z), 4))))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\8.001\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}6.0\\\\-1.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa2e6d68>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "A small perturbtion on matrix $A$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b = np.copy(a)\n",
      "b[1, 1] += .003\n",
      "\n",
      "fmt.displayMath(fmt.joinMath('=', sp.Matrix(b)*x, sp.Matrix(y)), \n",
      "                fmt.joinMath('=', x, sp.Matrix(np.round(np.linalg.solve(b, y), 4))))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 4.002\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\7.999\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}5.0\\\\-0.5\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa2d3e80>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "* How do we identify ill-conditioned linear system in practice?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Vector norms\n",
      "is a measure of the magnitude of the vector:\n",
      "* Positive: $\\Vert \\bs u\\Vert  \\ge 0$, $\\Vert \\bs u\\Vert  = 0 \\iff \\bs{u = 0}$\n",
      "* Homogeneous: $\\Vert a \\bs u \\Vert  = |a| \\Vert \\bs u\\Vert $\n",
      "* Triangle inequality: $\\Vert \\bs u + \\bs v\\Vert  \\le \\Vert \\bs u\\Vert  + \\Vert \\bs v\\Vert $\n",
      "\n",
      "##Common vector norms\n",
      "* L1: $\\Vert \\bs u\\Vert _1 = \\sum_i | u_i |$\n",
      "* L2 (Euclidean): $\\Vert \\bs u\\Vert _2 = (\\sum u_i^2)^{\\frac{1}{2}} = (\\bs u^T \\bs u)^\\frac{1}{2}$\n",
      "* Lp: $\\Vert \\bs u\\Vert _p = (\\sum u_i^p)^{\\frac{1}{p}}$\n",
      "* L${\\infty}$: $\\Vert \\bs u\\Vert _\\infty = \\max(u_1, u_2, ..., u_n)$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Vector norms comparison\n",
      "\n",
      "<center> <img src=\"img/vectornorms.png\" width=350 height=350> </center>\n",
      "\n",
      "Vectors with unit norms:\n",
      "* Unit L2 norm forms a perfect circle (sphere in high dimension)\n",
      "* Unit L1 and L${\\infty}$ norms are square boxes\n",
      "* The difference between different norms are not significant"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Matrix norms\n",
      "Defined to be the largest amount the linear transformation can stretch a vector:\n",
      "\n",
      "$$\\Vert A\\Vert  = \\max_{\\bs u \\ne 0}\\frac{\\Vert A\\bs u\\Vert }{\\Vert \\bs u\\Vert }$$\n",
      "\n",
      "The matrix norm definition depends on the vector norms. Only L1 and L$\\infty$ matrix norm have analytical formula:\n",
      "* L1: $\\Vert A\\Vert _1 = \\max_{j} \\sum_i |a_{ij}|$\n",
      "* L2: $\\Vert A \\Vert_2 =$ the largest singular value of $A$ \n",
      "* L${\\infty}$: $\\Vert A\\Vert _\\infty = \\max_i \\sum_j |a_{ij}|$\n",
      "\n",
      "## Inequalities:\n",
      "* $\\Vert A \\bs u \\Vert  \\le \\Vert A\\Vert  \\Vert u\\Vert $\n",
      "* $\\Vert b A\\Vert  = |b| \\Vert A\\Vert $ \n",
      "* $\\Vert A + B\\Vert  \\le \\Vert A\\Vert  + \\Vert B\\Vert $\n",
      "* $\\Vert AB\\Vert  \\le \\Vert A\\Vert \\Vert B\\Vert$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Matrix condition\n",
      "\n",
      "The propogation of errors in a linear system $\\bs y = A\\bs x$ with invertable $A$:\n",
      "* Consider a perturbation to $\\bs{ x' = x} + d\\bs x$, and corresponding $d \\bs y = A d\\bs x$:\n",
      "\n",
      "$$\\begin{array}\\\\\n",
      "\\Vert d \\bs y\\Vert &= \\Vert A d \\bs x\\Vert \\le \\Vert A\\Vert\\Vert d \\bs x\\Vert \n",
      "= \\Vert A\\Vert \\Vert \\bs x\\Vert \\frac{\\Vert d \\bs x\\Vert }{\\Vert \\bs x\\Vert } \\\\\n",
      " &= \\Vert A\\Vert \\Vert A^{-1} \\bs y\\Vert \\frac{\\Vert d \\bs x\\Vert }{\\Vert \\bs x\\Vert }\n",
      "\\le \\Vert A\\Vert \\Vert A^{-1} \\Vert \\Vert \\bs y\\Vert \\frac{\\Vert d \\bs x\\Vert }{\\Vert \\bs x\\Vert }\n",
      "\\end{array}$$\n",
      "\n",
      "$$\\frac{\\Vert d \\bs y\\Vert }{\\Vert \\bs y\\Vert } \\le \\Vert A\\Vert\\Vert A^{-1}\\Vert\\frac{\\Vert d \\bs x\\Vert }{\\Vert \\bs x\\Vert}$$\n",
      "\n",
      "* $k(A) = \\Vert A\\Vert\\Vert A^{-1}\\Vert$ is the condition number for the linear system $\\bs y = A\\bs x$, which defines the maximum possible magnification of the relative error."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Matrix perturbation\n",
      "\n",
      "What if we change the matrix itself? i.e. given $AB = C$, how would $B$ change under a small change in $A$ while holding $C$ constant?\n",
      "\n",
      "* we can no longer directly compute it via matrix calculus\n",
      "* perturbation is a powerful technique to solve this types of problem"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "We can write any $\\delta A = \\dot{A} \\epsilon$ and the resulting $\\delta B = \\dot{B} \\epsilon$: \n",
      "* $\\dot{A}, \\dot{B}$ are matrices representing the direction of the perturbation\n",
      "* $\\epsilon$ is a first order small scalar\n",
      "\n",
      "$$\\begin{array}\n",
      "\\\\ (A + \\delta A) (B + \\delta B) &= (A + \\dot{A} \\epsilon ) (B + \\dot{B} \\epsilon) = C \\\\\n",
      "AB + (\\dot{A}B + A\\dot{B})\\epsilon + \\dot{A}\\dot{B}\\epsilon^2 &= C \\\\\n",
      "(\\dot{A}B + A\\dot{B})\\epsilon + \\dot{A}\\dot{B}\\epsilon^2 &= 0\n",
      "\\end{array}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "Now we collect the first order terms of $\\epsilon$, $\\dot{A}B + A\\dot{B} = 0$:\n",
      "\n",
      "$$\\begin{array}\n",
      "\\\\ \\dot{B} &= -A^{-1}\\dot{A}B \\\\\n",
      "\\delta B &= -A^{-1}\\delta A B \\\\\n",
      "\\Vert \\delta B \\Vert &= \\Vert A^{-1}\\delta A B \\Vert \\le \\Vert A^{-1} \\Vert \\Vert \\delta A \\Vert \\Vert B \\Vert \\\\\n",
      "\\frac{\\Vert \\delta B \\Vert}{\\Vert B \\Vert} &\\le \\Vert A^{-1} \\Vert \\Vert \\delta A \\Vert \n",
      "= \\Vert A^{-1} \\Vert \\Vert A \\Vert \\frac{\\Vert \\delta A \\Vert}{\\Vert A \\Vert}\n",
      "\\end{array}$$\n",
      "\n",
      "We reach the same conclusion of $k(A) = \\Vert A^{-1} \\Vert \\Vert A \\Vert$ for a small change in $A$ under the linear system $AB = C$.\n",
      "\n",
      "* we will cover the condition number for non-square matrix in the next class."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "subslide"
      }
     },
     "source": [
      "## Numerical example\n",
      "\n",
      "Consider the ill-conditioned matrices from previous examples:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V = sp.MatrixSymbol('V', 3, 3)\n",
      "Vi = sp.MatrixSymbol('V^{-1}', 3, 3)\n",
      "fmt.displayMath(fmt.joinMath('=', V, sp.Matrix(cov*1e4).evalf(4)), fmt.joinMath('=', Vi, sp.Matrix(cov*1e4).inv().evalf(5)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ V=\\left(\\begin{matrix}8.023 & -0.1578 & 9.719\\\\-0.1578 & 7.27 & 8.525\\\\9.719 & 8.525 & 22.23\\end{matrix}\\right)\\;,\\;\\;\\;V^{{-1}}=\\left(\\begin{matrix}32585.0 & 31647.0 & -26385.0\\\\31647.0 & 30737.0 & -25626.0\\\\-26385.0 & -25626.0 & 21365.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa2eb5c0>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = sp.MatrixSymbol('A', 2, 2)\n",
      "Ai = sp.MatrixSymbol('A^{-1}', 2, 2)\n",
      "fmt.displayMath(fmt.joinMath('=', A, sp.Matrix(a)), fmt.joinMath('=', Ai, sp.Matrix(a).inv().evalf(4)))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$ A=\\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right)\\;,\\;\\;\\;A^{{-1}}=\\left(\\begin{matrix}-3999.0 & 2000.0\\\\2000.0 & -1000.0\\end{matrix}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0xa2d3cc0>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "their condition numbers are large because of the large elements in the inversion:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fmt.displayDF(pd.DataFrame([[np.linalg.cond(x, n) for n in (1, 2, inf)] for x in [a, cov*1e4]],\n",
      "             columns = [\"L-1\", \"L-2\", \"L-$\\infty$\"], index=['Condition number $A$', 'Condition number $V$']), \"4g\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<center><font size=2><table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>L-1</th>\n",
        "      <th>L-2</th>\n",
        "      <th>L-$\\infty$</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Condition number $A$</th>\n",
        "      <td>3.599e+04</td>\n",
        "      <td>2.499e+04</td>\n",
        "      <td>3.599e+04</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Condition number $V$</th>\n",
        "      <td>3.667e+06</td>\n",
        "      <td> 2.52e+06</td>\n",
        "      <td>3.667e+06</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table></font></center>"
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      "## Orthogonal transformation\n",
      "\n",
      "Orthogonal transformation is **unconditionally stable**:\n",
      " \n",
      "$$\\Vert Q \\bs u\\Vert_2^2 = (Q \\bs u)^T(Q \\bs u) \n",
      "= \\bs u^T Q^TQ \\bs u = \\bs u^T \\bs u =  \\Vert \\bs u \\Vert_2^2$$\n",
      " \n",
      "* therefore by definition: $\\Vert Q \\Vert_2 = \\Vert Q^{-1} \\Vert_2 = 1$\n",
      "* $k(Q) = \\Vert Q \\Vert_2 \\Vert Q^{-1} \\Vert_2 = 1$\n",
      "* the relative error does not grow under orthogonal transformation.\n",
      "* Orthogonal transformation is extremely important in numerical linear algebra."
     ]
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     "source": [
      "# Assignments\n",
      "\n",
      "Required reading:\n",
      "\n",
      "* Bindel and Goodman: Chapter 4, 5.1-5.4\n",
      "\n",
      "Homework:\n",
      "\n",
      "* Bindel and Goodman: Problem 4.4, 4.7, 4.8\n",
      "* Complete [homework set 2](http://yadongli.github.io/nyumath2048)\n",
      "* Problem 2, 4, 5 in the homework set and 4.4, 4.7 from the book are optional."
     ]
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