{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 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\n",
    "* Norm and condition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Review of Linear Algebra\n",
    "\n",
    "Richard Feynman: In fact, mathematics is, to a large extent, invention of better notations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Vector\n",
    "* Is a set of elements, which can be real or complex:\n",
    "\n",
    "\n",
    "$\\renewcommand{bs}{\\boldsymbol}$\n",
    "\n",
    "$$\n",
    "\\begin{matrix}\n",
    "\\bs u = \\left(\n",
    "\\begin{matrix}\n",
    "u_1 \\\\\n",
    "u_2 \\\\\n",
    "\\vdots \\\\\n",
    "u_n\n",
    "\\end{matrix}\n",
    "\\right) \n",
    "\\hspace{2cm}\n",
    "\\bs v = \\left(\n",
    "\\begin{matrix}\n",
    "v_1 \\\\\n",
    "v_2 \\\\\n",
    "\\vdots \\\\\n",
    "v_n\n",
    "\\end{matrix}\n",
    "\\right) \n",
    "\\end{matrix}\n",
    "$$\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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Notation\n",
    "\n",
    "We use the following notation throughout the class:\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: $A, B, P$\n",
    "* random variables: $\\tilde a, \\tilde{\\bs u}, \\tilde{\\bs v}^T$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Vector addition\n",
    "* Associativity: $\\bs{u + (v + w) = (u + v) + w}$\n",
    "* Commutativity: $\\bs{u + v = v + u}$\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 $\\forall \\bs v$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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",
    "* $\\bs {u, v}$ orthogonal if $\\langle \\bs{u, v} \\rangle = 0$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Dot product\n",
    "\n",
    "A special case of inner 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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Matrix multiplication\n",
    "\n",
    "<center><img src=\"img/matrixprod.png\"></center>\n",
    "\n",
    "$$\\begin{array}\n",
    "\\\\ \\bs u = A \\bs v &\\iff u_i = \\sum_{j=1}^{n} a_{ij}v_j \\\\\n",
    "C = AB &\\iff c_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} = a_i \\cdot b_j\n",
    "\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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 between finite dimensional vector space can be represented by a matrix multiplication, therefore we can write $L \\bs u$ instead of $L(\\bs u)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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: $AB \\ne BA$\n",
    "\n",
    "$\\renewcommand{id}{I}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Matrix definitions\n",
    "\n",
    "* Identity matrix $\\id$: $\\id A = A \\id = A$\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: $AA^{-1} = A^{-1}A = \\id$\n",
    "* Orthogonal matrix: $A^T = A^{-1} \\iff AA^T = \\id$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# LU Factorization\n",
    "\n",
    "Factorize: to resolve into factors."
   ]
  },
  {
   "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",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "lecture = 2\n",
    "\n",
    "import fmt\n",
    "import sympy as sp\n",
    "from IPython.display import display, HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  \\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\-22\\\\-3\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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",
    "\n",
    "fmt.displayMath(a, fmt.joinMath('=', x, y), sep=\"\", pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Gaussian elimination\n",
    "\n",
    "Eliminate the $x_1$ terms using the first row, this operation is a linear transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  L_{1} \\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=L_{1} \\left(\\begin{matrix}8\\\\-22\\\\-3\\end{matrix}\\right)\\;,\\;\\;\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 2 & 1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\2\\\\5\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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=\"\", \n",
    "                pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Use the 2nd equation (row) to eliminate the $x_2$ terms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  L_{2} \\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 2 & 1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=L_{2} \\left(\\begin{matrix}8\\\\2\\\\5\\end{matrix}\\right)\\;,\\;\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=\\left(\\begin{matrix}8\\\\2\\\\1\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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), \n",
    "                sep=\"\", pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "the $L_1$ and $L_2$ are both lower triangular matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  L_{1}=\\left(\\begin{matrix}1 & 0 & 0\\\\3 & 1 & 0\\\\1 & 0 & 1\\end{matrix}\\right)\\;,\\;\\;\\;L_{2}=\\left(\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & -2 & 1\\end{matrix}\\right)\\;,\\;\\;\\;U=\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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), \n",
    "                pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The resulting matrix $U = L_2L_1A$ is upper triangular"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## LU factorization\n",
    "\n",
    "The triangular matrix is easy to invert by variable replacement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  U^{-1}=\\left(\\begin{matrix}\\frac{1}{2} & - \\frac{1}{2} & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\;\\;,\\;\\;\\left(\\begin{matrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{matrix}\\right)=U^{-1}\\;\\left(\\begin{matrix}8\\\\2\\\\1\\end{matrix}\\right)=\\left(\\begin{matrix}2\\\\3\\\\-1\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y3 = l2*y2\n",
    "a3 = l2*a2\n",
    "ui = a3.inv()\n",
    "fmt.displayMath(fmt.joinMath('=', Ui, ui), \"\\;,\\;\", fmt.joinMath('=', x, Ui), \n",
    "                fmt.joinMath('=', l2*y2, ui*y3), sep=\"\\;\", pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "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",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  L=\\left(\\begin{matrix}1 & 0 & 0\\\\-3 & 1 & 0\\\\-1 & 2 & 1\\end{matrix}\\right)\\;,\\;\\;\\;U=\\left(\\begin{matrix}2 & 1 & -1\\\\0 & 1 & 1\\\\0 & 0 & -1\\end{matrix}\\right)\\;,\\;\\;\\;L U=\\left(\\begin{matrix}2 & 1 & -1\\\\-6 & -2 & 4\\\\-2 & 1 & 2\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "l =  l1.inv()*l2.inv()\n",
    "fmt.displayMath(fmt.joinMath('=', L, l), fmt.joinMath('=', U, a3), fmt.joinMath('=', L*U, l*a3), \n",
    "                pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "* The LU factorization is the matrix representation of Gaussian elimination\n",
    "* LU factorization can be used to compute matrix inversion\n",
    "  * triangular matrix can be inverted by simple substitution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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",
    "* Permuting rows is a linear operation, thus it can be expressed as a Matrix $P$\n",
    "\n",
    "$$ A = P \\cdot L \\cdot U $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "where the $P$ matrix represents the row permutation. The permutation (pivoting) also improve the numerical stability:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  A=\\left(\\begin{matrix}0 & 3 & 1 & 2\\\\4 & 0 & -3 & 1\\\\-3 & 1 & 0 & 2\\\\9 & 2 & 5 & 0\\end{matrix}\\right)\\;,\\;\\;\\;P=\\left(\\begin{matrix}0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0\\\\1.0 & 0.0 & 0.0 & 0.0\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize  P^{-1} A = \\left(\\begin{matrix}9.0 & 2.0 & 5.0 & 0\\\\0 & 3.0 & 1.0 & 2.0\\\\4.0 & 0 & -3.0 & 1.0\\\\-3.0 & 1.0 & 0 & 2.0\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayMath(sp.Eq (Pi*A, p.inv()*a), pre=\"\\\\scriptsize \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cholesky Decomposition\n",
    "\n",
    "a.k.a Cholesky Factorization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Covariance matrix\n",
    "\n",
    "The most important and ubiquitous matrix in quant Finance, \n",
    "\n",
    "* given random factors $\\bs{\\tilde r} = [r_1, ..., r_n]^T$ and their expectation: $\\bar{\\bs r} = \\mathbb{E}[\\bs {\\tilde r}]$\n",
    "\n",
    "The covariance matrix is:\n",
    "\n",
    "$$V = \\mathbb{E}[(\\bs {\\tilde r} - \\bar{\\bs r})(\\bs {\\tilde r} - \\bar{\\bs r})^T] = \\mathbb{E}[\\bs{\\tilde r} \\bs{\\tilde r}^T] - \\bar{\\bs r}\\bar{\\bs r}^T $$\n",
    "\n",
    "* The element $(i, j)$ in $V$ is: $\\text{cov}(r_i, r_j) = \\rho_{ij} \\sigma_i \\sigma_j$.\n",
    "\n",
    "Covariance of linear combinations of factors: \n",
    " \n",
    "$$\\begin{array}{l}\n",
    "\\text{cov}(\\bs x^T \\bs r, \\bs y^T \\bs r) &= \\mathbb{E}[(\\bs x^T \\bs r)(\\bs r^T \\bs y)] - \\mathbb{E}[\\bs x^T \\bs r]\\mathbb{E}[\\bs r^T \\bs y]\\\\\n",
    "&= \\bs x^T \\mathbb{E}[\\bs r \\bs r^T] \\bs y - \\bs x^T \\bar{\\bs r}\\bar{\\bs r}^T \\bs y = \\bs x^T V \\bs y\n",
    "\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Correlation matrix\n",
    "\n",
    "$\\renewcommand{Sigma}{\\mathcal{S}}$\n",
    "\n",
    "* $C = (\\rho_{ij})$ is the co-variance matrix of the normalized factors $\\bs {\\tilde s} = [\\frac{r_1}{\\sigma_1}, ..., \\frac{r_n}{\\sigma_n}]^T$\n",
    "* $V = \\Sigma C \\Sigma $, where $\\Sigma$ is a diagonal matrix of $\\sigma_i$\n",
    "* all elements in a correlation matrix are within [-1, 1]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Symmetric positive definite (SPD)\n",
    "\n",
    "Positive definite:\n",
    "* Matrix $A$ is positive definite if $\\bs x^T A \\bs x > 0$ for $\\forall \\bs{x \\ne 0}$\n",
    "* Matrix $A$ is semi positive definite if $\\bs x^T A \\bs {x \\ge 0}$ for $\\forall \\bs{x \\ne 0}$\n",
    "* Positive definite does not imply every element in the matrix is positive\n",
    "\n",
    "Both covariance and correlation matrices are symmetric (semi) positive definite (SPD):\n",
    "\n",
    "* $\\bs x^T V \\bs x = \\text{cov}[\\bs x^T \\bs {\\tilde r},\\bs x^T \\bs {\\tilde r}] = \\text{var}[\\bs x^T \\bs {\\tilde r}] \\ge 0$\n",
    "* $\\bs x^T C \\bs x = \\text{cov}[\\bs x^T \\bs {\\tilde s},\\bs x^T \\bs {\\tilde s}] = \\text{var}[\\bs x^T \\bs {\\tilde s}] \\ge 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Example: weekly price and returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "f3 = pd.read_csv('data/f3.csv', parse_dates=[0]).set_index('Date').sort_index()\n",
    "\n",
    "fig = figure(figsize=[12, 4])\n",
    "ax1 = fig.add_subplot(121)\n",
    "f3.plot(title='Historical Prices', ax=ax1);\n",
    "\n",
    "weeks_in_year = 52. #business weeks per year\n",
    "\n",
    "ax2 = fig.add_subplot(122)\n",
    "r = np.log(f3).diff()\n",
    "r.plot(title='Historical Returns', ax=ax2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Correlation and covariance matrix of weekly returns:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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:5px; 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:5px; 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>0.0418</td>\n",
       "      <td>0.0013</td>\n",
       "      <td>0.0292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GLD</th>\n",
       "      <td>0.0013</td>\n",
       "      <td>0.0379</td>\n",
       "      <td>0.0200</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>OIL</th>\n",
       "      <td>0.0292</td>\n",
       "      <td>0.0200</td>\n",
       "      <td>0.1159</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td></tr></table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cm = r.corr()\n",
    "cv = r.cov()\n",
    "    \n",
    "fmt.displayDFs(cm, cv*weeks_in_year, headers=['Correlation', 'Covariance'], fontsize=4, fmt=\"4f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Does it make sense to compute the covariance and correlation matrix of the price levels?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Cholesky decomposition\n",
    "\n",
    "If $A$ is symmetric semi positive definite (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 not unique if the matrix is semi positive definite, but unique when it is positive definite:\n",
    "\n",
    "* The rank is same between $L$ and $A$\n",
    "* Beware that $A = LL^T \\neq L^TL$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Examples of Cholesky decomposition:\n",
    "\n",
    "Previous correlation matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\tiny \\left(\\begin{matrix}1.0 & 0 & 0\\\\0.03215 & 0.9995 & 0\\\\0.4191 & 0.2878 & 0.8611\\end{matrix}\\right)\\left(\\begin{matrix}1.0 & 0.03215 & 0.4191\\\\0 & 0.9995 & 0.2878\\\\0 & 0 & 0.8611\\end{matrix}\\right)=\\left(\\begin{matrix}1.0 & 0.03215 & 0.4191\\\\0.03215 & 1.0 & 0.3011\\\\0.4191 & 0.3011 & 1.0\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lcm = np.linalg.cholesky(cm)\n",
    "lcv = np.linalg.cholesky(cv)\n",
    "fmt.displayMath(sp.Matrix(lcm).evalf(4), \n",
    "                fmt.joinMath('=', sp.Matrix(lcm.T).evalf(4), sp.Matrix(lcm.dot(lcm.T)).evalf(4)), \n",
    "                sep=\"\", pre=\"\\\\tiny\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Previous covariance matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\tiny \\left(\\begin{matrix}0.0284 & 0 & 0\\\\0.000868 & 0.027 & 0\\\\0.0198 & 0.0136 & 0.0406\\end{matrix}\\right)\\left(\\begin{matrix}0.0284 & 0.000868 & 0.0198\\\\0 & 0.027 & 0.0136\\\\0 & 0 & 0.0406\\end{matrix}\\right)=\\left(\\begin{matrix}0.000804 & 2.46 \\cdot 10^{-5} & 0.000561\\\\2.46 \\cdot 10^{-5} & 0.000729 & 0.000384\\\\0.000561 & 0.000384 & 0.00223\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayMath(sp.Matrix(lcv).evalf(3), \n",
    "                fmt.joinMath('=', sp.Matrix(lcv.T).evalf(3), sp.Matrix(lcv.dot(lcv.T)).evalf(3)), \n",
    "                sep=\"\", pre=\"\\\\tiny\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Recursive algorithm\n",
    "\n",
    "A SPD matrix $A$ and its Cholesky decomposition $L$ can be partitioned as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize A=\\left(\\begin{matrix}a_{11} & A_{21}^T\\\\A_{21} & A_{22}\\end{matrix}\\right)\\;,\\;\\;\\;L=\\left(\\begin{matrix}l_{11} & 0\\\\L_{21} & L_{22}\\end{matrix}\\right)\\;,\\;\\;\\;L^{T}=\\left(\\begin{matrix}l_{11} & L_{21}^T\\\\0 & L_{22}^T\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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)\n",
    "               , pre='\\\\scriptsize')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From $A = LL^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize \\left(\\begin{matrix}a_{11} & A_{21}^T\\\\A_{21} & A_{22}\\end{matrix}\\right)=\\left(\\begin{matrix}l_{11}^{2} & L_{21}^T l_{11}\\\\L_{21} l_{11} & L_{21} L_{21}^T + L_{22} L_{22}^T\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayMath(fmt.joinMath('=', A,  L*LT), pre='\\\\scriptsize')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We immediately have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize l_{11}=\\sqrt{a_{11}}\\;,\\;\\;\\;L_{21}=\\frac{A_{21}}{l_{11}}\\;,\\;\\;\\;A_{22} - L_{21} L_{21}^T=L_{22} L_{22}^T$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayMath(fmt.joinMath('=', l11, sp.sqrt(a11)), fmt.joinMath('=', L12,  A21/l11), \n",
    "                fmt.joinMath('=', A22 - L12*L12T, L22*L22T), pre='\\\\scriptsize')"
   ]
  },
  {
   "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 quantitative Finance\n",
    "\n",
    "* The most common processes for asset prices and risk factors\n",
    "\n",
    "Example: correlated n-dimensional Geometric Brownian motion:\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",
    "In vector form:\n",
    "$$ d \\bs x = X \\bs u dt + X \\Sigma d \\bs w, \\;\\;\\; d\\bs w d\\bs w^T = C dt$$\n",
    "\n",
    "where $X$ is a diagonal matrix of $x_i$, and $\\Sigma$ is a diagonal matrix of $\\sigma_i$ and $C$ is the correlation matrix of $\\rho_{ij}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Draw correlated Brownians\n",
    "\n",
    "Draw discretized version of $\\delta \\bs w = L \\bs z \\sqrt{\\delta t}$\n",
    "* where $\\bs z$ is a vector of independent standard normal random variables\n",
    "  * $\\mathbb{E}[\\bs z] = \\bs 0$, $\\mathbb{E}[\\bs z \\bs z^T] = I$ \n",
    "* $L$ is the Cholesky decomposition of the correlaton matrix $C = LL^T$\n",
    "\n",
    "$\\delta \\bs w$ have the desired correlation:\n",
    "$$\\mathbb{E}[\\delta \\bs w \\delta \\bs w^T] = \\mathbb{E}[L \\bs z \\bs z^T L^T] \\delta t = L \\mathbb{E}[\\bs z \\bs z^T] L^T \\delta t = LL^T \\delta t = C \\delta t$$\n",
    "\n",
    "Equivalently, we can draw $\\Sigma \\delta \\bs w = (\\Sigma L) \\bs z\\sqrt{\\delta t}$, where\n",
    "* $\\Sigma L$ is the Cholesky decomposition of the covariance matrix\n",
    "$$ C = LL^T \\iff V = \\Sigma C \\Sigma = \\Sigma L (\\Sigma L)^T $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Simulated paths"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the code below ignores the drifts and its correction\n",
    "\n",
    "nweeks = 1000\n",
    "e = np.random.normal(size=[3, nweeks])\n",
    "dw = (lcm.dot(e)).T\n",
    "ts = np.arange(nweeks)/weeks_in_year\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",
    "stdev = np.sqrt(np.diag(cv))\n",
    "\n",
    "dw = (lcm.dot(e)).T\n",
    "dws = dw*stdev\n",
    "wcm = np.exp(np.cumsum(dws, 0))\n",
    "plot(ts, wcm)\n",
    "xlabel(\"Year\")\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*stdev\n",
    "wcm = np.exp(np.cumsum(dws2, 0))\n",
    "plot(ts, wcm)\n",
    "xlabel(\"Year\")\n",
    "legend(cm.index, loc='best');\n",
    "title(('$L^T \\epsilon$'));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$L \\bs \\epsilon$ and $L^T \\bs \\epsilon$ are different, gives different correlation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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:5px; 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>-0.00805</td>\n",
       "      <td>0.4677</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GLD</th>\n",
       "      <td>-0.00805</td>\n",
       "      <td>1</td>\n",
       "      <td>0.2535</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>OIL</th>\n",
       "      <td>0.4677</td>\n",
       "      <td>0.2535</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td><td style=\"border:none; padding:5px; 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>0.09084</td>\n",
       "      <td>0.4258</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GLD</th>\n",
       "      <td>0.09084</td>\n",
       "      <td>1</td>\n",
       "      <td>0.2219</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>OIL</th>\n",
       "      <td>0.4258</td>\n",
       "      <td>0.2219</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td></tr></table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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=['$L \\epsilon$', '$L^T \\epsilon$'], fontsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Big correlation/covariance matrices\n",
    "\n",
    "\n",
    "In practice, we work with thousands of risk factors\n",
    "* Correlation matrix is easier to maintain than the covariance matrix, because it is \"normalized\"\n",
    "\n",
    "Very difficult to keep large correlation matrices semi positive definite (SPD)\n",
    " * Size of a few thousands is the practical limit\n",
    " * Small changes in few values can invalidate the whole matrix\n",
    " * Adding new entries can be extremely difficult\n",
    "\n",
    "Dimensionality reduction is required when dealing with very large number of factors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Complexity\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 common numerical linear algebra algorithms are of complexity of $O(n^3)$\n",
    "* matrix multiplication: $n^2$ elements in output, each element takes $O(n)$\n",
    "* LU decomposition: $n$ diagonal elements, to zero-out each column costs $O(n^2)$\n",
    "* Cholesky decomposition: $n$ recursive steps, each takes $O(n^2)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Matrix Calculus\n",
    "\n",
    "Morpheus: The Matrix is a system, Neo."
   ]
  },
  {
   "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 **always a row vector**, whenever the vector or matrix appears in the denominator of differentiation, the result is transposed.\n",
    "* Some time we use the notation: $\\frac{\\partial f}{\\partial \\bs x^T} = \\left(\\frac{\\partial f}{\\partial \\bs x}\\right)^T $ to denote a column vector."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Vector function \n",
    "\n",
    "$$\\renewcommand{p}{\\partial}\\bs y(\\bs x)  = [y_1(\\bs x), ..., y_n(\\bs x)]^T$$\n",
    "\n",
    "* Derivative to vector (Jacobian matrix): $\\frac{\\partial \\bs y}{\\partial \\bs x} = \\left[\\frac{\\partial{y_1}}{\\partial \\bs x}, \\frac{\\partial{y_2}}{\\partial \\bs x}, \\cdots, \\frac{\\partial{y_n}}{\\partial \\bs x} \\right]^T$\n",
    " *  $\\frac{\\partial \\bs y}{\\partial \\bs x}$ is a matrix of $\\bs y$ rows and $\\bs x$ columns\n",
    " * $\\frac{\\partial \\bs y}{\\partial \\bs x}\\frac{\\partial \\bs x}{\\partial \\bs y} = \\id$, even when $\\bs x$ and $\\bs y$ are of different dimension.\n",
    " * sometime we use the following notation:  $\\frac{\\partial \\bs y}{\\partial \\bs x^T} =  \\left(\\frac{\\partial \\bs y}{\\partial \\bs x}\\right)^T$\n",
    "* Derivative to scalar: $\\frac{\\partial \\bs y}{\\partial z} = [\\frac{\\partial y_1}{\\partial z}, \\cdots, \\frac{\\partial y_n}{\\partial z}]^T$\n",
    " * remains a column vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Vector differentiation cheatsheet\n",
    "\n",
    "\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",
    "* $O$ is the zero matrix, $I$ is the identity matrix\n",
    "\n",
    "| Expression | Results | Special Cases |\n",
    "| :------: | :-----: | :-----: |\n",
    "| $\\frac{\\p(a \\bs u + b \\bs v)}{\\p \\bs x}$ |  $a \\frac{\\p{\\bs u}}{\\p \\bs x} + b\\frac{\\p{\\bs v}}{\\p \\bs x}$ |$\\frac{\\p{\\bs c}}{\\p \\bs x} = O, \\frac{\\p{\\bs x}}{\\p \\bs x} = \\id$ |\n",
    "| $\\frac{\\p{A \\bs u}}{\\p \\bs x}$ |  $A\\frac{\\p{\\bs u}}{\\p \\bs x}$ | $\\frac{\\p{\\bs A x}}{\\p \\bs x} = A, \\frac{\\p{\\bs x^T A}}{\\p \\bs x} = A^T$ |\n",
    "| $\\frac{\\p{y \\bs u}}{\\p \\bs x}$ |  $y \\frac{\\p{\\bs u}}{\\p \\bs x} + \\bs u \\frac{\\p{y}}{\\p \\bs x}$ | - |\n",
    "| $\\frac{\\p \\bs{u}^T A \\bs v}{\\p \\bs x} $ |  $\\bs u^T A \\frac{\\p{\\bs v}}{\\p \\bs x}  +  \\bs v^T A^T \\frac{\\p{\\bs u}}{\\p \\bs x} $ |$\\frac{\\p \\bs{x}^T A \\bs x}{\\p \\bs x} = \\bs x^T (A + A^T) $, $\\frac{\\p \\bs{u^Tv}}{\\p \\bs x} =\\bs{u}^T\\frac{\\p \\bs{v}}{\\p \\bs x} + \\bs{v}^T\\frac{\\p \\bs{u}}{\\p \\bs x}$ |\n",
    "| $\\frac{\\p{\\bs g(\\bs u})}{\\p \\bs x}$ |  $\\frac{\\p{\\bs g}}{\\p \\bs u} \\frac{\\p{\\bs u}}{\\p \\bs x} $ | $\\frac{\\p \\bs y}{\\p \\bs x}\\frac{\\p \\bs x}{\\p \\bs y} = \\id$ , $\\frac{\\p \\bs z}{\\p \\bs y}\\frac{\\p \\bs y}{\\p \\bs x}  \\frac{\\p \\bs x} {\\p \\bs z}= \\id$, multi-step chain rules. |\n",
    "\n",
    "* Similar to  univariate calculus, a compact notation for multivariate 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 values (dollar) invested in each asset\n",
    "  * $\\bs w^T \\tilde{\\bs r}$ is the portfolio's P&L\n",
    "  * $\\sigma^2 = \\bs w^TV\\bs w$: the variance of the portfolio P&L"
   ]
  },
  {
   "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 **risky** assets\n",
    "  * $r_0$ is the risk free rate\n",
    "* $\\bs f$ is a view, which can be from:\n",
    "  * Fundamental research: earning forecasts, revenue growth etc\n",
    "  * Technical and quantitative analysis\n",
    "  * Your secret trading signal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Sharpe ratio\n",
    "\n",
    "Sharpe ratio of a portfolio $\\bs w$:\n",
    "$$s(\\bs w) = \\frac{\\bs w^T \\bs f}{\\sqrt{\\bs w^T V \\bs w}}$$\n",
    "\n",
    "Sharpe ratio is invariant under:\n",
    "* portfolio scaling: $s(a \\bs w) = s(\\bs w)$\n",
    "* leverage or deleverage: borrowing or lending money using the risk free asset "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Portfolio optimization\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 the portfolio $\\bs w^*$ with the maximum Sharpe ratio under the view $\\bs f$:\n",
    " * $\\bs w^*$ is a unique solution\n",
    " * all portfolios with optimal Sharpe ratio have the same risky asset mix\n",
    " * No need to constrain the total value because Sharp ratio is invariant by scaling the portfolio."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "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",
    "$$\\begin{eqnarray}\n",
    "l &=& \\bs w^TV \\bs w - 2 \\lambda (\\bs f^T \\bs w - 1) \\\\\n",
    "\\frac{\\partial l}{\\partial \\bs w^T} &=& 2 V \\bs w  - 2 \\lambda \\bs f = \\bs 0 \\\\\n",
    "\\bs w &=& \\lambda V^{-1} \\bs f\n",
    "\\end{eqnarray}$$\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",
    "$$\\bs w^*(\\bs f) = \\frac{V^{-1}\\bs f}{\\bs f^T V^{-1} \\bs f} \\propto V^{-1}\\bs f$$\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 unchanged 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",
    "* $\\bs w^*$ and $\\frac{1}{a} \\bs w^*$ defines the same risky asset mix, with identical Sharpe ratio\n",
    "\n",
    "\n",
    "Therefore, only the relative sizes of excess returns are important\n",
    "* e.g. excess return forecast of [1%, 2%] and [10%, 20%] of two assets would result in identical optimal (characteristic) portfolio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Benchmark portfolio and $\\bs \\beta$\n",
    "\n",
    "Benchmark portfolio $\\bs w_b$ is usually an index portfolio to measure the performance of active portfolio management. \n",
    "  * The beta of a portfolio $\\bs w$ to the benchmark is \n",
    "  $$\\frac{\\text{cov}(\\bs w^T \\tilde{\\bs r}, \\bs w_b^T \\tilde{\\bs r})}{\\sigma_b^2} = \\frac{\\bs w^T V \\bs w_b}{\\sigma_b^2} = \\bs w^T \\bs \\beta_b$$\n",
    "  * Define $\\bs \\beta_b = \\frac{V\\bs w_b}{\\sigma^2_b} $, the vector of individual assets' betas\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 unit beta\n",
    "  \n",
    "$\\bs w_m$ is 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": [
    "## 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 portfolio of \\$1 fully invested in risky assets\n",
    "* $\\bs w_e = \\frac{V^{-1}\\bs e}{\\bs e^TV^{-1}\\bs e}$ has the minimum variance of those fully-invested\n",
    "\n",
    "Beta to a benchmark portfolio: $\\bs f \\propto \\bs \\beta_b = \\frac{V\\bs w_b}{\\sigma_b^2}$:\n",
    "\n",
    "* $\\bs \\beta_b^T \\bs w_b = 1$ means the portfolio has a beta of 1 to the benchmark portfolio $\\bs w_b$\n",
    "* $\\bs w_{\\beta} = \\frac{V^{-1}\\bs \\beta_b}{\\bs \\beta_b^T V^{-1}\\bs \\beta_b} = \\frac{\\bs w_b}{\\bs{\\beta_b^T w_b}} = \\bs w_b$, i.e., the benchmark portfolio itself\n",
    "* the benchmark portfolio itself is optimal amongst those with unit beta."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Portfolio optimization example\n",
    "\n",
    "Given the following covariance matrix estimates and excess return forecasts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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\">Epected Excess Return</th></tr><tr style=\"border:none\"><td style=\"border:none; padding:5px; 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>8.043</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></td><td style=\"border:none; padding:5px; 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>Forcast (%)</th>\n",
       "      <td>5</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Naive(%)</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td></tr></table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "er = np.array([.05, .02, .01]).T\n",
    "flat_r = np.array([.01, .01, .01]).T\n",
    "df_er = pd.DataFrame(np.array([er, flat_r]).T*100, columns=[\"Forcast (%)\", \"Naive(%)\"], index = f3.columns).T\n",
    "\n",
    "\n",
    "fmt.displayDFs(cv*1e4, df_er, fontsize=4, headers=['Covariance', 'Epected Excess Return'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The (normalized) optimal portfolio for the given forecast is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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 (O)</th>\n",
       "      <td>0.8333</td>\n",
       "      <td>0.395</td>\n",
       "      <td>-0.2283</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Min Vol Portfolio (C)</th>\n",
       "      <td>0.5007</td>\n",
       "      <td>0.5433</td>\n",
       "      <td>-0.044</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cvi = np.linalg.inv(cv)\n",
    "w = cvi.dot(er.T)/er.T.dot(cvi).dot(er)\n",
    "w = w/np.sum(w)\n",
    "df_er.loc['Optimial Portfolio (O)', :] = w\n",
    "\n",
    "w2 = cvi.dot(flat_r.T)/er.T.dot(cvi).dot(flat_r)\n",
    "w2 = w2/np.sum(w2)\n",
    "df_er.loc['Min Vol Portfolio (C)', :] = w2\n",
    "\n",
    "fmt.displayDF(df_er[-2:], \"4g\", fontsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Efficient Frontier\n",
    "\n",
    "The simulated Sharpe ratios of all $1 portfolios fully invested in risky assets:\n",
    "* the optimal portfolio O has the optimal Sharpe Ratio\n",
    "* the min Variance portfolio C has the smallest variance\n",
    "* the line start from origin because we used excess return everywhere"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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6178mLy+PQCDADTfcwHvvvcfdd98dcXx+fj5vv/22oXCuuuoqrrqqwmUG1Ypj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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6ecc0c99b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rnd_w = np.random.uniform(size=[3, 10000]) - 0.5\n",
    "rnd_w = np.divide(rnd_w, np.sum(rnd_w, 0))\n",
    "\n",
    "rnd_r = er.dot(rnd_w)\n",
    "rnd_vol = np.sqrt([p.dot(cv).dot(p.T) for p in rnd_w.T])\n",
    "\n",
    "vol_o = sqrt(w.dot(cv).dot(w))\n",
    "r_o = er.dot(w)\n",
    "plot(rnd_vol, rnd_r, 'g.')\n",
    "plot([0, 10*vol_o], [0, 10*r_o], 'k');\n",
    "xlim(0, .06)\n",
    "ylim(-.01, .08)\n",
    "\n",
    "plot(vol_o, r_o, 'ro')\n",
    "text(vol_o-.001, r_o+.003, 'O', size=20)\n",
    "r_c = er.dot(w2)\n",
    "vol_c = sqrt(w2.dot(cv).dot(w2))\n",
    "plot(vol_c, r_c, 'r>')\n",
    "text(vol_c-.002, r_c-.008, 'C', size=20);\n",
    "\n",
    "legend(['Portfolios', 'Optimal Sharpe Ratio'], loc='best')\n",
    "\n",
    "ylabel('Portfolio Expected Excess Return')\n",
    "xlabel('Portfolio Return Volatility')\n",
    "title('Sharpe Ratios of $1 portfolios');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Implied view of a portfolio\n",
    "\n",
    "From any portfolio, we can back out its implied excess return forecast $\\bs f$:\n",
    "* Assuming the portfolio is mean variance optimal\n",
    "* It is nothing but the betas to the portfolio\n",
    "* Only meaningful in a relative sense as well\n",
    "\n",
    "Consider the market portfolio: the market collectively believe that $\\bs f \\propto \\bs \\beta_m$: \n",
    "\n",
    "$$\\mathbb{E}[\\tilde{\\bs r}] - r_0 = \\bs \\beta_m \\left(r_m - r_0\\right)$$\n",
    "\n",
    "* $r_m = \\frac{\\mathbb{E}[\\bs w_m^T \\bs {\\tilde r}]}{\\bs w_m^T \\bs 1} $ is the market portfolio's expected return\n",
    "* this is exactly the CAPM."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Estimate expected returns\n",
    "\n",
    "Estimate expected return from historical data is difficult\n",
    "* historical return is not a good indicator of future performance\n",
    "* even if we assume it is, it requires very long history, ~500 years\n",
    "\n",
    "The market implied return is a much better return estimate than historical data.\n",
    "* This is the key insight of Black-Litterman"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Implied views example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we are given the following portfolio:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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\", fontsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "We can compute its implied return forecast as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fmt.displayDF(df[1:], \"2f\", fontsize=4)"
   ]
  },
  {
   "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\n",
    "\n",
    "Robert Heinlein: Throughout history, poverty is the normal condition of man."
   ]
  },
  {
   "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",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "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))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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:5px; 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>0.5060</td>\n",
       "      <td>10.0011</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GLD</th>\n",
       "      <td>0.5060</td>\n",
       "      <td>7.2697</td>\n",
       "      <td>9.0361</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>OIL</th>\n",
       "      <td>10.0011</td>\n",
       "      <td>9.0361</td>\n",
       "      <td>22.2275</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td><td style=\"border:none; padding:5px; 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>0.0390</td>\n",
       "      <td>0.0342</td>\n",
       "      <td>0.0855</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Optimal Portfolio</th>\n",
       "      <td>0.2456</td>\n",
       "      <td>0.2158</td>\n",
       "      <td>0.5387</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td></tr></table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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.loc['Optimal Portfolio', :] = covi.dot(er.T)/np.sum(covi.dot(er.T))\n",
    "\n",
    "fmt.displayDFs(df_cov, pf, headers=[\"Covariance\", \"Optimized Portfolio\"], fontsize=4, fmt=\"4f\")"
   ]
  },
  {
   "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",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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:5px; 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>0.5062</td>\n",
       "      <td>10.0011</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GLD</th>\n",
       "      <td>0.5062</td>\n",
       "      <td>7.2697</td>\n",
       "      <td>9.0361</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>OIL</th>\n",
       "      <td>10.0011</td>\n",
       "      <td>9.0361</td>\n",
       "      <td>22.2275</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td><td style=\"border:none; padding:5px; 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>0.0390</td>\n",
       "      <td>0.0342</td>\n",
       "      <td>0.0855</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Optimal Portfolio</th>\n",
       "      <td>0.0462</td>\n",
       "      <td>0.0098</td>\n",
       "      <td>0.9441</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></td></tr></table></font></center>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pf1 = pd.DataFrame(np.array([er]), columns=f3.columns, index=['Expected Return'])\n",
    "covi1 = np.linalg.inv(cov1)\n",
    "pf1.loc['Optimal Portfolio', :] = covi1.dot(er.T)/np.sum(covi1.dot(er.T))\n",
    "\n",
    "fmt.displayDFs(df_cov1, pf1, headers=[\"Covariance\", \"Optimized Portfolio\"], fontsize=4, fmt=\"4f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "the optimal portfolio is totally different, how is it possible?"
   ]
  },
  {
   "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",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\7.999\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}2.0\\\\1.0\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A small perturbation on vector $\\bs y$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\8.001\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}6.0\\\\-1.0\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "A small perturbation on matrix $A$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 4.002\\end{matrix}\\right) x=\\left(\\begin{matrix}4.0\\\\7.999\\end{matrix}\\right)\\;,\\;\\;\\;x=\\left(\\begin{matrix}5.0\\\\-0.5\\end{matrix}\\right)$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "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))))"
   ]
  },
  {
   "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",
    "\n",
    "* Elementwise: based on looking at matrix as an elongated vector\n",
    "* Shatten norms: based on singular values\n",
    "* Vector norm induced: most common and introduced below\n",
    "\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${\\infty}$: $\\Vert A\\Vert _\\infty = \\max_i \\sum_j |a_{ij}|$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Norm 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 propagation of errors in a linear system $\\bs y = A\\bs x$ with invertible $A$:\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",
    "$$\\begin{array}\n",
    "\\\\ (A + \\delta A) (B + \\delta B) &= (A + \\dot{A} \\epsilon ) (B + \\dot{B} \\epsilon) = C \\\\\n",
    "AB + (\\dot{A}B + A\\dot{B})\\epsilon + \\dot{A}\\dot{B}\\epsilon^2 &= C \\\\\n",
    "(\\dot{A}B + A\\dot{B})\\epsilon + \\dot{A}\\dot{B}\\epsilon^2 &= 0\n",
    "\\end{array}$$"
   ]
  },
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   "cell_type": "markdown",
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   "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."
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Numerical example\n",
    "\n",
    "Consider the ill-conditioned matrices from previous examples:"
   ]
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   "cell_type": "code",
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    {
     "data": {
      "text/latex": [
       "$$\\scriptsize V=\\left(\\begin{matrix}8.023 & 0.506 & 10.0\\\\0.506 & 7.27 & 9.036\\\\10.0 & 9.036 & 22.23\\end{matrix}\\right)\\;,\\;\\;\\;V^{-1}=\\left(\\begin{matrix}32668.0 & 32335.0 & -27844.0\\\\32335.0 & 32006.0 & -27560.0\\\\-27844.0 & -27560.0 & 23732.0\\end{matrix}\\right)$$"
      ],
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       "<IPython.core.display.Math object>"
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   "source": [
    "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)), \n",
    "                fmt.joinMath('=', Vi, sp.Matrix(cov*1e4).inv().evalf(5)), pre=\"\\\\scriptsize\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\scriptsize A=\\left(\\begin{matrix}1.0 & 2.0\\\\2.0 & 3.999\\end{matrix}\\right)\\;,\\;\\;\\;A^{-1}=\\left(\\begin{matrix}-3999.0 & 2000.0\\\\2000.0 & -1000.0\\end{matrix}\\right)$$"
      ],
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       "<IPython.core.display.Math object>"
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   "source": [
    "A = sp.MatrixSymbol('A', 2, 2)\n",
    "Ai = sp.MatrixSymbol('A^{-1}', 2, 2)\n",
    "fmt.displayMath(fmt.joinMath('=', A, sp.Matrix(a)), \n",
    "                fmt.joinMath('=', Ai, sp.Matrix(a).inv().evalf(4)), pre=\"\\\\scriptsize\")"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "their condition numbers are large because of the large elements in the inversion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<center><font size=4><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>3.831e+06</td>\n",
       "      <td>2.688e+06</td>\n",
       "      <td>3.831e+06</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table></font></center>"
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       "<IPython.core.display.HTML object>"
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   "source": [
    "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 $V$']), \n",
    "              \"4g\", fontsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## 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."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Assignments\n",
    "\n",
    "Required reading:\n",
    "\n",
    "* Bindel and Goodman: Chapter 4, 5.1-5.4\n",
    "\n",
    "Highly recommended reading:\n",
    "\n",
    "Deflating Sharpe Ratio: http://www.davidhbailey.com/dhbpapers/deflated-sharpe.pdf\n",
    "\n",
    "Homework:\n",
    "\n",
    "* Complete [homework set 2](http://yadongli.github.io/nyumath2048)"
   ]
  }
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