pwlnyc_kalman.ipynb

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   "source": [
    "# A New Approach to Linear Filtering and Prediction Problems\n",
    "## a.k.a. Kalman Filters\n",
    "### Papers We Love NYC\n",
    "Elizabeth Ramirez"
   ]
  },
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   "source": [
    "%matplotlib inline\n",
    "import numpy\n",
    "from numpy.random import randn\n",
    "import matplotlib.pyplot as plt"
   ]
  },
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   "source": [
    "## Intro\n",
    "![Rudolf_Kalman_Obama](obama_kalman.jpg)  \n",
    "Original paper: [http://www.cs.unc.edu/~welch/kalman/media/pdf/Kalman1960.pdf]"
   ]
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   "source": [
    "## What is a Kalman Filter\n",
    "Prediction-correction technique  \n",
    "Calculate recursively system state $u_k$ at time $t_k$ given previous state $u_{k - 1}$ and new information $b_k$"
   ]
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   "source": [
    "## Kalman Filters for Rocket Science\n",
    "Used for Apollo Space Program of NASA in early 1960's  \n",
    "Transcription of the original code available at [http://www.ibiblio.org/apollo/]  \n",
    "Implemented in AGC4 assembly language.\n",
    "\n",
    "```\n",
    "# Copyright:\tPublic domain.\n",
    "# Filename:\t KALMAN_FILTER.agc\n",
    "# Purpose:\t  A section of Luminary 1C, revision 131.\n",
    "#\t\t       It is part of the source code for the Lunar Module's (LM)\n",
    "#\t\t       Apollo Guidance Computer (AGC) for Apollo 13 and Apollo 14.\n",
    "#\t\t       This file is intended to be a faithful transcription, except\n",
    "#\t\t       that the code format has been changed to conform to the\n",
    "#\t\t       requirements of the yaYUL assembler rather than the\n",
    "#\t\t       original YUL assembler.\n",
    "# Reference:\tpp. 1467-1468 of 1729.pdf.\n",
    "# Contact:\t  Ron Burkey <info@sandroid.org>.\n",
    "# Website:\t  www.ibiblio.org/apollo/index.html\n",
    "# Mod history:  06/08/03 RSB.\tBegan transcribing.\n",
    "#\t\t       05/14/05 RSB\tCorrected website reference above\n",
    "```"
   ]
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   "source": [
    "## Background\n",
    "- Random processes (Appendix)\n",
    "- Dynamic systems\n",
    "- Vector spaces\n",
    "- Least Squares and Normal Equation"
   ]
  },
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   "source": [
    "## Vector Spaces\n",
    "$\\mathcal{V}$ a.k.a. Linear Spaces. A space of vectors with a metric: norm, notion of distance, often represented by $|| \\cdot ||$ and the following properties:\n",
    "\n",
    "- $f, g \\in \\mathcal{V} \\rightarrow f + g = h \\in \\mathcal{V}$\n",
    "\n",
    "- $\\lambda \\in \\mathbb{R}, f \\in \\mathcal{V} \\rightarrow \\lambda f \\in \\mathcal{V}$"
   ]
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   "source": [
    "## Least Squares\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Linear System } \\textbf{A}\\textbf{u} = \\textbf{b} \\qquad A = m\\times n \\qquad \\textbf{u}, \\textbf{b} \n",
    "\\text{ are n-vectors} \\\\\n",
    "\\text{If } A \\text{ is square: } m = n \\rightarrow \\textbf{u} = A^{-1}\\textbf{b} \\\\~\\\\\n",
    "\\text{But if } A \\text{ is not square: } m > n \\\\\n",
    "\\text{System is } \\textbf{overdetermined} \\\\~\\\\\n",
    "\\text{Example: 100 points that fit } Cx + D\n",
    "$$"
   ]
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   "source": [
    "## Least Squares\n",
    "<br><br>\n",
    "$$\n",
    "\\textbf{Solution: } \\text{Find best estimate for state } \\textbf{u} \\text{ that minimizes:} \\\\\n",
    "E = \\left| \\textbf{b} - \\textbf{A}\\hat{\\textbf{u}} \\right|^2 \\\\~\\\\\n",
    "\\text{Solve for } \\hat{\\textbf{u}} \\text{ (an estimate) to minimize } E \\\\~\\\\\n",
    "\\textbf{Normal Equation: } \\textbf{A}^T\\textbf{A}\\hat{\\textbf{u}} = \\textbf{A}^T\\textbf{b}\\\\\n",
    "\\underset{n\\times m}{\\mathrm{A^T}} \\underset{m\\times n}{A} = \\underset{n\\times n}{A^TA}\\\\\n",
    "\\hat{u} = (A^TA)^{-1} A^T b\n",
    "$$"
   ]
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   "source": [
    "## Motivation\n",
    "Wiener Filter Problems:\n",
    "- Impulse response\n",
    "- Numerical determination"
   ]
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   "source": [
    "## Situation\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Given a signal: } \\textbf{y}(t) = \\textbf{x}_1(t) + \\textbf{x}_2(t) \\\\\n",
    "\\textbf{x}_1(t): \\text{true signal, }\\textbf{x}_2(t): \\text{noise} \\\\~\\\\\n",
    "\\textbf{y}(t_1) \\text{?} \\\\\n",
    "\\text{If }t_1 < t \\text{ : Smoothing} \\\\\n",
    "\\text{If }t_1 = t \\text{ : Filtering} \\\\ \n",
    "\\text{If }t_1 > t \\text{ : Prediction} \\\\\n",
    "$$"
   ]
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   "source": [
    "## Theorem 1\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Statistical Estimate } X_1(t_1|t) \\\\\n",
    "\\text{Estimation Error } \\epsilon = x_1(t_1) - X_1(t_1|t) \\\\~\\\\\n",
    "\\text{Loss Function } L(\\epsilon): \\\\\n",
    "L(0) = 0\\\\\n",
    "L(\\epsilon_2) \\geq L(\\epsilon_1) \\geq 0 \\text{ when } \\epsilon_2 \\geq \\epsilon_1 \\geq 0 \\\\\n",
    "L(\\epsilon) = L(-\\epsilon) \\\\~\\\\\n",
    "\\text{Minimize } E\\{L[x_1(t_1) - X_1(t_1)] | y(t_0) … y(t)\\}\n",
    "$$"
   ]
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   "source": [
    "## Theorem 1-a\n",
    "<br><br>\n",
    "$$\n",
    "\\text{If } L(\\epsilon) = \\epsilon^2 \\text{ , Theorem 1 is true without assumptions about symmetry and convexity around the mean.}\n",
    "$$\n",
    "\n",
    "## Problem Summary\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Given a vector-valued random process } {x(t)}\\\\\n",
    "\\text{and observed random variables } y(t_0) … y(t) \\\\\n",
    "\\text{Find an estimate } X(t_1) \\text{ that minimizes the expected loss } E\\{L||x(t_1) - X(t_1)||\\}\n",
    "$$ "
   ]
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    "## Orthogonal Projections\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Vector Space } \\mathcal{Y}(t) = \\sum_{i = t_0}^{t} a_i y(i) \\\\~\\\\\n",
    "\\text{Given } u, v \\in \\mathcal{Y}(t) \\text{ orthogonal if } E[u \\cdot v] = 0 \\\\~\\\\\n",
    "\\text{Orthonormal basis in } \\mathcal{Y}(t) = e_{t_0} ..., e_t \\text{ such that } \\\\\n",
    "\\bar{x} = \\sum_\n",
    "{i = t_0}^{t} a_i e_i \\\\~\\\\\n",
    "x = \\bar{x} + \\tilde{x}\n",
    "$$\n",
    "![orthoprojection](orthoproj.png)"
   ]
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   "source": [
    "From functional analysis, we know that our estimate X1(t1) might or might not live in this vector space Y(t). We would really like our solution to live here. If the vector space is too small, we want to minimize the distance between my estimate and the vector space.\n",
    "\n",
    "Given this, and given we can determine the coefficients of the linear combination, any random variable x, not necessarily in Y(t) can be uniquely decomposed into two parts: a part in Y(t) and a part orthogonal to Y(t) (orthogonal to every vector in Y(t)). \n",
    "\n",
    "See equation 10. The orthogonal projection of x in Y(t): x bar is that vector in Y(t) that minimizes the quadratic loss. This is the key part."
   ]
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    "## Theorem 2\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Let } \\{x(t)\\} \\text{ and } \\{y(t)\\} \\text{ be random processes with zero mean } E[x(t)] = E[y(t)] = 0 \\\\\n",
    "\\text{Observe } y(t_0) ... y(t) \\\\\n",
    "\\text{If either the random processes are gaussian or the loss function is restricted to be } L(\\epsilon)=\\epsilon^2 \\\\\n",
    "\\text{ then the optimal estimate of x(t1|t) is given by the orthogonal projection } \\bar{x} \\text{ of } x(t_1)\n",
    "\\text{ on } \\mathcal{Y}(t)\n",
    "$$"
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   "source": [
    "## Recursive Least Squares\n",
    "<br><br>\n",
    "$$\n",
    "\\hat{u_{99}}: \\text{Average of } b_1, b_2, \\dots , b_{99} \\\\~\\\\\n",
    "b_{100} \\rightarrow \\hat{u_{100}} \\\\~\\\\\n",
    "\\hat{u_{100}} = \\frac{99}{100} \\hat{u_{99}} + \\frac{1}{100}b_{100} \\\\\n",
    "\\qquad = \\hat{u_{99}} + \\frac{1}{100}(b_{100} - \\hat{u_{99}}) \\\\\n",
    "b_{100} - \\hat{u_{99}}: \\textbf{innovation} \\qquad\n",
    "\\frac{1}{100}: \\textbf{gain factor}\n",
    "$$"
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    "## Recursive Least Squares\n",
    "<br>\n",
    "Generalizing the Normal Equation:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\\begin{bmatrix} A_{old} \\\\ A_{new} \\end{bmatrix} \\begin{bmatrix} u \\end{bmatrix} = \n",
    "\\begin{bmatrix} b_{old} \\\\ b_{new} \\end{bmatrix} \\rightarrow \\hat{u_{new}}\n",
    "\\end{aligned}\n",
    "$$\n",
    "<br>\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&A^T = \\begin{bmatrix} A_{old}^T & A_{new}^T \\end{bmatrix}\\\\\n",
    "&A^TA = \\begin{bmatrix} A_{old}^T & A_{new}^T \\end{bmatrix} \\begin{bmatrix} A_{old}^T \\\\ A_{new}^T \\end{bmatrix} \\\\\n",
    "&\\qquad = A^T_{old} A_{old} + A^T_{new}A_{new} = (known) + (new)\n",
    "\\end{aligned}\n",
    "$$\n",
    "<br>\n",
    "Right-hand side\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&A^Tb =  \\begin{bmatrix} A_{old}^T & A_{new}^T \\end{bmatrix} \\begin{bmatrix} b_{old}^T \\\\ b_{new}^T \\end{bmatrix} \\\\\n",
    "&\\qquad = A^T_{old} b_{old} + A^T_{new} b_{new}\\\\\n",
    "&\\qquad = A^T_{old} A_{old} \\hat{u_{old}} + A^T_{new} b_{new}\n",
    "\\end{aligned}\n",
    "$$"
   ]
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   "source": [
    "## Recursive Least Squares \n",
    "<br><br>\n",
    "$$\n",
    "\\hat{u_{new}} = \\hat{u_{old}} + (A^TA)^{-1} A^T_{new}(b_{new} - A_{new} b_{old}) \\\\~\\\\\n",
    "(A^TA)^{-1} A^T_{new}: \\textbf {gain matrix, } K \\text{ (as in Kalman, duh)}\\\\~\\\\\n",
    "\\begin{bmatrix} 1 \\\\ 1 \\\\ \\vdots \\\\ 1 \\end{bmatrix} \\begin{bmatrix} u \\end{bmatrix} = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_{99} \\end{bmatrix}\\\\\n",
    "u = b_{100} = b_{new} \\rightarrow A_{new} = \\begin{bmatrix} 1 \\end{bmatrix}\\\\\n",
    "\\hat{u_{100}} = \\hat{u_{99}} + \\frac{1}{100}(b_{100} - \\hat{u_{99}})\n",
    "$$"
   ]
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    "## Covariance Matrix\n",
    "When errors are independent:\n",
    "<br><br>\n",
    "$$\n",
    "\\sigma_{ij} = \\sigma_{ji} = E[e_i e_j] = 0\n",
    "$$\n",
    "<br>\n",
    "$$\n",
    "\\Sigma_e = E(e e^T) = \n",
    "\\begin{bmatrix}\n",
    "\\sigma_1^2\\\\\n",
    "&\\sigma_2^2\\\\\n",
    "&& \\ddots\\\\\n",
    "&&&\\sigma_k^2\\\\\n",
    "\\end{bmatrix} \\\\~\\\\\n",
    "C = \\Sigma_e^{-1}: \\textbf{Weighting Matrix}\n",
    "$$\n",
    "### Weighted Normal Equation\n",
    "$$\n",
    "A^TCA\\hat{u} = A^TCb \\\\\n",
    "\\hat{u} = (A^TCA)^{-1} A^TCb = Lb\n",
    "$$"
   ]
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    "## The Kalman Filter\n",
    "<br><br>\n",
    "$$\n",
    "\\text{Time varying least squares problem: Estimate } \\hat{u_k} \\text{ at each time } t_k \\\\\n",
    "\\hat{u_{new}} = L \\hat{u_{old}} + Kb_{new}\\\\\n",
    "\\hat{u_{new}} = \\hat{u_{old}} + K(b_{new} - A_{new}\\hat{u_{old}}) \\\\\n",
    "\\text{innovation } b_{new} - A_{new}\\hat{u_{old}} \\\\\n",
    "\\text{covariance matrix } P = (A^TCA)^{-1}\n",
    "$$\n"
   ]
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   "source": [
    "## Algorithm\n",
    "### Prediction\n",
    "<br>\n",
    "$$\n",
    "\\hat{u_{k|k-1}} = F \\hat{u_{k-1|k-1}} + \\epsilon_i \\\\\n",
    "P_{k|k-1} = AP_{k-1|k-1}A^T + Q \\\\\n",
    "F: \\textbf{state transition matrix} \\\\\n",
    "Q: \\textbf{covariance matrix of random excitation}\n",
    "$$\n",
    "\n",
    "### Correction\n",
    "<br>\n",
    "$$\n",
    "K_k = P_{k|k-1}A^T(AP_{k|k-1}A^T + R)^{-1} \\\\\n",
    "\\hat{u_{k|k}} = \\hat{u_{k|k-1}}  + K_k(b_k - A\\hat{u_{k|k-1}}) \\\\\n",
    "P_{k|k} = (I - K_kA)P_{k|k-1}\n",
    "$$"
   ]
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   "source": [
    "F is called the state transition matrix. It establishes how the state vector changes from one time step to another.\n",
    "Q is the covariance matrix of system error"
   ]
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   "source": [
    "## Implementation\n",
    "- Predicted state and covariance (before the measurement)\n",
    "- Corrected state and covariance (after the measurement)\n",
    "- Innovation\n",
    "- Filter gain"
   ]
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    "## Prediction / Correction\n",
    "- Previous state vector $u$\n",
    "- Predicted state vector $\\hat u$\n",
    "- Matrix of observation equations $A$\n",
    "- Vector of observations $b$\n",
    "- Predicted covariance matrix $P$\n",
    "- Process noise matrix $Q$\n",
    "- Observations noise matrix $R$"
   ]
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   "source": [
    "def predict(u, P, F, Q):\n",
    "    u = numpy.dot(F, u)\n",
    "    P = numpy.dot(F, numpy.dot(P, F.T)) + Q\n",
    "    \n",
    "    return u, P"
   ]
  },
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   "execution_count": 3,
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    "def correct(u, A, b, P, Q, R):\n",
    "    C = numpy.dot(A, numpy.dot(P, A.T)) + R\n",
    "    K = numpy.dot(P, numpy.dot(A.T, numpy.linalg.inv(C)))\n",
    "    \n",
    "    u = u + numpy.dot(K, (b - numpy.dot(A, u)))\n",
    "    P = P - numpy.dot(K, numpy.dot(C, K.T))\n",
    "    \n",
    "    return u, P"
   ]
  },
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   "execution_count": 4,
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   "source": [
    "# Test data\n",
    "dt = 0.1\n",
    "A = numpy.array([[1, 0], [0, 1]])\n",
    "u = numpy.zeros((2, 1))\n",
    "# Random initial measurement centered at state value\n",
    "b = numpy.array([[u[0, 0] + randn(1)[0]], [u[1, 0] + randn(1)[0]]])\n",
    "\n",
    "P = numpy.diag((0.01, 0.01))\n",
    "F = numpy.array([[1.0, dt], [0.0, 1.0]])\n",
    "# Unit variance for the sake of simplicity\n",
    "Q = numpy.eye(u.shape[0])\n",
    "R = numpy.eye(b.shape[0])"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "predicted final estimate: -16.969732\n",
      "corrected final estimate: -16.994212\n",
      "measured state: -17.009443\n"
     ]
    }
   ],
   "source": [
    "N = 500\n",
    "predictions, corrections, measurements = [], [], []\n",
    "for k in numpy.arange(0, N):\n",
    "    u, P = predict(u, P, F, Q)\n",
    "    predictions.append(u)\n",
    "    u, P = correct(u, A, b, P, Q, R)\n",
    "    corrections.append(u)\n",
    "    measurements.append(b)\n",
    "    b = numpy.array([[u[0, 0] + randn(1)[0]],\n",
    "                     [u[1, 0] + randn(1)[0]]])\n",
    "\n",
    "print 'predicted final estimate: %f' % predictions[-1][0]\n",
    "print 'corrected final estimate: %f' % corrections[-1][0]\n",
    "print 'measured state: %f' % measurements[-1][0]"
   ]
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r9BjmGXmrUFDg+fljY2NRUJCLggLfrCNIRMEjZDNU39V953J5CSIiV9rqGdVR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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105282f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.arange(50, 100)\n",
    "fig = plt.figure(figsize=(15,15))\n",
    "\n",
    "axes = fig.add_subplot(2, 2, 1)\n",
    "axes.set_title(\"Kalman Filter\")\n",
    "axes.plot(t, numpy.array(predictions)[50:100, 0], 'o', label='prediction')\n",
    "axes.plot(t, numpy.array(corrections)[50:100, 0], 'x', label='correction')\n",
    "axes.plot(t, numpy.array(measurements)[50:100, 0], '^', label='measurement')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Conclusions\n",
    "- Kalman Filter is a generalized least squares problem for dynamical systems.\n",
    "- Computationally more efficient\n",
    "- Strang, Gilbert. *Computational Science and Engineering*"
   ]
  },
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   "source": [
    "## Thank you!\n",
    "eramirem@gmail.com\n",
    "<br>\n",
    "@eramirem"
   ]
  }
 ],
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