{
 "metadata": {
  "name": "",
  "signature": "sha256:a6e7e8e6dd048c1dfbedc22afe3d0202f5b899f4daa0f82572fb2ea998d52af0"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Filter Example\n",
      "\n",
      "This example demonstrates the connection between MKS and signal\n",
      "processing for a 1D filter. It shows that the filter is in fact the\n",
      "same as the influence coefficients and, thus, applying the `predict`\n",
      "method provided by the `MKSRegressionModel` is in essence just applying a filter."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "%load_ext autoreload\n",
      "%autoreload 2\n",
      "\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The autoreload extension is already loaded. To reload it, use:\n",
        "  %reload_ext autoreload\n"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we construct a filter, $F$, such that\n",
      "\n",
      "$$F\\left(x\\right) = e^{-|x|} \\cos{\\left(2\\pi x\\right)} $$\n",
      "\n",
      "We want to show that if $F$ is used to generate sample calibration\n",
      "data for the MKS, then the calculated influence coefficients are in\n",
      "fact just $F$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x0 = -10.\n",
      "x1 = 10.\n",
      "x = np.linspace(x0, x1, 1000)\n",
      "def F(x):\n",
      "    return np.exp(-abs(x)) * np.cos(2 * np.pi * x)\n",
      "p = plt.plot(x, F(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ySYFXXoHf+z145hkb+758efhwOXzYTuG7bNnFy666Kn7Q1+rslaV69BAu6Pv6bOPml8kU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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f11c6246050>"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we generate the sample data `(X, y)` using\n",
      "`scipy.ndimage.convolve`. This performs the convolution\n",
      "\n",
      "$$ p\\left[ i \\right] = F\\left[j\\right] X\\left[i - j\\right] $$\n",
      "\n",
      "for each sample."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.ndimage\n",
      "\n",
      "Nspace = 101\n",
      "Nsample = 50\n",
      "np.random.seed(201)\n",
      "x = np.linspace(x0, x1, Nspace)\n",
      "X = np.random.random((Nsample, Nspace))\n",
      "y = np.array([scipy.ndimage.convolve(xx, F(x), mode='wrap') for xx in X])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 71
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For this problem, a basis is unnecessary as no discretization is\n",
      "required in order to reproduce the convolution with the MKS.  Using\n",
      "the `ContinuousIndicatorBasis` with `n_states=2` is the equivalent of a\n",
      "non-discretized convolution in space."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from pymks import MKSRegressionModel\n",
      "from pymks import ContinuousIndicatorBasis\n",
      "\n",
      "basis = ContinuousIndicatorBasis(n_states=2, domain=[0, 1])\n",
      "model = MKSRegressionModel(basis=basis)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 72
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Fit the model using the data generated by $F$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model.fit(X, y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 73
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To check for internal consistency, we can compare the predicted\n",
      "output with the original for a few values"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y_pred = model.predict(X)\n",
      "print y[0, :4]\n",
      "print y_pred[0, :4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-0.41059557  0.20004566  0.61200171  0.5878077 ]\n",
        "[-0.41059557  0.20004566  0.61200171  0.5878077 ]\n"
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With a slight linear manipulation of the coefficients, they agree perfectly with the shape of the filter, $F$. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, F(x), 'r', label=r'$F$')\n",
      "plt.plot(x, -model.coeff[:,0] + model.coeff[:, 1], 'k--', label=r'$\\alpha$')\n",
      "l = plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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uukmIRx8V4sCBoPV7uFxCVFaKk19+Kc4cOiSaTpwQbpdLe/0hCvLfiPNqEfpO\nn9wdpNxXtN8zCuLDJ54Q2TqdeGrOHPHhE0+IU19+Kdw33CDEqlWa6t+Yni7ef+yxvgVutxBZWcED\n7FNPCbF0qXr5tGlCfPFF4HNkZwtRU6NenpkpRG1t4HOkpQmh9gdLCCFSU4VobFQvb2oSwmpVLX79\nn/9Z/OTCCwO3oatLiJQUIerq+hTtX7tWjEpIEC6nM/A5hBBi3TohCguFOHFCbPvtb8Vv77hDjDEY\nxAt33hm8LkUdQgz0kZhHvxry9MnVPsemBSmnQeSWX/4S7/72tzhbV4ff/Od/YtKMGVhdVQU89pim\n+nd961t4+623+hzv3LcP7uRk+YJfAMfMZvxl2zb1JwRKf+AVLA1CkKEbAHK5ShoE4Xbj+ZYWiACL\nrmCzAU6naoK1sqNHkaayurdHYiJw9dWAwvvRun49Hr/hBugDDP30uOMOOVnaxRfjjVWrcPTYMTzz\nyCN46N13g9cl6odo/3GkELldrpC+XtcfOyZydDrR6dcLfWHBAvHAhAlB6+9cvVpMS0pSaYxbCKNR\nCEkKfJKLLxbukhLV4h/qdMLV0RH4HAUFQhw7plgkNTYKo5bP8ujRQpSVKRb9oKBA/PH++4Of4+mn\nhViypPcxt1vuoe/eHby+fz0adBCFHj0NMTq9XnFxk5r0wkJUzJ4No98uSVs//xxXXndd0Pr5l1+O\nCpWedPvp01hnMATtjV9XUYHPd+1SLHO2t+NNIaBX2XPWq1SvR7vK4i3HmTOw6oJOMJO/eaikQaio\nr0d+sM1NANRNmYLH/L8hedNEXHpp3wqBaGkzDXoM9HRemL7/feDtt3seC7cb206fRtEPfxi07rCL\nLkKbEHDU1PQpqyguxuMaVnuaExPRprJBeEdTE4IM2gAAvlddjYMqM14cZ87AquWPX06OPNSkoKKt\nDfnTpwc9Rdp112F1XR3qfHedevtt4O67GbhJEQM9nR9z58pT755+GnA4cOiDD5CSkIC8K68MWlWn\n12O00YgTCnPQa48dQ06QOfIAYDWZ4FAJ9FJjIywaAqTFYICkkuGyrb4eVg1j48tPnMCrCtNRu1pb\ncdblwggNPXKD2Yyrs7Kw7ZVXIDo7gVdflae63n130Lo0NDHQ0/lhswE7dsiLu8aMwUtLluA67w5K\nGuTbbChXyGJZU16OYQESiXlZLRY4VHZ4kpqaYNHQGzcbDOhQuaDrqKvTFOiNNhtOnjrV93htLRpH\nj4Yh2AWgY9u6AAAMWElEQVRhjxumT8dvXnwR461W1LzxhpzuQMOwDw1NDPR0/kyaBLz1FrBtG+wZ\nGfjBIu0JTR+cNg0FCknFak6eRE6A9AdeVosFDpXeeEdLC8waAr0lMRGSyoKnjMREfH/48KDnyBk+\nXDkNQkUFLGPGBK3vNffHP8bItDT88YUXkLN9O3DNNZrr0tATiRQIRKG58EI8UVISUpXbi4oUk63V\n1tQgJ9jUSgDWpCTVQD/MasUzGoK0JTERkkqPfmxqKn6ssnLXV05eHjYqTRUNkONGyQVz5uDdQDti\nEflgoKfBoaAA8Cwb93WhwYCRARKJeT11yy3QqyQ2SzMYcIfafrM+xtrtSFIby3c45CX6QQwrKECN\n0h+Liopzm4ETRRgDPQ0O3nw3fu5JSZEX/gSRYLMBaimCOzrkPWGDePKqq4AJE5QL29oC5rnxyhk/\nHrVKGUYrKoCbbw5an6g/OEZPg4NKoNe0KhYIvDJWkoKvigUCrowNltDMK/+yy1DiyZLoq/P48ZCG\nbohCwUBPg0NWlpyT3v9iaG2tPDc9mGCBXkOPHhaL+uYbGgN9QmoqUg2GPj/HmC+/xKkgC7aI+ouB\nngYHnQ7PpKSgfMeOc8dcLvkCbbD8MEDkAr1Kj37roUMobm4Ofg6gz6KpjqYm1LndyJ2ilNyVKHwM\n9DRobBMCBz7//NyB+np5o5YAG4P3sFrhVpka+cnevXi9sjL4OQIM3bx74AB2qqx47cMvDULlF19g\nlMGgmkOeKFwM9DRo5A8bhvLDh3se1x46hHWBskX6+PLECVz55ZeKZQcqKrBHQ2+83u1Gpcp+ug5J\ngjXAxuC9+PXoK4qL5V22iAYIAz0NGvl5eagoL+95/PWuXXghUOphH8np6aobhHdIEswaxsfXHT6M\nJ3buVCwLNdCL6uqehxUHDyI/K0tbXaJ+YKCnQSN//HhU+ATImooK5GhIfwAA1sxMtKkEekmSYNEw\n68aSnAypq0uxzNHZCWtammKZv2ePHsWK997reVx/6hQK8/I01SXqDwZ6GjQmFRVhZ20tGj37ANee\nOoUchY3PlVizs+FQWTAlSRIsGi7GmpOT0aES6NucTlg1pGIAgNTsbFR70yB0duLx5mY8/tBDmuoS\n9QcDPQ0aE2+/HW8vWIDUJUsAtxs1NTUYpmXGDYDk7Gw4hPJeDR0dHbBoGOu3WK2QnE7FsvlWK/I0\n5qrJGT0aNd4Ea8uWyTtszZ2rqS5Rf3BlLA0q1//pT8D11wO//jVqGxowWeNGGyabDW7Im4wk+gX1\nhQUFSNGQRsGSkqIa6B+1WICxYzW1ZdiYMXIahL/9Tb4VFzOPPA0oBnoaXBIT5U02pk/HtU4npmgM\n9Dq9Hh02G/RdXYBfoL88JUVTnpn0zEyMVktFrHHBFADkTJiAWkkCfvQjYP16QOOQD1F/ceiGBp+8\nPOCVV7C4sRGTNWxc4qVPSVFeNKVxwdQlkyfjjREjlAtDCPTDJk1CpxAQS5cCV1yhqQ5RONijp8Hp\n298Gtm0DNAy59EhOVg70GpOaqa6Mdbm0r64FYE5Lw9r//m/oeAGWzhMGehq8NGws3otaGoRwk5q1\nt8vDQSFsmH71ww9rfi5RuDh0Q0OH1SqnE/YXZlKz1tpavBhCkCc63/jppCHDnZwMl0Kqg5+XlaFC\nZePwXlSGbmpPnMCzaumLiWIAAz0NGQsPHMB7mzf3Ob6usRFtKoupfAmTCcXt7X1yyWvdGJwoWhjo\naciwms1weBcq+ZBcLlg05KnRGY24XAg4/Xrvjvp6WLVk0CSKEgZ6GjLUNgiX3G6YNSYkswCQGhp6\nHXM0NMDKFMMUwxjoaciwJiejTWHWjeR2w2K3azqHRadDh9+3AkdjI6zcHYpiGAM9DRlWqxUOhUDf\nAcCiMfOkRa+H5BfoC202zNOY54YoGhjoachItloh+c+OcbvxKgCTxqEbs14PyW/4Z2pmJr43eXKE\nWkkUeZwqQEPGkltvhe6zz3of7OzE981mzYudpiYny/lyfIWQ/oAoGhjoacjQKeW60boq1uPt8eP7\nbkbOQE8xjkM3NHQopUAIIUcNAOXVsQz0FOMY6GnoUAr0WhOaeSmtjmWgpxjHQE9Dh1qPPoShG6XE\nZuuOHMERhdQKRLGCgZ6GDJGUhE6/GTPHSkvxSy15brwUhm5eP3oUpaGcg+g8Y6CnIaPK4UBBeXnv\nY6dP4xOljJYqKl0u1NbW9jrm6OzUvDE4UTQw0NOQYc3OhsMveZnU2gpLCAnJnj10CO98/nmvYw6n\nk4GeYhoDPQ0ZydnZaAMgfIJ9h8MBcwgJySxmM6T29l7HHE4nrBkZkWomUcQx0NOQYTCbYQQg+Yyn\nS21tsISQkMxsNqPDb4ze4XLBmpUVqWYSRRwDPQ0pVp0ObWfO9DyWHI6QAr3FYumTRmFxYiLS1TYN\nJ4oBXBlLQ4o9IQFtdXXw9r+vGz0aF4awwbjFYkGN38XYn7tcQE5OBFtJFFkM9DSklI4fD/hkqixM\nSUFhfr7m+iOzsyFVVZ070NUl7zjFfPQUwxjoaWjxXzQVYgqE+Vdf3XsePVfF0iDAMXoaWqxWwHfe\nfLgrY9vaGOgp5jHQ09ASZo++z8pY9uhpEGCgp6HFP9CHmdTsVFkZ/uifzZIoxjDQ05DSnZSELp+t\nAP9QXIytJ05oP4Hf0M3R0lK84be1IFGsYaCnIWV5cTGe/8c/eh5/WlWF0yHkumnX6XDAZ8GVo7ER\nVs64oRjHQE9DSnJycq8NwqWuLlhCGGMvr6/HgrKynseOpiYkm0wRbSNRpDHQ05AyfORIlJ882fO4\nw+mEOTlZc32LzQbJJ1dO+TffIDczM6JtJIo0BnoaUmbdey82VlT0JDaTnE5YUlI01zenpqLDJ9Bv\n/Oor3Hj77RFvJ1EkMdDTkFJ4ww0w6/U4sG4dAEDq7obFZtNc32K3n+vROxy4S5Jw/QMPDERTiSIm\nEoF+HoBZABaplD/tuVcrJzpvdHo95k2ahKMffAAA+PfsbIwbO1ZzfUtaGnrm3GzbhgeuugpW5rmh\nGBduoL/Uc7/Zcz9V4TmLABwFUKZQRnTePbtiBb5bXQ0AmGMwICM3V3NdU2oqLgEguruBDRuA2bMH\nqJVEkRNuoF8AwDvX7DiAGxWeswjAOABbwnwtosiYORP4/HN5PnyIC6Z0ej2+NJuh6+oCPv6YgZ4G\nhXADfRqABp/HStvspEMe2lka5msRRUZqKnDJJcCnn4aeAgGQn//NN0BDAzBlysC0kSiCIpG9Uhek\nfLXn/ibIAX+z/xNWrlzZ8++ioiIUFRVFoFlEAcyeLffIQ01qBsjPX78euOkmQM/5DDTwtm7diq1b\nt/a7frAgDShfRG0AsAbAKgAbIQfv+QAKADzrV9f73KUAmnAu8HsJIURorSYK1xdfAPffDxw+LA/f\nhLBBOAoL8VBLCxYuXoxrnnxy4NpIpEKn0wHa4jcAbT16/8Ds6x0A0yAH+gLIQR+Qh3SaII/b7/Yc\ny/ApJ4quadPwbnk5Vrvd2BhKkAfgMpvx1vHj+Pn8+QPUOKLICvd7Z4nnfhbkwL7X83iT534z5Au0\n8wDU+ZQTRVdCAl6zWrGpH98m32hrQ2piInKnKk0yI4o9mrv+A4hDNxQVT37nO/j53/6GUD9/Op0O\n03Ny8KVniibR+Rbq0A2vJNGQdedPf4oEXeh9nSmZmfjZww8PQIuIBgZ79EREgwx79ERE1AsDPRFR\nnGOgjyPhLKigvvh+Rhbfz+hhoI8j/EWKLL6fkcX3M3oY6ImI4hwDPRFRnIuF6ZV7AVwS7UYQEQ0i\n+wAwdSoRERER0WDytN/jYNu49oiVMfp+/wBEA4SfwcjhvtHhWwz5M+mlZRvXHrEQ6MP6AUgVf7n6\nj5/ByOK+0eF7GXLady8t27j2iIVAH9YPQKr4y9V//AxGFveNjjwt27j2iIVA7y+kH4BU8Zer//gZ\njCzuGz0wBn1Ss1iY9jnY8ZcrPPwMRs5qyMNgGZA/kxS+Jsi/4wBgB1Af6MmR2Bw8mEB7zioJ6Qcg\nVUE3ZSdV/AxGju++0fUAxoCfxUhQ28ZV0fkI9IH2nFUS0g8whAX6A7oY8i8Vf7n6h5/ByOG+0ZEx\nH/Jn8n4Ar0DexnUa+m7jGrPmQw5Q9/scWwRObQvHLACpnn+vAlfQ9Qc/g5Ezz3N7LNoNGao4Dhm/\nvFNWCwD8OpoNISIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIjpf/g+b2Duf8nScEwAAAABJRU5ErkJg\ngg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f11c60d4f10>"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Some manipulation of the coefficients is required to reproduce the filter. Remember the convolution for the MKS is\n",
      "\n",
      "$$ p \\left[i\\right] = \\sum_h^{n-1} \\alpha_h \\left[j\\right] m_h \\left[i - j\\right] $$\n",
      "\n",
      "However, the `MKSRegressionModel` solves a modified form of this. There are always redundant coefficients since\n",
      "\n",
      "$$ \\sum\\limits_{h=0}^{n-1} m_h \\left[i\\right] = 1 $$\n",
      "\n",
      "Thus, the convolution must be rewritten taking this into account. We can rewrite the convolution as\n",
      "\n",
      "$$ \\begin{split}\n",
      "p \\left[i\\right] &= \\sum\\limits_{h=0}^n \\alpha_h \\left[j\\right] m_h \\left[i - j\\right] \\\\\n",
      "        &= \\sum\\limits_{h=0}^{n - 2}  \\alpha_h \\left[j\\right] m_h \\left[i - j\\right] + \n",
      "           \\alpha_{n-1} \\left[j\\right] \\left( I\\left[i - j\\right] - \\sum\\limits_{h=0}^{n - 2} m_h \\left[i - j\\right] \\right) \\\\\n",
      "        &=  \\alpha_{n - 1} \\left[ j \\right]  I\\left[i - j\\right] +\n",
      "           \\sum\\limits_{h=0}^{n - 2} \\left(\\alpha_h\\left[j\\right] - \\alpha_{n-1}\\left[j\\right] \\right) m_h\\left[i - j\\right] \\\\\n",
      "        &= \\beta + \\sum\\limits_{h=0}^{n - 2} b_h \\left[ j \\right] m_h \\left[ i - j \\right] \n",
      "\\end{split}$$\n",
      "\n",
      "where $\\beta = \\alpha_{n - 1} \\left[ j \\right]  I\\left[i - j\\right]$ is just a constant and $I\\left[i\\right] = 1$ for all $i$.\n",
      "This removes the redundancies from the regression. Manipulation of the $\\beta$ and $b_0$ demonstrate that the filter is reproduced."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}