{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*This example is a Jupyter notebook. You can download it or run it interactively on mybinder.org.*\n",
    "\n",
    "# Linear regression\n",
    "\n",
    "## Data\n",
    "\n",
    "The true parameters of the linear regression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "k = 2 # slope\n",
    "c = 5 # bias\n",
    "s = 2 # noise standard deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "nbsphinx": "hidden"
   },
   "outputs": [],
   "source": [
    "# This cell content is hidden from Sphinx-generated documentation\n",
    "%matplotlib inline\n",
    "np.random.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.arange(10)\n",
    "y = k*x + c + s*np.random.randn(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model\n",
    "\n",
    "The regressors, that is, the input data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = np.vstack([x, np.ones(len(x))]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we added a column of ones to the regressor matrix for the bias term. We model the slope and the bias term in the same node so we do not factorize between them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from bayespy.nodes import GaussianARD\n",
    "B = GaussianARD(0, 1e-6, shape=(2,))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first element is the slope which multiplies x and the second element is the bias term which multiplies the constant ones. Now we compute the dot product of X and B:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from bayespy.nodes import SumMultiply\n",
    "F = SumMultiply('i,i', B, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The noise parameter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from bayespy.nodes import Gamma\n",
    "tau = Gamma(1e-3, 1e-3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The noisy observations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Y = GaussianARD(F, tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference\n",
    "\n",
    "Observe the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Y.observe(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct the variational Bayesian (VB) inference engine by giving all stochastic nodes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from bayespy.inference import VB\n",
    "Q = VB(Y, B, tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Iterate until convergence:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 1: loglike=-4.080970e+01 (0.003 seconds)\n",
      "Iteration 2: loglike=-4.057132e+01 (0.003 seconds)\n",
      "Iteration 3: loglike=-4.056572e+01 (0.003 seconds)\n",
      "Iteration 4: loglike=-4.056551e+01 (0.003 seconds)\n",
      "Converged at iteration 4.\n"
     ]
    }
   ],
   "source": [
    "Q.update(repeat=1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Results\n",
    "\n",
    "Create a simple predictive model for new inputs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xh = np.linspace(-5, 15, 100)\n",
    "Xh = np.vstack([xh, np.ones(len(xh))]).T\n",
    "Fh = SumMultiply('i,i', B, Xh)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we use the learned node B but create a new regressor array for predictions. Plot the predictive distribution of noiseless function values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9ivOi9nY016+zLy6Ogtdf59mvf42PtzdWzc29x33wAH75S7lCF/OGXKGL6fWi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\nE1LQhRBinpCCLoQQ88T/BcpRV5LmFP/wAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f9ba0a03240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import bayespy.plot as bpplt\n",
    "bpplt.pyplot.figure()\n",
    "bpplt.plot(Fh, x=xh, scale=2)\n",
    "bpplt.plot(y, x=x, color='r', marker='x', linestyle='None')\n",
    "bpplt.plot(k*xh+c, x=xh, color='r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the above plot shows two standard deviation of the posterior of the noiseless function, thus the data points may lie well outside this range. The red line shows the true linear function. Next, plot the distribution of the noise parameter and the true value, 2−2=0.25:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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77CvDVSiVUv7ABK1167TnbwBaa/3+I7bPAVzVWhdL5zWXXoWyQ4cOdOnShb59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\n+2lTV2MAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f9ba0a6ff28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bpplt.pyplot.figure()\n",
    "bpplt.pdf(tau, np.linspace(1e-6,1,100), color='k')\n",
    "bpplt.pyplot.axvline(s**(-2), color='r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The noise level is captured quite well, although the posterior has more mass on larger noise levels (smaller precision parameter values). Finally, plot the distribution of the regression parameters and mark the true value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DUVFRgbi4OAiFQrnx/eLiYtjY2GDs2LF17jbGMO+CBX7mk3D8+HEoKSnJ7beb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\nwBD8ABCY/wH0dew5Kx3OKQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f9ba095fa90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bpplt.pyplot.figure();\n",
    "bpplt.contour(B, np.linspace(1,3,1000), np.linspace(1,9,1000),\n",
    "              n=10, colors='k');\n",
    "bpplt.plot(c, x=k, color='r', marker='x', linestyle='None',\n",
    "           markersize=10, markeredgewidth=2)\n",
    "bpplt.pyplot.xlabel(r'$k$');\n",
    "bpplt.pyplot.ylabel(r'$c$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the true parameters are captured well by the posterior distribution.\n",
    "\n",
    "## Improving accuracy\n",
    "\n",
    "The model can be improved by not factorizing between B and tau but learning their joint posterior distribution. This requires a slight modification to the model by using GaussianGammaISO node:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from bayespy.nodes import GaussianGamma\n",
    "B_tau = GaussianGamma(np.zeros(2), 1e-6*np.identity(2), 1e-3, 1e-3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This node contains both the regression parameter vector and the noise parameter. We compute the dot product similarly as before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F_tau = SumMultiply('i,i', B_tau, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, Y is constructed as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Y = GaussianARD(F_tau, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because the noise parameter is already in F_tau we can give a constant one as the second argument. The total noise parameter for Y is the product of the noise parameter in F_tau and one. Now, inference is run similarly as before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 1: loglike=-4.130423e+01 (0.002 seconds)\n",
      "Iteration 2: loglike=-4.130423e+01 (0.002 seconds)\n",
      "Converged at iteration 2.\n"
     ]
    }
   ],
   "source": [
    "Y.observe(y)\n",
    "Q = VB(Y, B_tau)\n",
    "Q.update(repeat=1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the method converges immediately. This happens because there is only one unobserved stochastic node so there is no need for iteration and the result is actually the exact true posterior distribution, not an approximation. Currently, the main drawback of using this approach is that BayesPy does not yet contain any plotting utilities for nodes that contain both Gaussian and gamma variables jointly.\n",
    "\n",
    "## Further extensions\n",
    "\n",
    "The approach discussed in this example can easily be extended to non-linear regression and multivariate regression. For non-linear regression, the inputs are first transformed by some known non-linear functions and then linear regression is applied to this transformed data. For multivariate regression, X and B are concatenated appropriately: If there are more regressors, add more columns to both X and B. If there are more output dimensions, add plates to B."
   ]
  }
 ],
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