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{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"images/mcg.jpg\", style=\"width: 100px\">\n",
    "\n",
    "\n",
    "# Linear Gaussian Bayesian Networks (GBNs)\n",
    "\n",
    "## Generate $x_1$ $x_2$ and $Y$ from a Multivariate Gaussian Distribution with a Mean and a Variance.\n",
    "\n",
    "What if the inputs to the linear regression were correlated? This often happens in linear dynamical systems. Linear Gaussian Models are useful for modeling probabilistic PCA, factor analysis and linear dynamical systems. Linear Dynamical Systems have variety of uses such as tracking of moving objects. This is an area where Signal Processing methods have a high overlap with Machine Learning methods. When the problem is treated as a state-space problem with added stochasticity, then the future samples depend on the past. The latent parameters, $\\beta_i$ where $i \\in [1,...,k]$ provide a linear combination of the univariate gaussian distributions as shown in the figure. \n",
    "\n",
    "<img src=\"images/gbn.png\", style=\"width: 400px\">\n",
    "\n",
    "The observed variable, $y_{jx}$ can be described as a sample that is drawn from the conditional distribution:\n",
    "\n",
    "$$\\mathcal{N}(y_{jx} | \\sum_{i=1}^k \\beta_i^T x_i + \\beta_0; \\sigma^2)$$\n",
    "\n",
    "The latent parameters $\\beta_is$ and $\\sigma^2$ need to be determined. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('Y', ['X1', 'X2'])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# from pgmpy.factors.continuous import LinearGaussianCPD\n",
    "\n",
    "import sys\n",
    "import numpy as np\n",
    "import pgmpy\n",
    "\n",
    "sys.path.insert(0, \"../pgmpy/\")\n",
    "from pgmpy.factors.continuous import LinearGaussianCPD\n",
    "\n",
    "mu = np.array([7, 13])\n",
    "sigma = np.array([[4, 3], [3, 6]])\n",
    "\n",
    "cpd = LinearGaussianCPD(\n",
    "    \"Y\", evidence_mean=mu, evidence_variance=sigma, evidence=[\"X1\", \"X2\"]\n",
    ")\n",
    "cpd.variable, cpd.evidence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### import numpy as np\n",
    "%matplotlib inline\n",
    "\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "import numpy as np\n",
    "\n",
    "from matplotlib import cm\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from scipy.stats import multivariate_normal\n",
    "from matplotlib import pyplot\n",
    "\n",
    "# Obtain the X and Y which are jointly gaussian from the distribution\n",
    "mu_x = np.array([7, 13])\n",
    "sigma_x = np.array([[4, 3], [3, 6]])\n",
    "\n",
    "# Variables\n",
    "states = [\"X1\", \"X2\"]\n",
    "\n",
    "# Generate samples from the distribution\n",
    "X_Norm = multivariate_normal(mean=mu_x, cov=sigma_x)\n",
    "X_samples = X_Norm.rvs(size=10000)\n",
    "X_df = pd.DataFrame(X_samples, columns=states)\n",
    "\n",
    "# Generate\n",
    "X_df[\"P_X\"] = X_df.apply(X_Norm.pdf, axis=1)\n",
    "X_df.head()\n",
    "\n",
    "g = sns.jointplot(X_df[\"X1\"], X_df[\"X2\"], kind=\"kde\", height=10, space=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Gaussian Models - The Process\n",
    "\n",
    "The linear gaussian model in supervised learning scheme is nothing but a linear regression where inputs are drawn from a jointly gaussian distribution. \n",
    "\n",
    "Determining the Latent Parameters via Maximum Likelihood Estimation (MLE)\n",
    "\n",
    "The samples drawn from the conditional linear gaussian distributions are observed as:\n",
    "\n",
    "$$ p(Y|X) = \\cfrac{1}{\\sqrt(2\\pi\\sigma_c^2} \\times exp(\\cfrac{(\\sum_{i=1}^k \\beta_i^T x_i + \\beta_0 - x[m])^2}{2\\sigma^2})$$\n",
    "\n",
    "Taking log,\n",
    "\n",
    "$$ log(p(Y|X)) = (\\sum_{i=1}^k[-\\cfrac{1}{2}log(2\\pi\\sigma^2) - \\cfrac{1}{2\\sigma^2}( \\beta_i^T x_i + \\beta_0 - x[m])^2)]$$\n",
    "\n",
    "Differentiating w.r.t $\\beta_i$, we can get k+1 linear equations as shown below:\n",
    "\n",
    "\n",
    "### The Condtional Distribution p(Y|X)\n",
    "\n",
    "<img src=\"images/lgm.png\", style=\"width: 700px\">\n",
    "\n",
    "The betas can easily be estimated by inverting the coefficient matrix and multiplying it to the right-hand side."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/harishkashyap/Documents/MCG/pgmpy/venv/lib/python3.6/site-packages/scipy-1.1.0-py3.6-macosx-10.7-x86_64.egg/scipy/stats/stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.\n",
      "  return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta_vec = np.array([0.7, 0.3])\n",
    "beta_0 = 2\n",
    "sigma_c = 4\n",
    "\n",
    "\n",
    "def genYX(x):\n",
    "    x = [x[\"X1\"], x[\"X2\"]]\n",
    "    var_mean = np.dot(beta_vec.transpose(), x) + beta_0\n",
    "    Yx_sample = np.random.normal(var_mean, sigma_c, 1)\n",
    "    return Yx_sample[0]\n",
    "\n",
    "\n",
    "X_df[\"(Y|X)\"] = X_df.apply(genYX, axis=1)\n",
    "X_df.head()\n",
    "\n",
    "sns.distplot(X_df[\"(Y|X)\"])\n",
    "# X_df.to_csv('gbn_values.csv', index=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(Y|X)    108620.019971\n",
      "X1        70061.804718\n",
      "X2       130484.483348\n",
      "dtype: float64\n",
      "[108620.0199709961]\n",
      "         b0_coef  b1_coef      b2_coef\n",
      "0   10000.000000  70061.8       130484\n",
      "1   70061.804718   530593       943171\n",
      "2  130484.483348   943171  1.76157e+06\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([1.75405452, 0.69412373, 0.32531005]), 4.045369149779373)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cpd.fit(X_df, states=[\"(Y|X)\", \"X1\", \"X2\"], estimator=\"MLE\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "For any questions feel free to contact hkashyap [at] icloud.com. Thanks to Praveen Kaushik for the diagrams(diagram.ai), Kiran Byadarhaly and Karthik Chandrashekhar for proof reading the math."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "pgmpy",
   "language": "python",
   "name": "pgmpy"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
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