{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this post we discuss modeling covariances beyond the Gaussian approximation.\n",
    "\n",
    "<!-- END_TEASER -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>.grade {\n",
       "   background-color: #66FFCC;\n",
       "}\n",
       "</style>"
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       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<script id=\"MathJax-Element-48\" type=\"math/tex\">\\newcommand{\\vect}[1]{\\mathbf{#1}}\n",
       "\\newcommand{\\uvect}[1]{\\hat{#1}}\n",
       "\\newcommand{\\abs}[1]{\\lvert#1\\rvert}\n",
       "\\newcommand{\\norm}[1]{\\lVert#1\\rVert}\n",
       "\\newcommand{\\I}{\\mathrm{i}}\n",
       "\\newcommand{\\ket}[1]{\\left|#1\\right\\rangle}\n",
       "\\newcommand{\\bra}[1]{\\left\\langle#1\\right|}\n",
       "\\newcommand{\\braket}[1]{\\langle#1\\rangle}\n",
       "\\newcommand{\\Braket}[1]{\\left\\langle#1\\right\\rangle}\n",
       "\\newcommand{\\op}[1]{\\mathbf{#1}}\n",
       "\\newcommand{\\mat}[1]{\\mathbf{#1}}\n",
       "\\newcommand{\\d}{\\mathrm{d}}\n",
       "\\newcommand{\\pdiff}[3][]{\\frac{\\partial^{#1} #2}{\\partial {#3}^{#1}}}\n",
       "\\newcommand{\\diff}[3][]{\\frac{\\d^{#1} #2}{\\d {#3}^{#1}}}\n",
       "\\newcommand{\\ddiff}[3][]{\\frac{\\delta^{#1} #2}{\\delta {#3}^{#1}}}\n",
       "\\newcommand{\\floor}[1]{\\left\\lfloor#1\\right\\rfloor}\n",
       "\\newcommand{\\ceil}[1]{\\left\\lceil#1\\right\\rceil}\n",
       "\\DeclareMathOperator{\\Tr}{Tr}\n",
       "\\DeclareMathOperator{\\erf}{erf}\n",
       "\\DeclareMathOperator{\\erfi}{erfi}\n",
       "\\DeclareMathOperator{\\sech}{sech}\n",
       "\\DeclareMathOperator{\\sgn}{sgn}\n",
       "\\DeclareMathOperator{\\order}{O}\n",
       "\\DeclareMathOperator{\\diag}{diag}\n",
       "\n",
       "\\newcommand{\\mylabel}[1]{\\label{#1}\\tag{#1}}\n",
       "\\newcommand{\\degree}{\\circ}</script>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "<IPython.core.display.HTML object>"
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     "metadata": {},
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    {
     "data": {
      "text/html": [
       "\n",
       "<i>\n",
       "<p>This cell contains some definitions for equations and some CSS for styling\n",
       "  the notebook. If things look a bit strange, please try the following:\n",
       "<ul>\n",
       "  <li>Choose \"Trust Notebook\" from the \"File\" menu.</li>\n",
       "  <li>Re-execute this cell.</li>\n",
       "  <li>Reload the notebook.</li>\n",
       "</ul>\n",
       "</p>\n",
       "</i>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import mmf_setup;mmf_setup.nbinit(hgroot=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "ename": "TypeError",
     "evalue": "_parse_args() missing 1 required positional argument: 's'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-3b06b66c0832>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mscipy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mstats\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mstats\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlognorm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, *args, **kwds)\u001b[0m\n\u001b[1;32m    770\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    771\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__call__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 772\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfreeze\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    773\u001b[0m     \u001b[0m__call__\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__doc__\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfreeze\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__doc__\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    774\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py\u001b[0m in \u001b[0;36mfreeze\u001b[0;34m(self, *args, **kwds)\u001b[0m\n\u001b[1;32m    767\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    768\u001b[0m         \"\"\"\n\u001b[0;32m--> 769\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv_frozen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    770\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    771\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__call__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages/scipy/stats/_distn_infrastructure.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, dist, *args, **kwds)\u001b[0m\n\u001b[1;32m    435\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    436\u001b[0m         \u001b[0;31m# a, b may be set in _argcheck, depending on *args, **kwds. Ouch.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 437\u001b[0;31m         \u001b[0mshapes\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parse_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    438\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_argcheck\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mshapes\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    439\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: _parse_args() missing 1 required positional argument: 's'"
     ]
    }
   ],
   "source": [
    "%pylab inline --no-import-all\n",
    "from scipy import stats\n",
    "\n",
    "x = np.linspace\n",
    "f = stats.lognorm.pdf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Characterization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a random variable $X$ with probability distribution $f(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Addition of Stochastic Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $X_n$ be $N$ independent stochastic variables with joint probability density $P_X(x_1, x_2, \\dots, x_N) = P(\\vect{x})$.  Then, the probability distribution for the sum $Y = \\sum_n X_n$ is:\n",
    "\n",
    "$$\n",
    "  P_Y(y) = \\idotsint \\delta\\Bigl(\\sum_{n}x_n - y\\Bigr)P_X(\\vect{x})\\prod_{n=1}^{N}\\d{x_n}\n",
    "$$\n",
    "\n",
    "If the variables $X_n$ are independent, then $P_X(\\vect{x}) = \\prod_{n}P_{X_n}(x_n)$, and this simplifies:\n",
    "\n",
    "$$\n",
    "  P_Y(y) = \\idotsint \\delta\\left(\\sum_{n}x_n - y\\right)\\prod_{n=1}^{N}P_{X_n}(x_n)\\d{x_n}.\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Environment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Execute the following two cells to define a conda environment for running this notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting environment.soerp.yml\n"
     ]
    }
   ],
   "source": [
    "%%file environment.soerp.yml\n",
    "name: blog.soerp\n",
    "channels:\n",
    "  - defaults\n",
    "  - conda-forge\n",
    "dependencies:\n",
    "  - ipykernel\n",
    "  - notebook\n",
    "  - uncertainties\n",
    "  - scipy\n",
    "  - matplotlib\n",
    "  - pip:\n",
    "    - soerp\n",
    "    - mmf_setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving environment: done\n",
      "Preparing transaction: done\n",
      "Verifying transaction: done\n",
      "Executing transaction: done\n",
      "Looking in links: https://bitbucket.org/mforbes/mypi/\n",
      "Requirement already satisfied: soerp in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from -r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (0.9.6)\n",
      "Requirement already satisfied: mmf_setup in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from -r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.1.11)\n",
      "Requirement already satisfied: matplotlib in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (3.0.0)\n",
      "Requirement already satisfied: numpy in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.15.2)\n",
      "Requirement already satisfied: ad in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.3.2)\n",
      "Requirement already satisfied: scipy in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.1.0)\n",
      "Requirement already satisfied: python-hglib in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (2.6.1)\n",
      "Requirement already satisfied: nbstripout>=0.2.0 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.3.3)\n",
      "Requirement already satisfied: cycler>=0.10 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (0.10.0)\n",
      "Requirement already satisfied: kiwisolver>=1.0.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.0.1)\n",
      "Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (2.2.2)\n",
      "Requirement already satisfied: python-dateutil>=2.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (2.7.3)\n",
      "Requirement already satisfied: nbformat in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.4.0)\n",
      "Requirement already satisfied: six in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from cycler>=0.10->matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (1.11.0)\n",
      "Requirement already satisfied: setuptools in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from kiwisolver>=1.0.1->matplotlib->soerp->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 1)) (40.4.3)\n",
      "Requirement already satisfied: ipython-genutils in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (0.2.0)\n",
      "Requirement already satisfied: jsonschema!=2.5.0,>=2.4 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (2.6.0)\n",
      "Requirement already satisfied: traitlets>=4.1 in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.3.2)\n",
      "Requirement already satisfied: jupyter-core in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.4.0)\n",
      "Requirement already satisfied: decorator in /data/apps/conda/envs/blog.soerp/lib/python3.7/site-packages (from traitlets>=4.1->nbformat->nbstripout>=0.2.0->mmf_setup->-r /Users/mforbes/work/mmfbb/forbes_group_website/posts/draft/blog/condaenv.79jey1_z.requirements.txt (line 2)) (4.3.0)\n",
      "\u001b[31mmkl-random 1.0.1 requires cython, which is not installed.\u001b[0m\n",
      "#\n",
      "# To activate this environment, use\n",
      "#\n",
      "#     $ conda activate blog.soerp\n",
      "#\n",
      "# To deactivate an active environment, use\n",
      "#\n",
      "#     $ conda deactivate\n",
      "\n"
     ]
    }
   ],
   "source": [
    "!conda env update -f environment.soerp.yml"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:blog.soerp]",
   "language": "python",
   "name": "conda-env-blog.soerp-py"
  },
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