02-elementary_propagation_s.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Elementary Propagation\n",
    "S Singleton based an assignment by J Mills, R Estler"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Information\n",
    "- *An Introduction to Error Analysis*, Taylor, University Science Books, 1997, Ch 3.\n",
    "- Wikipedia [Propagation of Uncertainty](https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Simplification)\n",
    "- NIST's Engineering Statistics Handbook [Propagation of error considerations](http://itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm)\n",
    "- \"Error propagation: A functional approach,\" I.F. Hughes, T.P.A. Hase, *J. Chem. Ed.*, 89, 2012, p821,  [DOI: 10.1021/ed2004627](http://pubs.acs.org/doi/abs/10.1021/ed2004627)\n",
    "- *A Student's Guide to Data and Error Analysis*, Berendsen, Cambridge U. Press, 2011, Ch 3.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Learning Outcomes\n",
    "- Use the general formula for propagation of error\n",
    "- Use `sympy` to derive symbolic expressions useful in the propagation of error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task\n",
    "Apply the general uncertainty propagation equation to the measurement situation listed\n",
    "below to develop specific explicit uncertainty expressions.\n",
    "\n",
    "For example\n",
    "<center>Absorbance = absorptivity $\\times$ path length $\\times$ molarity</center>\n",
    "\n",
    "$$A = a \\times b \\times M$$\n",
    "\n",
    "The uncertainty in $A$ is $\\delta A$, and so on.\n",
    "\n",
    "\n",
    "$$\n",
    "  \\delta A = \\sqrt{\\left( \\frac{\\partial A}{\\partial a}\\right) ^2\n",
    "    _{bM} \\delta a^2 + \\left(\\frac{\\partial A}{\\partial b}\n",
    "    \\right)^2_{aM} \\delta b^2 + \\left(\\frac{\\partial A}{\\partial\n",
    "        M}\\right)^2_{ab} \\delta\n",
    "    M^2 }\n",
    "$$ \n",
    "\n",
    "Take the derivative of $A$ with respect to each variable and simplify:\n",
    "\n",
    "$$\n",
    "  = \\sqrt{(b^2 M^2 \\delta a^2 + a^2 M^2 \\delta b^2 + a^2 b^2 \\delta\n",
    "  M^2)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How it can be done"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a symbolic manipulation, so we use `sympy`. It's safe to '`import *`' because we won't be doing `numpy`-related work in this notebook.  The uncertainty terms (e.g, $\\delta a$) could be included, but I chose not to. In this case, I would \"manually\" include these terms in a written solution (sometimes the computer isn't the best tool for everything?)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# uncomment next line if running python2\n",
    "#from __future__ import print_function, division\n",
    "from sympy import var, sqrt, init_printing, diff\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# declare sympy variables\n",
    "var('a b M')\n",
    "\n",
    "# assign A an expression\n",
    "A = a * b * M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, calculate the partial derivatives with respect to each variable, e.g., $\\partial A / \\partial a$. I will show three ways to accomplish the task. There is usually a trade-off between readability and coding efficiency. Choose the former when in doubt."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Approach 1: compute each term individually"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAUBAMAAABltyL1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left ( M b, \\quad M a, \\quad a b\\right )$$"
      ],
      "text/plain": [
       "(M⋅b, M⋅a, a⋅b)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dA_da = diff(A, a)\n",
    "dA_db = diff(A, b)\n",
    "dA_dM = diff(A, M)\n",
    "\n",
    "dA_da, dA_db, dA_dM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAaBAMAAADfznSWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAmSK7q0TNEFTdiWZ2\n7zJQnLHkAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADgElEQVRIDdVUS2gUWRQ9Venuqv4k3SB+cJTU\nMKObLBI36kbolS7M0GFwNaK2g1RmYMTys1BEutCFIkKC6EYUOy4EGRjaIAwuxBJxI4rORGYYFNKI\nuE2jRuNn0t57X72qUtLueuGB3HfuO6f71Kv7OoBGf7u7aOkgmEMR7TZJjXQ7Ifr+1RHrOlnc9QQd\nYFU14zW3aGnYxiyph3xijadYB1ekR4Tthq8+pOolbAzbmCV1xXNOqqrYwq5Ij4i4D6nPhPU0ZhxF\nY/aJQZqs0/fiS65Ij4i4f6ear8wxfzDln8VgeCFjJjYu19o+1cL/67Mj5iysDePAAi6yiL7/v5DQ\nhsAiP5B6R6WvwiljHm8wYqZ6oFIn9myWiv0GyJSJLODiXdZ3acIrodDgOvCISr7C9DwXgWan9IZ5\no0z0WotKNgCKPhHtt2kjAdblTTEJkZV1ci0tf05RyZVCIWZRVP67Kj3QkSFyTNDfDo+K9n8WRXqa\nDqaMgMX0HheUVgLmjx+IXZWeS8SiKMN4CKwaDCiPH+hK0vVpFOu5cdLFCPMcW7mnqAf0KvPz9Cj1\ndBPm8hX/KsYa4qhCCwhuNoGfsJ1mcW8ZxM+mKOrbifue6JnhW74yYutucuTr7DMbY00cT9EXPT9x\nzMM2J9NSjMU4asCeh+1V6AL9cmIdrNdezRE/m3RU+m/sUXpt1HohRlLHGjRej315Z9C3/N4hYLrd\nhvUYxpAw1hJRk+YcDoIm2ttut/iSFevsF+ioIw0sUjpd4HdiJL13HDgpPgPFoAAeAsN4IxdMNZdd\n94Lryr1FCR8QmDxRhlFFrS4MOdf99R/XLVPX7+k7uQQW/4QEBfqBjAoz0Fs6Ch4Cg05XawpTJZpV\nCX/ZDk9UkClhhx/y+AVOwQoNO5GW684W6yXskngHkBltgIYgoGcdC6ksOspswD0Mnqig2OAjaIQv\nkA5ih4b3/Ho0+p2MI3wSxhPwEAQ1H2ttJ2xo0VF5BysDPrNCcYSGGkFHzaGnelR2Z0Fz0xisb1F0\nHIV59OkhZIL8mZz20KqjsvRfsomZIJRSjZ5S7NLX4jdcL9dlezPWxHJP62dpfngb2IG94e0mJVnD\nf3x/J3bpqNT0eaLHps94oW3FNwmTjjpw9+nwiOzvvRUfCulX5YS5I9Wn6mhgQUd1Ml1sdlKS+/uS\nTSduOZ0UtX/7y/JXqH4EY3hXr5YovSAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}}$$"
      ],
      "text/plain": [
       "   _______________________\n",
       "  ╱  2  2    2  2    2  2 \n",
       "╲╱  M ⋅a  + M ⋅b  + a ⋅b  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# square, sum, and square root these terms (sum in quadrature)\n",
    "deltaA = sqrt(dA_da**2 + dA_db**2 + dA_dM**2)\n",
    "deltaA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Approach 2: use a `for` loop and accumulation variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcMAAAAmBAMAAABKVu1oAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left ( M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}, \\quad \\sqrt{M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}}\\right )$$"
      ],
      "text/plain": [
       "⎛                          _______________________⎞\n",
       "⎜ 2  2    2  2    2  2    ╱  2  2    2  2    2  2 ⎟\n",
       "⎝M ⋅a  + M ⋅b  + a ⋅b , ╲╱  M ⋅a  + M ⋅b  + a ⋅b  ⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cumsum = 0  # a variable for the cumulative sum\n",
    "\n",
    "for v in ['a', 'b', 'M']:\n",
    "    cumsum = cumsum + diff(A, v)**2\n",
    "\n",
    "cumsum, sqrt(cumsum)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Approach 3: list comprehension\n",
    "(the melt-your-brain approach)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAaBAMAAADfznSWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTSduOZ0UtX/7y/JXqH4EY3hXr5YovSAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{M^{2} a^{2} + M^{2} b^{2} + a^{2} b^{2}}$$"
      ],
      "text/plain": [
       "   _______________________\n",
       "  ╱  2  2    2  2    2  2 \n",
       "╲╱  M ⋅a  + M ⋅b  + a ⋅b  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(sum(diff(A,v)**2 for v in['a', 'b', 'M']))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Your Turn\n",
    "Derive uncertainty relationships for the following expressions. Constants are noted; all other labels are variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Root-mean-square speed of a gas-phase molecule\n",
    "$$ \n",
    " v_{\\text{rms}} = \\sqrt{\\frac{3 RT}{M}}   \\qquad  R \\text{ constant}\n",
    " $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### van der Waals [equation of state](https://en.wikipedia.org/wiki/Van_der_Waals_equation) for real gases\n",
    " \n",
    "$$P = \\frac{RT}{V-b} - \\frac{a}{V^2} \\qquad  R \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compressibility of an ideal gas\n",
    "$$Z = \\frac{PV}{RT} \\qquad  R \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rotational partition function\n",
    " \n",
    "$$q_r  = \\frac{2IkT}{\\sigma h^2} \\qquad k,\\: \\sigma,\\: h\n",
    "  \\text{ constant}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Translational partition function\n",
    " \n",
    "$$ q_{tr} = \\left( \\frac{2 \\pi mkT}{h^2}\\right) ^{3/2} \\qquad\n",
    "  k, h \\text{ constant}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}