{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Shuttlecocks and the Akaike Information Criterion\n",
    "#A worksheet to help explore the project in Section 3.5.4.\n",
    "\n",
    "#The Data: First, the data for the shuttlecock's fall, in time (seconds)/distance (meters) pairs:\n",
    "shuttledata = [[0, 0], [0.347, 0.61], [0.47, 1.00], [0.519, 1.22], [0.582, 1.52], [0.650, 1.83], [0.674, 2.00], \n",
    "               [0.717, 2.13], [0.766, 2.44], [0.823, 2.74], [0.870, 3.00], [1.031, 4.00], [1.193, 5.00], \n",
    "               [1.354, 6.00], [1.501, 7.00], [1.726, 8.50], [1.873, 9.50]];\n",
    "N = len(shuttledata);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#A plot\n",
    "plt1 = scatter_plot(shuttledata,facecolor='red');\n",
    "show(plt1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#The Model: We might posit a model of the form v'(t) = g (no air resistance) and consider g as an unknown,\n",
    "#to be estimated. Then the governing ODE is (from equation (3.68) in the text)\n",
    "var('g t');\n",
    "v = function('v')(t)\n",
    "de = diff(v,t) == g\n",
    "vsol(t) = desolve(de,v,[0,0],t); #Solve with v(0) = 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Integrate to obtain the position in terms of t and k, taking x(0) = 0.\n",
    "var('tau');\n",
    "x(t,g) = integral(vsol(tau),tau,0,t);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Estimating Parameters: Form a sum of squares\n",
    "SS = function('SS')(g);\n",
    "SS(g) = add((x(shuttledata[i][0])-shuttledata[i][1])^2 for i in range(N));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#A quick plot of SS(g)\n",
    "plot(SS(g), (g,0,15)).show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#We can solve by setting dSS/dg = 0 and solving for g.\n",
    "criteqn = diff(SS,g) == 0;\n",
    "gbest = find_root(criteqn,5,15) #Look between 5 and 15."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#The residual is\n",
    "SS(gbest)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Plot the function x(t) with this best choice for g.\n",
    "plt2 = plot(x(t,gbest), t, 0, 1.873);\n",
    "pp = plt1 + plt2;\n",
    "show(plt1+plt2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#An alternative is to use Sage's \"model fitting\" ability, to adjust\n",
    "#k in the function x(t,g) to best fit the data.\n",
    "model(t) = x(t,g); #Specify the model to be fit, with \"t\" as the independent variable\n",
    "sol = find_fit(shuttledata,model,parameters=[g]) #Fit the model by adjusting k and b\n",
    "bestx(t) = model(g=sol[0].rhs()) #Define the best fit function of the form in \"model\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bestx(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.2",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}