LECTURE4_nonlinear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "16ba4d47-6066-48c0-af76-1f509a12c559",
   "metadata": {},
   "outputs": [],
   "source": [
    "% some housekeeping stuff\n",
    "register_graphics_toolkit (\"gnuplot\");\n",
    "available_graphics_toolkits ();\n",
    "graphics_toolkit (\"gnuplot\")\n",
    "clear\n",
    "% load packages\n",
    "pkg load statistics\n",
    "% end of housekeeping"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "725f486a",
   "metadata": {},
   "source": [
    "# Linear Regresion (reminder)\n",
    "\n",
    " - linear models have the form\n",
    " \n",
    " $$ y= \\beta_1 X_1 + \\beta_2 X_2 + ... + \\beta_n X_n$$\n",
    " \n",
    " - it is linear in the parameters (not necessarily in $X$)  \n",
    " - X can be transformation of the independent variable ($x$).  We did an example of $x^2$, but could be $\\sin(x)$ or $\\log(x)$ or whatever.  ($X$ can't be a function of $x$ and $y$ though, the only variable term in the expression for each $X$ must be $x$)\n",
    " - nonlinear models are everthing else\n",
    " - let's do an example with the first order decay equation\n",
    " \n",
    " $$ N = N_0 e^{-kt} $$\n",
    " \n",
    "  - let's assume an initial concentration of 0.01 M and a rate constant of 0.4 s$^{-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "edc3e534",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"62.92,384.00 534.81,384.00 534.81,18.02 62.92,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L75.52,384.00 M534.82,384.00 L522.22,384.00  \"/>\t<g transform=\"translate(54.53,387.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,323.00 L75.52,323.00 M534.82,323.00 L522.22,323.00  \"/>\t<g transform=\"translate(54.53,326.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,262.00 L75.52,262.00 M534.82,262.00 L522.22,262.00  \"/>\t<g transform=\"translate(54.53,265.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.004</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,201.00 L75.52,201.00 M534.82,201.00 L522.22,201.00  \"/>\t<g transform=\"translate(54.53,204.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.006</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,140.01 L75.52,140.01 M534.82,140.01 L522.22,140.01  \"/>\t<g transform=\"translate(54.53,143.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.008</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L75.52,79.01 M534.82,79.01 L522.22,79.01  \"/>\t<g transform=\"translate(54.53,82.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,18.01 L75.52,18.01 M534.82,18.01 L522.22,18.01  \"/>\t<g transform=\"translate(54.53,21.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.012</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L62.92,371.40 M62.92,18.01 L62.92,30.61  \"/>\t<g transform=\"translate(62.92,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M157.30,384.00 L157.30,371.40 M157.30,18.01 L157.30,30.61  \"/>\t<g transform=\"translate(157.30,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M251.68,384.00 L251.68,371.40 M251.68,18.01 L251.68,30.61  \"/>\t<g transform=\"translate(251.68,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M346.06,384.00 L346.06,371.40 M346.06,18.01 L346.06,30.61  \"/>\t<g transform=\"translate(346.06,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M440.44,384.00 L440.44,371.40 M440.44,18.01 L440.44,30.61  \"/>\t<g transform=\"translate(440.44,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M534.82,384.00 L534.82,371.40 M534.82,18.01 L534.82,30.61  \"/>\t<g transform=\"translate(534.82,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(62.92,76.89) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(110.11,189.26) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(157.30,252.79) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(204.49,288.68) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(251.68,322.62) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(298.87,345.27) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(346.06,354.75) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(393.25,359.83) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(440.44,374.24) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(487.63,371.92) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(534.82,381.66) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L110.11,179.56 L157.30,246.96 L204.49,292.14 L251.68,322.42 L298.87,342.72 L346.06,356.33 L393.25,365.45   L440.44,371.57 L487.63,375.67 L534.82,378.41  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M62.93,383.99 L534.83,383.99 M62.93,18.00 L534.83,18.00 M62.93,383.99 L62.93,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%plot -s 600,600 -f 'svg'\n",
    "\n",
    "% define the variables and constants\n",
    "N0=0.01; k=0.4; t=0:1:10;\n",
    "\n",
    "% define the function.  remember the anonymous function from Lecture 1?\n",
    "firstorderdecay = @(t,N0,k) (N0.*exp(-k*t));\n",
    "\n",
    "%calculate N as a function of time\n",
    "N=firstorderdecay(t,N0,k); Nnoerror=N;\n",
    "\n",
    "%plot it\n",
    "plot(t,N,'ko')\n",
    "\n",
    "%let's add some noise\n",
    "err=0.02*max(N);\n",
    "N=N+err*randn(1,length(N));\n",
    "\n",
    "%plot with error\n",
    "plot(t,N,'ko',t,Nnoerror,'k-')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0155c95b-fd0c-4f6c-ac3d-754a56b29f3d",
   "metadata": {},
   "source": [
    "### nonlinear optimization in matlab/octave\n",
    "\n",
    " - fundamentally nonlinear regression is searching for the minimum to the error furnction, defind as the sum of squares of the error\n",
    " - fminsearch is the command in Octave and Matlab to do this.  you will see an example below.\n",
    "  - fminsearch is based on the simplex algorithm (seperate lecture if we want to \"go there\").  Simplex has the advantage of not needing a gradient calculation and will even work for discontinuous functions\n",
    "  - with optimization toolbox in Matlab (and maybe in Octave?) there are other options for this search.  Gradient methods, constrained methods, ...\n",
    "  - first let's look at the error (SSE) surface"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "b3fa4b10-b69c-4330-848b-0ab9db32b8e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"46.14,384.00 534.81,384.00 534.81,18.02 46.14,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M46.14,322.16 L58.74,322.16 M534.82,322.16 L522.22,322.16  \"/>\t<g transform=\"translate(37.75,325.41)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M46.14,259.06 L58.74,259.06 M534.82,259.06 L522.22,259.06  \"/>\t<g transform=\"translate(37.75,262.31)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M46.14,195.96 L58.74,195.96 M534.82,195.96 L522.22,195.96  \"/>\t<g transform=\"translate(37.75,199.21)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M46.14,132.86 L58.74,132.86 M534.82,132.86 L522.22,132.86  \"/>\t<g transform=\"translate(37.75,136.11)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M46.14,69.75 L58.74,69.75 M534.82,69.75 L522.22,69.75  \"/>\t<g transform=\"translate(37.75,73.00)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">1</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M172.17,384.00 L172.17,371.40 M172.17,18.01 L172.17,30.61  \"/>\t<g transform=\"translate(172.17,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.005</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M300.77,384.00 L300.77,371.40 M300.77,18.01 L300.77,30.61  \"/>\t<g transform=\"translate(300.77,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M429.37,384.00 L429.37,371.40 M429.37,18.01 L429.37,30.61  \"/>\t<g transform=\"translate(429.37,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.015</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M290.49,257.80 L303.34,264.95 L312.70,257.80 L304.19,245.18 L303.34,244.72 L302.69,245.18 L290.49,257.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 72,  40, 120)\" d=\"M272.78,270.42 L277.62,275.58 L303.34,272.48 L306.83,270.42 L322.06,257.80 L322.89,245.18 L303.34,234.71 L288.30,245.18   L277.65,257.80 L277.62,257.88 L272.78,270.42  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_3a\"><title>gnuplot_plot_3a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 62,  74, 137)\" d=\"M262.31,283.04 L277.62,288.91 L295.82,283.04 L303.34,281.18 L321.58,270.42 L329.06,261.99 L333.73,257.80 L340.76,245.18   L340.23,232.56 L331.11,219.94 L329.06,218.23 L306.01,219.94 L303.34,220.05 L285.54,232.56 L277.62,239.82 L272.58,245.18   L262.80,257.80 L258.29,270.42 L262.31,283.04  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_4a\"><title>gnuplot_plot_4a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 49, 104, 142)\" d=\"M239.59,295.66 L251.90,302.60 L277.62,300.91 L292.90,295.66 L303.34,292.37 L321.95,283.04 L329.06,278.26 L340.64,270.42   L352.25,257.80 L354.78,252.62 L360.11,245.18 L364.22,232.56 L364.17,219.94 L360.09,207.32 L354.78,198.89 L348.76,194.69   L329.06,185.81 L303.34,192.82 L300.92,194.69 L284.55,207.32 L277.62,213.19 L271.46,219.94 L260.92,232.56 L253.03,245.18   L251.90,247.43 L245.11,257.80 L239.04,270.42 L236.17,283.04 L239.59,295.66  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_5a\"><title>gnuplot_plot_5a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38, 131, 142)\" d=\"M221.01,320.90 L226.18,322.14 L243.24,320.90 L251.90,320.42 L277.62,314.71 L296.95,308.28 L303.34,306.20 L327.72,295.66   L329.06,294.98 L350.74,283.04 L354.78,279.92 L368.26,270.42 L380.04,257.80 L380.50,257.05 L389.90,245.18 L396.03,232.56   L399.34,219.94 L400.46,207.32 L399.60,194.69 L396.76,182.07 L391.91,169.45 L385.00,156.83 L380.50,150.03 L374.36,144.21   L358.34,131.59 L354.78,129.02 L329.06,125.29 L314.22,131.59 L303.34,135.93 L294.06,144.21 L280.50,156.83 L277.62,159.51   L269.71,169.45 L260.26,182.07 L251.90,194.59 L251.83,194.69 L243.59,207.32 L236.10,219.94 L229.53,232.56 L226.18,239.78   L223.27,245.18 L216.86,257.80 L211.46,270.42 L207.32,283.04 L204.81,295.66 L205.48,308.28 L221.01,320.90  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_6a\"><title>gnuplot_plot_6a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 31, 157, 137)\" d=\"M268.90,18.01 L262.43,30.63 L256.09,43.25 L251.90,51.68 L250.16,55.87 L244.82,68.49 L239.56,81.11 L234.39,93.73   L229.35,106.35 L226.18,114.33 L224.43,118.97 L219.56,131.59 L214.76,144.21 L210.06,156.83 L205.49,169.45 L201.09,182.07   L200.46,183.84 L196.37,194.69 L191.66,207.32 L187.08,219.94 L182.69,232.56 L178.52,245.18 L174.74,257.35 L174.58,257.80   L170.10,270.42 L165.98,283.04 L162.37,295.66 L159.55,308.28 L158.08,320.90 L159.68,333.52 L173.39,346.14 L174.74,346.55   L200.46,346.85 L204.57,346.14 L226.18,342.83 L251.90,337.30 L266.10,333.52 L277.62,330.58 L303.34,322.70 L308.81,320.90   L329.06,313.67 L342.90,308.28 L354.78,302.90 L370.35,295.66 L380.50,289.73 L392.39,283.04 L406.22,272.69 L409.54,270.42   L423.68,257.80 L431.94,247.39 L433.97,245.18 L442.51,232.56 L448.43,219.94 L452.39,207.32 L454.81,194.69 L455.98,182.07   L456.07,169.45 L455.22,156.83 L453.50,144.21 L451.02,131.59 L447.85,118.97 L444.07,106.35 L439.76,93.73 L435.00,81.11   L431.94,73.44 L429.70,68.49 L423.75,55.87 L417.45,43.25 L410.89,30.63 L406.22,21.77 L403.61,18.01  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_7a\"><title>gnuplot_plot_7a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 53, 183, 121)\" d=\"M111.09,384.00 L97.58,379.24 L94.41,371.38 L93.67,358.76 L94.59,346.14 L96.26,333.52 L97.58,325.30 L98.23,320.90   L100.18,308.28 L102.20,295.66 L104.28,283.04 L106.37,270.42 L108.47,257.80 L110.56,245.18 L112.63,232.56 L114.67,219.94   L116.68,207.32 L118.64,194.69 L120.56,182.07 L122.44,169.45 L123.30,163.43 L124.19,156.83 L125.83,144.21 L127.44,131.59   L129.02,118.97 L130.58,106.35 L132.10,93.73 L133.58,81.11 L135.04,68.49 L136.45,55.87 L137.84,43.25 L139.18,30.63   L140.50,18.01  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_8a\"><title>gnuplot_plot_8a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 53, 183, 121)\" d=\"M126.32,384.00 L149.02,379.39 L174.74,373.64 L183.84,371.38 L200.46,367.44 L226.18,361.20 L235.87,358.76 L251.90,354.77   L277.62,348.09 L285.02,346.14 L303.34,341.12 L329.06,333.64 L329.47,333.52 L354.78,325.55 L369.01,320.90 L380.50,316.60   L402.34,308.28 L406.22,306.51 L430.36,295.66 L431.94,294.77 L453.88,283.04 L457.66,280.45 L473.38,270.42 L483.38,262.13   L489.06,257.80 L502.04,245.18 L509.10,236.22 L512.32,232.56 L520.92,219.94 L527.51,207.32 L532.51,194.69 L534.82,186.84    \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_9a\"><title>gnuplot_plot_9a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 53, 183, 121)\" d=\"M531.60,18.01 L533.72,30.63 L534.82,37.59  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_10a\"><title>gnuplot_plot_10a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(108, 206,  89)\" d=\"M207.02,384.00 L226.18,378.98 L251.90,372.62 L257.19,371.38 L277.62,366.42 L303.34,360.28 L309.94,358.76 L329.06,354.08   L354.78,347.76 L361.58,346.14 L380.50,341.16 L406.22,334.31 L409.30,333.52 L431.94,326.85 L452.01,320.90 L457.66,318.93   L483.38,310.15 L488.98,308.28 L509.10,300.21 L520.71,295.66 L534.82,288.90  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_11a\"><title>gnuplot_plot_11a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(180, 221,  44)\" d=\"M281.34,384.00 L303.34,379.32 L329.06,374.07 L342.92,371.38 L354.78,368.91 L380.50,363.76 L405.91,358.76 L406.22,358.69   L431.94,353.37 L457.66,348.08 L467.43,346.14 L483.38,342.52 L509.10,336.78 L524.02,333.52 L534.82,330.76  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_12a\"><title>gnuplot_plot_12a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(253, 231,  37)\" d=\"M383.93,384.00 L406.22,380.33 L431.94,376.25 L457.66,372.32 L464.10,371.38 L483.38,368.27 L509.10,364.24 L534.82,360.31    \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(253, 231,  37)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M46.15,383.99 L534.83,383.99 M46.15,18.00 L534.83,18.00 M46.15,383.99 L46.15,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%plot -s 600,600 -f 'svg'\n",
    "\n",
    "% make parameter ranges based on % of the parameter values\n",
    "% you could \"hard code\" this with specific numbers but using \n",
    "% percent let's you span a good range I think.\n",
    "\n",
    "N0range=0.01*N0:0.1*N0:2*N0;\n",
    "krange=0.01*k:0.1*k:3*k;\n",
    "\n",
    "% now a loop to calcualte the error\n",
    "\n",
    "for i=1:length(N0range)\n",
    "    for j=1:length(krange)\n",
    "        model=firstorderdecay(t,N0range(i),krange(j)); \n",
    "        SSE(j,i)=sum((model-N).^2);\n",
    "    end\n",
    "end\n",
    "\n",
    "% and contour it.  try also contour as SSE and you will see\n",
    "% how log of the error helps\n",
    "% and also look at the mesh\n",
    "\n",
    "contour(N0range,krange,log10(SSE),10)\n",
    "%mesh(N0range,krange,log10(SSE))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "13486c6c-f270-437e-9e99-3e0ee9d7f9d4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "err = -5.1849\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"62.92,384.00 534.81,384.00 534.81,18.02 62.92,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L75.52,384.00 M534.82,384.00 L522.22,384.00  \"/>\t<g transform=\"translate(54.53,387.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,323.00 L75.52,323.00 M534.82,323.00 L522.22,323.00  \"/>\t<g transform=\"translate(54.53,326.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,262.00 L75.52,262.00 M534.82,262.00 L522.22,262.00  \"/>\t<g transform=\"translate(54.53,265.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.004</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,201.00 L75.52,201.00 M534.82,201.00 L522.22,201.00  \"/>\t<g transform=\"translate(54.53,204.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.006</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,140.01 L75.52,140.01 M534.82,140.01 L522.22,140.01  \"/>\t<g transform=\"translate(54.53,143.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.008</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L75.52,79.01 M534.82,79.01 L522.22,79.01  \"/>\t<g transform=\"translate(54.53,82.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,18.01 L75.52,18.01 M534.82,18.01 L522.22,18.01  \"/>\t<g transform=\"translate(54.53,21.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.012</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L62.92,371.40 M62.92,18.01 L62.92,30.61  \"/>\t<g transform=\"translate(62.92,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M157.30,384.00 L157.30,371.40 M157.30,18.01 L157.30,30.61  \"/>\t<g transform=\"translate(157.30,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M251.68,384.00 L251.68,371.40 M251.68,18.01 L251.68,30.61  \"/>\t<g transform=\"translate(251.68,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M346.06,384.00 L346.06,371.40 M346.06,18.01 L346.06,30.61  \"/>\t<g transform=\"translate(346.06,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M440.44,384.00 L440.44,371.40 M440.44,18.01 L440.44,30.61  \"/>\t<g transform=\"translate(440.44,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M534.82,384.00 L534.82,371.40 M534.82,18.01 L534.82,30.61  \"/>\t<g transform=\"translate(534.82,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(62.92,76.89) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(110.11,189.26) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(157.30,252.79) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(204.49,288.68) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(251.68,322.62) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(298.87,345.27) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(346.06,354.75) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(393.25,359.83) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(440.44,374.24) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(487.63,371.92) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(534.82,381.66) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L110.11,158.06 L157.30,216.62 L204.49,260.00 L251.68,292.14 L298.87,315.95 L346.06,333.59 L393.25,346.65   L440.44,356.33 L487.63,363.50 L534.82,368.82  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M62.93,383.99 L534.83,383.99 M62.93,18.00 L534.83,18.00 M62.93,383.99 L62.93,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% so to optmize it we need to first define the error function\n",
    "% this is what will get minimized\n",
    "\n",
    "% initial guess\n",
    "\n",
    "errorfunction = @(p,N,t) (log10(sum((N-p(1).*exp(-p(2)*t)).^2)));\n",
    "\n",
    "ktst=0.3; N0tst=0.01; ptst=[N0tst ktst];\n",
    "\n",
    "err=errorfunction(ptst,N,t)\n",
    "\n",
    "Ntst=firstorderdecay(t,N0tst,ktst); \n",
    "\n",
    "plot(t,N,'ko',t,Ntst,'k-'); \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "89c8ee51-1469-4d68-8834-593a8810d428",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x0) = -5.1849e+00\n",
      "Iter.  1,  how = initial  , nf =   3,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  2,  how = shrink,    nf =   7,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  3,  how = shrink,    nf =  11,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  4,  how = shrink,    nf =  15,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  5,  how = shrink,    nf =  19,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  6,  how = shrink,    nf =  23,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  7,  how = contract,  nf =  25,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  8,  how = contract,  nf =  27,  f = -5.1849e+00  (0.0%)\n",
      "Iter.  9,  how = shrink,    nf =  31,  f = -5.1849e+00  (0.0%)\n",
      "Iter. 10,  how = contract,  nf =  33,  f = -5.2726e+00  (1.7%)\n",
      "Iter. 11,  how = contract,  nf =  35,  f = -5.2735e+00  (0.0%)\n",
      "Iter. 12,  how = contract,  nf =  37,  f = -5.4351e+00  (3.1%)\n",
      "Iter. 13,  how = contract,  nf =  39,  f = -5.4587e+00  (0.4%)\n",
      "Iter. 14,  how = contract,  nf =  41,  f = -5.4851e+00  (0.5%)\n",
      "Iter. 15,  how = contract,  nf =  43,  f = -5.4851e+00  (0.0%)\n",
      "Iter. 16,  how = reflect,   nf =  44,  f = -5.4851e+00  (0.0%)\n",
      "Iter. 17,  how = contract,  nf =  46,  f = -5.5124e+00  (0.5%)\n",
      "Iter. 18,  how = reflect,   nf =  47,  f = -5.5124e+00  (0.0%)\n",
      "Iter. 19,  how = contract,  nf =  49,  f = -5.5124e+00  (0.0%)\n",
      "Iter. 20,  how = reflect,   nf =  50,  f = -5.5124e+00  (0.0%)\n",
      "Iter. 21,  how = expand,    nf =  52,  f = -5.5348e+00  (0.4%)\n",
      "Iter. 22,  how = reflect,   nf =  53,  f = -5.5348e+00  (0.0%)\n",
      "Iter. 23,  how = expand,    nf =  55,  f = -5.5666e+00  (0.6%)\n",
      "Iter. 24,  how = expand,    nf =  57,  f = -5.5969e+00  (0.5%)\n",
      "Iter. 25,  how = expand,    nf =  59,  f = -5.7020e+00  (1.9%)\n",
      "Iter. 26,  how = expand,    nf =  61,  f = -5.7402e+00  (0.7%)\n",
      "Iter. 27,  how = expand,    nf =  63,  f = -6.0887e+00  (6.1%)\n",
      "Iter. 28,  how = reflect,   nf =  64,  f = -6.0887e+00  (0.0%)\n",
      "Iter. 29,  how = reflect,   nf =  65,  f = -6.0887e+00  (0.0%)\n",
      "Iter. 30,  how = expand,    nf =  67,  f = -6.5038e+00  (6.8%)\n",
      "Iter. 31,  how = contract,  nf =  69,  f = -6.5038e+00  (0.0%)\n",
      "Iter. 32,  how = reflect,   nf =  71,  f = -6.5724e+00  (1.1%)\n",
      "Iter. 33,  how = contract,  nf =  73,  f = -6.6234e+00  (0.8%)\n",
      "Iter. 34,  how = contract,  nf =  75,  f = -6.6318e+00  (0.1%)\n",
      "Iter. 35,  how = contract,  nf =  77,  f = -6.6629e+00  (0.5%)\n",
      "Iter. 36,  how = contract,  nf =  79,  f = -6.6706e+00  (0.1%)\n",
      "Iter. 37,  how = reflect,   nf =  80,  f = -6.6706e+00  (0.0%)\n",
      "Iter. 38,  how = contract,  nf =  82,  f = -6.6782e+00  (0.1%)\n",
      "Iter. 39,  how = contract,  nf =  84,  f = -6.6782e+00  (0.0%)\n",
      "Iter. 40,  how = contract,  nf =  86,  f = -6.6791e+00  (0.0%)\n",
      "Iter. 41,  how = contract,  nf =  88,  f = -6.6801e+00  (0.0%)\n",
      "Iter. 42,  how = contract,  nf =  90,  f = -6.6801e+00  (0.0%)\n",
      "Iter. 43,  how = contract,  nf =  92,  f = -6.6802e+00  (0.0%)\n",
      "Iter. 44,  how = contract,  nf =  94,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 45,  how = contract,  nf =  96,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 46,  how = contract,  nf =  98,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 47,  how = reflect,   nf = 100,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 48,  how = contract,  nf = 102,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 49,  how = contract,  nf = 104,  f = -6.6803e+00  (0.0%)\n",
      "Iter. 50,  how = shrink,    nf = 108,  f = -6.6803e+00  (0.0%)\n",
      "Algorithm converged.  Simplex size 6.9842e-05 <= 1.0000e-04 and step in function value 3.9425e-06 <= 1.0000e-04\n",
      "pbest =\n",
      "\n",
      "   9.9233e-03   4.0381e-01\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"62.92,384.00 534.81,384.00 534.81,18.02 62.92,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L75.52,384.00 M534.82,384.00 L522.22,384.00  \"/>\t<g transform=\"translate(54.53,387.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,323.00 L75.52,323.00 M534.82,323.00 L522.22,323.00  \"/>\t<g transform=\"translate(54.53,326.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,262.00 L75.52,262.00 M534.82,262.00 L522.22,262.00  \"/>\t<g transform=\"translate(54.53,265.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.004</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,201.00 L75.52,201.00 M534.82,201.00 L522.22,201.00  \"/>\t<g transform=\"translate(54.53,204.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.006</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,140.01 L75.52,140.01 M534.82,140.01 L522.22,140.01  \"/>\t<g transform=\"translate(54.53,143.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.008</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L75.52,79.01 M534.82,79.01 L522.22,79.01  \"/>\t<g transform=\"translate(54.53,82.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,18.01 L75.52,18.01 M534.82,18.01 L522.22,18.01  \"/>\t<g transform=\"translate(54.53,21.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.012</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L62.92,371.40 M62.92,18.01 L62.92,30.61  \"/>\t<g transform=\"translate(62.92,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M157.30,384.00 L157.30,371.40 M157.30,18.01 L157.30,30.61  \"/>\t<g transform=\"translate(157.30,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M251.68,384.00 L251.68,371.40 M251.68,18.01 L251.68,30.61  \"/>\t<g transform=\"translate(251.68,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M346.06,384.00 L346.06,371.40 M346.06,18.01 L346.06,30.61  \"/>\t<g transform=\"translate(346.06,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M440.44,384.00 L440.44,371.40 M440.44,18.01 L440.44,30.61  \"/>\t<g transform=\"translate(440.44,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M534.82,384.00 L534.82,371.40 M534.82,18.01 L534.82,30.61  \"/>\t<g transform=\"translate(534.82,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(62.92,76.89) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(110.11,189.26) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(157.30,252.79) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(204.49,288.68) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(251.68,322.62) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(298.87,345.27) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(346.06,354.75) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(393.25,359.83) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(440.44,374.24) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(487.63,371.92) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(534.82,381.66) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0, 255)\" d=\"M62.92,81.35 L110.11,181.90 L157.30,249.04 L204.49,293.88 L251.68,323.82 L298.87,343.81 L346.06,357.16 L393.25,366.08   L440.44,372.03 L487.63,376.01 L534.82,378.66  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0, 255)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M62.93,383.99 L534.83,383.99 M62.93,18.00 L534.83,18.00 M62.93,383.99 L62.93,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% now best fit\n",
    "\n",
    "options = optimset('Display','iter','TolFun',1e-4,'TolX',1e-4);\n",
    "\n",
    "pbest=fminsearch(@(p) errorfunction(p,N,t),ptst,options)\n",
    "\n",
    "Nbst=firstorderdecay(t,pbest(1),pbest(2)); \n",
    "\n",
    "plot(t,N,'ko',t,Nbst,'b-')\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "8c9f8080-e5c7-4d22-8ba4-0833ddd40ba6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"79.70,384.00 534.81,384.00 534.81,18.02 79.70,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,384.00 L92.30,384.00 M534.82,384.00 L522.22,384.00  \"/>\t<g transform=\"translate(71.31,387.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">-0.0003</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,323.00 L92.30,323.00 M534.82,323.00 L522.22,323.00  \"/>\t<g transform=\"translate(71.31,326.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">-0.0002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,262.00 L92.30,262.00 M534.82,262.00 L522.22,262.00  \"/>\t<g transform=\"translate(71.31,265.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">-0.0001</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,201.00 L92.30,201.00 M534.82,201.00 L522.22,201.00  \"/>\t<g transform=\"translate(71.31,204.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,140.01 L92.30,140.01 M534.82,140.01 L522.22,140.01  \"/>\t<g transform=\"translate(71.31,143.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0001</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,79.01 L92.30,79.01 M534.82,79.01 L522.22,79.01  \"/>\t<g transform=\"translate(71.31,82.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,18.01 L92.30,18.01 M534.82,18.01 L522.22,18.01  \"/>\t<g transform=\"translate(71.31,21.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0003</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,384.00 L79.70,371.40 M79.70,18.01 L79.70,30.61  \"/>\t<g transform=\"translate(79.70,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M170.72,384.00 L170.72,371.40 M170.72,18.01 L170.72,30.61  \"/>\t<g transform=\"translate(170.72,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M261.75,384.00 L261.75,371.40 M261.75,18.01 L261.75,30.61  \"/>\t<g transform=\"translate(261.75,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M352.77,384.00 L352.77,371.40 M352.77,18.01 L352.77,30.61  \"/>\t<g transform=\"translate(352.77,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M443.80,384.00 L443.80,371.40 M443.80,18.01 L443.80,30.61  \"/>\t<g transform=\"translate(443.80,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M534.82,384.00 L534.82,371.40 M534.82,18.01 L534.82,30.61  \"/>\t<g transform=\"translate(534.82,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(79.70,111.92) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(125.21,348.32) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(170.72,276.08) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(216.24,97.09) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(261.75,176.94) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(307.26,230.07) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(352.77,152.73) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(398.28,76.03) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(443.80,245.16) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(489.31,119.26) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(534.82,260.99) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M79.70,201.00 L125.21,201.00 L170.72,201.00 L216.24,201.00 L261.75,201.00 L307.26,201.00 L352.77,201.00 L398.28,201.00   L443.80,201.00 L489.31,201.00 L534.82,201.00  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M79.71,383.99 L534.83,383.99 M79.71,18.00 L534.83,18.00 M79.71,383.99 L79.71,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% plot the residuals\n",
    "\n",
    "plot(t,N-Nbst,'ko',t,zeros(size(N)),'k-')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dab8d0f-c149-445b-9464-96af38b91766",
   "metadata": {},
   "source": [
    "### now monte carlo analysis for parameter uncertainty estimation\n",
    "\n",
    "to estimate uncertainty in the parameters just simulate 1000 datasets and fit them all.  Like we did in Lecture 3 (but we did it with a linear example that time to illustrate the method)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "d91b430a-2a4f-4723-b83a-820cbdac2889",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"62.92,384.00 534.81,384.00 534.81,18.02 62.92,18.02 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L75.52,384.00 M534.82,384.00 L522.22,384.00  \"/>\t<g transform=\"translate(54.53,387.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,323.00 L75.52,323.00 M534.82,323.00 L522.22,323.00  \"/>\t<g transform=\"translate(54.53,326.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.002</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,262.00 L75.52,262.00 M534.82,262.00 L522.22,262.00  \"/>\t<g transform=\"translate(54.53,265.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.004</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,201.00 L75.52,201.00 M534.82,201.00 L522.22,201.00  \"/>\t<g transform=\"translate(54.53,204.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.006</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,140.01 L75.52,140.01 M534.82,140.01 L522.22,140.01  \"/>\t<g transform=\"translate(54.53,143.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.008</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,79.01 L75.52,79.01 M534.82,79.01 L522.22,79.01  \"/>\t<g transform=\"translate(54.53,82.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,18.01 L75.52,18.01 M534.82,18.01 L522.22,18.01  \"/>\t<g transform=\"translate(54.53,21.26)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.012</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M62.92,384.00 L62.92,371.40 M62.92,18.01 L62.92,30.61  \"/>\t<g transform=\"translate(62.92,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M157.30,384.00 L157.30,371.40 M157.30,18.01 L157.30,30.61  \"/>\t<g transform=\"translate(157.30,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">2</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M251.68,384.00 L251.68,371.40 M251.68,18.01 L251.68,30.61  \"/>\t<g transform=\"translate(251.68,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M346.06,384.00 L346.06,371.40 M346.06,18.01 L346.06,30.61  \"/>\t<g transform=\"translate(346.06,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">6</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M440.44,384.00 L440.44,371.40 M440.44,18.01 L440.44,30.61  \"/>\t<g transform=\"translate(440.44,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">8</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M534.82,384.00 L534.82,371.40 M534.82,18.01 L534.82,30.61  \"/>\t<g transform=\"translate(534.82,405.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(62.92,76.89) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(110.11,189.26) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(157.30,252.79) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(204.49,288.68) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(251.68,322.62) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(298.87,345.27) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(346.06,354.75) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(393.25,359.83) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(440.44,374.24) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(487.63,371.92) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" transform=\"translate(534.82,381.66) scale(9.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(62.92,80.73) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(110.11,177.99) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(157.30,249.04) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(204.49,300.23) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(251.68,327.92) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(298.87,352.65) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(346.06,357.16) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(393.25,366.08) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(440.44,372.03) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(487.63,376.01) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "\t<use xlink:href=\"#gpPt6\" transform=\"translate(534.82,373.28) scale(3.00)\" color=\"rgb(  0,   0,   0)\"/>\n",
       "</g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M62.93,383.99 L534.83,383.99 M62.93,18.00 L534.83,18.00 M62.93,383.99 L62.93,18.00 M534.83,383.99 L534.83,18.00    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "residuals=N-Nbst; sestimate=std(residuals);\n",
    "\n",
    "% need to make simulated dataset\n",
    "\n",
    "count = 1:length(N); % an integer vector with an entry in order per each point in the data\n",
    "\n",
    "% replacing all data seems to work better\n",
    "\n",
    "%noreplace=round(length(N)/3); % replace 1/3 of the datapoints\n",
    "noreplace=round(length(N)/2); % replace 1/2 of the datapoints\n",
    "%noreplace=round(length(N)/1); % replace all of the datapoints\n",
    "\n",
    "indexreplace = randsample(count,noreplace); % pick at random wich of the 1/3 of the points to replace.\n",
    "\n",
    "Nnew=Nbst; % initialize the simulated data just as the original data.\n",
    "\n",
    "for i=1:length(indexreplace)\n",
    "    Nnew(indexreplace(i))=Nbst(indexreplace(i))+randn(1,1)*sestimate;\n",
    "end\n",
    "\n",
    "plot(t,N,'ko',t,Nnew,'k.')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "af3a210d-083f-44a9-86f5-4a5dec79ce55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600px\" height=\"600px\" viewBox=\"0 0 560 420\">\n",
       "\n",
       "<title>Gnuplot</title>\n",
       "<desc>Produced by GNUPLOT 5.4 patchlevel 3 </desc>\n",
       "\n",
       "<g id=\"gnuplot_canvas\">\n",
       "\n",
       "<rect x=\"0\" y=\"0\" width=\"560\" height=\"420\" fill=\"none\"/>\n",
       "<defs>\n",
       "\n",
       "\t<circle id=\"gpDot\" r=\"0.5\" stroke-width=\"0.5\" stroke=\"currentColor\"/>\n",
       "\t<path id=\"gpPt0\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 h2 M0,-1 v2\"/>\n",
       "\t<path id=\"gpPt1\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,-1 L1,1 M1,-1 L-1,1\"/>\n",
       "\t<path id=\"gpPt2\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M-1,0 L1,0 M0,-1 L0,1 M-1,-1 L1,1 M-1,1 L1,-1\"/>\n",
       "\t<rect id=\"gpPt3\" stroke-width=\"0.222\" stroke=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<rect id=\"gpPt4\" stroke-width=\"0.222\" stroke=\"currentColor\" fill=\"currentColor\" x=\"-1\" y=\"-1\" width=\"2\" height=\"2\"/>\n",
       "\t<circle id=\"gpPt5\" stroke-width=\"0.222\" stroke=\"currentColor\" cx=\"0\" cy=\"0\" r=\"1\"/>\n",
       "\t<use xlink:href=\"#gpPt5\" id=\"gpPt6\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt7\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,-1.33 L-1.33,0.67 L1.33,0.67 z\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt8\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt7\" id=\"gpPt9\" stroke=\"currentColor\" transform=\"rotate(180)\"/>\n",
       "\t<use xlink:href=\"#gpPt9\" id=\"gpPt10\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<use xlink:href=\"#gpPt3\" id=\"gpPt11\" stroke=\"currentColor\" transform=\"rotate(45)\"/>\n",
       "\t<use xlink:href=\"#gpPt11\" id=\"gpPt12\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<path id=\"gpPt13\" stroke-width=\"0.222\" stroke=\"currentColor\" d=\"M0,1.330 L1.265,0.411 L0.782,-1.067 L-0.782,-1.076 L-1.265,0.411 z\"/>\n",
       "\t<use xlink:href=\"#gpPt13\" id=\"gpPt14\" fill=\"currentColor\" stroke=\"none\"/>\n",
       "\t<filter id=\"textbox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"bgnd\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "\t<filter id=\"greybox\" filterUnits=\"objectBoundingBox\" x=\"0\" y=\"0\" height=\"1\" width=\"1\">\n",
       "\t  <feFlood flood-color=\"lightgrey\" flood-opacity=\"1\" result=\"grey\"/>\n",
       "\t  <feComposite in=\"SourceGraphic\" in2=\"grey\" operator=\"atop\"/>\n",
       "\t</filter>\n",
       "</defs>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"72.80,173.94 506.78,173.94 506.78,31.52 72.80,31.52 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"155.05,173.94 155.05,171.09 189.26,171.09 189.26,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M155.05,173.94 L155.05,171.09 L189.26,171.09 L189.26,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M155.05,173.94 L155.05,171.09 L189.26,171.09 L189.26,173.94 L155.05,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_3a\"><title>gnuplot_plot_3a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"189.26,173.94 189.26,168.24 223.48,168.24 223.48,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M189.26,173.94 L189.26,168.24 L223.48,168.24 L223.48,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_4a\"><title>gnuplot_plot_4a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M189.26,173.94 L189.26,168.24 L223.48,168.24 L223.48,173.94 L189.26,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_5a\"><title>gnuplot_plot_5a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"223.48,173.94 223.48,159.70 257.69,159.70 257.69,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M223.48,173.94 L223.48,159.70 L257.69,159.70 L257.69,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_6a\"><title>gnuplot_plot_6a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M223.48,173.94 L223.48,159.70 L257.69,159.70 L257.69,173.94 L223.48,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_7a\"><title>gnuplot_plot_7a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"257.69,173.94 257.69,151.15 291.90,151.15 291.90,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M257.69,173.94 L257.69,151.15 L291.90,151.15 L291.90,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_8a\"><title>gnuplot_plot_8a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M257.69,173.94 L257.69,151.15 L291.90,151.15 L291.90,173.94 L257.69,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_9a\"><title>gnuplot_plot_9a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"291.90,173.94 291.90,131.21 326.12,131.21 326.12,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M291.90,173.94 L291.90,131.21 L326.12,131.21 L326.12,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_10a\"><title>gnuplot_plot_10a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M291.90,173.94 L291.90,131.21 L326.12,131.21 L326.12,173.94 L291.90,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_11a\"><title>gnuplot_plot_11a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"326.12,173.94 326.12,151.15 360.33,151.15 360.33,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M326.12,173.94 L326.12,151.15 L360.33,151.15 L360.33,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_12a\"><title>gnuplot_plot_12a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M326.12,173.94 L326.12,151.15 L360.33,151.15 L360.33,173.94 L326.12,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_13a\"><title>gnuplot_plot_13a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"360.33,173.94 360.33,51.45 394.55,51.45 394.55,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M360.33,173.94 L360.33,51.45 L394.55,51.45 L394.55,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_14a\"><title>gnuplot_plot_14a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M360.33,173.94 L360.33,51.45 L394.55,51.45 L394.55,173.94 L360.33,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_15a\"><title>gnuplot_plot_15a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"394.55,173.94 394.55,148.30 428.76,148.30 428.76,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M394.55,173.94 L394.55,148.30 L428.76,148.30 L428.76,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_16a\"><title>gnuplot_plot_16a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M394.55,173.94 L394.55,148.30 L428.76,148.30 L428.76,173.94 L394.55,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_17a\"><title>gnuplot_plot_17a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"428.76,173.94 428.76,162.55 462.98,162.55 462.98,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M428.76,173.94 L428.76,162.55 L462.98,162.55 L462.98,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_18a\"><title>gnuplot_plot_18a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M428.76,173.94 L428.76,162.55 L462.98,162.55 L462.98,173.94 L428.76,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_19a\"><title>gnuplot_plot_19a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"462.98,173.94 462.98,159.70 497.19,159.70 497.19,173.94 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M462.98,173.94 L462.98,159.70 L497.19,159.70 L497.19,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_20a\"><title>gnuplot_plot_20a</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M462.98,173.94 L462.98,159.70 L497.19,159.70 L497.19,173.94 L462.98,173.94  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_21a\"><title>gnuplot_plot_21a</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,173.94 L506.79,173.94  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,173.94 L85.40,173.94 M506.79,173.94 L494.19,173.94  \"/>\t<g transform=\"translate(64.41,177.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,145.45 L85.40,145.45 M506.79,145.45 L494.19,145.45  \"/>\t<g transform=\"translate(64.41,148.70)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,116.97 L85.40,116.97 M506.79,116.97 L494.19,116.97  \"/>\t<g transform=\"translate(64.41,120.22)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">20</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,88.48 L85.40,88.48 M506.79,88.48 L494.19,88.48  \"/>\t<g transform=\"translate(64.41,91.73)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">30</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,60.00 L85.40,60.00 M506.79,60.00 L494.19,60.00  \"/>\t<g transform=\"translate(64.41,63.25)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">40</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,31.51 L85.40,31.51 M506.79,31.51 L494.19,31.51  \"/>\t<g transform=\"translate(64.41,34.76)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">50</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,173.94 L72.80,161.34 M72.80,31.51 L72.80,44.11  \"/>\t<g transform=\"translate(72.80,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.37</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M159.60,173.94 L159.60,161.34 M159.60,31.51 L159.60,44.11  \"/>\t<g transform=\"translate(159.60,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.38</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M246.40,173.94 L246.40,161.34 M246.40,31.51 L246.40,44.11  \"/>\t<g transform=\"translate(246.40,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.39</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M333.19,173.94 L333.19,161.34 M333.19,31.51 L333.19,44.11  \"/>\t<g transform=\"translate(333.19,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.4</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M419.99,173.94 L419.99,161.34 M419.99,31.51 L419.99,44.11  \"/>\t<g transform=\"translate(419.99,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.41</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M506.79,173.94 L506.79,161.34 M506.79,31.51 L506.79,44.11  \"/>\t<g transform=\"translate(506.79,195.19)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.42</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M72.81,173.93 L506.80,173.93 M72.81,31.50 L506.80,31.50 M72.81,173.93 L72.81,31.50 M506.80,173.93 L506.80,31.50    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"72.80,373.80 506.78,373.80 506.78,231.38 72.80,231.38 \"/>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "\t<g id=\"gnuplot_plot_1b\"><title>gnuplot_plot_1b</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"97.59,373.80 97.59,362.41 138.46,362.41 138.46,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M97.59,373.80 L97.59,362.41 L138.46,362.41 L138.46,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_2b\"><title>gnuplot_plot_2b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M97.59,373.80 L97.59,362.41 L138.46,362.41 L138.46,373.80 L97.59,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_3b\"><title>gnuplot_plot_3b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"138.46,373.80 138.46,333.92 179.33,333.92 179.33,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M138.46,373.80 L138.46,333.92 L179.33,333.92 L179.33,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_4b\"><title>gnuplot_plot_4b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M138.46,373.80 L138.46,333.92 L179.33,333.92 L179.33,373.80 L138.46,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_5b\"><title>gnuplot_plot_5b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"179.33,373.80 179.33,333.92 220.20,333.92 220.20,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M179.33,373.80 L179.33,333.92 L220.20,333.92 L220.20,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_6b\"><title>gnuplot_plot_6b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M179.33,373.80 L179.33,333.92 L220.20,333.92 L220.20,373.80 L179.33,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_7b\"><title>gnuplot_plot_7b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"220.20,373.80 220.20,276.95 261.07,276.95 261.07,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M220.20,373.80 L220.20,276.95 L261.07,276.95 L261.07,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_8b\"><title>gnuplot_plot_8b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M220.20,373.80 L220.20,276.95 L261.07,276.95 L261.07,373.80 L220.20,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_9b\"><title>gnuplot_plot_9b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"261.07,373.80 261.07,265.55 301.94,265.55 301.94,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M261.07,373.80 L261.07,265.55 L301.94,265.55 L301.94,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_10b\"><title>gnuplot_plot_10b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M261.07,373.80 L261.07,265.55 L301.94,265.55 L301.94,373.80 L261.07,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_11b\"><title>gnuplot_plot_11b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"301.94,373.80 301.94,231.37 342.81,231.37 342.81,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M301.94,373.80 L301.94,231.37 L342.81,231.37 L342.81,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_12b\"><title>gnuplot_plot_12b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M301.94,373.80 L301.94,231.37 L342.81,231.37 L342.81,373.80 L301.94,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_13b\"><title>gnuplot_plot_13b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"342.81,373.80 342.81,311.13 383.68,311.13 383.68,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M342.81,373.80 L342.81,311.13 L383.68,311.13 L383.68,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_14b\"><title>gnuplot_plot_14b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M342.81,373.80 L342.81,311.13 L383.68,311.13 L383.68,373.80 L342.81,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_15b\"><title>gnuplot_plot_15b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"383.68,373.80 383.68,328.22 424.55,328.22 424.55,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M383.68,373.80 L383.68,328.22 L424.55,328.22 L424.55,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_16b\"><title>gnuplot_plot_16b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M383.68,373.80 L383.68,328.22 L424.55,328.22 L424.55,373.80 L383.68,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_17b\"><title>gnuplot_plot_17b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"424.55,373.80 424.55,368.10 465.42,368.10 465.42,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M424.55,373.80 L424.55,368.10 L465.42,368.10 L465.42,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_18b\"><title>gnuplot_plot_18b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M424.55,373.80 L424.55,368.10 L465.42,368.10 L465.42,373.80 L424.55,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_19b\"><title>gnuplot_plot_19b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<g stroke=\"none\" shape-rendering=\"crispEdges\">\n",
       "\t\t<polygon fill=\"rgb( 68,   1,  84)\" points=\"465.42,373.80 465.42,356.71 506.29,356.71 506.29,373.80 \"/>\n",
       "\t</g>\n",
       "\t<path stroke=\"rgb( 68,   1,  84)\" d=\"M465.42,373.80 L465.42,356.71 L506.29,356.71 L506.29,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_20b\"><title>gnuplot_plot_20b</title>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb( 68,   1,  84)\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M465.42,373.80 L465.42,356.71 L506.29,356.71 L506.29,373.80 L465.42,373.80  \"/></g>\n",
       "\t</g>\n",
       "\t<g id=\"gnuplot_plot_21b\"><title>gnuplot_plot_21b</title>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,373.80 L506.79,373.80  \"/></g>\n",
       "\t</g>\n",
       "<g fill=\"none\" color=\"white\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,373.80 L85.40,373.80 M506.79,373.80 L494.19,373.80  \"/>\t<g transform=\"translate(64.41,377.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,345.31 L85.40,345.31 M506.79,345.31 L494.19,345.31  \"/>\t<g transform=\"translate(64.41,348.56)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">5</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,316.83 L85.40,316.83 M506.79,316.83 L494.19,316.83  \"/>\t<g transform=\"translate(64.41,320.08)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">10</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,288.34 L85.40,288.34 M506.79,288.34 L494.19,288.34  \"/>\t<g transform=\"translate(64.41,291.59)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">15</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,259.86 L85.40,259.86 M506.79,259.86 L494.19,259.86  \"/>\t<g transform=\"translate(64.41,263.11)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">20</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,231.37 L85.40,231.37 M506.79,231.37 L494.19,231.37  \"/>\t<g transform=\"translate(64.41,234.62)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"end\">\n",
       "\t\t<text><tspan font-family=\"Arial\">25</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M72.80,373.80 L72.80,361.20 M72.80,231.37 L72.80,243.97  \"/>\t<g transform=\"translate(72.80,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0096</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M145.13,373.80 L145.13,361.20 M145.13,231.37 L145.13,243.97  \"/>\t<g transform=\"translate(145.13,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0097</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M217.46,373.80 L217.46,361.20 M217.46,231.37 L217.46,243.97  \"/>\t<g transform=\"translate(217.46,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0098</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M289.80,373.80 L289.80,361.20 M289.80,231.37 L289.80,243.97  \"/>\t<g transform=\"translate(289.80,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0099</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M362.13,373.80 L362.13,361.20 M362.13,231.37 L362.13,243.97  \"/>\t<g transform=\"translate(362.13,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.01</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M434.46,373.80 L434.46,361.20 M434.46,231.37 L434.46,243.97  \"/>\t<g transform=\"translate(434.46,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0101</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb(  0,   0,   0)\" d=\"M506.79,373.80 L506.79,361.20 M506.79,231.37 L506.79,243.97  \"/>\t<g transform=\"translate(506.79,395.05)\" stroke=\"none\" fill=\"rgb(38,38,38)\" font-family=\"Arial\" font-size=\"10.00\" text-anchor=\"middle\">\n",
       "\t\t<text><tspan font-family=\"Arial\">0.0102</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb(  0,   0,   0)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "\t<path stroke=\"rgb( 38,  38,  38)\" d=\"M72.81,373.79 L506.80,373.79 M72.81,231.36 L506.80,231.36 M72.81,373.79 L72.81,231.36 M506.80,373.79 L506.80,231.36    \"/></g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"rgb( 38,  38,  38)\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% monte carlo analysis\n",
    "\n",
    "%Nbst=Nnoerror; %if set that the first fit was exactly correct we get the mean value as the true values.\n",
    "\n",
    "for j=1:100\n",
    "\n",
    "    Nnew=Nbst; % initialize the simulated data just as the original data.\n",
    "\n",
    "    for i=1:length(indexreplace)\n",
    "        Nnew(indexreplace(i))=Nbst(indexreplace(i))+randn(1,1)*sestimate;\n",
    "    end\n",
    "    \n",
    "    MCbetas(:,j)=fminsearch(@(p) errorfunction(p,Nnew,t),pbest);\n",
    "    \n",
    "end\n",
    "\n",
    "MCk=MCbetas(2,:); MCN0=MCbetas(1,:);\n",
    "\n",
    "subplot(211); \n",
    "hist(MCk,10);\n",
    "\n",
    "subplot(212)\n",
    "hist(MCN0,10);\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7d6b9f15-db43-4267-8d31-2e8eec29ea9b",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "3ce013ad-5fc3-4a98-ac84-353e9c595f22",
   "metadata": {},
   "source": [
    "# excercise 6\n",
    "\n",
    "repeat this simulation and fitting exercise for Michaelis-Menten kinetics\n",
    "\n",
    "$$v=V_{max}\\dfrac{[\\mathrm{S}]}{K_M+[\\mathrm{S}]}$$\n",
    "\n",
    "where the measured reaction rate ($v$) is a function of the subtrate concentration ($[\\mathrm{S}]$) and the two parameters, maximum rate $V_m$ and Michaelis constant ($K_M$).\n",
    "\n",
    "Let's do this for carbonic anhydrase. with CO2 as the substrate.  $K_M = 12$ mM. For bovine CA II the $V_{max}$ value is 0.839 mM/s.  (I just googled some random values).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "da26663e-2c1d-4173-b16f-9daf03e45f06",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "6.4.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}