4_cheminformatics_reactionYield.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chemometrics\n",
    "<br>\n",
    "**Julien Wist / 2017 / Universidad del Valle**\n",
    "<br>\n",
    "**Andrés Bernal / 2017 / ???**\n",
    "\n",
    "An up-to-date version of this notebook can be found here: https://github.com/jwist/chemometrics/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To best understand the following notebook, please read chapter 2 of Brereton, R. G. (2003). Chemometrics. Technometrics. Chichester, UK: John Wiley & Sons, Ltd. http://doi.org/10.1002/0470863242.\n",
    "\n",
    "Use the following commands to create the dataset of Table 2.6 of Brereton."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "rm(list=ls(all=TRUE)) # to clean the variable space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=4) # change these setting to plot larger figures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>ph</th><th scope=col>temp</th><th scope=col>conc</th><th scope=col>ph2</th><th scope=col>temp2</th><th scope=col>conc2</th><th scope=col>phTemp</th><th scope=col>phConc</th><th scope=col>tempConc</th><th scope=col>yield</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>   6.000</td><td>  60.000</td><td>   4.000</td><td>  36.000</td><td>3600.000</td><td>  16.000</td><td> 360.000</td><td>  24.000</td><td> 240.000</td><td>  34.841</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>   6.000</td><td>  60.000</td><td>   2.000</td><td>  36.000</td><td>3600.000</td><td>   4.000</td><td> 360.000</td><td>  12.000</td><td> 120.000</td><td>  16.567</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>  6.000</td><td> 20.000</td><td>  4.000</td><td> 36.000</td><td>400.000</td><td> 16.000</td><td>120.000</td><td> 24.000</td><td> 80.000</td><td> 45.396</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>  6.000</td><td> 20.000</td><td>  2.000</td><td> 36.000</td><td>400.000</td><td>  4.000</td><td>120.000</td><td> 12.000</td><td> 40.000</td><td> 27.939</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>   4.000</td><td>  60.000</td><td>   4.000</td><td>  16.000</td><td>3600.000</td><td>  16.000</td><td> 240.000</td><td>  16.000</td><td> 240.000</td><td>  19.825</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>   4.000</td><td>  60.000</td><td>   2.000</td><td>  16.000</td><td>3600.000</td><td>   4.000</td><td> 240.000</td><td>   8.000</td><td> 120.000</td><td>   1.444</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>  4.000</td><td> 20.000</td><td>  4.000</td><td> 16.000</td><td>400.000</td><td> 16.000</td><td> 80.000</td><td> 16.000</td><td> 80.000</td><td> 37.673</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>  4.000</td><td> 20.000</td><td>  2.000</td><td> 16.000</td><td>400.000</td><td>  4.000</td><td> 80.000</td><td>  8.000</td><td> 40.000</td><td> 23.131</td></tr>\n",
       "\t<tr><th scope=row>9</th><td>   6.000</td><td>  40.000</td><td>   3.000</td><td>  36.000</td><td>1600.000</td><td>   9.000</td><td> 240.000</td><td>  18.000</td><td> 120.000</td><td>  23.088</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>   4.000</td><td>  40.000</td><td>   3.000</td><td>  16.000</td><td>1600.000</td><td>   9.000</td><td> 160.000</td><td>  12.000</td><td> 120.000</td><td>  12.325</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>   5.000</td><td>  60.000</td><td>   3.000</td><td>  25.000</td><td>3600.000</td><td>   9.000</td><td> 300.000</td><td>  15.000</td><td> 180.000</td><td>  16.461</td></tr>\n",
       "\t<tr><th scope=row>12</th><td>  5.000</td><td> 20.000</td><td>  3.000</td><td> 25.000</td><td>400.000</td><td>  9.000</td><td>100.000</td><td> 15.000</td><td> 60.000</td><td> 33.489</td></tr>\n",
       "\t<tr><th scope=row>13</th><td>   5.000</td><td>  40.000</td><td>   4.000</td><td>  25.000</td><td>1600.000</td><td>  16.000</td><td> 200.000</td><td>  20.000</td><td> 160.000</td><td>  26.189</td></tr>\n",
       "\t<tr><th scope=row>14</th><td>   5.000</td><td>  40.000</td><td>   2.000</td><td>  25.000</td><td>1600.000</td><td>   4.000</td><td> 200.000</td><td>  10.000</td><td>  80.000</td><td>   8.337</td></tr>\n",
       "\t<tr><th scope=row>15</th><td>   5.000</td><td>  40.000</td><td>   3.000</td><td>  25.000</td><td>1600.000</td><td>   9.000</td><td> 200.000</td><td>  15.000</td><td> 120.000</td><td>  19.192</td></tr>\n",
       "\t<tr><th scope=row>16</th><td>   5.000</td><td>  40.000</td><td>   3.000</td><td>  25.000</td><td>1600.000</td><td>   9.000</td><td> 200.000</td><td>  15.000</td><td> 120.000</td><td>  16.579</td></tr>\n",
       "\t<tr><th scope=row>17</th><td>   5.000</td><td>  40.000</td><td>   3.000</td><td>  25.000</td><td>1600.000</td><td>   9.000</td><td> 200.000</td><td>  15.000</td><td> 120.000</td><td>  17.794</td></tr>\n",
       "\t<tr><th scope=row>18</th><td>   5.00</td><td>  40.00</td><td>   3.00</td><td>  25.00</td><td>1600.00</td><td>   9.00</td><td> 200.00</td><td>  15.00</td><td> 120.00</td><td>  16.65</td></tr>\n",
       "\t<tr><th scope=row>19</th><td>   5.000</td><td>  40.000</td><td>   3.000</td><td>  25.000</td><td>1600.000</td><td>   9.000</td><td> 200.000</td><td>  15.000</td><td> 120.000</td><td>  16.799</td></tr>\n",
       "\t<tr><th scope=row>20</th><td>   5.000</td><td>  40.000</td><td>   3.000</td><td>  25.000</td><td>1600.000</td><td>   9.000</td><td> 200.000</td><td>  15.000</td><td> 120.000</td><td>  16.635</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllll}\n",
       "  & ph & temp & conc & ph2 & temp2 & conc2 & phTemp & phConc & tempConc & yield\\\\\n",
       "\\hline\n",
       "\t1 &    6.000 &   60.000 &    4.000 &   36.000 & 3600.000 &   16.000 &  360.000 &   24.000 &  240.000 &   34.841\\\\\n",
       "\t2 &    6.000 &   60.000 &    2.000 &   36.000 & 3600.000 &    4.000 &  360.000 &   12.000 &  120.000 &   16.567\\\\\n",
       "\t3 &   6.000 &  20.000 &   4.000 &  36.000 & 400.000 &  16.000 & 120.000 &  24.000 &  80.000 &  45.396\\\\\n",
       "\t4 &   6.000 &  20.000 &   2.000 &  36.000 & 400.000 &   4.000 & 120.000 &  12.000 &  40.000 &  27.939\\\\\n",
       "\t5 &    4.000 &   60.000 &    4.000 &   16.000 & 3600.000 &   16.000 &  240.000 &   16.000 &  240.000 &   19.825\\\\\n",
       "\t6 &    4.000 &   60.000 &    2.000 &   16.000 & 3600.000 &    4.000 &  240.000 &    8.000 &  120.000 &    1.444\\\\\n",
       "\t7 &   4.000 &  20.000 &   4.000 &  16.000 & 400.000 &  16.000 &  80.000 &  16.000 &  80.000 &  37.673\\\\\n",
       "\t8 &   4.000 &  20.000 &   2.000 &  16.000 & 400.000 &   4.000 &  80.000 &   8.000 &  40.000 &  23.131\\\\\n",
       "\t9 &    6.000 &   40.000 &    3.000 &   36.000 & 1600.000 &    9.000 &  240.000 &   18.000 &  120.000 &   23.088\\\\\n",
       "\t10 &    4.000 &   40.000 &    3.000 &   16.000 & 1600.000 &    9.000 &  160.000 &   12.000 &  120.000 &   12.325\\\\\n",
       "\t11 &    5.000 &   60.000 &    3.000 &   25.000 & 3600.000 &    9.000 &  300.000 &   15.000 &  180.000 &   16.461\\\\\n",
       "\t12 &   5.000 &  20.000 &   3.000 &  25.000 & 400.000 &   9.000 & 100.000 &  15.000 &  60.000 &  33.489\\\\\n",
       "\t13 &    5.000 &   40.000 &    4.000 &   25.000 & 1600.000 &   16.000 &  200.000 &   20.000 &  160.000 &   26.189\\\\\n",
       "\t14 &    5.000 &   40.000 &    2.000 &   25.000 & 1600.000 &    4.000 &  200.000 &   10.000 &   80.000 &    8.337\\\\\n",
       "\t15 &    5.000 &   40.000 &    3.000 &   25.000 & 1600.000 &    9.000 &  200.000 &   15.000 &  120.000 &   19.192\\\\\n",
       "\t16 &    5.000 &   40.000 &    3.000 &   25.000 & 1600.000 &    9.000 &  200.000 &   15.000 &  120.000 &   16.579\\\\\n",
       "\t17 &    5.000 &   40.000 &    3.000 &   25.000 & 1600.000 &    9.000 &  200.000 &   15.000 &  120.000 &   17.794\\\\\n",
       "\t18 &    5.00 &   40.00 &    3.00 &   25.00 & 1600.00 &    9.00 &  200.00 &   15.00 &  120.00 &   16.65\\\\\n",
       "\t19 &    5.000 &   40.000 &    3.000 &   25.000 & 1600.000 &    9.000 &  200.000 &   15.000 &  120.000 &   16.799\\\\\n",
       "\t20 &    5.000 &   40.000 &    3.000 &   25.000 & 1600.000 &    9.000 &  200.000 &   15.000 &  120.000 &   16.635\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "   ph temp conc ph2 temp2 conc2 phTemp phConc tempConc  yield\n",
       "1   6   60    4  36  3600    16    360     24      240 34.841\n",
       "2   6   60    2  36  3600     4    360     12      120 16.567\n",
       "3   6   20    4  36   400    16    120     24       80 45.396\n",
       "4   6   20    2  36   400     4    120     12       40 27.939\n",
       "5   4   60    4  16  3600    16    240     16      240 19.825\n",
       "6   4   60    2  16  3600     4    240      8      120  1.444\n",
       "7   4   20    4  16   400    16     80     16       80 37.673\n",
       "8   4   20    2  16   400     4     80      8       40 23.131\n",
       "9   6   40    3  36  1600     9    240     18      120 23.088\n",
       "10  4   40    3  16  1600     9    160     12      120 12.325\n",
       "11  5   60    3  25  3600     9    300     15      180 16.461\n",
       "12  5   20    3  25   400     9    100     15       60 33.489\n",
       "13  5   40    4  25  1600    16    200     20      160 26.189\n",
       "14  5   40    2  25  1600     4    200     10       80  8.337\n",
       "15  5   40    3  25  1600     9    200     15      120 19.192\n",
       "16  5   40    3  25  1600     9    200     15      120 16.579\n",
       "17  5   40    3  25  1600     9    200     15      120 17.794\n",
       "18  5   40    3  25  1600     9    200     15      120 16.650\n",
       "19  5   40    3  25  1600     9    200     15      120 16.799\n",
       "20  5   40    3  25  1600     9    200     15      120 16.635"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we create the matrix manually\n",
    "d <- c(6,60,4,34.841,6,60,2,16.567,6,20,4,45.396,6,20,2,27.939,4,60,4,19.825,4,60,2,1.444,4,20,4,37.673,4,20,2,23.131,6,40,3,23.088,4,40,3,12.325,5,60,3,16.461,5,20,3,33.489,5,40,4,26.189,5,40,2,8.337,5,40,3,19.192,5,40,3,16.579,5,40,3,17.794,5,40,3,16.65,5,40,3,16.799,5,40,3,16.635)\n",
    "dim(d) <- c(4,20) # we force the data into a mtrix of dimension 4 x 20\n",
    "# we create the data frame\n",
    "datas <- data.frame(\"ph\"=d[1,],\"temp\"=d[2,],\"conc\"=d[3,], \"ph2\"=d[1,]^2, \"temp2\"=d[2,]^2, \"conc2\"=d[3,]^2, \"phTemp\"=d[1,]*d[2,], \"phConc\"=d[1,]*d[3,], \"tempConc\"=d[2,]*d[3,], \"yield\"=d[4,])\n",
    "datas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we can perform a manual regression using the following sequence. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>coeffs</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>intercept</th><td>58.808</td></tr>\n",
       "\t<tr><th scope=row>ph</th><td>-6.091</td></tr>\n",
       "\t<tr><th scope=row>temp</th><td>-2.603</td></tr>\n",
       "\t<tr><th scope=row>conc</th><td>4.805</td></tr>\n",
       "\t<tr><th scope=row>ph2</th><td>0.598</td></tr>\n",
       "\t<tr><th scope=row>temp2</th><td>0.02</td></tr>\n",
       "\t<tr><th scope=row>conc2</th><td>0.154</td></tr>\n",
       "\t<tr><th scope=row>phTemp</th><td>0.11</td></tr>\n",
       "\t<tr><th scope=row>phConc</th><td>0.351</td></tr>\n",
       "\t<tr><th scope=row>tempConc</th><td>0.029</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|l}\n",
       "  & coeffs\\\\\n",
       "\\hline\n",
       "\tintercept & 58.808\\\\\n",
       "\tph & -6.091\\\\\n",
       "\ttemp & -2.603\\\\\n",
       "\tconc & 4.805\\\\\n",
       "\tph2 & 0.598\\\\\n",
       "\ttemp2 & 0.02\\\\\n",
       "\tconc2 & 0.154\\\\\n",
       "\tphTemp & 0.11\\\\\n",
       "\tphConc & 0.351\\\\\n",
       "\ttempConc & 0.029\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "          coeffs\n",
       "intercept 58.808\n",
       "ph        -6.091\n",
       "temp      -2.603\n",
       "conc       4.805\n",
       "ph2        0.598\n",
       "temp2      0.020\n",
       "conc2      0.154\n",
       "phTemp     0.110\n",
       "phConc     0.351\n",
       "tempConc   0.029"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we define the matrix of data (X)\n",
    "M <- cbind(\"intercept\"=rep(1,20),datas[,1:9]) # adds intercept\n",
    "M <- as.matrix(M)\n",
    "\n",
    "# we define y, the response\n",
    "y <- datas[,10]\n",
    "\n",
    "# we calculate the regression coefficient manually\n",
    "coeffs = data.frame(\"coeffs\"=solve(t(M) %*% M) %*% t(M) %*% y)\n",
    "round(coeffs, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where the first coefficient would be the intercept, as we already discussed in the previous example. \n",
    "\n",
    "We can check the data and the prediction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
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   ],
   "source": [
    "# we can check that predictions are just projection of our data on the coefficients\n",
    "predictions <- M %*% as.matrix(coeffs)\n",
    "\n",
    "plot(predictions,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By looking at the coefficient it looks like pH is the most important factor, but concentration and temperature are also significant. The other cross terms are, to the contrary, very small and seems not to influence the yield.\n",
    "\n",
    "The problem with this analysis is that the coefficient are obtained by solving the linear model $y=D*c$. where $D$ is the experimental design matrix and $c$ is for the coefficients. Therefore if the values in the D matrix are large, as for the temperature, the corresponding coefficient will be small to maintain the product constant. It is not possible to compare directly the coefficient if the parameters are not scaled.\n",
    "\n",
    "Before scaling the matrix $D$, here described as M, we can try to use statistics to help find out if what are the relevant parameters and how good is our model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "5"
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      "text/latex": [
       "5"
      ],
      "text/markdown": [
       "5"
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      "text/plain": [
       "[1] 5"
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     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "20"
      ],
      "text/latex": [
       "20"
      ],
      "text/markdown": [
       "20"
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      "text/plain": [
       "[1] 20"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "10"
      ],
      "text/latex": [
       "10"
      ],
      "text/markdown": [
       "10"
      ],
      "text/plain": [
       "[1] 10"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "5"
      ],
      "text/latex": [
       "5"
      ],
      "text/markdown": [
       "5"
      ],
      "text/plain": [
       "[1] 5"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we start the evaluation of our model\n",
    "Rep <- duplicated(datas[,1:3])\n",
    "R <- sum( Rep ) \n",
    "N <- dim(datas)[1]\n",
    "P <- length(coeffs[[1]])\n",
    "D <- N - P - R\n",
    "D\n",
    "N\n",
    "P\n",
    "R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 5 degrees of freedom, so the experimental design should allow to rule out or not the accuracy of the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>mean.replicate</th><th scope=col>mean.residuals</th><th scope=col>mean.total</th><th scope=col>mean.predicted</th><th scope=col>mean.lack.of.fit</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>   0.21</td><td>   0.80</td><td> 565.40</td><td>1130.00</td><td>   1.39</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllll}\n",
       "  & mean.replicate & mean.residuals & mean.total & mean.predicted & mean.lack.of.fit\\\\\n",
       "\\hline\n",
       "\t1 &    0.21 &    0.80 &  565.40 & 1130.00 &    1.39\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "  mean.replicate mean.residuals mean.total mean.predicted mean.lack.of.fit\n",
       "1           0.21            0.8      565.4           1130             1.39"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_replicate <- sum( (mean(datas$yield[Rep]) - datas$yield[Rep])^2 ) # R degrees of freedom\n",
    "S_residual <- sum ( (predictions - y)^2 ) # N-P degrees of freedom (# or S_total - S_predicted)\n",
    "S_total <- sum( datas$yield^2 ) # N degrees of freedom\n",
    "S_predicted <- sum( predictions^2 ) \n",
    "S_lackOfFit <- S_residual - S_replicate # N-P-R degrees of freedom\n",
    "Rsquare <- S_predicted / S_total\n",
    "\n",
    "meanS_replicate <- S_replicate / R\n",
    "meanS_residual <- S_residual / (N-P)\n",
    "meanS_total <- S_total / N\n",
    "meanS_predicted <- S_predicted / P\n",
    "meanS_lackOfFit <- S_lackOfFit / D\n",
    "\n",
    "result <- data.frame(\"mean replicate\"=meanS_replicate,\"mean residuals\"=meanS_residual,\"mean total\"=meanS_total, \"mean predicted\"=meanS_predicted, \"mean lack of fit\"=meanS_lackOfFit)\n",
    "round(result,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>coeffs</th><th scope=col>t</th><th scope=col>1-t</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>intercept</th><td>4.544</td><td>0.001</td><td>0.999</td></tr>\n",
       "\t<tr><th scope=row>ph</th><td>-1.104</td><td> 0.295</td><td> 0.705</td></tr>\n",
       "\t<tr><th scope=row>temp</th><td>-18.27</td><td>  0.00</td><td>  1.00</td></tr>\n",
       "\t<tr><th scope=row>conc</th><td>1.311</td><td>0.219</td><td>0.781</td></tr>\n",
       "\t<tr><th scope=row>ph2</th><td>1.109</td><td>0.293</td><td>0.707</td></tr>\n",
       "\t<tr><th scope=row>temp2</th><td>14.597</td><td> 0.000</td><td> 1.000</td></tr>\n",
       "\t<tr><th scope=row>conc2</th><td>0.287</td><td>0.780</td><td>0.220</td></tr>\n",
       "\t<tr><th scope=row>phTemp</th><td>6.966</td><td>0.000</td><td>1.000</td></tr>\n",
       "\t<tr><th scope=row>phConc</th><td>1.111</td><td>0.293</td><td>0.707</td></tr>\n",
       "\t<tr><th scope=row>tempConc</th><td>1.842</td><td>0.095</td><td>0.905</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & coeffs & t & 1-t\\\\\n",
       "\\hline\n",
       "\tintercept & 4.544 & 0.001 & 0.999\\\\\n",
       "\tph & -1.104 &  0.295 &  0.705\\\\\n",
       "\ttemp & -18.27 &   0.00 &   1.00\\\\\n",
       "\tconc & 1.311 & 0.219 & 0.781\\\\\n",
       "\tph2 & 1.109 & 0.293 & 0.707\\\\\n",
       "\ttemp2 & 14.597 &  0.000 &  1.000\\\\\n",
       "\tconc2 & 0.287 & 0.780 & 0.220\\\\\n",
       "\tphTemp & 6.966 & 0.000 & 1.000\\\\\n",
       "\tphConc & 1.111 & 0.293 & 0.707\\\\\n",
       "\ttempConc & 1.842 & 0.095 & 0.905\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "           coeffs coeffs coeffs\n",
       "intercept   4.544  0.001  0.999\n",
       "ph         -1.104  0.295  0.705\n",
       "temp      -18.270  0.000  1.000\n",
       "conc        1.311  0.219  0.781\n",
       "ph2         1.109  0.293  0.707\n",
       "temp2      14.597  0.000  1.000\n",
       "conc2       0.287  0.780  0.220\n",
       "phTemp      6.966  0.000  1.000\n",
       "phConc      1.111  0.293  0.707\n",
       "tempConc    1.842  0.095  0.905"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we calculate t-values and their probability associated \n",
    "T <- as.matrix(coeffs) / sqrt( diag( solve(t(M) %*% M) ) * meanS_residual )\n",
    "result <- data.frame(\"T\"=T)\n",
    "result[\"t\"] <- 2*pt(abs(T), df = N-P, lower.tail = FALSE)\n",
    "result[\"1-t\"] <- 1-(2 * pt(abs(T), df = N-P, lower.tail = FALSE))\n",
    "round(result, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>F</th><th scope=col>qf</th><th scope=col>pf</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>6.64</td><td>5.05</td><td>0.97</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & F & qf & pf\\\\\n",
       "\\hline\n",
       "\t1 & 6.64 & 5.05 & 0.97\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "     F   qf   pf\n",
       "1 6.64 5.05 0.97"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result <- data.frame( \"F\"=meanS_lackOfFit/meanS_replicate )\n",
    "result[\"qf\"] <- qf(0.95,D,R) # F-factor for 95% confidence\n",
    "result[\"pf\"] <- pf(result$F,D,R,lower.tail = TRUE)\n",
    "round(result,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because p-value is larger than 0.05 we can conclude that the model is accurate. Or more precisely, we cannot rule out our model as inaccurate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## relation between the importance of variables and their magnitude\n",
    "We can scale the data. Sometime this is referred to as encoding the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>ph</th><th scope=col>temp</th><th scope=col>conc</th><th scope=col>ph2</th><th scope=col>temp2</th><th scope=col>conc2</th><th scope=col>phTemp</th><th scope=col>phConc</th><th scope=col>tempConc</th><th scope=col>yield</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>34.841</td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 1.000</td><td> 1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td>-1.000</td><td>16.567</td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td> 1.000</td><td>-1.000</td><td>45.396</td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 1.000</td><td>-1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td>-1.000</td><td> 1.000</td><td>27.939</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td>-1.000</td><td> 1.000</td><td>19.825</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>-1.000</td><td> 1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td> 1.000</td><td>-1.000</td><td> 1.444</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>-1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>-1.000</td><td>-1.000</td><td>37.673</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>-1.000</td><td>-1.000</td><td>-1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td> 1.000</td><td>23.131</td></tr>\n",
       "\t<tr><th scope=row>9</th><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>23.088</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>-1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>12.325</td></tr>\n",
       "\t<tr><th scope=row>11</th><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>16.461</td></tr>\n",
       "\t<tr><th scope=row>12</th><td> 0.000</td><td>-1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>33.489</td></tr>\n",
       "\t<tr><th scope=row>13</th><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>26.189</td></tr>\n",
       "\t<tr><th scope=row>14</th><td> 0.000</td><td> 0.000</td><td>-1.000</td><td> 0.000</td><td> 0.000</td><td> 1.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 8.337</td></tr>\n",
       "\t<tr><th scope=row>15</th><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>19.192</td></tr>\n",
       "\t<tr><th scope=row>16</th><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>16.579</td></tr>\n",
       "\t<tr><th scope=row>17</th><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>17.794</td></tr>\n",
       "\t<tr><th scope=row>18</th><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td> 0.00</td><td>16.65</td></tr>\n",
       "\t<tr><th scope=row>19</th><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>16.799</td></tr>\n",
       "\t<tr><th scope=row>20</th><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td> 0.000</td><td>16.635</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllll}\n",
       "  & ph & temp & conc & ph2 & temp2 & conc2 & phTemp & phConc & tempConc & yield\\\\\n",
       "\\hline\n",
       "\t1 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 & 34.841\\\\\n",
       "\t2 &  1.000 &  1.000 & -1.000 &  1.000 &  1.000 &  1.000 &  1.000 & -1.000 & -1.000 & 16.567\\\\\n",
       "\t3 &  1.000 & -1.000 &  1.000 &  1.000 &  1.000 &  1.000 & -1.000 &  1.000 & -1.000 & 45.396\\\\\n",
       "\t4 &  1.000 & -1.000 & -1.000 &  1.000 &  1.000 &  1.000 & -1.000 & -1.000 &  1.000 & 27.939\\\\\n",
       "\t5 & -1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 & -1.000 & -1.000 &  1.000 & 19.825\\\\\n",
       "\t6 & -1.000 &  1.000 & -1.000 &  1.000 &  1.000 &  1.000 & -1.000 &  1.000 & -1.000 &  1.444\\\\\n",
       "\t7 & -1.000 & -1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 & -1.000 & -1.000 & 37.673\\\\\n",
       "\t8 & -1.000 & -1.000 & -1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 &  1.000 & 23.131\\\\\n",
       "\t9 &  1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 23.088\\\\\n",
       "\t10 & -1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 12.325\\\\\n",
       "\t11 &  0.000 &  1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 &  0.000 & 16.461\\\\\n",
       "\t12 &  0.000 & -1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 &  0.000 & 33.489\\\\\n",
       "\t13 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 & 26.189\\\\\n",
       "\t14 &  0.000 &  0.000 & -1.000 &  0.000 &  0.000 &  1.000 &  0.000 &  0.000 &  0.000 &  8.337\\\\\n",
       "\t15 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 19.192\\\\\n",
       "\t16 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 16.579\\\\\n",
       "\t17 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 17.794\\\\\n",
       "\t18 &  0.00 &  0.00 &  0.00 &  0.00 &  0.00 &  0.00 &  0.00 &  0.00 &  0.00 & 16.65\\\\\n",
       "\t19 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 16.799\\\\\n",
       "\t20 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 &  0.000 & 16.635\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "   ph temp conc ph2 temp2 conc2 phTemp phConc tempConc  yield\n",
       "1   1    1    1   1     1     1      1      1        1 34.841\n",
       "2   1    1   -1   1     1     1      1     -1       -1 16.567\n",
       "3   1   -1    1   1     1     1     -1      1       -1 45.396\n",
       "4   1   -1   -1   1     1     1     -1     -1        1 27.939\n",
       "5  -1    1    1   1     1     1     -1     -1        1 19.825\n",
       "6  -1    1   -1   1     1     1     -1      1       -1  1.444\n",
       "7  -1   -1    1   1     1     1      1     -1       -1 37.673\n",
       "8  -1   -1   -1   1     1     1      1      1        1 23.131\n",
       "9   1    0    0   1     0     0      0      0        0 23.088\n",
       "10 -1    0    0   1     0     0      0      0        0 12.325\n",
       "11  0    1    0   0     1     0      0      0        0 16.461\n",
       "12  0   -1    0   0     1     0      0      0        0 33.489\n",
       "13  0    0    1   0     0     1      0      0        0 26.189\n",
       "14  0    0   -1   0     0     1      0      0        0  8.337\n",
       "15  0    0    0   0     0     0      0      0        0 19.192\n",
       "16  0    0    0   0     0     0      0      0        0 16.579\n",
       "17  0    0    0   0     0     0      0      0        0 17.794\n",
       "18  0    0    0   0     0     0      0      0        0 16.650\n",
       "19  0    0    0   0     0     0      0      0        0 16.799\n",
       "20  0    0    0   0     0     0      0      0        0 16.635"
      ]
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maxs <- apply(d[1:3,],1,max); mins <- apply(d[1:3,],1,min)\n",
    "d[1:3,] <- t( scale(t(d)[,1:3],center = (mins + maxs)/2, scale = (maxs - mins)/2) )\n",
    "datas <- data.frame(\"ph\"=d[1,],\"temp\"=d[2,],\"conc\"=d[3,], \"ph2\"=d[1,]^2, \"temp2\"=d[2,]^2, \"conc2\"=d[3,]^2, \"phTemp\"=d[1,]*d[2,], \"phConc\"=d[1,]*d[3,], \"tempConc\"=d[2,]*d[3,], \"yield\"=d[4,])\n",
    "datas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we perform the regression again, the magnitude of the coefficient will not depend any more on their absolute value. This simplifies the analysis of finding the most important parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>F</th><th scope=col>mean.replicate</th><th scope=col>mean.residuals</th><th scope=col>mean.total</th><th scope=col>mean.predicted</th><th scope=col>mean.lack.of.fit</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>   6.64</td><td>   0.21</td><td>   0.80</td><td> 565.40</td><td>1130.00</td><td>   1.39</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       "  & F & mean.replicate & mean.residuals & mean.total & mean.predicted & mean.lack.of.fit\\\\\n",
       "\\hline\n",
       "\t1 &    6.64 &    0.21 &    0.80 &  565.40 & 1130.00 &    1.39\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "     F mean.replicate mean.residuals mean.total mean.predicted mean.lack.of.fit\n",
       "1 6.64           0.21            0.8      565.4           1130             1.39"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>coeffs</th><th scope=col>t</th><th scope=col>t.prob</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row></th><td> 17.20834</td><td> 56.01061</td><td>100.00000</td></tr>\n",
       "\t<tr><th scope=row>ph</th><td>  5.34330</td><td> 18.90673</td><td>100.00000</td></tr>\n",
       "\t<tr><th scope=row>temp</th><td> -7.84900</td><td>-27.77289</td><td>100.00000</td></tr>\n",
       "\t<tr><th scope=row>conc</th><td>  8.65060</td><td> 30.60927</td><td>100.00000</td></tr>\n",
       "\t<tr><th scope=row>ph2</th><td> 0.5979091</td><td> 1.1094514</td><td>70.6797496</td></tr>\n",
       "\t<tr><th scope=row>temp2</th><td>  7.866409</td><td> 14.596530</td><td> 99.999995</td></tr>\n",
       "\t<tr><th scope=row>conc2</th><td> 0.1544091</td><td> 0.2865141</td><td>21.9667327</td></tr>\n",
       "\t<tr><th scope=row>phTemp</th><td> 2.201000</td><td> 6.965813</td><td>99.996128</td></tr>\n",
       "\t<tr><th scope=row>phConc</th><td> 0.351000</td><td> 1.110859</td><td>70.737550</td></tr>\n",
       "\t<tr><th scope=row>tempConc</th><td> 0.582000</td><td> 1.841937</td><td>90.470412</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & coeffs & t & t.prob\\\\\n",
       "\\hline\n",
       "\t &  17.20834 &  56.01061 & 100.00000\\\\\n",
       "\tph &   5.34330 &  18.90673 & 100.00000\\\\\n",
       "\ttemp &  -7.84900 & -27.77289 & 100.00000\\\\\n",
       "\tconc &   8.65060 &  30.60927 & 100.00000\\\\\n",
       "\tph2 &  0.5979091 &  1.1094514 & 70.6797496\\\\\n",
       "\ttemp2 &   7.866409 &  14.596530 &  99.999995\\\\\n",
       "\tconc2 &  0.1544091 &  0.2865141 & 21.9667327\\\\\n",
       "\tphTemp &  2.201000 &  6.965813 & 99.996128\\\\\n",
       "\tphConc &  0.351000 &  1.110859 & 70.737550\\\\\n",
       "\ttempConc &  0.582000 &  1.841937 & 90.470412\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "             coeffs           t    t.prob\n",
       "         17.2083364  56.0106113 100.00000\n",
       "ph        5.3433000  18.9067255 100.00000\n",
       "temp     -7.8490000 -27.7728909 100.00000\n",
       "conc      8.6506000  30.6092713 100.00000\n",
       "ph2       0.5979091   1.1094514  70.67975\n",
       "temp2     7.8664091  14.5965303 100.00000\n",
       "conc2     0.1544091   0.2865141  21.96673\n",
       "phTemp    2.2010000   6.9658127  99.99613\n",
       "phConc    0.3510000   1.1108588  70.73755\n",
       "tempConc  0.5820000   1.8419369  90.47041"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we define the matrix of data (X)\n",
    "M <- as.matrix(datas[,1:9])\n",
    "M <- cbind(rep(1,20),M) # adds intercept\n",
    "# we define y, the response\n",
    "y <- datas[,10]\n",
    "# we calculate the regression coefficient manually\n",
    "coeffs <- solve(t(M) %*% M) %*% t(M) %*% y\n",
    "# we can check that predictions are just projection of our data on the coefficients\n",
    "predictions <- M %*% coeffs\n",
    "\n",
    "# we start the evaluation of our model\n",
    "Rep <- duplicated(datas[,1:3])\n",
    "R <- sum( Rep ) \n",
    "N <- dim(datas)[1]\n",
    "P <- length(coeffs)\n",
    "D <- N - P - R\n",
    "\n",
    "S_replicate <- sum( (mean(datas$yield[Rep]) - datas$yield[Rep])^2 ) # R degrees of freedom\n",
    "S_residual <- sum ( (predictions - y)^2 ) # N-P degrees of freedom (# or S_total - S_predicted)\n",
    "S_total <- sum( datas$yield^2 ) # N degrees of freedom\n",
    "S_predicted <- sum( predictions^2 ) \n",
    "S_lackOfFit <- S_residual - S_replicate # N-P-R degrees of freedom\n",
    "Rsquare <- S_predicted / S_total\n",
    "\n",
    "meanS_replicate <- S_replicate / R\n",
    "meanS_residual <- S_residual / (N-P)\n",
    "meanS_total <- S_total / N\n",
    "meanS_predicted <- S_predicted / P\n",
    "meanS_lackOfFit <- S_lackOfFit / D\n",
    "\n",
    "F <- meanS_lackOfFit / meanS_replicate\n",
    "\n",
    "result = data.frame(\"F\"=F,\"mean replicate\"=meanS_replicate,\"mean residuals\"=meanS_residual,\"mean total\"=meanS_total, \"mean predicted\"=meanS_predicted, \"mean lack of fit\"=meanS_lackOfFit)\n",
    "round(result,2)\n",
    "\n",
    "# we calculate t-values and their probability associated \n",
    "ttest <- coeffs / sqrt( diag( solve(t(M) %*% M) ) * meanS_residual )\n",
    "tprob <- 1-(2 * pt(abs(ttest), df = N-P, lower.tail = FALSE))\n",
    "table2.11 <- data.frame(\"coeffs\"=coeffs,\"t\"=ttest,\"t-prob\"=tprob*100)\n",
    "table2.11"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scaling the data allows us to directly compare the coefficients, since the significance of the variable is a product of its magnitude by the coefficients. Now, all variable weight the same (their norm is equal to 1), so their coefficient directly reflects their importance. By comparing results obtained with and without scaling the variables we inmediatly notice important changes. By not scaling the data you might have wrongly concluded that only the first 3 parameters were relevant, while a proper scaling of the data shows that the $T^2$ and pH$ \\times T$ might be relevant as well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Leverage\n",
    "\n",
    "Finally we can test different experimental designs (See Table 2.15) and see how the prediction confidence depends on it. For example the accuracy of the prediction may change if points for sampling are distributed in a different manner.\n",
    "\n",
    "<mark>Not only the number of point, and the number of replicate have an influence of the model accuracy, the way sample points are chosen also affect the result.</mark>\n",
    "\n",
    "Here we calculate the leverate for designs of Table 2.15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.23</li>\n",
       "\t<li>0.23</li>\n",
       "\t<li>0.23</li>\n",
       "\t<li>0.13</li>\n",
       "\t<li>0.13</li>\n",
       "\t<li>0.09</li>\n",
       "\t<li>0.13</li>\n",
       "\t<li>0.13</li>\n",
       "\t<li>0.23</li>\n",
       "\t<li>0.23</li>\n",
       "\t<li>0.23</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.23\n",
       "\\item 0.23\n",
       "\\item 0.23\n",
       "\\item 0.13\n",
       "\\item 0.13\n",
       "\\item 0.09\n",
       "\\item 0.13\n",
       "\\item 0.13\n",
       "\\item 0.23\n",
       "\\item 0.23\n",
       "\\item 0.23\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.23\n",
       "2. 0.23\n",
       "3. 0.23\n",
       "4. 0.13\n",
       "5. 0.13\n",
       "6. 0.09\n",
       "7. 0.13\n",
       "8. 0.13\n",
       "9. 0.23\n",
       "10. 0.23\n",
       "11. 0.23\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.23 0.23 0.23 0.13 0.13 0.09 0.13 0.13 0.23 0.23 0.23"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.29</li>\n",
       "\t<li>0.29</li>\n",
       "\t<li>0.14</li>\n",
       "\t<li>0.14</li>\n",
       "\t<li>0.09</li>\n",
       "\t<li>0.09</li>\n",
       "\t<li>0.09</li>\n",
       "\t<li>0.14</li>\n",
       "\t<li>0.14</li>\n",
       "\t<li>0.29</li>\n",
       "\t<li>0.29</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.29\n",
       "\\item 0.29\n",
       "\\item 0.14\n",
       "\\item 0.14\n",
       "\\item 0.09\n",
       "\\item 0.09\n",
       "\\item 0.09\n",
       "\\item 0.14\n",
       "\\item 0.14\n",
       "\\item 0.29\n",
       "\\item 0.29\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.29\n",
       "2. 0.29\n",
       "3. 0.14\n",
       "4. 0.14\n",
       "5. 0.09\n",
       "6. 0.09\n",
       "7. 0.09\n",
       "8. 0.14\n",
       "9. 0.14\n",
       "10. 0.29\n",
       "11. 0.29\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.29 0.29 0.14 0.14 0.09 0.09 0.09 0.14 0.14 0.29 0.29"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.18</li>\n",
       "\t<li>0.18</li>\n",
       "\t<li>0.18</li>\n",
       "\t<li>0.18</li>\n",
       "\t<li>0.1</li>\n",
       "\t<li>0.1</li>\n",
       "\t<li>0.1</li>\n",
       "\t<li>0.12</li>\n",
       "\t<li>0.12</li>\n",
       "\t<li>0.26</li>\n",
       "\t<li>0.5</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.18\n",
       "\\item 0.18\n",
       "\\item 0.18\n",
       "\\item 0.18\n",
       "\\item 0.1\n",
       "\\item 0.1\n",
       "\\item 0.1\n",
       "\\item 0.12\n",
       "\\item 0.12\n",
       "\\item 0.26\n",
       "\\item 0.5\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.18\n",
       "2. 0.18\n",
       "3. 0.18\n",
       "4. 0.18\n",
       "5. 0.1\n",
       "6. 0.1\n",
       "7. 0.1\n",
       "8. 0.12\n",
       "9. 0.12\n",
       "10. 0.26\n",
       "11. 0.5\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.18 0.18 0.18 0.18 0.10 0.10 0.10 0.12 0.12 0.26 0.50"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A <- cbind(rep(1,11), c(1, 1, 1, 2, 2, 3, 4, 4, 5, 5, 5) )\n",
    "round(diag( A %*% solve(t(A) %*% A) %*% t(A) ),2)\n",
    "\n",
    "A <- cbind(rep(1,11), c(1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5) )\n",
    "round(diag( A %*% solve(t(A) %*% A) %*% t(A) ),2)\n",
    "\n",
    "A <- cbind(rep(1,11), c(1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5) )\n",
    "round(diag( A %*% solve(t(A) %*% A) %*% t(A) ),2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For more on experimental design, please consult Brereton, R. G. (2003). Chemometrics. Technometrics. Chichester, UK: John Wiley & Sons, Ltd. http://doi.org/10.1002/0470863242"
   ]
  }
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