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Step 1. Input the data, look at the variables, remove constant or otherwise doubtfully useful variables

Remember that variables that may be constant in this data set could become useful in a larger data set where they vary more.

When editing the data file in Excel prior to importing the data into R, use the following formula to make a version substituting “NA� in the blank cells: =IF(ISBLANK(cell reference),“NA�, cell reference)

Read in the data to R from a text file:


martinx<-read.table("Cyperus margaritaceus dataset2_SimoneditSep22.txt", header=TRUE)

Look at the data to see if there are constant variables and missing values. Get a general idea by looking at the column names:


names(martinx)
 [1] "Specimen" "Taxon"    "Lflen"    "Lfwid"    "Cullen"   "Culwid"  
 [7] "Numbrac"  "Lenbrac"  "Numspk"   "Spklen"   "Spkwid"   "Gllen"   
[13] "Glwid"    "Nutlen"   "Nutwid"   "Numgl"    "Bshpers"  "Bshtext" 
[19] "Bshglos" 

Look at the first six lines with function “head�


head(martinx)
NA

Look to see how many missing values there are (NA) with function “summary�:


summary(martinx)
        Specimen               Taxon         Lflen      
 M.Swa.1592 :  2   margaritaceus  :201   Min.   : 1.50  
 M.Za.9883  :  2   obtusiflorus   : 72   1st Qu.:10.12  
 Nd.Za.1450 :  2   sphaerocephalus: 64   Median :16.00  
 Sph.Sa.18  :  2   nduru          : 60   Mean   :17.25  
 ka.Bu.21193:  1   tisserantii    : 47   3rd Qu.:22.50  
 ka.Gh.1975 :  1   karlschumannii : 14   Max.   :90.00  
 (Other)    :479   (Other)        : 31   NA's   :22     
     Lfwid            Cullen           Culwid         Numbrac     
 Min.   : 0.200   Min.   :  3.50   Min.   :0.400   Min.   :1.000  
 1st Qu.: 0.600   1st Qu.: 18.12   1st Qu.:0.800   1st Qu.:2.000  
 Median : 0.900   Median : 27.75   Median :1.000   Median :3.000  
 Mean   : 1.131   Mean   : 31.62   Mean   :1.092   Mean   :3.022  
 3rd Qu.: 1.500   3rd Qu.: 41.35   3rd Qu.:1.300   3rd Qu.:3.500  
 Max.   :13.000   Max.   :115.70   Max.   :2.500   Max.   :6.000  
 NA's   :6        NA's   :10       NA's   :3       NA's   :2      
    Lenbrac           Numspk           Spklen           Spkwid      
 Min.   : 0.160   Min.   : 1.000   Min.   : 4.000   Min.   : 2.000  
 1st Qu.: 2.500   1st Qu.: 3.000   1st Qu.: 8.815   1st Qu.: 4.300  
 Median : 3.750   Median : 5.000   Median :10.650   Median : 5.200  
 Mean   : 4.244   Mean   : 6.964   Mean   :11.988   Mean   : 5.414  
 3rd Qu.: 5.500   3rd Qu.: 9.000   3rd Qu.:13.800   3rd Qu.: 6.500  
 Max.   :25.000   Max.   :40.000   Max.   :40.000   Max.   :10.000  
 NA's   :17       NA's   :7        NA's   :3        NA's   :2       
     Gllen            Glwid           Nutlen          Nutwid     
 Min.   : 2.500   Min.   :1.000   Min.   :1.000   Min.   :0.500  
 1st Qu.: 4.500   1st Qu.:1.500   1st Qu.:1.800   1st Qu.:1.200  
 Median : 5.000   Median :2.000   Median :2.100   Median :1.400  
 Mean   : 5.247   Mean   :2.096   Mean   :2.104   Mean   :1.374  
 3rd Qu.: 6.000   3rd Qu.:2.500   3rd Qu.:2.400   3rd Qu.:1.500  
 Max.   :10.000   Max.   :4.500   Max.   :3.500   Max.   :2.500  
 NA's   :3        NA's   :2       NA's   :223     NA's   :223    
     Numgl           Bshpers      Bshtext     Bshglos   
 Min.   : 1.3   fibrous  : 19   firm  : 53   dull :357  
 1st Qu.: 8.5   flattened:457   hard  : 70   shiny: 99  
 Median :11.0   NA's     : 13   papery:335   NA's : 33  
 Mean   :12.4                   NA's  : 31              
 3rd Qu.:14.6                                           
 Max.   :41.5                                           
 NA's   :2                                              

The “summary� function also summary statistics for each variable.

Two characters have very large proportions of missing values and for these interpolation is inadvisable and so they have to be deleted (nutlet length and width):


martinx.edit<-martinx[,-c(14:15)]
names(martinx.edit)
 [1] "Specimen" "Taxon"    "Lflen"    "Lfwid"    "Cullen"   "Culwid"  
 [7] "Numbrac"  "Lenbrac"  "Numspk"   "Spklen"   "Spkwid"   "Gllen"   
[13] "Glwid"    "Numgl"    "Bshpers"  "Bshtext"  "Bshglos" 

Step 2. Imputation of missing variables

Now we can interpolate values for the missing data using the “missForest� package. This will provide a complete matrix for analysis. This package handles mixed continuous and categorical data sets, and is non-parametric.

Load the “missForest� package:


library(missForest)
Loading required package: randomForest
randomForest 4.6-14
Type rfNews() to see new features/changes/bug fixes.
Loading required package: foreach
Loading required package: itertools
Loading required package: iterators

Delete the first two columns of factors (specimen and taxon names), set seed and carry out the interpolation with the function “missForest�.


martinx.edit.2<-martinx.edit[,-c(1:2)]
set.seed(810)
martinx.edit.2.imp<-missForest(martinx.edit.2, verbose = TRUE)
  missForest iteration 1 in progress...done!
    estimated error(s): 0.3799116 0.08545668 
    difference(s): 0.005567988 0.00408998 
    time: 1.81 seconds

  missForest iteration 2 in progress...done!
    estimated error(s): 0.3834789 0.08692505 
    difference(s): 0.000424447 0.0006816633 
    time: 1.76 seconds

  missForest iteration 3 in progress...done!
    estimated error(s): 0.3766948 0.07962788 
    difference(s): 7.394146e-05 0.0006816633 
    time: 1.76 seconds

  missForest iteration 4 in progress...done!
    estimated error(s): 0.374215 0.09202605 
    difference(s): 0.0001042416 0.0006816633 
    time: 1.75 seconds

Look at the structure of the output object “martinx.edit.2.imp�:


str(martinx.edit.2.imp)
List of 2
 $ ximp    :'data.frame':   489 obs. of  15 variables:
  ..$ Lflen  : num [1:489] 21 12 11.5 19 24 16 21 17.5 36.5 20 ...
  ..$ Lfwid  : num [1:489] 1.5 1 0.7 1 0.75 1 2.5 1.2 0.7 1.5 ...
  ..$ Cullen : num [1:489] 49 33 27 28 28 40 40.8 27.3 54.5 62.2 ...
  ..$ Culwid : num [1:489] 0.8 1 0.6 1 0.8 1 1.8 0.9 1.1 1.3 ...
  ..$ Numbrac: num [1:489] 3 2 2 3 2.5 2 4 3 2 3 ...
  ..$ Lenbrac: num [1:489] 4.8 3.5 3.44 5.5 12.5 ...
  ..$ Numspk : num [1:489] 2.5 1.5 2 3.5 2.5 4.5 6 5 2 4 ...
  ..$ Spklen : num [1:489] 16.2 10.1 15 14.6 9.8 16 14.8 13 19.8 14.5 ...
  ..$ Spkwid : num [1:489] 5.8 6.8 7 5.7 5.6 6.25 6.6 5.2 5.3 6.6 ...
  ..$ Gllen  : num [1:489] 5.7 4.5 5.5 5.5 6 6.2 5.4 5 5 5.5 ...
  ..$ Glwid  : num [1:489] 3 3 2.5 2.5 4 3.2 3 3 3 4 ...
  ..$ Numgl  : num [1:489] 12.7 16.7 17 14.8 10 ...
  ..$ Bshpers: Factor w/ 2 levels "fibrous","flattened": 2 2 2 2 2 2 2 2 2 2 ...
  ..$ Bshtext: Factor w/ 3 levels "firm","hard",..: 3 3 3 3 3 3 3 3 3 3 ...
  ..$ Bshglos: Factor w/ 2 levels "dull","shiny": 1 1 1 1 1 1 1 1 1 1 ...
 $ OOBerror: Named num [1:2] 0.3767 0.0796
  ..- attr(*, "names")= chr [1:2] "NRMSE" "PFC"
 - attr(*, "class")= chr "missForest"

The completed data frame with imputed values is the object “martinx.edit.2.imp$ximp�:


head(martinx.edit.2.imp$ximp)
NA

Now we add back the factor columns we removed (columns 1 and 2) and give the imputed matrix a simpler name:


martinx.imp<-data.frame(martinx.edit[, c(1:2)], martinx.edit.2.imp$ximp)

head(martinx.imp)
NA

Write the new imputed matrix back into Excel so it can be saved separately. This script writes the file into the working directory:


write.csv(martinx.imp, file="Cyperus margaritaceus imputed.csv", row.names = FALSE)

Now have a careful look at the Excel file to see if there are any peculiarities that need to be addressed. Compare with the original Excel file.

Step 3. Principal Coordinate Analysis (PCoA

Because the data frame has both quantitative variable and qualitative characters, we will use Gower’s distance coefficient to compute a distance matrix.

The cells of the distance matrix show the Gower’s morphological distance between every pair of individuals.

The PCoA is a computational approach which attempts to represent these distances as faithfully as possible using orthogonal axes (analogous to PCA axes).

To compute the Gower distance matrix we use the function daisy from the package cluster.


library(cluster)

The computation of Gower’s distance standardizes the variables, so there is no need to standardize them first.


martin.gower<-daisy(martinx.imp[-c(1:2)], metric="gower")

Carry out the PCoA using the function cmdscale in the “stats� package that is loaded automatically with R:


pcoa<-cmdscale(martin.gower, k=10, eig=TRUE)

summary(pcoa)
       Length Class  Mode   
points 4890   -none- numeric
eig     489   -none- numeric
x         0   -none- NULL   
ac        1   -none- numeric
GOF       2   -none- numeric

The object “pcoa$points� has the coordinates of the scores on each of the principal coordinates (PCos). The call was for 10 PCos, which means that the scores for the first 10 PCos are printed.

The object eig gives the values of all 414 eigenvalues in descending order. There are as many eigenvalues as objects in the distance matrix. Printing eig shows how many are positive and how many negative. Only positive eigenvalues (and their corresponding PCos) are meaningful.


pcoa$eig
  [1]  4.244259e+00  3.173968e+00  1.065240e+00  7.932528e-01
  [5]  5.628640e-01  4.392599e-01  3.959920e-01  3.440904e-01
  [9]  3.037265e-01  2.503792e-01  2.319920e-01  2.168423e-01
 [13]  1.927479e-01  1.646096e-01  1.517877e-01  1.470159e-01
 [17]  1.390535e-01  1.273764e-01  1.006129e-01  9.652160e-02
 [21]  9.434178e-02  8.596209e-02  8.103600e-02  7.937024e-02
 [25]  7.686824e-02  7.128964e-02  6.310731e-02  6.209858e-02
 [29]  5.759248e-02  5.491411e-02  5.299950e-02  4.687674e-02
 [33]  4.466907e-02  4.287627e-02  3.995105e-02  3.758260e-02
 [37]  3.694669e-02  3.393739e-02  3.251374e-02  3.176206e-02
 [41]  2.909629e-02  2.865197e-02  2.573298e-02  2.516563e-02
 [45]  2.454557e-02  2.409919e-02  2.295154e-02  2.199292e-02
 [49]  2.023421e-02  1.952380e-02  1.867106e-02  1.797300e-02
 [53]  1.663856e-02  1.569946e-02  1.521185e-02  1.462230e-02
 [57]  1.306318e-02  1.266385e-02  1.242249e-02  1.175119e-02
 [61]  1.131447e-02  1.053885e-02  1.008696e-02  9.471585e-03
 [65]  9.118144e-03  8.415913e-03  8.088253e-03  7.899138e-03
 [69]  7.527452e-03  7.496477e-03  7.137608e-03  6.960198e-03
 [73]  6.377337e-03  5.884372e-03  5.533017e-03  5.454866e-03
 [77]  5.011755e-03  4.903804e-03  4.859934e-03  4.535768e-03
 [81]  4.416824e-03  4.231235e-03  4.137643e-03  3.841396e-03
 [85]  3.427142e-03  3.378176e-03  3.017528e-03  2.918212e-03
 [89]  2.722487e-03  2.609034e-03  2.518789e-03  2.271621e-03
 [93]  2.148236e-03  2.051238e-03  1.887642e-03  1.735942e-03
 [97]  1.608233e-03  1.539720e-03  1.443297e-03  1.366625e-03
[101]  1.302038e-03  1.210318e-03  1.112911e-03  1.043919e-03
[105]  9.965999e-04  8.862354e-04  8.216288e-04  6.964927e-04
[109]  6.206712e-04  5.232006e-04  4.762193e-04  3.952441e-04
[113]  3.785104e-04  2.986538e-04  2.671530e-04  2.113794e-04
[117]  1.534363e-04  1.093472e-04  7.669815e-05  2.158849e-05
[121]  2.775558e-17 -2.550674e-05 -6.523860e-05 -9.116807e-05
[125] -1.431669e-04 -1.873903e-04 -2.031885e-04 -2.300432e-04
[129] -2.443387e-04 -2.972826e-04 -3.056029e-04 -3.365362e-04
[133] -3.777453e-04 -4.112787e-04 -4.447016e-04 -4.649513e-04
[137] -4.745252e-04 -4.881963e-04 -5.226172e-04 -5.592212e-04
[141] -5.766836e-04 -6.163620e-04 -6.203184e-04 -6.478443e-04
[145] -6.555357e-04 -6.855215e-04 -7.044100e-04 -7.081698e-04
[149] -7.303650e-04 -7.390374e-04 -7.476732e-04 -7.575619e-04
[153] -7.840543e-04 -8.060396e-04 -8.174363e-04 -8.382559e-04
[157] -8.497114e-04 -8.618062e-04 -8.729267e-04 -9.013596e-04
[161] -9.062107e-04 -9.159225e-04 -9.411694e-04 -9.610851e-04
[165] -9.755973e-04 -1.010917e-03 -1.017257e-03 -1.046977e-03
[169] -1.051747e-03 -1.061857e-03 -1.091973e-03 -1.097687e-03
[173] -1.107245e-03 -1.131032e-03 -1.144608e-03 -1.165058e-03
[177] -1.187734e-03 -1.200830e-03 -1.210359e-03 -1.224088e-03
[181] -1.229099e-03 -1.264417e-03 -1.275568e-03 -1.285518e-03
[185] -1.315358e-03 -1.335821e-03 -1.344381e-03 -1.359120e-03
[189] -1.366938e-03 -1.379697e-03 -1.387230e-03 -1.418340e-03
[193] -1.426083e-03 -1.456446e-03 -1.471890e-03 -1.477788e-03
[197] -1.504407e-03 -1.516667e-03 -1.521371e-03 -1.536069e-03
[201] -1.564475e-03 -1.580368e-03 -1.585794e-03 -1.621367e-03
[205] -1.624634e-03 -1.638950e-03 -1.650526e-03 -1.680749e-03
[209] -1.701321e-03 -1.713365e-03 -1.751104e-03 -1.768158e-03
[213] -1.790585e-03 -1.831733e-03 -1.838004e-03 -1.844192e-03
[217] -1.873476e-03 -1.895872e-03 -1.912144e-03 -1.945025e-03
[221] -1.950108e-03 -1.967734e-03 -1.984179e-03 -1.995005e-03
[225] -2.010586e-03 -2.029487e-03 -2.031994e-03 -2.053296e-03
[229] -2.074300e-03 -2.087567e-03 -2.125527e-03 -2.135541e-03
[233] -2.189118e-03 -2.190727e-03 -2.198023e-03 -2.214908e-03
[237] -2.230486e-03 -2.268535e-03 -2.291847e-03 -2.312243e-03
[241] -2.329168e-03 -2.348215e-03 -2.361153e-03 -2.402889e-03
[245] -2.427377e-03 -2.440798e-03 -2.445314e-03 -2.484042e-03
[249] -2.493179e-03 -2.521582e-03 -2.560208e-03 -2.582064e-03
[253] -2.624078e-03 -2.650615e-03 -2.676032e-03 -2.699435e-03
[257] -2.727183e-03 -2.748361e-03 -2.774237e-03 -2.801458e-03
[261] -2.838893e-03 -2.851463e-03 -2.866315e-03 -2.894944e-03
[265] -2.902159e-03 -2.918049e-03 -2.981492e-03 -3.003366e-03
[269] -3.037144e-03 -3.060796e-03 -3.084182e-03 -3.095895e-03
[273] -3.128297e-03 -3.173526e-03 -3.194930e-03 -3.205263e-03
[277] -3.241370e-03 -3.273891e-03 -3.305683e-03 -3.336924e-03
[281] -3.343132e-03 -3.375187e-03 -3.383750e-03 -3.429806e-03
[285] -3.473284e-03 -3.503079e-03 -3.531455e-03 -3.553633e-03
[289] -3.598149e-03 -3.617062e-03 -3.657897e-03 -3.673377e-03
[293] -3.745989e-03 -3.759814e-03 -3.800429e-03 -3.821012e-03
[297] -3.866854e-03 -3.900946e-03 -3.929614e-03 -3.955516e-03
[301] -3.994187e-03 -4.051024e-03 -4.107279e-03 -4.118227e-03
[305] -4.171557e-03 -4.209810e-03 -4.223324e-03 -4.283607e-03
[309] -4.340416e-03 -4.366020e-03 -4.378551e-03 -4.440503e-03
[313] -4.482116e-03 -4.547933e-03 -4.564028e-03 -4.612708e-03
[317] -4.649907e-03 -4.698543e-03 -4.710476e-03 -4.788004e-03
[321] -4.800426e-03 -4.843616e-03 -4.877196e-03 -4.931099e-03
[325] -4.967732e-03 -5.025881e-03 -5.072611e-03 -5.151579e-03
[329] -5.189556e-03 -5.231932e-03 -5.276806e-03 -5.332301e-03
[333] -5.369760e-03 -5.431945e-03 -5.480724e-03 -5.517574e-03
[337] -5.620018e-03 -5.677121e-03 -5.760451e-03 -5.792143e-03
[341] -5.842003e-03 -5.862578e-03 -5.970124e-03 -5.987550e-03
[345] -6.073581e-03 -6.167259e-03 -6.185092e-03 -6.244635e-03
[349] -6.265631e-03 -6.407388e-03 -6.415749e-03 -6.465985e-03
[353] -6.495097e-03 -6.590136e-03 -6.670360e-03 -6.710346e-03
[357] -6.789286e-03 -6.887198e-03 -6.911581e-03 -6.942788e-03
[361] -7.049548e-03 -7.089427e-03 -7.183878e-03 -7.226286e-03
[365] -7.315986e-03 -7.488770e-03 -7.526437e-03 -7.679019e-03
[369] -7.710868e-03 -7.757474e-03 -7.932557e-03 -7.957283e-03
[373] -8.092033e-03 -8.168503e-03 -8.230399e-03 -8.323442e-03
[377] -8.438209e-03 -8.472540e-03 -8.546880e-03 -8.669975e-03
[381] -8.873451e-03 -8.910139e-03 -8.917740e-03 -9.112978e-03
[385] -9.262132e-03 -9.329742e-03 -9.416222e-03 -9.541235e-03
[389] -9.636501e-03 -9.789663e-03 -9.994689e-03 -1.001568e-02
[393] -1.007838e-02 -1.033799e-02 -1.035657e-02 -1.051687e-02
[397] -1.072252e-02 -1.086099e-02 -1.101685e-02 -1.106863e-02
[401] -1.120132e-02 -1.143265e-02 -1.162336e-02 -1.168029e-02
[405] -1.182179e-02 -1.201426e-02 -1.209987e-02 -1.230504e-02
[409] -1.238868e-02 -1.252550e-02 -1.275634e-02 -1.296922e-02
[413] -1.323681e-02 -1.335526e-02 -1.342488e-02 -1.368185e-02
[417] -1.398655e-02 -1.420258e-02 -1.452596e-02 -1.475708e-02
[421] -1.484249e-02 -1.513289e-02 -1.536927e-02 -1.574063e-02
[425] -1.601145e-02 -1.616073e-02 -1.640332e-02 -1.660931e-02
[429] -1.688632e-02 -1.718898e-02 -1.779875e-02 -1.796789e-02
[433] -1.821736e-02 -1.845657e-02 -1.894686e-02 -1.901240e-02
[437] -1.949613e-02 -1.997751e-02 -2.008348e-02 -2.058428e-02
[441] -2.081246e-02 -2.119970e-02 -2.195006e-02 -2.210595e-02
[445] -2.271940e-02 -2.347715e-02 -2.377401e-02 -2.443264e-02
[449] -2.485684e-02 -2.590615e-02 -2.627284e-02 -2.649268e-02
[453] -2.767302e-02 -2.845276e-02 -2.899381e-02 -2.970586e-02
[457] -3.033085e-02 -3.050061e-02 -3.223162e-02 -3.260044e-02
[461] -3.337235e-02 -3.447256e-02 -3.538942e-02 -3.902985e-02
[465] -4.038473e-02 -4.101068e-02 -4.184464e-02 -4.251955e-02
[469] -4.683444e-02 -4.897365e-02 -4.990785e-02 -5.165909e-02
[473] -5.692140e-02 -6.100464e-02 -6.192402e-02 -7.008225e-02
[477] -7.232298e-02 -7.355198e-02 -7.848086e-02 -8.661082e-02
[481] -9.579114e-02 -9.773078e-02 -1.105718e-01 -1.230960e-01
[485] -1.445094e-01 -1.605858e-01 -2.264697e-01 -2.526160e-01
[489] -5.938851e-01

Only the first 121 eigenvalues are positive. We need to visualize them:


barplot(pcoa$eig[1:121])

It would be more useful to show these as percentages:


eig.percent<-(pcoa$eig[1:121]*100)/sum(pcoa$eig[1:121])
round(eig.percent, 3)
  [1] 28.541 21.344  7.163  5.334  3.785  2.954  2.663  2.314  2.042
 [10]  1.684  1.560  1.458  1.296  1.107  1.021  0.989  0.935  0.857
 [19]  0.677  0.649  0.634  0.578  0.545  0.534  0.517  0.479  0.424
 [28]  0.418  0.387  0.369  0.356  0.315  0.300  0.288  0.269  0.253
 [37]  0.248  0.228  0.219  0.214  0.196  0.193  0.173  0.169  0.165
 [46]  0.162  0.154  0.148  0.136  0.131  0.126  0.121  0.112  0.106
 [55]  0.102  0.098  0.088  0.085  0.084  0.079  0.076  0.071  0.068
 [64]  0.064  0.061  0.057  0.054  0.053  0.051  0.050  0.048  0.047
 [73]  0.043  0.040  0.037  0.037  0.034  0.033  0.033  0.031  0.030
 [82]  0.028  0.028  0.026  0.023  0.023  0.020  0.020  0.018  0.018
 [91]  0.017  0.015  0.014  0.014  0.013  0.012  0.011  0.010  0.010
[100]  0.009  0.009  0.008  0.007  0.007  0.007  0.006  0.006  0.005
[109]  0.004  0.004  0.003  0.003  0.003  0.002  0.002  0.001  0.001
[118]  0.001  0.001  0.000  0.000

Still more useful is a cumulative sum of the eigenvalues so we can see where a good cut-off point is:


eig.percent.cumul<-cumsum(eig.percent)
eig.percent.cumul
  [1]  28.54149  49.88557  57.04902  62.38343  66.16854  69.12244
  [7]  71.78538  74.09929  76.14177  77.82551  79.38559  80.84379
 [13]  82.13997  83.24693  84.26766  85.25630  86.19140  87.04797
 [19]  87.72456  88.37364  89.00807  89.58614  90.13108  90.66483
 [25]  91.18175  91.66115  92.08553  92.50312  92.89042  93.25970
 [31]  93.61611  93.93134  94.23173  94.52006  94.78872  95.04145
 [37]  95.28991  95.51813  95.73678  95.95037  96.14603  96.33871
 [43]  96.51176  96.68099  96.84605  97.00811  97.16245  97.31035
 [49]  97.44642  97.57771  97.70327  97.82413  97.93602  98.04160
 [55]  98.14389  98.24222  98.33007  98.41523  98.49877  98.57779
 [61]  98.65388  98.72475  98.79258  98.85628  98.91759  98.97419
 [67]  99.02858  99.08170  99.13232  99.18273  99.23073  99.27753
 [73]  99.32042  99.35999  99.39720  99.43388  99.46758  99.50056
 [79]  99.53324  99.56374  99.59345  99.62190  99.64973  99.67556
 [85]  99.69860  99.72132  99.74161  99.76124  99.77955  99.79709
 [91]  99.81403  99.82931  99.84375  99.85755  99.87024  99.88191
 [97]  99.89273  99.90308  99.91279  99.92198  99.93073  99.93887
[103]  99.94636  99.95338  99.96008  99.96604  99.97156  99.97625
[109]  99.98042  99.98394  99.98714  99.98980  99.99235  99.99435
[115]  99.99615  99.99757  99.99860  99.99934  99.99985 100.00000
[121] 100.00000

We need 36 eigenvalues to capture 95% of the variance:


barplot(pcoa$eig[1:36])

What this means is that the ordinations of the first two principal coordinates will only express 49.9% of the total variation.

We might need to try Non-Metric Multidimensional Scaling (NMDS)to improve on this - it finds the best 2 dimensional plot by an iteration process.

Plot the PCoA ordinations:


scores<-pcoa$points
plot(scores[,1],scores[,2], pch=16, cex=0.8, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey")[martinx.imp[,2]], asp=1, xlab="PCo1", ylab="PCo2", main="PCoA Cyperus margaritaceus complex", cex.main=0.8)

legend("bottomleft", levels(martinx.imp[,2]), pch=16, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.8, pt.cex = 1.0)

NA
NA

Carry out a Principal Component Analysis (PCA) using the quantitative variables on the imputed data set martinx.imp

Run the PCA


martinx.imp.pca<-prcomp(martinx.imp[,3:14], scale=TRUE)

summary(martinx.imp.pca)
Importance of components:
                          PC1    PC2     PC3     PC4     PC5     PC6
Standard deviation     2.1980 1.4456 1.03308 0.92074 0.89322 0.76397
Proportion of Variance 0.4026 0.1741 0.08894 0.07065 0.06649 0.04864
Cumulative Proportion  0.4026 0.5767 0.66567 0.73631 0.80280 0.85144
                          PC7     PC8     PC9    PC10    PC11    PC12
Standard deviation     0.7039 0.66439 0.51426 0.50551 0.46846 0.32604
Proportion of Variance 0.0413 0.03679 0.02204 0.02129 0.01829 0.00886
Cumulative Proportion  0.8927 0.92952 0.95156 0.97285 0.99114 1.00000

Compute the number of significant principal components using the function “evplot.R� which is obtainable via the Springer book web page for Borcard et al’s “Numerical Ecology� (Borcard et al., 2011)

First run the function:

evplot() Plot eigenvalues and percentages of variation Kaiser rule and broken stick model License: GPL-2 Author: Francois Gillet, Octber 2007


evplot = function(ev) {
# Broken stick model (MacArthur 1957)
n = length(ev)
bsm = data.frame(j=seq(1:n), p=0)
bsm$p[1] = 1/n
for (i in 2:n) {
bsm$p[i] = bsm$p[i-1] + (1/(n + 1 - i))
}
bsm$p = 100*bsm$p/n

# Plot eigenvalues and % of variation for each axis

op = par(mfrow=c(2,1))
barplot(ev, main="Eigenvalues", col="bisque", las=2)
abline(h=mean(ev), col="red")
legend("topright", "Average eigenvalue", lwd=1, col=2, bty="n")
barplot(t(cbind(100*ev/sum(ev), bsm$p[n:1])), beside=T, main="% variation", col=c("bisque",2), las=2)
legend("topright", c("% eigenvalue", "Broken stick model"),pch=15, col=c("bisque",2), bty="n")
par(op)
}

Use the function:


ev<-(martinx.imp.pca$sdev)^2
evplot(ev)

This shows significance for the first 3 PCs, which express 66.6% of total variance.

Plot the first three PCs:


par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)


par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

NA
NA

par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,2], martinx.imp.pca$x[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

NA
NA

Replot with convex hulls

To plot convex hulls around the species groups we can use the function Plot ConvexHull from Ken Takagi’s website Chit Chat R to overplot convex hulls onto the original plot.


### Plotting function to plot convex hulls

### Filename: Plot_ConvexHull.R
### Notes:
############################################################################
# INPUTS:
# xcoords: x-coordinates of point data
# ycoords: y-coordinates of point data
# lcolor: line color
# OUTPUTS:
# convex hull around data points in a particular color (specified by lcolor)
# FUNCTION:
Plot_ConvexHull<-function(xcoord, ycoord, lcolor){
hpts <- chull(x = xcoord, y = ycoord)
hpts <- c(hpts, hpts[1])
lines(xcoord[hpts], ycoord[hpts], col = lcolor)
}
# END OF FUNCTION

par(mfrow=c(1,1))

plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="karlschumannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="karlschumannii"], lcolor = "red")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="ledermannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="ledermannii"], lcolor = "blue")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="margaritaceus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="margaritaceus"], lcolor = "green")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="nduru"],martinx.imp.pca$x[,2][martinx.imp[,2]=="nduru"], lcolor = "black")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="niveus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="niveus"], lcolor = "magenta")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="obtusiflorus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="somaliensis"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "sienna")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="sphaerocephalus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "khaki")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="tisserantii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="tisserantii"], lcolor = "grey")

NA
NA

Now replot with 95% confidence ellipses.

Now plot with 95% confidence ellipses using using the ELLI function from Julien Claude’s book (Morphometrics with R) website:

Run ELLI function:


#run the ELLI function:
ELLI<-function(x,y,conf=0.95,np=50)
{centroid<-apply(cbind(x,y),2,mean)
ang <- seq(0,2*pi,length=np)
z<-cbind(cos(ang),sin(ang))
radiuscoef<-qnorm((1-conf)/2, lower.tail=F)
vcvxy<-var(cbind(x,y))
r<-cor(x,y)
M1<-matrix(c(1,1,-1,1),2,2)
M2<-matrix(c(var(x), var(y)),2,2)
M3<-matrix(c(1+r, 1-r),2,2, byrow=T)
ellpar<-M1*sqrt(M2*M3/2)
t(centroid + radiuscoef * ellpar %*% t(z))}

par(mfrow=c(1,1))

plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

#plot the ellipses:
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="karlschumannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="karlschumannii"]), col = "red")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="ledermannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="ledermannii"]), col = "blue")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="margaritaceus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="margaritaceus"]), col = "green")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="nduru"],martinx.imp.pca$x[,2][martinx.imp[,2]=="nduru"]), col = "black")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="niveus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="niveus"]), col = "magenta")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="obtusiflorus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="somaliensis"],martinx.imp.pca$x[,2][martinx.imp[,2]=="somaliensis"]), col = "sienna")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="sphaerocephalus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="tisserantii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="tisserantii"]), col = "grey")

NA
NA

Plot the loadings for the first two principal components


par(mfrow=c(1,2))
barplot(sort(martinx.imp.pca$rotation[,1]), las=2, cex.names=0.7, main="PC1", ylab="Loadings")
barplot(sort(martinx.imp.pca$rotation[,2]), las=2, cex.names=0.7, main="PC2", ylab="Loadings")
par(mfrow=c(1,1))

Plot the biplot


biplot(martinx.imp.pca, cex=0.5)

Linear discriminant analysis (LDA)

Following Legendre & Legendre (2012, Numerical Ecology, 3rd English Edition), we should analyse untransformed data for testing the allocation of individuals to populations, and standardized data for computing the contribution of each variable to the discriminant function axes. But first we have to test the data for homoscedasticity (data is homoscedastic if the species/groups/categorical variables have equal/homogeneous covariance matrices) with “betadisper� function in “vegan� package.

Make sure you have installed the vegan package (command is install.packages(“vegan�))

Test for multivariate homogeneity of variances using function “betadisper� in package “vegan�. Start by checking the for homoscedasticity (multivariate homogeneity of within-group covariance matrices) of the multivariate data ([Borcard et al., 2011], page 208). We use the “betadisper� function of the “vegan� package, implementing Marti Anderson’s testing method. First compute the Euclidean distance matrix between pairs of objects:


library(vegan)
Loading required package: permute
Loading required package: lattice
This is vegan 2.5-7
means.dist<- dist(martinx.imp[, 3:14])
means.MHV<-betadisper(means.dist, martinx.imp[,2])
means.MHV

    Homogeneity of multivariate dispersions

Call: betadisper(d = means.dist, group = martinx.imp[, 2])

No. of Positive Eigenvalues: 12
No. of Negative Eigenvalues: 0

Average distance to median:
 karlschumannii     ledermannii   margaritaceus           nduru 
         25.749          12.144          18.280          10.931 
         niveus    obtusiflorus     somaliensis sphaerocephalus 
         17.545          17.513           3.723          12.399 
    tisserantii 
         10.932 

Eigenvalues for PCoA axes:
(Showing 8 of 12 eigenvalues)
   PCoA1    PCoA2    PCoA3    PCoA4    PCoA5    PCoA6    PCoA7    PCoA8 
213818.5  22662.7  18618.4  11950.6   2434.7   2094.1    928.3    370.1 
anova(means.MHV)
Analysis of Variance Table

Response: Distances
           Df Sum Sq Mean Sq F value    Pr(>F)    
Groups      8   7184  897.96  8.4702 8.417e-11 ***
Residuals 480  50886  106.01                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
permutest(means.MHV)

Permutation test for homogeneity of multivariate dispersions
Permutation: free
Number of permutations: 999

Response: Distances
           Df Sum Sq Mean Sq      F N.Perm Pr(>F)    
Groups      8   7184  897.96 8.4702    999  0.001 ***
Residuals 480  50886  106.01                         
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

In the above, the Analysis of Variance is a parametric test. The permutation test is non-parametric.

These results show that the species groups do not have homogeneous covariance matrices, i.e. they are heteroscadistic. This is not surprising since they are different species. For the purpose of plotting the ordinations of the species, this is not really a problem. It is a problem if you want to use multivariate analysis of variance (MANOVA) to compute the significance of the differences between the species group centroids (means). However, we can get round this using non-parametric tests.

Now we compute the LDA, after loading the MASS package:


library(MASS)

lda<-lda(martinx.imp[, 3:14], martinx.imp[,2])
lda
Call:
lda(martinx.imp[, 3:14], martinx.imp[, 2])

Prior probabilities of groups:
 karlschumannii     ledermannii   margaritaceus           nduru 
     0.02862986      0.02658487      0.41104294      0.12269939 
         niveus    obtusiflorus     somaliensis sphaerocephalus 
     0.02453988      0.14723926      0.01226994      0.13087935 
    tisserantii 
     0.09611452 

Group means:
                    Lflen     Lfwid    Cullen    Culwid  Numbrac
karlschumannii  37.128643 1.7035714 72.671367 1.7428571 2.607143
ledermannii     14.743312 2.7461538 38.309267 1.6000000 3.807692
margaritaceus   22.146063 1.2690124 38.258418 1.1181169 2.677363
nduru            5.997958 0.5658333 19.923333 0.8886667 2.675000
niveus          17.177927 1.1186542 34.341837 1.1086694 3.375000
obtusiflorus    17.561627 1.3098611 31.125694 1.1773611 3.451389
somaliensis      4.583333 0.4000000  6.133333 0.5166667 2.416667
sphaerocephalus 15.218403 1.0339813 26.860937 1.1351563 3.371094
tisserantii     11.990000 0.6308511 17.636596 0.7997872 3.702128
                 Lenbrac    Numspk    Spklen   Spkwid    Gllen    Glwid
karlschumannii  5.975998  3.535714 26.382143 6.778571 5.821429 4.092857
ledermannii     4.757692 12.846154 11.584615 5.792308 6.969231 1.484615
margaritaceus   5.412128  4.146353 14.359368 5.833215 5.293510 2.591835
nduru           1.715500  3.835000  8.312500 4.310500 4.411000 1.853333
niveus          4.954167  7.625000 11.709167 4.907500 5.011667 1.516667
obtusiflorus    4.615773 15.974861  9.681111 4.897778 5.340694 1.376389
somaliensis     2.041667  8.250000  5.166667 2.683333 3.283333 1.083333
sphaerocephalus 3.384063  9.028125 10.898438 6.631250 6.092187 1.820312
tisserantii     3.018085  5.459805  8.198389 4.099792 4.461523 1.605128
                    Numgl
karlschumannii  26.469286
ledermannii     12.038462
margaritaceus   14.741902
nduru            8.504577
niveus          13.895833
obtusiflorus    10.577361
somaliensis      7.166667
sphaerocephalus 10.789062
tisserantii      8.511277

Coefficients of linear discriminants:
                 LD1         LD2         LD3          LD4          LD5
Lflen   -0.023525716  0.04721197  0.03339859  0.087453512  0.020389863
Lfwid   -0.050499480  0.11003994 -0.03215027 -0.411868760 -0.510151118
Cullen  -0.001049291 -0.01240766  0.02637771 -0.034887998 -0.006978838
Culwid  -0.193396314  0.67623511 -1.25282683 -1.650969331  0.704954296
Numbrac  0.317217302 -0.20156809  0.12812841 -0.002648659  0.997125842
Lenbrac -0.057260622 -0.05334681  0.02657255 -0.003301194 -0.356712335
Numspk   0.138681816  0.14654102  0.12413233  0.096384224  0.034244679
Spklen  -0.045497741  0.03549820  0.08874904  0.064787702  0.160510267
Spkwid   0.077392289 -0.07153027 -0.56378902  0.623529349 -0.073350292
Gllen    0.295913140  0.58331145 -0.48217096 -0.490178211 -0.133578753
Glwid   -1.152399505  0.10950581  0.44181671  0.275730541  0.623281770
Numgl   -0.032087894  0.07627901 -0.05164136 -0.093020425 -0.057264312
                LD6          LD7         LD8
Lflen    0.04948426  0.008739104  0.02828168
Lfwid    0.30095690  0.470288628  0.43134241
Cullen  -0.01189099 -0.011423317 -0.05421764
Culwid  -0.61746631 -0.869524369 -0.82136196
Numbrac  0.84727743  0.138985759 -0.22485825
Lenbrac  0.11331662 -0.043187061 -0.15412531
Numspk  -0.13405577  0.056804396  0.02626085
Spklen  -0.02325600  0.047768333  0.04356984
Spkwid  -0.16202133 -0.196410804 -0.14100907
Gllen    0.20259878  0.194052825  0.23645061
Glwid   -0.76148447  1.140571960 -0.07692226
Numgl    0.07367221 -0.141219816  0.13327458

Proportion of trace:
   LD1    LD2    LD3    LD4    LD5    LD6    LD7    LD8 
0.5553 0.2309 0.0852 0.0469 0.0425 0.0331 0.0044 0.0017 

The proportion of trace table shows that 78.62% (0.7862) of the variance is expressed in the first two axes

Now we compute the scores for plotting the ordination. What happens here is that we multiply the matrix of variables in the martinx.imp data frame by the eigenvector matrix, which is a sub-object of the lda object (lda$scaling) and described in the above output as “Coefficients of linear discriminants�:


scores<-as.matrix(martinx.imp[,3:14])%*%lda$scaling
head(scores)
           LD1      LD2       LD3         LD4       LD5       LD6
[1,] -2.217331 5.374388 -2.143290 -0.07282965 3.8080457 1.6511650
[2,] -2.512219 4.669681 -4.120335 -0.07095028 2.4433946 0.2929680
[3,] -1.675271 5.207487 -4.122514  0.70947675 2.6909460 0.8847473
[4,] -1.548064 5.668314 -3.173801  0.05446960 3.4403451 2.0208105
[5,] -3.530078 5.252783 -2.453074  1.04374979 0.8760719 1.5570799
[6,] -2.391148 6.273904 -3.220752 -0.37658305 2.4234024 0.4511635
           LD7        LD8
[1,] 2.3599009 -0.7293124
[2,] 0.6297856 -0.2547761
[3,] 0.7021037  0.7830545
[4,] 1.2388024  0.1052282
[5,] 3.1792555 -1.5148822
[6,] 1.6469284 -0.2528704

Now plot the first two discriminant axes and add legend:


plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Repeat for axes 2 and 3:


plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)


plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

To plot convex hulls around the species groups we can use the function “Plot ConvexHull� from Ken Takagi’s website “Chit Chat R� to overplot convex hulls onto the original plot.

Load the convex hulls function:

### Plotting function to plot convex hulls
### Filename: Plot_ConvexHull.R
### Notes:
############################################################################
# INPUTS:
# xcoords: x-coordinates of point data
# ycoords: y-coordinates of point data
# lcolor: line color
# OUTPUTS:
# convex hull around data points in a particular color (specified by lcolor)
# FUNCTION:

Plot_ConvexHull<-function(xcoord, ycoord, lcolor){
hpts <- chull(x = xcoord, y = ycoord)
hpts <- c(hpts, hpts[1])
lines(xcoord[hpts], ycoord[hpts], col = lcolor)
}

# END OF FUNCTION

Replot the LDA ordinations and then the convex hulls:


plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,2][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,2][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,2][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="nduru"],
scores[,2][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="niveus"],
scores[,2][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,2][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,2][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,2][martinx.imp[,2]=="tisserantii"], lcolor = "grey")



plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"], lcolor = "grey")


plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"], lcolor = "grey")

Now plot with 95% confidence ellipses using using the ELLI function from Julien Claude’s book (Morphometrics with R) website:

Run ELLI function:


ELLI<-function(x,y,conf=0.95,np=50)
{centroid<-apply(cbind(x,y),2,mean)
ang <- seq(0,2*pi,length=np)
z<-cbind(cos(ang),sin(ang))
radiuscoef<-qnorm((1-conf)/2, lower.tail=F)
vcvxy<-var(cbind(x,y))
r<-cor(x,y)
M1<-matrix(c(1,1,-1,1),2,2)
M2<-matrix(c(var(x), var(y)),2,2)
M3<-matrix(c(1+r, 1-r),2,2, byrow=T)
ellpar<-M1*sqrt(M2*M3/2)
t(centroid + radiuscoef * ellpar %*% t(z))}

Replot now using ELLI to plot the ellipses:


plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,2][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,2][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,2][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,1][martinx.imp[,2]=="nduru"],
scores[,2][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,1][martinx.imp[,2]=="niveus"],
scores[,2][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,2][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,2][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,2][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,2][martinx.imp[,2]=="tisserantii"]), col = "grey")


plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,1][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,1][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"]), col = "grey")


plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,7))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,2][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,2][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,2][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,2][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,2][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,2][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,2][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,2][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,2][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"]), col = "grey")

Now plot the loadings on the first two discriminant axes to see which characters are most influential in separating the species:


par(mfrow=c(2,1))

barplot(sort(lda$scaling[,1]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="A", ylab="loadings")
barplot(sort(lda$scaling[,2]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="B", ylab="loadings")

These bar charts tell us which characters are most influential in the scores on each axis. Individuals that have more negative scores on a given discriminant axis are tend to have higher values for the characters (bars) towards the left, and individuals with more positive scores have higher values for the characters (bars) towards the right.

LDA using scaled data

Now we repeat these LDA analyses using scaled data to see what difference this makes.

First make a data frame of the scaled data using the function “scale� , which centres the data (in each column, the mean of the column is subtracted from the individual values) and then the variance of each column is set to zero (by dividing the centred values of each column by the column standard deviation).


martinx.imp.sc<-scale(martinx.imp[, 3:14])
head(martinx.imp.sc)
          Lflen      Lfwid      Cullen     Culwid     Numbrac    Lenbrac
[1,]  0.3438436  0.3826877  0.90977461 -0.8126153 -0.02570601  0.1793059
[2,] -0.5407290 -0.1401218  0.05446955 -0.2586966 -1.17893911 -0.2794728
[3,] -0.5898719 -0.4538075 -0.26626985 -1.3665339 -1.17893911 -0.2998003
[4,]  0.1472719 -0.1401218 -0.21281329 -0.2586966 -0.02570601  0.4263406
[5,]  0.6387011 -0.4015265 -0.21281329 -0.8126153 -0.60232256  2.8966878
[6,] -0.1475856 -0.1401218  0.42866551 -0.2586966 -1.17893911  0.7792474
         Numspk     Spklen    Spkwid      Gllen     Glwid       Numgl
[1,] -0.7805886  0.8115038 0.2497099  0.4151647 1.1385881  0.04564896
[2,] -0.9554338 -0.3604367 0.8917976 -0.6793429 1.1385881  0.74092434
[3,] -0.8680112  0.5809581 1.0202151  0.2327468 0.5094702  0.80002274
[4,] -0.6057433  0.5041096 0.1855011  0.2327468 0.5094702  0.40893034
[5,] -0.7805886 -0.4180731 0.1212923  0.6887916 2.3968239 -0.41670916
[6,] -0.4308980  0.7730795 0.5386493  0.8712096 1.3902353  0.65053854

Now run the LDas on the scaled data frame:


lda.sc<-lda(martinx.imp.sc, martinx.imp[,2])
lda.sc
Call:
lda(martinx.imp.sc, grouping = martinx.imp[, 2])

Prior probabilities of groups:
 karlschumannii     ledermannii   margaritaceus           nduru 
     0.02862986      0.02658487      0.41104294      0.12269939 
         niveus    obtusiflorus     somaliensis sphaerocephalus 
     0.02453988      0.14723926      0.01226994      0.13087935 
    tisserantii 
     0.09611452 

Group means:
                       Lflen       Lfwid      Cullen      Culwid
karlschumannii   1.929060809  0.59554580  2.17516459  1.79871554
ledermannii     -0.271100277  1.68568973  0.33828471  1.40305935
margaritaceus    0.456485384  0.14116269  0.33556651  0.06843919
nduru           -1.130644714 -0.59409470 -0.64456415 -0.56704468
niveus          -0.031812053 -0.01605477  0.12619957  0.04227354
obtusiflorus     0.005900256  0.18387484 -0.04572439  0.23252152
somaliensis     -1.269682266 -0.76749318 -1.38173020 -1.59733339
sphaerocephalus -0.224405501 -0.10459037 -0.27370366  0.11563122
tisserantii     -0.541711813 -0.52611093 -0.76680529 -0.81320456
                   Numbrac    Lenbrac     Numspk      Spklen     Spkwid
karlschumannii  -0.4787619  0.5943234 -0.5994988  2.76771138  0.8780386
ledermannii      0.9057515  0.1643752  1.0283874 -0.07521033  0.2447707
margaritaceus   -0.3977815  0.3953301 -0.4927315  0.45787896  0.2710368
nduru           -0.4005068 -0.9092349 -0.5471701 -0.70385373 -0.7066798
niveus           0.4067564  0.2337124  0.1154934 -0.05128137 -0.3233534
obtusiflorus     0.4948506  0.1142910  1.5754270 -0.44091424 -0.3295960
somaliensis     -0.6984253 -0.7941285  0.2247717 -1.30823563 -1.7514635
sphaerocephalus  0.4022516 -0.3203880  0.3608231 -0.20703979  0.7834453
tisserantii      0.7840108 -0.4495439 -0.2630807 -0.72577682 -0.8419729
                      Gllen      Glwid       Numgl
karlschumannii   0.52591846  2.5136601  2.44596304
ledermannii      1.57281700 -0.7681230 -0.06238614
margaritaceus    0.04440944  0.6250201  0.40752279
nduru           -0.76051889 -0.3041889 -0.67664182
niveus          -0.21265702 -0.7277950  0.26046008
obtusiflorus     0.08744582 -0.9042975 -0.31635292
somaliensis     -1.78905202 -1.2730305 -0.90919589
sphaerocephalus  0.77287488 -0.3457369 -0.27955523
tisserantii     -0.71443701 -0.6164902 -0.67547735

Coefficients of linear discriminants:
                LD1         LD2         LD3          LD4        LD5
Lflen   -0.23936018  0.48035369  0.33981078  0.889787508  0.2074547
Lfwid   -0.04829626  0.10523905 -0.03074759 -0.393899482 -0.4878939
Cullen  -0.01962885 -0.23210726  0.49344186 -0.652641949 -0.1305516
Culwid  -0.06982842  0.24416405 -0.45235047 -0.596105331  0.2545335
Numbrac  0.27506781 -0.17478521  0.11110365 -0.002296725  0.8646351
Lenbrac -0.16225426 -0.15116404  0.07529623 -0.009354295 -1.0107836
Numspk   0.79316887  0.83811836  0.70995539  0.551254435  0.1958571
Spklen  -0.23681766  0.18476962  0.46194250  0.337222720  0.8354627
Spkwid   0.12053227 -0.11140264 -0.87805607  0.971096826 -0.1142372
Gllen    0.32443426  0.63953300 -0.52864424 -0.537423260 -0.1464535
Glwid   -0.91588518  0.08703123  0.35113984  0.219140596  0.4953617
Numgl   -0.18460538  0.43884202 -0.29709875 -0.535157311 -0.3294482
               LD6         LD7         LD8
Lflen    0.5034730 -0.08891519 -0.28774926
Lfwid    0.2878266 -0.44977057 -0.41252353
Cullen  -0.2224421  0.21369343  1.01423722
Culwid  -0.2229448  0.31395381  0.29656410
Numbrac  0.7346975 -0.12051836  0.19498075
Lenbrac  0.3210951  0.12237528  0.43673101
Numspk  -0.7667109 -0.32488383 -0.15019479
Spklen  -0.1210484 -0.24863619 -0.22678286
Spkwid  -0.2523352  0.30589404  0.21961029
Gllen    0.2221259 -0.21275630 -0.25924053
Glwid   -0.6052001 -0.90648507  0.06113501
Numgl    0.4238448  0.81245400 -0.76674415

Proportion of trace:
   LD1    LD2    LD3    LD4    LD5    LD6    LD7    LD8 
0.5553 0.2309 0.0852 0.0469 0.0425 0.0331 0.0044 0.0017 
scores<-as.matrix(martinx.imp.sc)%*%lda.sc$scaling
head(scores)
           LD1        LD2        LD3        LD4         LD5         LD6
[1,] -1.795831 -0.3742138  0.7004206  0.1320457  0.34028774  0.16048506
[2,] -2.090720 -1.0789210 -1.2766238  0.1339250 -1.02436335 -1.19771192
[3,] -1.253771 -0.5411153 -1.2788028  0.9143521 -0.77681192 -0.60593267
[4,] -1.126564 -0.0802878 -0.3300897  0.2593449 -0.02741286  0.53013059
[5,] -3.108579 -0.4958188  0.3906370  1.2486251 -2.59168606  0.06639996
[6,] -1.969649  0.5253025 -0.3770413 -0.1717077 -1.04435558 -1.03951640
             LD7         LD8
[1,] -1.19350294  0.41324563
[2,]  0.53661237 -0.06129069
[3,]  0.46429431 -1.09912129
[4,] -0.07240445 -0.42129500
[5,] -2.01285754  1.19881542
[6,] -0.48053043 -0.06319639
plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

par(mfrow=c(2,1))

barplot(sort(lda.sc$scaling[,1]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="A", ylab="loadings")
barplot(sort(lda.sc$scaling[,2]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="B", ylab="loadings")

Scaling the data has no effect on the topology of the ordinations, but it changes the loadings on the discriminant axes (bar chart above) that indicate which variables/characters are influencing the score values the most and the least (the central bars are the least influential). It is best do use the scaled data result to estimate the influence of each character on the discriminant analyses.

LDA using cross-validation

When the data violate the basic assumptions of the parametric LDA approach (data overall with a normal distribution, and group covariance matrices homoscedastic), the best way to estimate the significance of the intergroup differences is to use cross-validation. This is a process of allocating each individual to the group to which it is nearest, as estimated by its (Mahalanobis) distance from each group centroid, as computed from its scores on the discriminant axes. For this it is best to use the untransformed data.

To use cross-validation, we run the “lda� function with the argument “CV=TRUE�:


lda.cv<-lda(martinx.imp[, 3:14], martinx.imp[,2], CV=TRUE)
summary(lda.cv)
          Length Class  Mode   
class      489   factor numeric
posterior 4401   -none- numeric
call         4   -none- call   

We make a table of the assignments of the cross-validation (also called a Jackknife test):


cv.jack.table<-table(martinx.imp[,2], lda.cv$class)
cv.jack.table
                 
                  karlschumannii ledermannii margaritaceus nduru niveus
  karlschumannii               9           0             5     0      0
  ledermannii                  0           9             0     0      0
  margaritaceus                9           3           175     5      2
  nduru                        0           0             5    44      0
  niveus                       0           0             4     1      0
  obtusiflorus                 0           0             4     2      2
  somaliensis                  0           0             0     1      0
  sphaerocephalus              0           3             5     3      1
  tisserantii                  0           0             4     6      0
                 
                  obtusiflorus somaliensis sphaerocephalus tisserantii
  karlschumannii             0           0               0           0
  ledermannii                1           0               3           0
  margaritaceus              3           0               2           2
  nduru                      0           0               1          10
  niveus                     2           0               1           4
  obtusiflorus              50           0              10           4
  somaliensis                0           3               0           2
  sphaerocephalus            5           0              45           2
  tisserantii                1           0               0          36

This looks quite successful, apart from niveus, where none of its individuals were successfully allocated. This analysis uses only the quantitative variables. The result for margaritaceus is especially notable, with most individuals assigned to it.


# calculate total errors in cross-validation

# run function "errors.cv"

errors.cv<-function(square.tab) {
    errors<-round(sum(square.tab) - sum(diag(square.tab)), 1)
    total<-round(sum(square.tab), 0)
    errors.prop<-round(errors/total, 4)
    result<-c(errors, total, errors.prop)
    names(result)<-c("errors", "total", "errors.proportion")    
    result
}

# Use function "errors.cv" to compute overall errors of cross validation

errors.cv(cv.jack.table)
           errors             total errors.proportion 
         118.0000          489.0000            0.2413 
# Compute percentage correct assignments for each population

round(diag(cv.jack.table)/rowSums(cv.jack.table),3)*100
 karlschumannii     ledermannii   margaritaceus           nduru 
           64.3            69.2            87.1            73.3 
         niveus    obtusiflorus     somaliensis sphaerocephalus 
            0.0            69.4            50.0            70.3 
    tisserantii 
           76.6 
# pegar a matriz de probabilidades posteriores de cada individuo e plotar em barras

indiv.probs<-t(lda.cv$posterior)
colnames(indiv.probs)<-martinx.imp$Specimen

barplot(indiv.probs, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), border=NA, space=0, las=2, cex.names=0.4, ylab = "posterior probabilities")

## plot the eigenvectors (= loadings)

par(mfrow=c(1,2))

barplot(sort(lda$scaling[,1]), las=2, cex.names=0.7, cex=0.9, cex.main=0.8, main="discriminant function 1", ylab="pesos")
barplot(sort(lda$scaling[,2]), las=2, cex.names=0.7, cex=0.9, cex.main=0.8, main="discriminant function 2", ylab="pesos")
par(mfrow=c(1,1))

Now we turn this table into proportions and make a bar plot:


proportions<-prop.table(cv.jack.table,2)
proportions
                 
                  karlschumannii ledermannii margaritaceus      nduru
  karlschumannii      0.50000000  0.00000000    0.02475248 0.00000000
  ledermannii         0.00000000  0.60000000    0.00000000 0.00000000
  margaritaceus       0.50000000  0.20000000    0.86633663 0.08064516
  nduru               0.00000000  0.00000000    0.02475248 0.70967742
  niveus              0.00000000  0.00000000    0.01980198 0.01612903
  obtusiflorus        0.00000000  0.00000000    0.01980198 0.03225806
  somaliensis         0.00000000  0.00000000    0.00000000 0.01612903
  sphaerocephalus     0.00000000  0.20000000    0.02475248 0.04838710
  tisserantii         0.00000000  0.00000000    0.01980198 0.09677419
                 
                      niveus obtusiflorus somaliensis sphaerocephalus
  karlschumannii  0.00000000   0.00000000  0.00000000      0.00000000
  ledermannii     0.00000000   0.01612903  0.00000000      0.04838710
  margaritaceus   0.40000000   0.04838710  0.00000000      0.03225806
  nduru           0.00000000   0.00000000  0.00000000      0.01612903
  niveus          0.00000000   0.03225806  0.00000000      0.01612903
  obtusiflorus    0.40000000   0.80645161  0.00000000      0.16129032
  somaliensis     0.00000000   0.00000000  1.00000000      0.00000000
  sphaerocephalus 0.20000000   0.08064516  0.00000000      0.72580645
  tisserantii     0.00000000   0.01612903  0.00000000      0.00000000
                 
                  tisserantii
  karlschumannii   0.00000000
  ledermannii      0.00000000
  margaritaceus    0.03333333
  nduru            0.16666667
  niveus           0.06666667
  obtusiflorus     0.06666667
  somaliensis      0.03333333
  sphaerocephalus  0.03333333
  tisserantii      0.60000000
#check the columns sum to 1
apply(prop.table(cv.jack.table,2),2,sum)
 karlschumannii     ledermannii   margaritaceus           nduru 
              1               1               1               1 
         niveus    obtusiflorus     somaliensis sphaerocephalus 
              1               1               1               1 
    tisserantii 
              1 

Now barplot the proportions table:


par(mar=c(5,4,4,2))

barplot(proportions, main="LDA: assignment of original populations by cross-validation using untransformed data",cex=0.8, cex.lab=0.8, cex.main=0.8, cex.names=0.7, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), ylab="proportion of populations assigned")

legend("topright", title="original", cex=0.7, row.names(proportions),fill=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), bg="white", inset=0.045)

This shows the proportions of correctly assigned individuals in the classes produced by cross validation. Each bar is the result of the cross-validation, and the colours are those of the original populations. In this analysis, margaritaceus has a successful result, considering its wide range.

Now add the LDA cross-validation assignments to the data frame martinx.imp as a new column and name the new data frame:


martinx.imp.lda.crossval<-data.frame(martinx.imp, lda.cv$class)
head(martinx.imp.lda.crossval)
NA

Extract the individuals and their LDA assignments


LDA.assignments<-martinx.imp.lda.crossval[,c(1:2,18)]
LDA.assignments

#write this file to Excel csv
write.csv(LDA.assignments, file="LDA assignments to species.csv", row.names = FALSE)

Carry out a Classification Tree analysis

load the package rpart


library(rpart)

Compute and plot the tree


tree1<-rpart(Taxon~ Lflen + Lfwid + Cullen + Culwid + Numbrac +  Lenbrac + Numspk +  Spklen + Spkwid + Gllen + Glwid + Numgl + Bshpers + Bshtext + Bshglos, data=martinx.imp, method="class")
tree1
n= 489 

node), split, n, loss, yval, (yprob)
      * denotes terminal node

 1) root 489 288 margaritaceus (0.029 0.027 0.41 0.12 0.025 0.15 0.012 0.13 0.096)  
   2) Glwid>=1.95 306 113 margaritaceus (0.046 0.0065 0.63 0.12 0.0098 0.039 0 0.11 0.039)  
     4) Bshtext=firm,papery 261  74 margaritaceus (0.054 0.0077 0.72 0 0.011 0.046 0 0.13 0.034)  
       8) Numspk< 7.165 217  37 margaritaceus (0.065 0 0.83 0 0.014 0.014 0 0.037 0.041)  
        16) Cullen>=70.34458 15   5 karlschumannii (0.67 0 0.33 0 0 0 0 0 0) *
        17) Cullen< 70.34458 202  27 margaritaceus (0.02 0 0.87 0 0.015 0.015 0 0.04 0.045) *
       9) Numspk>=7.165 44  18 sphaerocephalus (0 0.045 0.16 0 0 0.2 0 0.59 0)  
        18) Spkwid< 5.25 9   4 margaritaceus (0 0 0.56 0 0 0.44 0 0 0) *
        19) Spkwid>=5.25 35   9 sphaerocephalus (0 0.057 0.057 0 0 0.14 0 0.74 0) *
     5) Bshtext=hard 45   9 nduru (0 0 0.13 0.8 0 0 0 0 0.067)  
      10) Lflen>=14.25 8   2 margaritaceus (0 0 0.75 0.12 0 0 0 0 0.12) *
      11) Lflen< 14.25 37   2 nduru (0 0 0 0.95 0 0 0 0 0.054) *
   3) Glwid< 1.95 183 123 obtusiflorus (0 0.06 0.044 0.13 0.049 0.33 0.033 0.16 0.19)  
     6) Numspk>=11.25 66  19 obtusiflorus (0 0.12 0.045 0 0.03 0.71 0 0.076 0.015)  
      12) Lfwid>=2.35 10   4 ledermannii (0 0.6 0.2 0 0 0.2 0 0 0) *
      13) Lfwid< 2.35 56  11 obtusiflorus (0 0.036 0.018 0 0.036 0.8 0 0.089 0.018) *
     7) Numspk< 11.25 117  83 tisserantii (0 0.026 0.043 0.21 0.06 0.11 0.051 0.21 0.29)  
      14) Lenbrac< 1.325 24   6 nduru (0 0 0 0.75 0 0.042 0.083 0.042 0.083) *
      15) Lenbrac>=1.325 93  61 tisserantii (0 0.032 0.054 0.065 0.075 0.13 0.043 0.26 0.34)  
        30) Spklen>=9.09 41  20 sphaerocephalus (0 0.049 0.073 0.049 0.073 0.17 0 0.51 0.073) *
        31) Spklen< 9.09 52  23 tisserantii (0 0.019 0.038 0.077 0.077 0.096 0.077 0.058 0.56) *
plot(tree1)
text(tree1, cex=0.7, xpd=TRUE)

Plot with package rpart.plot


library(rpart.plot)
rpart.plot(tree1)
All boxes will be white (the box.palette argument will be ignored) because
the number of classes in the response 9 is greater than length(box.palette) 6.
To silence this warning use box.palette=0 or trace=-1.

Make a bar chart of the assignments

#check that order of assigned individuals is the same as in the data frame 
data.frame(rownames(martinx.imp),tree1$where)

#add assignments to data frame 
martinx.imp.rpart.assign<-data.frame(martinx.imp, tree1$where)
head(martinx.imp.rpart.assign)
NA

Find CART assignments within each original species


#make a simplified data frame with just the three factors
d<-martinx.imp.rpart.assign[,c(1:2,18)]
head(d)

#table the species and assignments:
rpart.counts<-table(d$Taxon,d$tree1.where)
rpart.counts
                 
                    5   6   8   9  11  12  15  16  18  20  21
  karlschumannii   10   4   0   0   0   0   0   0   0   0   0
  ledermannii       0   0   0   2   0   0   6   2   0   2   1
  margaritaceus     5 175   5   2   6   0   2   1   0   3   2
  nduru             0   0   0   0   1  35   0   0  18   2   4
  niveus            0   3   0   0   0   0   0   2   0   3   4
  obtusiflorus      0   3   4   5   0   0   2  45   1   7   5
  somaliensis       0   0   0   0   0   0   0   0   2   0   4
  sphaerocephalus   0   8   0  26   0   0   0   5   1  21   3
  tisserantii       0   9   0   0   1   2   0   1   2   3  29
#check that columns sum to species totals:
data.frame(apply(rpart.counts,1,sum), table(d$Taxon))
NA

Find out which are the species names of the columns; the programme assigns to the column the species with the most individuals assigned to it


assign<-rep(0,11)
for(i in 1:11) {
assign[i]<-names(which.max(rpart.counts[,i]))
}
assign
 [1] "karlschumannii"  "margaritaceus"   "margaritaceus"  
 [4] "sphaerocephalus" "margaritaceus"   "nduru"          
 [7] "ledermannii"     "obtusiflorus"    "nduru"          
[10] "sphaerocephalus" "tisserantii"    

Make the new column names for the counts matrix, avoiding duplication of names


colnames(rpart.counts)<-c("karl.crt","marg1.crt","marg2.crt","sphaer1", "marg3.crt","nduru1.crt","leder.crt","obtus.crt", "nduru2.crt", "sphaer2", "tiss")
rpart.counts
                 
                  karl.crt marg1.crt marg2.crt sphaer1 marg3.crt
  karlschumannii        10         4         0       0         0
  ledermannii            0         0         0       2         0
  margaritaceus          5       175         5       2         6
  nduru                  0         0         0       0         1
  niveus                 0         3         0       0         0
  obtusiflorus           0         3         4       5         0
  somaliensis            0         0         0       0         0
  sphaerocephalus        0         8         0      26         0
  tisserantii            0         9         0       0         1
                 
                  nduru1.crt leder.crt obtus.crt nduru2.crt sphaer2 tiss
  karlschumannii           0         0         0          0       0    0
  ledermannii              0         6         2          0       2    1
  margaritaceus            0         2         1          0       3    2
  nduru                   35         0         0         18       2    4
  niveus                   0         0         2          0       3    4
  obtusiflorus             0         2        45          1       7    5
  somaliensis              0         0         0          2       0    4
  sphaerocephalus          0         0         5          1      21    3
  tisserantii              2         0         1          2       3   29

Make this table into a table of proportions for each row.


#make table of proportions and table as percentages
proportions<-prop.table(rpart.counts,2)
round(proportions*100, 2)
                 
                  karl.crt marg1.crt marg2.crt sphaer1 marg3.crt
  karlschumannii     66.67      1.98      0.00    0.00      0.00
  ledermannii         0.00      0.00      0.00    5.71      0.00
  margaritaceus      33.33     86.63     55.56    5.71     75.00
  nduru               0.00      0.00      0.00    0.00     12.50
  niveus              0.00      1.49      0.00    0.00      0.00
  obtusiflorus        0.00      1.49     44.44   14.29      0.00
  somaliensis         0.00      0.00      0.00    0.00      0.00
  sphaerocephalus     0.00      3.96      0.00   74.29      0.00
  tisserantii         0.00      4.46      0.00    0.00     12.50
                 
                  nduru1.crt leder.crt obtus.crt nduru2.crt sphaer2
  karlschumannii        0.00      0.00      0.00       0.00    0.00
  ledermannii           0.00     60.00      3.57       0.00    4.88
  margaritaceus         0.00     20.00      1.79       0.00    7.32
  nduru                94.59      0.00      0.00      75.00    4.88
  niveus                0.00      0.00      3.57       0.00    7.32
  obtusiflorus          0.00     20.00     80.36       4.17   17.07
  somaliensis           0.00      0.00      0.00       8.33    0.00
  sphaerocephalus       0.00      0.00      8.93       4.17   51.22
  tisserantii           5.41      0.00      1.79       8.33    7.32
                 
                   tiss
  karlschumannii   0.00
  ledermannii      1.92
  margaritaceus    3.85
  nduru            7.69
  niveus           7.69
  obtusiflorus     9.62
  somaliensis      7.69
  sphaerocephalus  5.77
  tisserantii     55.77
#check the columns sum to 100%:
apply(proportions*100,2,sum)
  karl.crt  marg1.crt  marg2.crt    sphaer1  marg3.crt nduru1.crt 
       100        100        100        100        100        100 
 leder.crt  obtus.crt nduru2.crt    sphaer2       tiss 
       100        100        100        100        100 

Barplot proportions of original species individuals assigned to each “species� resulting from CART analysis


barplot(as.matrix(proportions), names.arg=colnames(rpart.counts), cex=0.8, cex.names=0.7,
main="tree1, eleven leaves, with imputed data",
cex.lab=0.8, cex.main=0.8, xlab="CART species",
ylab="proportion of species individuals assigned to each column (crt)",
col=rainbow(length(rpart.counts[,1])))

legend("bottomleft", title="orig. species", cex=0.8, rownames(rpart.counts),
fill=rainbow(9), bg="white",
inset=0.05)

NA
NA

Now show the individual assignments made by CART:


res<-rep(0,489)
ass<-as.character(tree1$where)
cartnames<- colnames(rpart.counts)
for(i in 1:length(tree1$where))if(ass[i]=="5"){res[i]<-cartnames[1]}
for(i in 1:length(tree1$where))if(ass[i]=="6"){res[i]<-cartnames[2]}
for(i in 1:length(tree1$where))if(ass[i]=="8"){res[i]<-cartnames[3]}
for(i in 1:length(tree1$where))if(ass[i]=="9"){res[i]<-cartnames[4]}
for(i in 1:length(tree1$where))if(ass[i]=="11"){res[i]<-cartnames[5]}
for(i in 1:length(tree1$where))if(ass[i]=="12"){res[i]<-cartnames[6]}
for(i in 1:length(tree1$where))if(ass[i]=="15"){res[i]<-cartnames[7]}
for(i in 1:length(tree1$where))if(ass[i]=="16"){res[i]<-cartnames[8]}
for(i in 1:length(tree1$where))if(ass[i]=="18"){res[i]<-cartnames[9]}
for(i in 1:length(tree1$where))if(ass[i]=="20"){res[i]<-cartnames[10]}
for(i in 1:length(tree1$where))if(ass[i]=="21"){res[i]<-cartnames[11]}
res
  [1] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
  [6] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [11] "marg1.crt"  "sphaer1"    "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [16] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [21] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [26] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [31] "sphaer1"    "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [36] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [41] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [46] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [51] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [56] "marg2.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "karl.crt"  
 [61] "marg1.crt"  "karl.crt"   "marg1.crt"  "marg1.crt"  "marg2.crt" 
 [66] "leder.crt"  "sphaer2"    "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [71] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [76] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "leder.crt" 
 [81] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [86] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [91] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
 [96] "karl.crt"   "marg1.crt"  "karl.crt"   "marg1.crt"  "marg1.crt" 
[101] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[106] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[111] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[116] "marg1.crt"  "marg1.crt"  "marg1.crt"  "karl.crt"   "marg1.crt" 
[121] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[126] "marg1.crt"  "marg3.crt"  "marg1.crt"  "marg3.crt"  "marg1.crt" 
[131] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[136] "marg3.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[141] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg2.crt" 
[146] "marg1.crt"  "marg1.crt"  "marg2.crt"  "marg1.crt"  "marg1.crt" 
[151] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg3.crt"  "marg3.crt" 
[156] "marg1.crt"  "marg3.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[161] "marg1.crt"  "marg1.crt"  "marg1.crt"  "sphaer2"    "marg1.crt" 
[166] "marg1.crt"  "marg1.crt"  "obtus.crt"  "marg2.crt"  "marg1.crt" 
[171] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[176] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[181] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[186] "tiss"       "marg1.crt"  "marg1.crt"  "tiss"       "marg1.crt" 
[191] "marg1.crt"  "marg1.crt"  "marg1.crt"  "sphaer2"    "marg1.crt" 
[196] "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt"  "marg1.crt" 
[201] "marg1.crt"  "marg1.crt"  "karl.crt"   "karl.crt"   "marg1.crt" 
[206] "karl.crt"   "karl.crt"   "marg1.crt"  "karl.crt"   "karl.crt"  
[211] "marg1.crt"  "karl.crt"   "karl.crt"   "karl.crt"   "karl.crt"  
[216] "sphaer2"    "sphaer1"    "obtus.crt"  "leder.crt"  "tiss"      
[221] "sphaer2"    "sphaer1"    "leder.crt"  "obtus.crt"  "leder.crt" 
[226] "leder.crt"  "leder.crt"  "leder.crt"  "tiss"       "tiss"      
[231] "tiss"       "tiss"       "tiss"       "tiss"       "marg1.crt" 
[236] "marg1.crt"  "tiss"       "tiss"       "marg1.crt"  "tiss"      
[241] "tiss"       "sphaer2"    "tiss"       "tiss"       "tiss"      
[246] "nduru1.crt" "tiss"       "tiss"       "tiss"       "marg3.crt" 
[251] "tiss"       "tiss"       "marg1.crt"  "tiss"       "obtus.crt" 
[256] "tiss"       "marg1.crt"  "sphaer2"    "tiss"       "tiss"      
[261] "tiss"       "tiss"       "tiss"       "marg1.crt"  "marg1.crt" 
[266] "tiss"       "nduru2.crt" "marg1.crt"  "tiss"       "nduru2.crt"
[271] "tiss"       "nduru1.crt" "sphaer2"    "marg1.crt"  "tiss"      
[276] "tiss"       "tiss"       "marg1.crt"  "tiss"       "obtus.crt" 
[281] "tiss"       "marg1.crt"  "sphaer2"    "sphaer2"    "marg1.crt" 
[286] "sphaer2"    "obtus.crt"  "nduru1.crt" "nduru1.crt" "nduru1.crt"
[291] "nduru1.crt" "nduru1.crt" "nduru1.crt" "nduru1.crt" "nduru1.crt"
[296] "nduru1.crt" "nduru1.crt" "tiss"       "nduru1.crt" "nduru1.crt"
[301] "sphaer2"    "nduru1.crt" "nduru1.crt" "nduru1.crt" "tiss"      
[306] "nduru1.crt" "nduru1.crt" "nduru1.crt" "nduru2.crt" "nduru1.crt"
[311] "nduru2.crt" "nduru2.crt" "nduru2.crt" "nduru1.crt" "nduru1.crt"
[316] "nduru2.crt" "nduru2.crt" "nduru2.crt" "nduru1.crt" "nduru1.crt"
[321] "tiss"       "nduru2.crt" "nduru2.crt" "nduru1.crt" "nduru1.crt"
[326] "nduru1.crt" "nduru2.crt" "sphaer2"    "nduru2.crt" "nduru2.crt"
[331] "nduru1.crt" "nduru1.crt" "nduru2.crt" "nduru1.crt" "nduru2.crt"
[336] "nduru2.crt" "tiss"       "nduru2.crt" "nduru2.crt" "nduru1.crt"
[341] "nduru1.crt" "nduru1.crt" "nduru2.crt" "nduru1.crt" "nduru1.crt"
[346] "marg3.crt"  "nduru1.crt" "leder.crt"  "obtus.crt"  "obtus.crt" 
[351] "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt" 
[356] "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt" 
[361] "obtus.crt"  "obtus.crt"  "tiss"       "obtus.crt"  "nduru2.crt"
[366] "obtus.crt"  "marg2.crt"  "tiss"       "obtus.crt"  "obtus.crt" 
[371] "tiss"       "sphaer2"    "obtus.crt"  "obtus.crt"  "sphaer2"   
[376] "obtus.crt"  "tiss"       "obtus.crt"  "obtus.crt"  "obtus.crt" 
[381] "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt" 
[386] "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt" 
[391] "marg1.crt"  "sphaer1"    "marg1.crt"  "sphaer2"    "sphaer2"   
[396] "sphaer2"    "sphaer1"    "sphaer1"    "sphaer1"    "sphaer2"   
[401] "obtus.crt"  "marg2.crt"  "obtus.crt"  "marg2.crt"  "marg2.crt" 
[406] "marg1.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "sphaer1"   
[411] "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt"  "obtus.crt" 
[416] "sphaer2"    "obtus.crt"  "leder.crt"  "tiss"       "sphaer2"   
[421] "sphaer2"    "sphaer2"    "sphaer2"    "sphaer2"    "obtus.crt" 
[426] "sphaer2"    "sphaer1"    "sphaer1"    "sphaer2"    "sphaer2"   
[431] "sphaer2"    "sphaer2"    "sphaer2"    "sphaer2"    "sphaer2"   
[436] "sphaer2"    "sphaer2"    "sphaer2"    "sphaer2"    "marg1.crt" 
[441] "sphaer2"    "sphaer1"    "marg1.crt"  "sphaer1"    "tiss"      
[446] "sphaer1"    "sphaer1"    "obtus.crt"  "sphaer1"    "sphaer1"   
[451] "obtus.crt"  "sphaer1"    "marg1.crt"  "sphaer1"    "sphaer1"   
[456] "obtus.crt"  "sphaer1"    "sphaer1"    "tiss"       "sphaer1"   
[461] "sphaer1"    "obtus.crt"  "sphaer2"    "sphaer2"    "sphaer1"   
[466] "marg1.crt"  "sphaer1"    "marg1.crt"  "marg1.crt"  "marg1.crt" 
[471] "sphaer1"    "sphaer2"    "marg1.crt"  "sphaer1"    "sphaer1"   
[476] "sphaer1"    "sphaer1"    "sphaer1"    "sphaer1"    "sphaer1"   
[481] "sphaer1"    "tiss"       "nduru2.crt" "nduru2.crt" "tiss"      
[486] "tiss"       "tiss"       "nduru2.crt" "tiss"      

Put assignments in the righthand columns


d.names<-data.frame(d,res)
d.names

#write this file to Excel csv
write.csv(d.names, file="CART assignments to species.csv", row.names = FALSE)

REPEAT THE CART ANALYSIS WITH NON-DEFAULT SETTINGS FOR MINSPLIT AND MINBUCKET

# Compute the tree with minsplit=10 (minimum number of individuals in a node to permit further splitting) and minbucket=5 (minimum number of individuals in a terminal node).

tree3<-rpart(Taxon~ Lflen + Lfwid + Cullen + Culwid + Numbrac +  Lenbrac + Numspk +  Spklen + Spkwid + Gllen + Glwid + Numgl + Bshpers + Bshtext + Bshglos, data=martinx.imp, method="class", control= rpart.control(minsplit = 10, minbucket = 9))
plot(tree3)
text(tree3, cex=0.7, xpd=TRUE)

printcp(tree3)

Classification tree:
rpart(formula = Taxon ~ Lflen + Lfwid + Cullen + Culwid + Numbrac + 
    Lenbrac + Numspk + Spklen + Spkwid + Gllen + Glwid + Numgl + 
    Bshpers + Bshtext + Bshglos, data = martinx.imp, method = "class", 
    control = rpart.control(minsplit = 10, minbucket = 9))

Variables actually used in tree construction:
[1] Bshtext Cullen  Glwid   Lenbrac Lflen   Lfwid   Numspk  Spklen 
[9] Spkwid 

Root node error: 288/489 = 0.58896

n= 489 

        CP nsplit rel error  xerror     xstd
1 0.180556      0   1.00000 1.00000 0.037779
2 0.104167      1   0.81944 0.83681 0.038387
3 0.072917      2   0.71528 0.74306 0.038091
4 0.065972      3   0.64236 0.71875 0.037937
5 0.059028      4   0.57639 0.68750 0.037691
6 0.017361      6   0.45833 0.53125 0.035602
7 0.013889      8   0.42361 0.54167 0.035788
8 0.010000     10   0.39583 0.55208 0.035967
plotcp(tree3)


tree4<- prune(tree3, cp=0.032)
plot(tree4)
text(tree4, cex=0.7, xpd=TRUE)

print(tree4)
n= 489 

node), split, n, loss, yval, (yprob)
      * denotes terminal node

 1) root 489 288 margaritaceus (0.029 0.027 0.41 0.12 0.025 0.15 0.012 0.13 0.096)  
   2) Glwid>=1.95 306 113 margaritaceus (0.046 0.0065 0.63 0.12 0.0098 0.039 0 0.11 0.039)  
     4) Bshtext=firm,papery 261  74 margaritaceus (0.054 0.0077 0.72 0 0.011 0.046 0 0.13 0.034)  
       8) Numspk< 7.165 217  37 margaritaceus (0.065 0 0.83 0 0.014 0.014 0 0.037 0.041) *
       9) Numspk>=7.165 44  18 sphaerocephalus (0 0.045 0.16 0 0 0.2 0 0.59 0) *
     5) Bshtext=hard 45   9 nduru (0 0 0.13 0.8 0 0 0 0 0.067) *
   3) Glwid< 1.95 183 123 obtusiflorus (0 0.06 0.044 0.13 0.049 0.33 0.033 0.16 0.19)  
     6) Numspk>=11.25 66  19 obtusiflorus (0 0.12 0.045 0 0.03 0.71 0 0.076 0.015) *
     7) Numspk< 11.25 117  83 tisserantii (0 0.026 0.043 0.21 0.06 0.11 0.051 0.21 0.29)  
      14) Lenbrac< 1.325 24   6 nduru (0 0 0 0.75 0 0.042 0.083 0.042 0.083) *
      15) Lenbrac>=1.325 93  61 tisserantii (0 0.032 0.054 0.065 0.075 0.13 0.043 0.26 0.34)  
        30) Spklen>=9.09 41  20 sphaerocephalus (0 0.049 0.073 0.049 0.073 0.17 0 0.51 0.073) *
        31) Spklen< 9.09 52  23 tisserantii (0 0.019 0.038 0.077 0.077 0.096 0.077 0.058 0.56) *
tree5<- prune(tree3, cp=0.016)
plot(tree5)
text(tree5, cex=0.7, xpd=TRUE)

print(tree5)
n= 489 

node), split, n, loss, yval, (yprob)
      * denotes terminal node

 1) root 489 288 margaritaceus (0.029 0.027 0.41 0.12 0.025 0.15 0.012 0.13 0.096)  
   2) Glwid>=1.95 306 113 margaritaceus (0.046 0.0065 0.63 0.12 0.0098 0.039 0 0.11 0.039)  
     4) Bshtext=firm,papery 261  74 margaritaceus (0.054 0.0077 0.72 0 0.011 0.046 0 0.13 0.034)  
       8) Numspk< 7.165 217  37 margaritaceus (0.065 0 0.83 0 0.014 0.014 0 0.037 0.041)  
        16) Cullen>=70.34458 15   5 karlschumannii (0.67 0 0.33 0 0 0 0 0 0) *
        17) Cullen< 70.34458 202  27 margaritaceus (0.02 0 0.87 0 0.015 0.015 0 0.04 0.045) *
       9) Numspk>=7.165 44  18 sphaerocephalus (0 0.045 0.16 0 0 0.2 0 0.59 0)  
        18) Spkwid< 5.25 9   4 margaritaceus (0 0 0.56 0 0 0.44 0 0 0) *
        19) Spkwid>=5.25 35   9 sphaerocephalus (0 0.057 0.057 0 0 0.14 0 0.74 0) *
     5) Bshtext=hard 45   9 nduru (0 0 0.13 0.8 0 0 0 0 0.067) *
   3) Glwid< 1.95 183 123 obtusiflorus (0 0.06 0.044 0.13 0.049 0.33 0.033 0.16 0.19)  
     6) Numspk>=11.25 66  19 obtusiflorus (0 0.12 0.045 0 0.03 0.71 0 0.076 0.015) *
     7) Numspk< 11.25 117  83 tisserantii (0 0.026 0.043 0.21 0.06 0.11 0.051 0.21 0.29)  
      14) Lenbrac< 1.325 24   6 nduru (0 0 0 0.75 0 0.042 0.083 0.042 0.083) *
      15) Lenbrac>=1.325 93  61 tisserantii (0 0.032 0.054 0.065 0.075 0.13 0.043 0.26 0.34)  
        30) Spklen>=9.09 41  20 sphaerocephalus (0 0.049 0.073 0.049 0.073 0.17 0 0.51 0.073) *
        31) Spklen< 9.09 52  23 tisserantii (0 0.019 0.038 0.077 0.077 0.096 0.077 0.058 0.56) *
# Adjust the rpart.plot arguments *type*, *tweak* and *extra* to give a more useful and informative display of the tree 

rpart.plot(tree4, type=0, tweak=0.9, extra=2+100)


rpart.plot(tree5, type=0, tweak=0.9, extra=2+100)

Make a bar chart of the assignments

#check that order of assigned individuals is the same as in the data frame 
data.frame(rownames(martinx.imp),tree4$where)

#add assignments to data frame 
martinx.imp.rpart.assign.tree4<-data.frame(martinx.imp, tree4$where)
head(martinx.imp.rpart.assign.tree4)
NA

Find CART assignments within each original species


#make a simplified data frame with just the three factors
dtree4<-martinx.imp.rpart.assign.tree4[,c(1:2,18)]
head(dtree4)

#table the species and assignments:
rpart.counts.tree4<-table(dtree4$Taxon,dtree4$tree4.where)
rpart.counts.tree4
                 
                    4   5   6   8  10  12  13
  karlschumannii   14   0   0   0   0   0   0
  ledermannii       0   2   0   8   0   2   1
  margaritaceus   180   7   6   3   0   3   2
  nduru             0   0  36   0  18   2   4
  niveus            3   0   0   2   0   3   4
  obtusiflorus      3   9   0  47   1   7   5
  somaliensis       0   0   0   0   2   0   4
  sphaerocephalus   8  26   0   5   1  21   3
  tisserantii       9   0   3   1   2   3  29
#check that rows sum to species totals:
data.frame(apply(rpart.counts.tree4,1,sum), table(dtree4$Taxon))
NA

Find out which are the species names of the columns; the programme assigns to the column the species with the most individuals assigned to it


assign.tree4<-rep(0,7)
for(i in 1:7) {
assign.tree4[i]<-names(which.max(rpart.counts.tree4[,i]))
}
assign.tree4
[1] "margaritaceus"   "sphaerocephalus" "nduru"          
[4] "obtusiflorus"    "nduru"           "sphaerocephalus"
[7] "tisserantii"    

Make the new column names for the counts matrix, avoiding duplication of names


colnames(rpart.counts.tree4)<-c("marg.crt","sphaer1.crt","nduru1.crt","obtusi.crt","nduru2.crt","sphaer2.crt","tisser.crt")
rpart.counts.tree4
                 
                  marg.crt sphaer1.crt nduru1.crt obtusi.crt nduru2.crt
  karlschumannii        14           0          0          0          0
  ledermannii            0           2          0          8          0
  margaritaceus        180           7          6          3          0
  nduru                  0           0         36          0         18
  niveus                 3           0          0          2          0
  obtusiflorus           3           9          0         47          1
  somaliensis            0           0          0          0          2
  sphaerocephalus        8          26          0          5          1
  tisserantii            9           0          3          1          2
                 
                  sphaer2.crt tisser.crt
  karlschumannii            0          0
  ledermannii               2          1
  margaritaceus             3          2
  nduru                     2          4
  niveus                    3          4
  obtusiflorus              7          5
  somaliensis               0          4
  sphaerocephalus          21          3
  tisserantii               3         29

Make this table into a table of proportions for each row.


#make table of proportions and table as percentages
proportions<-prop.table(rpart.counts.tree4,2)
round(proportions*100, 2)
                 
                  marg.crt sphaer1.crt nduru1.crt obtusi.crt nduru2.crt
  karlschumannii      6.45        0.00       0.00       0.00       0.00
  ledermannii         0.00        4.55       0.00      12.12       0.00
  margaritaceus      82.95       15.91      13.33       4.55       0.00
  nduru               0.00        0.00      80.00       0.00      75.00
  niveus              1.38        0.00       0.00       3.03       0.00
  obtusiflorus        1.38       20.45       0.00      71.21       4.17
  somaliensis         0.00        0.00       0.00       0.00       8.33
  sphaerocephalus     3.69       59.09       0.00       7.58       4.17
  tisserantii         4.15        0.00       6.67       1.52       8.33
                 
                  sphaer2.crt tisser.crt
  karlschumannii         0.00       0.00
  ledermannii            4.88       1.92
  margaritaceus          7.32       3.85
  nduru                  4.88       7.69
  niveus                 7.32       7.69
  obtusiflorus          17.07       9.62
  somaliensis            0.00       7.69
  sphaerocephalus       51.22       5.77
  tisserantii            7.32      55.77
#check the columns sum to 100%:
apply(proportions*100,2,sum)
   marg.crt sphaer1.crt  nduru1.crt  obtusi.crt  nduru2.crt sphaer2.crt 
        100         100         100         100         100         100 
 tisser.crt 
        100 

barplot proportions of original species individuals assigned to each “species� resulting from CART analysis


barplot(as.matrix(proportions), names.arg=colnames(rpart.counts.tree4), cex=0.8, cex.names=0.7,
main="tree4, seven leaves, with imputed data",
cex.lab=0.8, cex.main=0.8, xlab="CART species",
ylab="proportion of species individuals assigned to each column (crt)",
col=rainbow(length(rpart.counts.tree4[,1])))

legend("bottomleft", title="orig. species", cex=0.8, rownames(rpart.counts.tree4),
fill=rainbow(7), bg="white",
inset=0.05)

Now show the individual assignments made by CART:


res<-rep(0,489)
ass<-as.character(tree4$where)
cartnames<- colnames(rpart.counts.tree4)
for(i in 1:length(tree1$where))if(ass[i]=="4"){res[i]<-cartnames[1]}
for(i in 1:length(tree1$where))if(ass[i]=="5"){res[i]<-cartnames[2]}
for(i in 1:length(tree1$where))if(ass[i]=="6"){res[i]<-cartnames[3]}
for(i in 1:length(tree1$where))if(ass[i]=="8"){res[i]<-cartnames[4]}
for(i in 1:length(tree1$where))if(ass[i]=="10"){res[i]<-cartnames[5]}
for(i in 1:length(tree1$where))if(ass[i]=="12"){res[i]<-cartnames[6]}
for(i in 1:length(tree1$where))if(ass[i]=="13"){res[i]<-cartnames[7]}

res
  [1] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
  [5] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
  [9] "marg.crt"    "marg.crt"    "marg.crt"    "sphaer1.crt"
 [13] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [17] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [21] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [25] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [29] "marg.crt"    "marg.crt"    "sphaer1.crt" "marg.crt"   
 [33] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [37] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [41] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [45] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [49] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [53] "marg.crt"    "marg.crt"    "marg.crt"    "sphaer1.crt"
 [57] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [61] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [65] "sphaer1.crt" "obtusi.crt"  "sphaer2.crt" "marg.crt"   
 [69] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [73] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [77] "marg.crt"    "marg.crt"    "marg.crt"    "obtusi.crt" 
 [81] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [85] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [89] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [93] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
 [97] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[101] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[105] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[109] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[113] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[117] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[121] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[125] "marg.crt"    "marg.crt"    "nduru1.crt"  "marg.crt"   
[129] "nduru1.crt"  "marg.crt"    "marg.crt"    "marg.crt"   
[133] "marg.crt"    "marg.crt"    "marg.crt"    "nduru1.crt" 
[137] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[141] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[145] "sphaer1.crt" "marg.crt"    "marg.crt"    "sphaer1.crt"
[149] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[153] "marg.crt"    "nduru1.crt"  "nduru1.crt"  "marg.crt"   
[157] "nduru1.crt"  "marg.crt"    "marg.crt"    "marg.crt"   
[161] "marg.crt"    "marg.crt"    "marg.crt"    "sphaer2.crt"
[165] "marg.crt"    "marg.crt"    "marg.crt"    "obtusi.crt" 
[169] "sphaer1.crt" "marg.crt"    "marg.crt"    "marg.crt"   
[173] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[177] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[181] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[185] "marg.crt"    "tisser.crt"  "marg.crt"    "marg.crt"   
[189] "tisser.crt"  "marg.crt"    "marg.crt"    "marg.crt"   
[193] "marg.crt"    "sphaer2.crt" "marg.crt"    "marg.crt"   
[197] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[201] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[205] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[209] "marg.crt"    "marg.crt"    "marg.crt"    "marg.crt"   
[213] "marg.crt"    "marg.crt"    "marg.crt"    "sphaer2.crt"
[217] "sphaer1.crt" "obtusi.crt"  "obtusi.crt"  "tisser.crt" 
[221] "sphaer2.crt" "sphaer1.crt" "obtusi.crt"  "obtusi.crt" 
[225] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[229] "tisser.crt"  "tisser.crt"  "tisser.crt"  "tisser.crt" 
[233] "tisser.crt"  "tisser.crt"  "marg.crt"    "marg.crt"   
[237] "tisser.crt"  "tisser.crt"  "marg.crt"    "tisser.crt" 
[241] "tisser.crt"  "sphaer2.crt" "tisser.crt"  "tisser.crt" 
[245] "tisser.crt"  "nduru1.crt"  "tisser.crt"  "tisser.crt" 
[249] "tisser.crt"  "nduru1.crt"  "tisser.crt"  "tisser.crt" 
[253] "marg.crt"    "tisser.crt"  "obtusi.crt"  "tisser.crt" 
[257] "marg.crt"    "sphaer2.crt" "tisser.crt"  "tisser.crt" 
[261] "tisser.crt"  "tisser.crt"  "tisser.crt"  "marg.crt"   
[265] "marg.crt"    "tisser.crt"  "nduru2.crt"  "marg.crt"   
[269] "tisser.crt"  "nduru2.crt"  "tisser.crt"  "nduru1.crt" 
[273] "sphaer2.crt" "marg.crt"    "tisser.crt"  "tisser.crt" 
[277] "tisser.crt"  "marg.crt"    "tisser.crt"  "obtusi.crt" 
[281] "tisser.crt"  "marg.crt"    "sphaer2.crt" "sphaer2.crt"
[285] "marg.crt"    "sphaer2.crt" "obtusi.crt"  "nduru1.crt" 
[289] "nduru1.crt"  "nduru1.crt"  "nduru1.crt"  "nduru1.crt" 
[293] "nduru1.crt"  "nduru1.crt"  "nduru1.crt"  "nduru1.crt" 
[297] "nduru1.crt"  "tisser.crt"  "nduru1.crt"  "nduru1.crt" 
[301] "sphaer2.crt" "nduru1.crt"  "nduru1.crt"  "nduru1.crt" 
[305] "tisser.crt"  "nduru1.crt"  "nduru1.crt"  "nduru1.crt" 
[309] "nduru2.crt"  "nduru1.crt"  "nduru2.crt"  "nduru2.crt" 
[313] "nduru2.crt"  "nduru1.crt"  "nduru1.crt"  "nduru2.crt" 
[317] "nduru2.crt"  "nduru2.crt"  "nduru1.crt"  "nduru1.crt" 
[321] "tisser.crt"  "nduru2.crt"  "nduru2.crt"  "nduru1.crt" 
[325] "nduru1.crt"  "nduru1.crt"  "nduru2.crt"  "sphaer2.crt"
[329] "nduru2.crt"  "nduru2.crt"  "nduru1.crt"  "nduru1.crt" 
[333] "nduru2.crt"  "nduru1.crt"  "nduru2.crt"  "nduru2.crt" 
[337] "tisser.crt"  "nduru2.crt"  "nduru2.crt"  "nduru1.crt" 
[341] "nduru1.crt"  "nduru1.crt"  "nduru2.crt"  "nduru1.crt" 
[345] "nduru1.crt"  "nduru1.crt"  "nduru1.crt"  "obtusi.crt" 
[349] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[353] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[357] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[361] "obtusi.crt"  "obtusi.crt"  "tisser.crt"  "obtusi.crt" 
[365] "nduru2.crt"  "obtusi.crt"  "sphaer1.crt" "tisser.crt" 
[369] "obtusi.crt"  "obtusi.crt"  "tisser.crt"  "sphaer2.crt"
[373] "obtusi.crt"  "obtusi.crt"  "sphaer2.crt" "obtusi.crt" 
[377] "tisser.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[381] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[385] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "obtusi.crt" 
[389] "obtusi.crt"  "obtusi.crt"  "marg.crt"    "sphaer1.crt"
[393] "marg.crt"    "sphaer2.crt" "sphaer2.crt" "sphaer2.crt"
[397] "sphaer1.crt" "sphaer1.crt" "sphaer1.crt" "sphaer2.crt"
[401] "obtusi.crt"  "sphaer1.crt" "obtusi.crt"  "sphaer1.crt"
[405] "sphaer1.crt" "marg.crt"    "obtusi.crt"  "obtusi.crt" 
[409] "obtusi.crt"  "sphaer1.crt" "obtusi.crt"  "obtusi.crt" 
[413] "obtusi.crt"  "obtusi.crt"  "obtusi.crt"  "sphaer2.crt"
[417] "obtusi.crt"  "obtusi.crt"  "tisser.crt"  "sphaer2.crt"
[421] "sphaer2.crt" "sphaer2.crt" "sphaer2.crt" "sphaer2.crt"
[425] "obtusi.crt"  "sphaer2.crt" "sphaer1.crt" "sphaer1.crt"
[429] "sphaer2.crt" "sphaer2.crt" "sphaer2.crt" "sphaer2.crt"
[433] "sphaer2.crt" "sphaer2.crt" "sphaer2.crt" "sphaer2.crt"
[437] "sphaer2.crt" "sphaer2.crt" "sphaer2.crt" "marg.crt"   
[441] "sphaer2.crt" "sphaer1.crt" "marg.crt"    "sphaer1.crt"
[445] "tisser.crt"  "sphaer1.crt" "sphaer1.crt" "obtusi.crt" 
[449] "sphaer1.crt" "sphaer1.crt" "obtusi.crt"  "sphaer1.crt"
[453] "marg.crt"    "sphaer1.crt" "sphaer1.crt" "obtusi.crt" 
[457] "sphaer1.crt" "sphaer1.crt" "tisser.crt"  "sphaer1.crt"
[461] "sphaer1.crt" "obtusi.crt"  "sphaer2.crt" "sphaer2.crt"
[465] "sphaer1.crt" "marg.crt"    "sphaer1.crt" "marg.crt"   
[469] "marg.crt"    "marg.crt"    "sphaer1.crt" "sphaer2.crt"
[473] "marg.crt"    "sphaer1.crt" "sphaer1.crt" "sphaer1.crt"
[477] "sphaer1.crt" "sphaer1.crt" "sphaer1.crt" "sphaer1.crt"
[481] "sphaer1.crt" "tisser.crt"  "nduru2.crt"  "nduru2.crt" 
[485] "tisser.crt"  "tisser.crt"  "tisser.crt"  "nduru2.crt" 
[489] "tisser.crt" 

Put assignments in the righthand columns


d.names.tree4<-data.frame(dtree4,res)
d.names.tree4
#write this file to Excel csv
write.csv(d.names.tree4, file="CART assignments to species.csv tree4", row.names = FALSE)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Add a new chunk by clicking the Insert Chunk button on the toolbar or by pressing Ctrl+Alt+I.(Simon’s keyboard customized to Ctrl+Shift+I)

When you save the notebook, an HTML file containing the code and output will be saved alongside it (click the Preview button or press Ctrl+Shift+K to preview the HTML file).

---
title: "Martin Xanthos Cyperus margaritaceus complex with nine species"
output:
  html_notebook: default
  pdf_document: default
date: "9 September 2022"
---


This is an [R Markdown](http://rmarkdown.rstudio.com) Notebook. When you execute code within the notebook, the results appear beneath the code. 

Try executing this chunk by clicking the *Run* button within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


**Step 1. Input the data, look at the variables, remove constant or otherwise doubtfully useful variables**

Remember that variables that may be constant in this data set could become useful in a larger data set where they vary more.

When editing the data file in Excel prior to importing the data into R, use the following formula to make a version substituting "NA" in the blank cells:
=IF(ISBLANK(*cell reference*),"NA", *cell reference*)

Read in the data to R from a text file:


```{r data input}

martinx<-read.table("Cyperus margaritaceus dataset2_SimoneditSep22.txt", header=TRUE)

```

Look at the data to see if there are constant variables and missing values. Get a general idea by looking at the column names:

```{r column names}

names(martinx)

```

Look at the first six lines with function "head"

```{r head martinx}

head(martinx)

```

Look to see how many missing values there are (NA) with function "summary":

```{r summary martinx}

summary(martinx)

```


The "summary" function also summary statistics for each variable.

Two characters have very large proportions of missing values and for these interpolation is inadvisable and so they have to be deleted (nutlet length and width):

```{r martinx.edit}

martinx.edit<-martinx[,-c(14:15)]
names(martinx.edit)

```

**Step 2. Imputation of missing variables**

Now we can interpolate values for the missing data using the  "missForest" package. This will provide a complete matrix for analysis. This package handles mixed continuous and categorical data sets, and is non-parametric.

Load the "missForest" package:

```{r load missForest}

library(missForest)

```

Delete the first two columns of factors (specimen and taxon names), set seed and carry out the interpolation with the function "missForest". 

```{r imputation}

martinx.edit.2<-martinx.edit[,-c(1:2)]
set.seed(810)
martinx.edit.2.imp<-missForest(martinx.edit.2, verbose = TRUE)

```

Look at the structure of the output object "martinx.edit.2.imp":


```{r martinx.edit.2.imp structure}

str(martinx.edit.2.imp)

```

The completed data frame with imputed values is the object "martinx.edit.2.imp$ximp":

```{r imputed matrix object}

head(martinx.edit.2.imp$ximp)

```

Now we add back the factor columns we removed (columns 1 and 2) and give the imputed matrix a simpler name:

```{r renamed imputed matrix}

martinx.imp<-data.frame(martinx.edit[, c(1:2)], martinx.edit.2.imp$ximp)

head(martinx.imp)

```

Write the new imputed matrix back into Excel so it can be saved separately. This script writes the file into the working directory:

```{r write imputed matrix to Excel}

write.csv(martinx.imp, file="Cyperus margaritaceus imputed.csv", row.names = FALSE)

```

Now have a careful look at the Excel file to see if there are any peculiarities that need to be addressed. Compare with the original Excel file. 

**Step 3. Principal Coordinate Analysis (PCoA**

Because the data frame has both quantitative variable and qualitative characters, we will use Gower's distance coefficient to compute a distance matrix. 

The cells of the distance matrix show the Gower's morphological distance between every pair of individuals. 

The PCoA is a computational approach which attempts to represent these distances as faithfully as possible using orthogonal axes (analogous to PCA axes). 

To compute the Gower distance matrix we use the function *daisy* from the package *cluster*.

```{r load cluster package}

library(cluster)

```

The computation of Gower's distance standardizes the variables, so there is no need to standardize them first.

```{r compute Gower distance matrix}

martin.gower<-daisy(martinx.imp[-c(1:2)], metric="gower")

```

Carry out the PCoA using the function *cmdscale* in the "stats" package that is loaded automatically with R:

```{r compute PCoA}

pcoa<-cmdscale(martin.gower, k=10, eig=TRUE)

summary(pcoa)

```

The object "pcoa$points" has the coordinates of the scores on each of the principal coordinates (PCos). The call was for 10 PCos, which means that the scores for the first 10 PCos are printed. 

The object *eig* gives the values of all 414 eigenvalues in descending order. There are as many eigenvalues as objects in the distance matrix. Printing *eig* shows how many are positive and how many negative. Only positive eigenvalues (and their corresponding PCos) are meaningful.

```{r eigenvalues}

pcoa$eig

```

Only the first 121 eigenvalues are positive. We need to visualize them:

```{r barplot eigenvalues}

barplot(pcoa$eig[1:121])

```

It would be more useful to show these as percentages:

```{r eigenvalues percentages}

eig.percent<-(pcoa$eig[1:121]*100)/sum(pcoa$eig[1:121])
round(eig.percent, 3)

```

Still more useful is a cumulative sum of the eigenvalues so we can see where a good cut-off point is:

```{r cumulative percentage eigenvalues}

eig.percent.cumul<-cumsum(eig.percent)
eig.percent.cumul

```

We need 36 eigenvalues to capture 95% of the variance: 

```{r barplot eigval 36}

barplot(pcoa$eig[1:36])

```

What this means is that the ordinations of the first two principal coordinates will only express 49.9% of the total variation.

We might need to try Non-Metric Multidimensional Scaling (NMDS)to improve on this - it finds the best 2 dimensional plot by an iteration process. 

Plot the PCoA ordinations:

```{r plot PCos 1 and 2}

scores<-pcoa$points
plot(scores[,1],scores[,2], pch=16, cex=0.8, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey")[martinx.imp[,2]], asp=1, xlab="PCo1", ylab="PCo2", main="PCoA Cyperus margaritaceus complex", cex.main=0.8)

legend("bottomleft", levels(martinx.imp[,2]), pch=16, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.8, pt.cex = 1.0)


```



**Carry out a Principal Component Analysis (PCA) using the quantitative variables on the imputed data set *martinx.imp***


Run the PCA

```{r PCA of imputed matrix}

martinx.imp.pca<-prcomp(martinx.imp[,3:14], scale=TRUE)

summary(martinx.imp.pca)

```


Compute the number of significant principal components using the function "evplot.R" which is obtainable via the Springer book web page for Borcard et al's "Numerical Ecology" (Borcard et al., 2011)

First run the function:

evplot()
Plot eigenvalues and percentages of variation
Kaiser rule and broken stick model
License: GPL-2
Author: Francois Gillet, Octber 2007

```{r  function evplot}

evplot = function(ev) {
# Broken stick model (MacArthur 1957)
n = length(ev)
bsm = data.frame(j=seq(1:n), p=0)
bsm$p[1] = 1/n
for (i in 2:n) {
bsm$p[i] = bsm$p[i-1] + (1/(n + 1 - i))
}
bsm$p = 100*bsm$p/n

# Plot eigenvalues and % of variation for each axis

op = par(mfrow=c(2,1))
barplot(ev, main="Eigenvalues", col="bisque", las=2)
abline(h=mean(ev), col="red")
legend("topright", "Average eigenvalue", lwd=1, col=2, bty="n")
barplot(t(cbind(100*ev/sum(ev), bsm$p[n:1])), beside=T, main="% variation", col=c("bisque",2), las=2)
legend("topright", c("% eigenvalue", "Broken stick model"),pch=15, col=c("bisque",2), bty="n")
par(op)
}

```

Use the function:

```{r  use evplot}

ev<-(martinx.imp.pca$sdev)^2
evplot(ev)

```


This shows significance for the first 3 PCs, which express 66.6% of total variance.

Plot the first three PCs:

```{r pca plot PCs 1 and 2}

par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

```


```{r pca plot PCs 1 and 3}

par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)


```



```{r  pca plot PCs 2 and 3}

par(mfrow=c(1,1))
plot(martinx.imp.pca$x[,2], martinx.imp.pca$x[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)


```




Replot with convex hulls

To plot convex hulls around the species groups we can use the function *Plot ConvexHull* from Ken Takagi's website *Chit Chat R* to overplot convex hulls onto the original plot.

```{r pca no outliers with convex hulls}

### Plotting function to plot convex hulls

### Filename: Plot_ConvexHull.R
### Notes:
############################################################################
# INPUTS:
# xcoords: x-coordinates of point data
# ycoords: y-coordinates of point data
# lcolor: line color
# OUTPUTS:
# convex hull around data points in a particular color (specified by lcolor)
# FUNCTION:
Plot_ConvexHull<-function(xcoord, ycoord, lcolor){
hpts <- chull(x = xcoord, y = ycoord)
hpts <- c(hpts, hpts[1])
lines(xcoord[hpts], ycoord[hpts], col = lcolor)
}
# END OF FUNCTION

par(mfrow=c(1,1))

plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="karlschumannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="karlschumannii"], lcolor = "red")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="ledermannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="ledermannii"], lcolor = "blue")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="margaritaceus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="margaritaceus"], lcolor = "green")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="nduru"],martinx.imp.pca$x[,2][martinx.imp[,2]=="nduru"], lcolor = "black")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="niveus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="niveus"], lcolor = "magenta")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="obtusiflorus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="somaliensis"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "sienna")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="sphaerocephalus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "khaki")
Plot_ConvexHull(martinx.imp.pca$x[,1][martinx.imp[,2]=="tisserantii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="tisserantii"], lcolor = "grey")


```

Now replot with 95% confidence ellipses.  

Now plot with 95% confidence ellipses using using the ELLI function from Julien Claude's book (Morphometrics with R) website:

Run ELLI function:

```{r PCA with ellipses with outliers removed}

#run the ELLI function:
ELLI<-function(x,y,conf=0.95,np=50)
{centroid<-apply(cbind(x,y),2,mean)
ang <- seq(0,2*pi,length=np)
z<-cbind(cos(ang),sin(ang))
radiuscoef<-qnorm((1-conf)/2, lower.tail=F)
vcvxy<-var(cbind(x,y))
r<-cor(x,y)
M1<-matrix(c(1,1,-1,1),2,2)
M2<-matrix(c(var(x), var(y)),2,2)
M3<-matrix(c(1+r, 1-r),2,2, byrow=T)
ellpar<-M1*sqrt(M2*M3/2)
t(centroid + radiuscoef * ellpar %*% t(z))}

par(mfrow=c(1,1))

plot(martinx.imp.pca$x[,1], martinx.imp.pca$x[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

#plot the ellipses:
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="karlschumannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="karlschumannii"]), col = "red")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="ledermannii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="ledermannii"]), col = "blue")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="margaritaceus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="margaritaceus"]), col = "green")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="nduru"],martinx.imp.pca$x[,2][martinx.imp[,2]=="nduru"]), col = "black")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="niveus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="niveus"]), col = "magenta")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="obtusiflorus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="somaliensis"],martinx.imp.pca$x[,2][martinx.imp[,2]=="somaliensis"]), col = "sienna")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="sphaerocephalus"],martinx.imp.pca$x[,2][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")
lines(ELLI(martinx.imp.pca$x[,1][martinx.imp[,2]=="tisserantii"],martinx.imp.pca$x[,2][martinx.imp[,2]=="tisserantii"]), col = "grey")


```


*Plot the loadings for the first two principal components*


```{r pc1 and 2 loadings barchart}

par(mfrow=c(1,2))
barplot(sort(martinx.imp.pca$rotation[,1]), las=2, cex.names=0.7, main="PC1", ylab="Loadings")
barplot(sort(martinx.imp.pca$rotation[,2]), las=2, cex.names=0.7, main="PC2", ylab="Loadings")
par(mfrow=c(1,1))

```

Plot the biplot

```{r PC1 and 2 biplot}

biplot(martinx.imp.pca, cex=0.5)

```




**Linear discriminant analysis (LDA)**

Following Legendre & Legendre (2012, Numerical Ecology, 3rd English Edition), we should analyse untransformed data for testing the allocation of individuals to populations, and standardized data for computing the contribution of each variable to the discriminant function axes. But first we have to test the data for homoscedasticity (data is homoscedastic if the species/groups/categorical variables have equal/homogeneous covariance matrices) with "betadisper" function in "vegan" package. 

Make sure you have installed the vegan package (command is install.packages("vegan"))

**Test for multivariate homogeneity of variances** using function "betadisper" in package "vegan". Start by checking the for homoscedasticity (multivariate homogeneity of within-group covariance matrices) of the multivariate data ([Borcard et al., 2011], page 208). We use the "betadisper" function of the "vegan" package, implementing Marti Anderson's testing method. First compute the Euclidean distance matrix between pairs of objects:

```{r betadisper analysis}

library(vegan)
means.dist<- dist(martinx.imp[, 3:14])
means.MHV<-betadisper(means.dist, martinx.imp[,2])
means.MHV
anova(means.MHV)
permutest(means.MHV)

```

In the above, the Analysis of Variance is a parametric test. The permutation test is non-parametric.

These results show that the species groups do not have homogeneous covariance matrices, i.e. they are heteroscadistic. This is not surprising since they are different species. For the purpose of plotting the ordinations of the species, this is not really a problem. It is a problem if you want to use multivariate analysis of variance (MANOVA) to compute the significance of the differences between the species group centroids (means). However, we can get round this using non-parametric tests.

Now we compute the LDA, after loading the MASS package:

```{r lda martinx.imp}

library(MASS)

lda<-lda(martinx.imp[, 3:14], martinx.imp[,2])
lda

```


The proportion of trace table shows that 78.62% (0.7862) of the variance is expressed in the first two axes

Now we compute the scores for plotting the ordination. What happens here is that we multiply the matrix of variables in the martinx.imp data frame by the eigenvector matrix, which is a sub-object of the lda object (lda$scaling) and described in the above output as "Coefficients of linear discriminants":

```{r scores lda martinx.imp}

scores<-as.matrix(martinx.imp[,3:14])%*%lda$scaling
head(scores)

```


Now plot the first two discriminant axes and add legend:

```{r lda ordination LD 1 and LD 2}

plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

```


Repeat for axes 2 and 3:

```{r lda ordination LD 1 and LD 3}

plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

```


```{r lda ordination LD 2 and LD 3}

plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("bottomleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

```


To plot convex hulls around the species groups we can use the function "Plot ConvexHull" from Ken Takagi's website "Chit Chat R" to overplot convex hulls onto the original plot.

Load the convex hulls function:

```{r Plot_ConvexHull.R script}
### Plotting function to plot convex hulls
### Filename: Plot_ConvexHull.R
### Notes:
############################################################################
# INPUTS:
# xcoords: x-coordinates of point data
# ycoords: y-coordinates of point data
# lcolor: line color
# OUTPUTS:
# convex hull around data points in a particular color (specified by lcolor)
# FUNCTION:

Plot_ConvexHull<-function(xcoord, ycoord, lcolor){
hpts <- chull(x = xcoord, y = ycoord)
hpts <- c(hpts, hpts[1])
lines(xcoord[hpts], ycoord[hpts], col = lcolor)
}

# END OF FUNCTION

```

Replot the LDA ordinations and then the convex hulls:

```{r replot lda LD1 and LD2 with convex hulls}

plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,2][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,2][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,2][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="nduru"],
scores[,2][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="niveus"],
scores[,2][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,2][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,2][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,2][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,2][martinx.imp[,2]=="tisserantii"], lcolor = "grey")

```


```{r replot lda LD1 and LD3 with convex hulls}


plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"], lcolor = "grey")

```


```{r replot lda LD2 and LD3 with convex hulls}

plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"], lcolor = "red")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"], lcolor = "blue")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"], lcolor = "green")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"], lcolor = "black")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"], lcolor = "magenta")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"], lcolor = "cyan3")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"], lcolor = "sienna")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"], lcolor = "khaki")

Plot_ConvexHull(scores[,2][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"], lcolor = "grey")

```


Now plot with 95% confidence ellipses using using the ELLI function from Julien Claude's book (Morphometrics with R) website:

Run ELLI function:

```{r ELLI function}

ELLI<-function(x,y,conf=0.95,np=50)
{centroid<-apply(cbind(x,y),2,mean)
ang <- seq(0,2*pi,length=np)
z<-cbind(cos(ang),sin(ang))
radiuscoef<-qnorm((1-conf)/2, lower.tail=F)
vcvxy<-var(cbind(x,y))
r<-cor(x,y)
M1<-matrix(c(1,1,-1,1),2,2)
M2<-matrix(c(var(x), var(y)),2,2)
M3<-matrix(c(1+r, 1-r),2,2, byrow=T)
ellpar<-M1*sqrt(M2*M3/2)
t(centroid + radiuscoef * ellpar %*% t(z))}

```

Replot now using ELLI to plot the ellipses:

```{r replot lda LD1 and LD2 with 95 per cent ellipses}

plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,2][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,2][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,2][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,1][martinx.imp[,2]=="nduru"],
scores[,2][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,1][martinx.imp[,2]=="niveus"],
scores[,2][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,2][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,2][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,2][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,2][martinx.imp[,2]=="tisserantii"]), col = "grey")

```


```{r replot lda LD1 and LD3 with 95 per cent ellipses}

plot(scores[,1], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,1][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,1][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,1][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,1][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,1][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,1][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,1][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,1][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,1][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"]), col = "grey")

```



```{r replot lda LD2 and LD3 with 95 per cent ellipses}

plot(scores[,2], scores[,3],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

legend("topleft",inset=0.025, legend=c("karlschumannii","ledermannii","margaritaceus",      "nduru","niveus","obtusiflorus","somaliensis", "sphaerocephalus","tisserantii"),pch=c(rep(16,7))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), cex=0.7)

lines(ELLI(scores[,2][martinx.imp[,2]=="karlschumannii"],
scores[,3][martinx.imp[,2]=="karlschumannii"]), col= "red")

lines(ELLI(scores[,2][martinx.imp[,2]=="ledermannii"],
scores[,3][martinx.imp[,2]=="ledermannii"]), col= "blue")

lines(ELLI(scores[,2][martinx.imp[,2]=="margaritaceus"],
scores[,3][martinx.imp[,2]=="margaritaceus"]), col = "green")

lines(ELLI(scores[,2][martinx.imp[,2]=="nduru"],
scores[,3][martinx.imp[,2]=="nduru"]), col = "black")

lines(ELLI(scores[,2][martinx.imp[,2]=="niveus"],
scores[,3][martinx.imp[,2]=="niveus"]), col = "magenta")

lines(ELLI(scores[,2][martinx.imp[,2]=="obtusiflorus"],
scores[,3][martinx.imp[,2]=="obtusiflorus"]), col = "cyan3")

lines(ELLI(scores[,2][martinx.imp[,2]=="somaliensis"],
scores[,3][martinx.imp[,2]=="somaliensis"]), col = "sienna")

lines(ELLI(scores[,2][martinx.imp[,2]=="sphaerocephalus"],
scores[,3][martinx.imp[,2]=="sphaerocephalus"]), col = "khaki")

lines(ELLI(scores[,2][martinx.imp[,2]=="tisserantii"],
scores[,3][martinx.imp[,2]=="tisserantii"]), col = "grey")

```


Now plot the loadings on the first two discriminant axes to see which characters are most influential in separating the species:

```{r loadings barcharts fir LDA}

par(mfrow=c(2,1))

barplot(sort(lda$scaling[,1]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="A", ylab="loadings")
barplot(sort(lda$scaling[,2]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="B", ylab="loadings")

```

These bar charts tell us which characters are most influential in the scores on each axis. Individuals that have more negative scores on a given discriminant axis are tend to have higher values for the characters (bars) towards the left, and individuals with more positive scores have higher values for the characters (bars) towards the right.

**LDA using scaled data**

Now we repeat these LDA analyses using scaled data to see what difference this makes. 

First make a data frame of the scaled data using the function "scale"
, which centres the data (in each column, the mean of the column is subtracted from the individual values) and then the variance of each column is set to zero (by dividing the centred values of each column by the column standard deviation).

```{r scale data frame martinx.imp}

martinx.imp.sc<-scale(martinx.imp[, 3:14])
head(martinx.imp.sc)

```


Now run the LDas on the scaled data frame:

```{r lda scaled data}

lda.sc<-lda(martinx.imp.sc, martinx.imp[,2])
lda.sc
scores<-as.matrix(martinx.imp.sc)%*%lda.sc$scaling
head(scores)
plot(scores[,1], scores[,2],pch=c(rep(16,9))[martinx.imp[,2]], col= c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey") [martinx.imp[,2]], cex=1, asp=1)

par(mfrow=c(2,1))
barplot(sort(lda.sc$scaling[,1]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="A", ylab="loadings")
barplot(sort(lda.sc$scaling[,2]), cex=0.8, cex.main=1.2, las=2, cex.names=0.8, main="B", ylab="loadings")

```


Scaling the data has no effect on the topology of the ordinations, but it changes the loadings on the discriminant axes (bar chart above) that indicate which variables/characters are influencing the score values the most and the least (the central bars are the least influential). It is best do use the scaled data result to estimate the influence of each character on the discriminant analyses. 


**LDA using cross-validation**

When the data violate the basic assumptions of the parametric LDA approach (data overall with a normal distribution, and group covariance matrices homoscedastic), the best way to estimate the significance of the intergroup differences is to use cross-validation. This is a process of allocating each individual to the group to which it is nearest, as estimated by its (Mahalanobis) distance from each group centroid, as computed from its scores on the discriminant axes. For this it is best to use the untransformed data.

To use cross-validation, we run the "lda" function with the argument "CV=TRUE":

```{r lda with cross-validation}

lda.cv<-lda(martinx.imp[, 3:14], martinx.imp[,2], CV=TRUE)
summary(lda.cv)



```


We make a table of the assignments of the cross-validation (also called a Jackknife test):

```{r CV assignment table}

cv.jack.table<-table(martinx.imp[,2], lda.cv$class)
cv.jack.table

```

This looks quite successful, apart from niveus, where none of its individuals were successfully allocated. This analysis uses only the quantitative variables. The result for margaritaceus is especially notable, with most individuals assigned to it.


```{r cross valid LDA results on nine species}

# calculate total errors in cross-validation

# run function "errors.cv"

errors.cv<-function(square.tab) {
	errors<-round(sum(square.tab) - sum(diag(square.tab)), 1)
	total<-round(sum(square.tab), 0)
	errors.prop<-round(errors/total, 4)
	result<-c(errors, total, errors.prop)
	names(result)<-c("errors", "total", "errors.proportion")	
	result
}

# Use function "errors.cv" to compute overall errors of cross validation

errors.cv(cv.jack.table)

# Compute percentage correct assignments for each population

round(diag(cv.jack.table)/rowSums(cv.jack.table),3)*100

# pegar a matriz de probabilidades posteriores de cada individuo e plotar em barras

indiv.probs<-t(lda.cv$posterior)
colnames(indiv.probs)<-martinx.imp$Specimen

barplot(indiv.probs, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), border=NA, space=0, las=2, cex.names=0.4, ylab = "posterior probabilities")

## plot the eigenvectors (= loadings)

par(mfrow=c(1,2))
barplot(sort(lda$scaling[,1]), las=2, cex.names=0.7, cex=0.9, cex.main=0.8, main="discriminant function 1", ylab="pesos")
barplot(sort(lda$scaling[,2]), las=2, cex.names=0.7, cex=0.9, cex.main=0.8, main="discriminant function 2", ylab="pesos")
par(mfrow=c(1,1))

```



Now we turn this table into proportions and make a bar plot:

```{r CV proportions table}

proportions<-prop.table(cv.jack.table,2)
proportions

#check the columns sum to 1
apply(prop.table(cv.jack.table,2),2,sum)

```


Now barplot the proportions table:

```{r barplot CV proportions}

par(mar=c(5,4,4,2))

barplot(proportions, main="LDA: assignment of original populations by cross-validation using untransformed data",cex=0.8, cex.lab=0.8, cex.main=0.8, cex.names=0.7, col=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), ylab="proportion of populations assigned")

legend("topright", title="original", cex=0.7, row.names(proportions),fill=c("red","blue","green","black","magenta","cyan3","sienna", "khaki", "grey"), bg="white", inset=0.045)

```

This shows the proportions of correctly assigned individuals in the classes produced by cross validation. Each bar is the result of the cross-validation, and the colours are those of the original populations. In this analysis, margaritaceus has a successful result, considering its wide range.

Now add the LDA cross-validation assignments to the data frame **martinx.imp** as a new column and name the new data frame:

```{r add LDA cross validation assignments}

martinx.imp.lda.crossval<-data.frame(martinx.imp, lda.cv$class)
head(martinx.imp.lda.crossval)

```

Extract the individuals and their LDA assignments

```{r LDA assignments}

LDA.assignments<-martinx.imp.lda.crossval[,c(1:2,18)]
LDA.assignments

#write this file to Excel csv
write.csv(LDA.assignments, file="LDA assignments to species.csv", row.names = FALSE)

```


**Carry out a Classification Tree analysis**

load the package rpart

```{r load rpart}

library(rpart)

```


Compute and plot the tree

```{r CART tree1}

tree1<-rpart(Taxon~ Lflen + Lfwid + Cullen + Culwid + Numbrac +  Lenbrac + Numspk +  Spklen + Spkwid + Gllen + Glwid + Numgl + Bshpers + Bshtext + Bshglos, data=martinx.imp, method="class")
tree1

plot(tree1)
text(tree1, cex=0.7, xpd=TRUE)

```


Plot with package rpart.plot

```{r rpart.plot}

library(rpart.plot)
rpart.plot(tree1)


```

Make a bar chart of the assignments

```{r CART bar chart assignments}
#check that order of assigned individuals is the same as in the data frame 
data.frame(rownames(martinx.imp),tree1$where)

#add assignments to data frame 
martinx.imp.rpart.assign<-data.frame(martinx.imp, tree1$where)
head(martinx.imp.rpart.assign)

```

Find CART assignments within each original species  


```{r counts matrix of CART assignments}

#make a simplified data frame with just the three factors
d<-martinx.imp.rpart.assign[,c(1:2,18)]
head(d)

#table the species and assignments:
rpart.counts<-table(d$Taxon,d$tree1.where)
rpart.counts

#check that columns sum to species totals:
data.frame(apply(rpart.counts,1,sum), table(d$Taxon))

```

Find out which are the species names of the columns; the programme assigns to the column the species with the most individuals assigned to it

```{r assignment function}

assign<-rep(0,11)
for(i in 1:11) {
assign[i]<-names(which.max(rpart.counts[,i]))
}
assign

```

Make the new column names for the counts matrix, avoiding duplication of names

```{r new names}

colnames(rpart.counts)<-c("karl.crt","marg1.crt","marg2.crt","sphaer1", "marg3.crt","nduru1.crt","leder.crt","obtus.crt", "nduru2.crt", "sphaer2", "tiss")
rpart.counts


```

Make this table into a table of proportions for each row.

```{r proportion table}

#make table of proportions and table as percentages
proportions<-prop.table(rpart.counts,2)
round(proportions*100, 2)

#check the columns sum to 100%:
apply(proportions*100,2,sum)
```


Barplot proportions of original species individuals assigned to each "species" resulting from CART analysis

```{r Cart assignment barplot}

barplot(as.matrix(proportions), names.arg=colnames(rpart.counts), cex=0.8, cex.names=0.7,
main="tree1, eleven leaves, with imputed data",
cex.lab=0.8, cex.main=0.8, xlab="CART species",
ylab="proportion of species individuals assigned to each column (crt)",
col=rainbow(length(rpart.counts[,1])))

legend("bottomleft", title="orig. species", cex=0.8, rownames(rpart.counts),
fill=rainbow(9), bg="white",
inset=0.05)


```

Now show the individual assignments made by CART:

```{r individual CART assignments to species}

res<-rep(0,489)
ass<-as.character(tree1$where)
cartnames<- colnames(rpart.counts)
for(i in 1:length(tree1$where))if(ass[i]=="5"){res[i]<-cartnames[1]}
for(i in 1:length(tree1$where))if(ass[i]=="6"){res[i]<-cartnames[2]}
for(i in 1:length(tree1$where))if(ass[i]=="8"){res[i]<-cartnames[3]}
for(i in 1:length(tree1$where))if(ass[i]=="9"){res[i]<-cartnames[4]}
for(i in 1:length(tree1$where))if(ass[i]=="11"){res[i]<-cartnames[5]}
for(i in 1:length(tree1$where))if(ass[i]=="12"){res[i]<-cartnames[6]}
for(i in 1:length(tree1$where))if(ass[i]=="15"){res[i]<-cartnames[7]}
for(i in 1:length(tree1$where))if(ass[i]=="16"){res[i]<-cartnames[8]}
for(i in 1:length(tree1$where))if(ass[i]=="18"){res[i]<-cartnames[9]}
for(i in 1:length(tree1$where))if(ass[i]=="20"){res[i]<-cartnames[10]}
for(i in 1:length(tree1$where))if(ass[i]=="21"){res[i]<-cartnames[11]}
res


```

Put assignments in the righthand columns

```{r final assignments from CART with names and numbers of CART groups}

d.names<-data.frame(d,res)
d.names

#write this file to Excel csv
write.csv(d.names, file="CART assignments to species.csv", row.names = FALSE)

```


**REPEAT THE CART ANALYSIS WITH NON-DEFAULT SETTINGS FOR MINSPLIT AND MINBUCKET**

```{r rpart tree3}
# Compute the tree with minsplit=10 (minimum number of individuals in a node to permit further splitting) and minbucket=5 (minimum number of individuals in a terminal node).

tree3<-rpart(Taxon~ Lflen + Lfwid + Cullen + Culwid + Numbrac +  Lenbrac + Numspk +  Spklen + Spkwid + Gllen + Glwid + Numgl + Bshpers + Bshtext + Bshglos, data=martinx.imp, method="class", control= rpart.control(minsplit = 10, minbucket = 9))
plot(tree3)
text(tree3, cex=0.7, xpd=TRUE)
printcp(tree3)
plotcp(tree3)

```



```{r prune tree3 to tree4}

tree4<- prune(tree3, cp=0.032)
plot(tree4)
text(tree4, cex=0.7, xpd=TRUE)
print(tree4)

tree5<- prune(tree3, cp=0.016)
plot(tree5)
text(tree5, cex=0.7, xpd=TRUE)
print(tree5)

# Adjust the rpart.plot arguments *type*, *tweak* and *extra* to give a more useful and informative display of the tree 

rpart.plot(tree4, type=0, tweak=0.9, extra=2+100)

rpart.plot(tree5, type=0, tweak=0.9, extra=2+100)

```


Make a bar chart of the assignments

```{r CART bar chart assignments tree4}
#check that order of assigned individuals is the same as in the data frame 
data.frame(rownames(martinx.imp),tree4$where)

#add assignments to data frame 
martinx.imp.rpart.assign.tree4<-data.frame(martinx.imp, tree4$where)
head(martinx.imp.rpart.assign.tree4)

```


Find CART assignments within each original species  


```{r counts matrix of CART assignments tree4}

#make a simplified data frame with just the three factors
dtree4<-martinx.imp.rpart.assign.tree4[,c(1:2,18)]
head(dtree4)

#table the species and assignments:
rpart.counts.tree4<-table(dtree4$Taxon,dtree4$tree4.where)
rpart.counts.tree4

#check that rows sum to species totals:
data.frame(apply(rpart.counts.tree4,1,sum), table(dtree4$Taxon))

```


Find out which are the species names of the columns; the programme assigns to the column the species with the most individuals assigned to it

```{r assignment function tree4}

assign.tree4<-rep(0,7)
for(i in 1:7) {
assign.tree4[i]<-names(which.max(rpart.counts.tree4[,i]))
}
assign.tree4

```


Make the new column names for the counts matrix, avoiding duplication of names

```{r new names tree4}

colnames(rpart.counts.tree4)<-c("marg.crt","sphaer1.crt","nduru1.crt","obtusi.crt","nduru2.crt","sphaer2.crt","tisser.crt")
rpart.counts.tree4


```


Make this table into a table of proportions for each row.

```{r proportion table tree4}

#make table of proportions and table as percentages
proportions<-prop.table(rpart.counts.tree4,2)
round(proportions*100, 2)

#check the columns sum to 100%:
apply(proportions*100,2,sum)
```


barplot proportions of original species individuals assigned to each "species" resulting from CART analysis

```{r Cart assignment barplot tree4}

barplot(as.matrix(proportions), names.arg=colnames(rpart.counts.tree4), cex=0.8, cex.names=0.7,
main="tree4, seven leaves, with imputed data",
cex.lab=0.8, cex.main=0.8, xlab="CART species",
ylab="proportion of species individuals assigned to each column (crt)",
col=rainbow(length(rpart.counts.tree4[,1])))

legend("bottomleft", title="orig. species", cex=0.8, rownames(rpart.counts.tree4),
fill=rainbow(7), bg="white",
inset=0.05)

```



Now show the individual assignments made by CART:

```{r individual CART assignments to species tree4}

res<-rep(0,489)
ass<-as.character(tree4$where)
cartnames<- colnames(rpart.counts.tree4)
for(i in 1:length(tree1$where))if(ass[i]=="4"){res[i]<-cartnames[1]}
for(i in 1:length(tree1$where))if(ass[i]=="5"){res[i]<-cartnames[2]}
for(i in 1:length(tree1$where))if(ass[i]=="6"){res[i]<-cartnames[3]}
for(i in 1:length(tree1$where))if(ass[i]=="8"){res[i]<-cartnames[4]}
for(i in 1:length(tree1$where))if(ass[i]=="10"){res[i]<-cartnames[5]}
for(i in 1:length(tree1$where))if(ass[i]=="12"){res[i]<-cartnames[6]}
for(i in 1:length(tree1$where))if(ass[i]=="13"){res[i]<-cartnames[7]}

res


```

Put assignments in the righthand columns

```{r final assignments from CART with names and numbers of CART groups tree4}

d.names.tree4<-data.frame(dtree4,res)
d.names.tree4
#write this file to Excel csv
write.csv(d.names.tree4, file="CART assignments to species.csv tree4", row.names = FALSE)

```




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