Q & K Box 14.1
Deriving components (modified from Jackson 2003)
There are two strategies for extracting eigenvectors (components) and their eigenvalues from a multivariate data set of n objects by p variables. First, we can use an eigenanalysis (spectral decomposition) of a p x p association matrix between variables. Second, we can use a singular value decomposition (SVD) of the n x p data matrix, with variables standardized as necessary.
Consider the matrix (Y) of raw data from Clevenger and Waltho (2000), who recorded the numbers of seven taxa of wildlife for eleven underpasses in Alberta, Canada.
Load the vegan package
library(tidyverse)
library(vegan)#datasets: clevenger.csv
clevenger <- read_csv("../data/clevenger.csv")Rows: 11 Columns: 22── Column specification ────────────────────────────────────────────────────────────
Delimiter: ","
chr (1): underpass
dbl (21): width, height, length, openness, noise, egdist, forcov, draindist, rai...
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.head(clevenger,10)Illustrate centroid using relationship between human activity index (hui) and elk numbers
Plots used for QK use the ggplot classic theme, with some tweaks. Tweaks are consolidated into theme_QK; use this theme for figures and tweak the theme to avoid repetitive code changes.
Fig 14.2
f14_2<-ggplot(data=clevenger,aes(x=hui, y=elk))+
geom_point(color=sc, size=ss, alpha= 1) +
theme_qk()+
geom_point(aes(x=mean(hui),y=mean(elk)), pch=8, size=3)
f14_2
Get centered and standardized raw data matrices
# centered
clevenger.cen <- scale(clevenger[,c('bbear','gbear','cougar','wolf','deer','elk','moose')], scale=F)
clevenger.cen bbear gbear cougar wolf deer elk
[1,] -7.545455 -0.6363636 -5.636364 -27.2727273 334.818182 10.545455
[2,] 2.454545 -0.6363636 18.363636 -21.2727273 -177.181818 -613.454545
[3,] 25.454545 -0.6363636 -7.636364 -25.2727273 74.818182 -483.454545
[4,] 19.454545 1.3636364 19.363636 -0.2727273 33.818182 384.545455
[5,] -4.545455 -0.6363636 -3.636364 -25.2727273 -4.181818 247.545455
[6,] -9.545455 -0.6363636 -10.636364 -23.2727273 -198.181818 -347.454545
[7,] -17.545455 -0.6363636 -6.636364 -27.2727273 -158.181818 761.545455
[8,] -13.545455 -0.6363636 -6.636364 -23.2727273 118.818182 707.545455
[9,] -9.545455 -0.6363636 9.363636 48.7272727 68.818182 6.545455
[10,] 16.454545 4.3636364 4.363636 117.7272727 71.818182 -131.454545
[11,] -1.545455 -0.6363636 -10.636364 6.7272727 -165.181818 -542.454545
moose
[1,] 0.8181818
[2,] -0.1818182
[3,] 0.8181818
[4,] -0.1818182
[5,] -0.1818182
[6,] -0.1818182
[7,] -0.1818182
[8,] -0.1818182
[9,] -0.1818182
[10,] -0.1818182
[11,] -0.1818182
attr(,"scaled:center")
bbear gbear cougar wolf deer elk moose
17.5454545 0.6363636 10.6363636 28.2727273 219.1818182 814.4545455 0.1818182 # get standardized raw data
clevenger.stan <- scale(clevenger[,c('bbear','gbear','cougar','wolf','deer','elk','moose')], scale=T)
clevenger.stan bbear gbear cougar wolf deer elk
[1,] -0.5261348 -0.4061812 -0.5068669 -0.601179227 2.05348246 0.02183648
[2,] 0.1711523 -0.4061812 1.6514052 -0.468919797 -1.08667861 -1.27028099
[3,] 1.7749127 -0.4061812 -0.6867229 -0.557092751 0.45886942 -1.00108985
[4,] 1.3565404 0.8703883 1.7413332 -0.006011792 0.20741121 0.79627869
[5,] -0.3169487 -0.4061812 -0.3270109 -0.557092751 -0.02564762 0.51259264
[6,] -0.6655923 -0.4061812 -0.9565070 -0.513006274 -1.21547428 -0.71947450
[7,] -1.2234220 -0.4061812 -0.5967949 -0.601179227 -0.97014920 1.57693299
[8,] -0.9445071 -0.4061812 -0.5967949 -0.513006274 0.72872701 1.46511513
[9,] -0.6655923 -0.4061812 0.8420531 1.074106886 0.42207066 0.01355368
[10,] 1.1473543 2.7852425 0.3924131 2.595090331 0.44047004 -0.27220307
[11,] -0.1077626 -0.4061812 -0.9565070 0.148290876 -1.01308109 -1.12326121
moose
[1,] 2.0225996
[2,] -0.4494666
[3,] 2.0225996
[4,] -0.4494666
[5,] -0.4494666
[6,] -0.4494666
[7,] -0.4494666
[8,] -0.4494666
[9,] -0.4494666
[10,] -0.4494666
[11,] -0.4494666
attr(,"scaled:center")
bbear gbear cougar wolf deer elk moose
17.5454545 0.6363636 10.6363636 28.2727273 219.1818182 814.4545455 0.1818182
attr(,"scaled:scale")
bbear gbear cougar wolf deer elk moose
14.3412945 1.5666989 11.1200065 45.3653853 163.0489609 482.9282273 0.4045199 Generate covariance and correlation matrices
Covariance matrix for wildlife
clevenger.cov <- cov(clevenger[,c('bbear','gbear','cougar','wolf','deer','elk','moose')])Correlation matrix for wildlife
clevenger.cor <- cor(clevenger[,c('bbear','gbear','cougar','wolf','deer','elk','moose')])Derive eigenvectors and eigenvalues based on correlation matrix
clevenger.eigen <- eigen(clevenger.cor)
clevenger.eigeneigen() decomposition
$values
[1] 2.48262264 1.79202546 1.33144952 0.81062441 0.42903804 0.08769838 0.06654156
$vectors
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.4480426 -0.32853418 0.28726775 -0.195436794 0.586304172 -0.47611634
[2,] -0.5734521 0.04251404 -0.20081526 0.255898326 0.277980195 0.61766855
[3,] -0.3798802 0.16723268 0.02083960 -0.817320247 -0.324352619 0.19419176
[4,] -0.5355751 0.11425393 -0.14514937 0.430423505 -0.377378599 -0.28984791
[5,] -0.0895651 -0.53785185 -0.53718087 -0.082108683 -0.345693443 -0.27316305
[6,] 0.1440767 0.19264407 -0.75205162 -0.190408179 0.460246579 -0.06783799
[7,] 0.1023762 -0.72307729 0.03876746 -0.006827542 -0.004932548 0.43687126
[,7]
[1,] 0.01286228
[2,] -0.32364656
[3,] 0.12791605
[4,] 0.51705536
[5,] -0.46180770
[6,] 0.35194368
[7,] 0.52368693Do PCA
clevenger.pca <- princomp(~bbear+gbear+cougar+wolf+deer+elk+moose, cor=T, data=clevenger)
summary(clevenger.pca)Importance of components:
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
Standard deviation 1.5756340 1.3386656 1.1538845 0.9003468 0.65500995
Proportion of Variance 0.3546604 0.2560036 0.1902071 0.1158035 0.06129115
Cumulative Proportion 0.3546604 0.6106640 0.8008711 0.9166746 0.97796572
Comp.6 Comp.7
Standard deviation 0.29613912 0.257956501
Proportion of Variance 0.01252834 0.009505937
Cumulative Proportion 0.99049406 1.000000000Calculate dissimilarities for raw and standardized (by object and variables) and presence/absence
Calculate Euclidean, Chisq and Bray-Curtis measures
# raw data
clev1 <- clevenger[c(16:22)]
vegdist(clev1, method="euclidean") 1 2 3 4 5 6 7
2 807.60944
3 559.22536 285.70264
4 482.24994 1020.42148 869.77008
5 413.65203 878.52092 735.87975 148.14520
6 642.10513 268.67452 307.02443 769.36467 625.89057
7 898.41638 1375.51699 1267.34723 426.34610 536.74389 1109.76439
8 729.73899 1354.08345 1192.45293 337.41962 476.25938 1101.61064 282.27115
9 277.08843 670.85095 497.11870 383.99870 262.83074 449.65431 792.24365
10 333.25666 560.41324 380.28016 530.91148 412.79050 374.65050 934.17129
11 746.34241 82.61961 250.68706 948.85405 806.90954 200.19491 1304.56621
8 9 10
2
3
4
5
6
7
8
9 706.65197
10 852.67637 156.65248
11 1282.26986 598.65265 487.83706vegdist(clev1, method="chisq") 1 2 3 4 5 6 7
2 1.21475615
3 0.47318577 1.17635422
4 0.61384857 0.86798529 0.78818807
5 0.58227734 1.05366620 0.83599435 0.21674323
6 0.89187108 1.12775158 1.09118378 0.39319617 0.31798393
7 0.91245764 1.17435640 1.16260729 0.45273730 0.34119097 0.14487655
8 0.55123078 1.11106385 0.84408915 0.27447668 0.07803388 0.35930254 0.36290581
9 0.55537285 0.99969109 0.77917948 0.37637843 0.45788835 0.64890817 0.69433371
10 0.87912364 1.14433514 0.93156102 0.73629009 0.86127129 0.96885842 1.04763957
11 0.86727424 1.09145007 0.91740896 0.53301980 0.63188889 0.64730790 0.75690369
8 9 10
2
3
4
5
6
7
8
9 0.45353095
10 0.86766247 0.46840925
11 0.65817925 0.43340441 0.42203679vegdist(clev1, method="bray") 1 2 3 4 5 6 7
2 0.69439528
3 0.38194109 0.44763860
4 0.25704584 0.67640693 0.43615108
5 0.21661721 0.66729206 0.42784810 0.08739909
6 0.47601476 0.41250000 0.38265306 0.51121951 0.44586341
7 0.41342989 0.74446162 0.65817868 0.20714510 0.23317471 0.54363042
8 0.28296115 0.76427256 0.50156986 0.14377557 0.18815002 0.58129739 0.09644381
9 0.13946360 0.63251818 0.32980413 0.18204850 0.16149562 0.41574344 0.37885154
10 0.22957198 0.61303462 0.28393726 0.25449871 0.25545675 0.40179104 0.46803977
11 0.61985336 0.21301775 0.34410646 0.61578401 0.59212880 0.30296128 0.67607727
8 9 10
2
3
4
5
6
7
8
9 0.27308066
10 0.35214966 0.10301508
11 0.70222222 0.53614079 0.51386202# standardized by object totals
clev1$total <- rowSums(clev1)
clev2 <- clev1
clev2$bbeart <- clev2$bbear/clev2$total
clev2$gbeart <- clev2$gbear/clev2$total
clev2$cougart <- clev2$cougar/clev2$total
clev2$wolft <- clev2$wolf/clev2$total
clev2$deert <- clev2$deer/clev2$total
clev2$elkt <- clev2$elk/clev2$total
clev2$mooset <- clev2$moose/clev2$total
clev3 <- clev2[c(9:15)]
vegdist(clev3, method="euclidean") 1 2 3 4 5 6 7
2 0.29179246
3 0.12178440 0.35929117
4 0.29812496 0.13708413 0.39569783
5 0.32348508 0.18337700 0.42722446 0.04981133
6 0.49248814 0.29906244 0.59366647 0.20059515 0.16923915
7 0.51524001 0.32757067 0.61901148 0.22713767 0.19227850 0.03372006
8 0.30979659 0.18687072 0.41558090 0.05280824 0.01784560 0.18366346 0.20546479
9 0.19197240 0.14525707 0.28422021 0.13185757 0.16974574 0.32684568 0.35299841
10 0.19530845 0.19611057 0.24342704 0.23556628 0.27826144 0.42261729 0.45121044
11 0.30182196 0.13136818 0.38360445 0.09734980 0.13739012 0.24941037 0.27993555
8 9 10
2
3
4
5
6
7
8
9 0.16038740
10 0.27108275 0.11528079
11 0.13945202 0.11532916 0.17839581vegdist(clev3, method="chisq") 1 2 3 4 5 6 7
2 1.04480761
3 0.39517291 1.04001877
4 0.60305175 0.70863697 0.76350072
5 0.58507556 0.86215541 0.80156474 0.18679083
6 0.89540748 0.94138220 1.07368158 0.37617696 0.31635064
7 0.91715806 0.98417343 1.13041746 0.42787726 0.33966677 0.12188311
8 0.55354775 0.90889652 0.79773472 0.23202001 0.06650063 0.35481352 0.36496751
9 0.52716675 0.82688142 0.71506346 0.35025941 0.42619090 0.62838501 0.67298864
10 0.81600325 0.98389894 0.86660229 0.68588704 0.80144200 0.92462096 0.99390701
11 0.82056558 0.90283478 0.88548124 0.47730196 0.56841064 0.60119998 0.69403213
8 9 10
2
3
4
5
6
7
8
9 0.41801288
10 0.80255881 0.43516350
11 0.58842686 0.38209782 0.40990737vegdist(clev3, method="bray") 1 2 3 4 5 6 7
2 0.25709624
3 0.10060384 0.29656881
4 0.23423329 0.12596108 0.31352318
5 0.23217985 0.16959866 0.32749288 0.04492874
6 0.35922996 0.25989493 0.44730095 0.15808796 0.12885306
7 0.36883090 0.28756431 0.46943475 0.18575734 0.14288204 0.03010544
8 0.22357908 0.18035023 0.32222974 0.05568031 0.01543636 0.14067860 0.14750545
9 0.16090572 0.14081535 0.25691866 0.11925162 0.14405652 0.26523724 0.28352834
10 0.15889643 0.21261103 0.22390939 0.19886655 0.23515136 0.35036401 0.37803340
11 0.25791006 0.12143928 0.31950879 0.09331596 0.12297082 0.21065032 0.24075576
8 9 10
2
3
4
5
6
7
8
9 0.13632334
10 0.23082839 0.09820268
11 0.13047339 0.11047068 0.15319319# standardized by variable totals
clev1$bbeart <- clev1$bbear/sum(clev2$bbear)
clev1$gbeart <- clev1$gbear/sum(clev2$gbear)
clev1$cougart <- clev1$cougar/sum(clev2$cougar)
clev1$wolft <- clev1$wolf/sum(clev2$wolf)
clev1$deert <- clev1$deer/sum(clev2$deer)
clev1$elkt <- clev1$elk/sum(clev2$elk)
clev1$mooset <- clev1$moose/sum(clev2$moose)
clev4 <- clev1[c(9:15)]
vegdist(clev4, method="euclidean") 1 2 3 4 5 6 7 8
2 0.5874372
3 0.2103305 0.5699869
4 0.6494049 0.3377750 0.6340280
5 0.5206197 0.2263239 0.5320355 0.3777360
6 0.5500617 0.2574895 0.5446442 0.4374485 0.1231964
7 0.5491779 0.2835227 0.5730657 0.4282729 0.1122824 0.1365675
8 0.5150571 0.2989734 0.5558359 0.4101989 0.0900685 0.1809769 0.1176054
9 0.5817512 0.2750346 0.6030030 0.3719280 0.2669713 0.3108749 0.3100064 0.2815680
10 1.0062311 0.8620352 0.9929078 0.5898117 0.8607697 0.8737626 0.8868434 0.8705775
11 0.5581777 0.2646868 0.5392290 0.4210031 0.1632272 0.1080943 0.2029463 0.2183625
9 10
2
3
4
5
6
7
8
9
10 0.7613418
11 0.2497030 0.8212754vegdist(clev4, method="chisq") 1 2 3 4 5 6 7 8
2 2.1918656
3 0.6060261 2.1062286
4 1.7774812 1.2249504 1.7410878
5 1.6550584 1.4861911 1.7333325 1.0365979
6 1.9594176 1.9177901 1.8499636 1.3528694 0.8747182
7 2.2679736 2.2799015 2.4479938 1.8581292 1.2332715 1.3157014
8 1.7397774 1.9752053 1.9751799 1.4191550 0.5881129 1.1561780 1.0587556
9 1.8312148 1.3956048 1.9103379 1.1027533 1.1541210 1.4626882 1.8867908 1.3639082
10 2.0304452 1.9180098 2.0047472 0.8446477 1.6407999 1.7194328 2.2843090 1.8454187
11 2.0549749 1.9295504 1.8915082 1.4251156 1.4681142 1.1908051 2.2198998 1.7491817
9 10
2
3
4
5
6
7
8
9
10 1.2995151
11 1.0384181 1.3874031vegdist(clev4, method="bray") 1 2 3 4 5 6 7 8
2 0.7935939
3 0.1946466 0.7313399
4 0.7025767 0.4394077 0.6273209
5 0.5585673 0.5340500 0.6373176 0.5103013
6 0.7967183 0.6667775 0.8130980 0.7995520 0.5163939
7 0.7327162 0.7631796 0.8423259 0.6979275 0.3784111 0.6411470
8 0.5534611 0.7211135 0.6688852 0.5708830 0.2495338 0.6089989 0.2493285
9 0.6247022 0.4935050 0.7063851 0.4243562 0.4254924 0.7001292 0.6606665 0.4637913
10 0.7737168 0.7196663 0.7162999 0.3730783 0.7021629 0.8687325 0.8555458 0.7406492
11 0.8154067 0.5611676 0.7505849 0.6556519 0.5621685 0.4729512 0.7700434 0.7153495
9 10
2
3
4
5
6
7
8
9
10 0.4797086
11 0.5501897 0.7432297create pres/abs datafile
clev5 <- clevenger[c(16:22)]
clev5[clev5>0] <-1
clev5vegdist(clev5, method="jaccard") 1 2 3 4 5 6 7 8
2 0.1666667
3 0.0000000 0.1666667
4 0.2857143 0.1666667 0.2857143
5 0.1666667 0.0000000 0.1666667 0.1666667
6 0.3333333 0.2000000 0.3333333 0.3333333 0.2000000
7 0.3333333 0.2000000 0.3333333 0.3333333 0.2000000 0.4000000
8 0.1666667 0.0000000 0.1666667 0.1666667 0.0000000 0.2000000 0.2000000
9 0.1666667 0.0000000 0.1666667 0.1666667 0.0000000 0.2000000 0.2000000 0.0000000
10 0.2857143 0.1666667 0.2857143 0.0000000 0.1666667 0.3333333 0.3333333 0.1666667
11 0.3333333 0.2000000 0.3333333 0.3333333 0.2000000 0.0000000 0.4000000 0.2000000
9 10
2
3
4
5
6
7
8
9
10 0.1666667
11 0.2000000 0.3333333vegdist(clev5, method="bray") 1 2 3 4 5 6 7
2 0.09090909
3 0.00000000 0.09090909
4 0.16666667 0.09090909 0.16666667
5 0.09090909 0.00000000 0.09090909 0.09090909
6 0.20000000 0.11111111 0.20000000 0.20000000 0.11111111
7 0.20000000 0.11111111 0.20000000 0.20000000 0.11111111 0.25000000
8 0.09090909 0.00000000 0.09090909 0.09090909 0.00000000 0.11111111 0.11111111
9 0.09090909 0.00000000 0.09090909 0.09090909 0.00000000 0.11111111 0.11111111
10 0.16666667 0.09090909 0.16666667 0.00000000 0.09090909 0.20000000 0.20000000
11 0.20000000 0.11111111 0.20000000 0.20000000 0.11111111 0.00000000 0.25000000
8 9 10
2
3
4
5
6
7
8
9 0.00000000
10 0.09090909 0.09090909
11 0.11111111 0.11111111 0.20000000Check outliers with Mahalanobis distances - no particularly unusual objects
# Problem with code; needs to be fixed
#mahalanobis(clev1, colMeans(clev1), cov(clev1))Fig 14.3 - ordination with BC MDS
clev6 <- clevenger[c(1,16:22)]
clev6.stand <- decostand(clev6[,2:8], "total")
clev6.stand.bc <- vegdist(clev6.stand,"bray")
clev6.mds <- metaMDS(clev6.stand.bc,k=2)Run 0 stress 0.03076215
Run 1 stress 0.05949932
Run 2 stress 0.0541
Run 3 stress 0.05411455
Run 4 stress 0.3407335
Run 5 stress 0.03076228
... Procrustes: rmse 0.0001732853 max resid 0.0002983285
... Similar to previous best
Run 6 stress 0.03076217
... Procrustes: rmse 0.0001250452 max resid 0.0002150335
... Similar to previous best
Run 7 stress 0.03076224
... Procrustes: rmse 0.0002129152 max resid 0.0003668702
... Similar to previous best
Run 8 stress 0.05410005
Run 9 stress 0.03076216
... Procrustes: rmse 3.156241e-05 max resid 5.317318e-05
... Similar to previous best
Run 10 stress 0.03076228
... Procrustes: rmse 0.0001713665 max resid 0.0002951362
... Similar to previous best
Run 11 stress 0.05949931
Run 12 stress 0.03076222
... Procrustes: rmse 0.0001142774 max resid 0.0001958333
... Similar to previous best
Run 13 stress 0.05409998
Run 14 stress 0.03076215
... Procrustes: rmse 2.215246e-05 max resid 3.596323e-05
... Similar to previous best
Run 15 stress 0.05410003
Run 16 stress 0.03076223
... Procrustes: rmse 0.0001307716 max resid 0.000224776
... Similar to previous best
Run 17 stress 0.03076222
... Procrustes: rmse 0.0001136536 max resid 0.0001950247
... Similar to previous best
Run 18 stress 0.03076229
... Procrustes: rmse 0.0001758204 max resid 0.0003030112
... Similar to previous best
Run 19 stress 0.05410003
Run 20 stress 0.05410004
*** Best solution repeated 10 timesplot(clev6.mds$points,type="n", )
text(clev6.mds$points,lab=clev6$underpass, col="black")
mdsplot <- as.data.frame(scores(clev6.mds))
mdsplot$underpass<-clev6$underpass
f14_3<-ggplot(data=mdsplot,aes(x=NMDS1, y=NMDS2, label=underpass))+
geom_text()+
theme_qk()+
theme(axis.text = element_blank())
f14_3
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