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Introduction to Multivariate Analysis and Multivariate Distances. Hal Whitehead BIOL4062/5062. Data matrices Problems with data matrices missing values outliers Matrices used in multivariate analysis Multivariate distances Association matrices. The Data Matrix. Variables:. Units:. - PowerPoint PPT Presentation
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Introduction to Multivariate Analysis and
Multivariate Distances
Hal WhiteheadBIOL40625062
bull Data matricesbull Problems with data matrices
ndash missing valuesndash outliers
bull Matrices used in multivariate analysisbull Multivariate distancesbull Association matrices
The Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Data Matrix
Subject Units VariablesAnimal behaviour Animals Scores on different measures
Communityecology
Plots Species counts
Palaentology Specimens Measurements on bones
Marine ecology Stations Temperature salinityspecies counts etc
Visualize Data Matrix asPoints in multidimensional space
RodentiaPrimatesMarsupialiaLagomorphaInsectivoraEdentataChiropteraCarnivora
ORDER
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
bull Data matricesbull Problems with data matrices
ndash missing valuesndash outliers
bull Matrices used in multivariate analysisbull Multivariate distancesbull Association matrices
The Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Data Matrix
Subject Units VariablesAnimal behaviour Animals Scores on different measures
Communityecology
Plots Species counts
Palaentology Specimens Measurements on bones
Marine ecology Stations Temperature salinityspecies counts etc
Visualize Data Matrix asPoints in multidimensional space
RodentiaPrimatesMarsupialiaLagomorphaInsectivoraEdentataChiropteraCarnivora
ORDER
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
The Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Data Matrix
Subject Units VariablesAnimal behaviour Animals Scores on different measures
Communityecology
Plots Species counts
Palaentology Specimens Measurements on bones
Marine ecology Stations Temperature salinityspecies counts etc
Visualize Data Matrix asPoints in multidimensional space
RodentiaPrimatesMarsupialiaLagomorphaInsectivoraEdentataChiropteraCarnivora
ORDER
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
The Data Matrix
Subject Units VariablesAnimal behaviour Animals Scores on different measures
Communityecology
Plots Species counts
Palaentology Specimens Measurements on bones
Marine ecology Stations Temperature salinityspecies counts etc
Visualize Data Matrix asPoints in multidimensional space
RodentiaPrimatesMarsupialiaLagomorphaInsectivoraEdentataChiropteraCarnivora
ORDER
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Visualize Data Matrix asPoints in multidimensional space
RodentiaPrimatesMarsupialiaLagomorphaInsectivoraEdentataChiropteraCarnivora
ORDER
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Problems with Data Matrix
bull Missing valuesbull Outliersbull Units not independentbull Many zerosbull Not multivariate normal
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Missing DataOften present in ecological or other biological data
bull delete columns of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrix
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missing
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Missing DataOften present in ecological or other biological data
bull delete columns of data matrixbull delete rows of data matrixbull just delete pairs of elements where one is missingbull interpolate
Date Year Mon Area Clan Shitr M24 M12 M3 Clan Area$23-Feb-1985 1985 2 1 1 179 Reg Galapagos24-Feb-1985 1985 2 1 1 010 554 547 143 Reg Galapagos25-Feb-1985 1985 2 1 1 268 183 102 Reg Galapagos07-Mar-1985 1985 3 1 1 014 386 261 109 Reg Galapagos08-Mar-1985 1985 3 1 1 553 318 84 Reg Galapagos
012
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Outliersbull Statistical packages often indicate ldquooutliersrdquo
WARNING Case 86 has large leverage (Leverage = 0252)
bull If plausiblyndash the result of biological or other processes outside the scope of
the model being usedndash or the results of measurement or coding errorndash they may be discarded
bull Otherwise they should be retainedndash (perhaps use a different model)
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Problems with Data Matrixbull Missing valuesbull Outliersbull Units not independent
ndash Not a problem unless doing testsbull Many zeros
ndash Special methods (eg correspondence analysis)bull Not multivariate normal
ndash Transform if possible
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Uses of Multivariate Analysisbull Large data sets
ndash simplifyndash summarizendash find patterns
bull Analyze groupings of units
bull Find groupings of unitsbull Examine relationships
between variables
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Some Matrices Used inMultivariate Analysis
bull Data matrix rectangularndash units i=1hellipnndash variables j k
bull Covariance matrix between variables symmetric (squaretriangular)ndash cjk= Σ (xij-xj) (xik-xk) (n-1) [xk = mean(xik)]
bull Correlation matrix between variables symmetric (squaretriangular)ndash rjk=cjk(Sj Sk) [Sk = SD(xik)]
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Data MatrixLMASS LFAT LCNS LMUSCLELHEART LBONE
895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Covariance Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Correlation Matrix
LMASS LFAT LCNS LMUSCLELHEART LBONE895 702 465 838 332 645872 660 440 819 324 631914 633 445 857 439 678521 113 190 465 063 309694 419 289 637 203 439871 692 407 798 359 625240 -151 -105 171 -190 031370 133 -004 289 -076 150415 183 019 337 -030 209198 -139 -099 135 -230 -037
hellip hellip hellip hellip hellip hellip
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 538LFAT 649 823LCNS 415 493 333LMUSCLE 544 651 421 551LHEART 467 563 363 472 412LBONE 525 631 405 531 457 516
LMASS LFAT LCNS LMUSCLELHEART LBONELMASS 1LFAT 097 1LCNS 098 094 1LMUSCLE 1 097 098 1LHEART 099 097 098 099 1LBONE 1 097 098 1 099 1
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Multivariate distancesbetween units or groups of units
1 Euclidean distance
d x xij ik jk
p
=
2
1
321
SPECIES
p variables
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Multivariate distancesbetween units or groups of units
2 Penrose distance
p variablesSk
2 variance of xik
321
SPECIES
P x x p Sij ik j kk
p
=
2 ( )2
1
Corrects fordifferent unitsdifferent ranges
of units ofvariables
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Multivariate distancesbetween units or groups of units
3 Mahalanobis distance
p variablesvrs elements of inverse of covariance matrix
D x x v x xij ir jr rss
p
r
p
is js2
11 =
321
SPECIESCorrects forcorrelations
between variables
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
3 species of iris 4 measurementsbull Euclidean distances
A 0B 32 0 C 48 16 0
A B C
bull Penrose distancesA 0B 28 0 C 39 15 0
A B C
bull Mahalanobis distancesA 0B 899 0 C 1794 172 0
A B C
CBA
SPECIES
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
The Standard Data Matrix
A B C D E F G hellip1234567
hellip
Variables
Units
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
The Association Matrix
A B C D E F G hellipABCDEFGhellip
Units
Units
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Association matricesbull Social structure
ndash association between individualsbull Community ecology
ndash similarity between species sitesndash dissimilarities between species sites
bull Genetic distancesbull Correlation matricesbull Covariance matricesbull Distance matrices
ndash Euclidean Penrose Mahalanobis
SimilarityDissimilarity
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Association matricesDissimilaritySimilarity
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Mahalanobis distances between iris species
A 0B 899 0 C 1794 172 0
A B C
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
Association matricesSymmetricAsymmetric
Genetic relatedness among bottlenose dolphins (Krutzen et
al 2003)
Grooming ratesof capuchinmonkeys(Perry 1996)
GRI -024VAX 002 008KRI 002 -004 -019
MYR -027 044 -003 -011WOW 022 011 032 -010 010HOB -004 011 -017 -013 -008 -012WBE 015 007 -008 008 -008 023 013HOR -008 021 -014 -023 018 012 011 026AJA -024 023 -004 -016 -001 -016 007 025 032PIK -011 035 -007 004 002 -005 009 060 021 027ANV -005 -023 -039 -039 -021 -013 -041 011 011 002 -006VEE 014 002 015 -011 -008 000 -009 -005 006 001 -017 -017
LAT GRI VAX KRI MYR WOW HOB WBE HOR AJA PIK ANV
Recipient
Actor A S N D W T
A - 58 35 21 23 004
S 416 - 286 181 90 74
N 103 255 - 96 99 43
D 233 93 105 - 134 69
W 212 152 146 251 - 104
T 25 29 37 36 53 -
