Multivariate Description. What Technique? Response variable(s)... Predictors(s) No Predictors(s) Yes... is one distribution summary regression models...

Multivariate Description

What Technique?

Response variable(s) ...

Predictors(s)

No

Predictors(s)

Yes

... is one • distribution summary • regression models

... are many • indirect gradient analysis

(PCA, CA, DCA, MDS)

• cluster analysis

• direct gradient analysis

• constrained cluster analysis

• discriminant analysis (CVA)

a) Rotate the Variable Space

Raw Data

65 70 75 80 85

81

01

21

41

61

82

0

Height

Dia

me

ter

Linear Regression

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Dia

me

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Two Regressions

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Dia

me

ter

Principal Components

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Dia

me

ter

Gulls Variables

Weight

400 420 440 105 115 125 135

700

900

1100

400

420

440

Wing

Bill

1618

2022

700 800 900 1100

105

115

125

135

16 17 18 19 20 21 22

H.and.B

Scree Plot

Comp.1 Comp.2 Comp.3 Comp.4

gulls.pca2V

ari

an

ces

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Output

> gulls.pca2$loadings

Loadings:

Comp.1 Comp.2 Comp.3 Comp.4Weight -0.505 -0.343 0.285 0.739Wing -0.490 0.852 -0.143 0.116Bill -0.500 -0.381 -0.742 -0.232H.and.B -0.505 -0.107 0.589 -0.622

> summary(gulls.pca2)

Importance of components:

Comp.1 Comp.2 Comp.3 Standard deviation 1.8133342 0.52544623 0.47501980 Proportion of Variance 0.8243224 0.06921464 0.05656722 Cumulative Proportion 0.8243224 0.89353703 0.95010425

Bi-Plot

-0.15 -0.10 -0.05 0.00 0.05 0.10

-0.1

5-0

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50

.00

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.10

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mp

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-20 -10 0 10

-20

-10

01

0

Weight

Wing

Bill

H.and.B

Male or Female?

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

PC1

PC

2

Linear Discriminant

> gulls.lda <- lda(Sex ~ Wing + Weight + H.and.B + Bill, gulls)

lda(Sex ~ Wing + Weight + H.and.B + Bill, data = gulls)

Prior probabilities of groups:

0 1 0.5801105 0.4198895

Group means:

Wing Weight H.and.B Bill0 410.0381 871.7619 115.1143 17.625241 430.6118 1054.3092 125.9474 19.50789

Coefficients of linear discriminants:

LD1Wing 0.045512619Weight 0.001887236H.and.B 0.138127194Bill 0.444847743

Discriminating

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

group 0

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

group 1

Relationship between PCA and LDA

-4 -2 0 2 4

-4-2

02

4

gulls.pca2$scores[, 1]

gu

lls.p

red

$x

CVA

CVA

b) Use Distance or Dissimilarity (Multi-Dimensional Scaling)

A Distance Matrix

Uses of Distances

Distance/Dissimilarity can be used to:-

• Explore dimensionality in data(using PCO)

• As a basis for clustering/classification

UK Wet Deposition Network

-400 -200 0 200 400

-40

0-2

00

02

00

40

0

Dim1

Dim

2

Goonhilly

Lough Navar

Achanarras

Flatford Mill

Strathvaich Dam

Yarner WoodBarcombe Mills

Stoke Ferry

Hillsborough Forest

Tycanol Wood

Allt a MharcaidhGlen Dye

Driby

Woburn

Balquhidder 2

Compton

High Muffles

Bottesford

Whiteadder

Pumlumon

Loch Dee Redesdale

Wardlow Hay Cop

Cow Green ReservoirBannisdale

Fitting Environmental Variables

-400 -200 0 200 400

-40

0-2

00

02

00

40

0

Dim1

Dim

2

Goonhilly

Lough Navar

Achanarras

Flatford Mill

Strathvaich Dam

Yarner WoodBarcombe Mills

Stoke Ferry

Hillsborough Forest

Tycanol Wood

Allt a MharcaidhGlen Dye

Driby

Woburn

Balquhidder 2

Compton

High Muffles

Bottesford

Whiteadder

Pumlumon

Loch Dee Redesdale

Wardlow Hay Cop

Cow Green ReservoirBannisdale

A Map based on Measured Variables

-3 -2 -1 0 1 2

-3-2

-10

1

CA1

CA

2

AchanarrasGoonhilly

Stoke Ferry

Flatford Mill

Woburn

Barcombe Mills

Pumlumon

Driby

Balquhidder 2

Lough Navar

Eskdalemuir

Compton

Tycanol Wood

Preston Montford

Allt a Mharcaidh

High Muffles

Whiteadder

Glen Dye

Fitting Environmental Variables

-3 -2 -1 0 1 2

-3-2

-10

1

CA1

CA

2

Goonhilly

Yarner Wood

Barcombe Mills

ComptonFlatford Mill

Woburn

Tycanol Wood

Llyn Brianne

Pumlumon

Stoke Ferry

Preston Montford

Bottesford

Wardlow Hay CopDriby

High MufflesBannisdale

Hillsborough Forest

Lough Navar

Cow Green ReservoirLoch Dee

Redesdale Eskdalemuir

Whiteadder

Balquhidder 2Glen Dye

Allt a Mharcaidh

Strathvaich Dam

Achanarras

c) Summarise by Weighted Averages

Species and Sites as

Weighted Averages of each other

SITES

1 1 1111 1 2111 SPP. 23466185750198304927 Bel per 3.2....2..2..22..... Jun buf .3..........4..…42.. Jun art ...3..4..3..4..4.... Air pra ........2........3.. Ele pal ...8..4..5.....44... Rum ace ....6..5....2..…23.. Vic lat ..........12.1...... Bra rut ..246.22.4242624.342 Ran fla .2.2..2..2.....42... Hyp rad ........2..2.....5.. Leo aut 522.3.33223525222623 Pot pal .........2......2... Poa pra 424.34421.44435.…4.. Cal cus ...3...........34... Tri pra ....5..2........…2.. Tri rep 521.5.22.163322.6232 Ant odo ....3..44.4......4.2 Sal rep .............3.5.3.. Ach mil 3...21.22.4.....…2.. Poa tri 79524246..4.5.6…45.. Ely rep 4.4..4.4....6.4..... Sag pro .25...2....22....34. Pla lan ....5..52.33.3..…5.. Agr sto .587..4..4..3.454.4. Lol per 5.5.6742..67226.…6.. Alo gen 2524..5.....3.7...8. Bro hor 4.3....2..4.....…2..

Species and Sites as Weighted Averages of each other

-2 -1 0 1 2

-10

12

3

CA1

CA

2

Belper

Empnig

Junbuf

Junart

Airpra

Elepal

Rumace

ViclatBrarut Ranfla

Cirarv

Hyprad

LeoautPotpal

Poapra

Calcus

TripraTrirep

Antodo

Salrep

Achmil

Poatri

ChealbElyrep

Sagpro

Plalan

AgrstoLolper

Alogen

Brohor

213

4

166

1

85

17

15

10

11

9

18

3

20

14

19

12

7

Reciprocal Averaging - unimodal

Site A B C D E F Species

Prunus serotina 6 3 4 6 5 1 Tilia americana 2 0 7 0 6 6 Acer saccharum 0 0 8 0 4 9 Quercus velutina 0 8 0 8 0 0 Juglans nigra 3 2 3 0 6 0


Site A B C D E F Species ScoreSpecies Iteration 1

Prunus serotina 6 3 4 6 5 1 1.00 Tilia americana 2 0 7 0 6 6 0.63 Acer saccharum 0 0 8 0 4 9 0.63 Quercus velutina 0 8 0 8 0 0 0.18 Juglans nigra 3 2 3 0 6 0 0.00

Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site Score


Site A B C D E F Species ScoreSpecies Iteration 1 2

Prunus serotina 6 3 4 6 5 1 1.00 0.68 Tilia americana 2 0 7 0 6 6 0.63 0.84 Acer saccharum 0 0 8 0 4 9 0.63 0.87 Quercus velutina 0 8 0 8 0 0 0.18 0.30 Juglans nigra 3 2 3 0 6 0 0.00 0.67

Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score


Site A B C D E F Species ScoreSpecies Iteration 1 2 3

Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66

Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00


Site A B C D E F Species ScoreSpecies Iteration 1 2 3 9

Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 0.48Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 0.85Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 0.91Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 0.00Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66 0.65

Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00 9 0.59 0.01 0.87 0.00 0.78 1.00

Reordered Sites and Species

Site A C E B D F Species Species Score

Quercus velutina 8 8 0 0 0 0 0.004Prunus serotina 6 3 6 5 4 1 0.477Juglans nigra 0 2 3 6 3 0 0.647Tilia americana 0 0 2 6 7 6 0.845Acer saccharum 0 0 0 4 8 9 0.909

Site Score 0.000 0.008 0.589 0.778 0.872 1.000

Managing Dimensionality (but not acronyms)

PCA, CA, RDA, CCA, MDS, NMDS, DCA, DCCA, pRDA, pCCA

Type of Data Matrix

species

site

s

attributes

spec

ies

attributes

site

s

attributes

indi

vidu

als

desertmacrophinverts

uses

watervarrain

gulls

Ordination Techniques

Linear methods Weighted averaging

unconstrained Principal Components

Analysis (PCA)

Correspondence Analysis

(CA)

constrained Redundancy Analysis (RDA)

Canonical

Correspondence Analysis (CCA)

Models of Species Response

There are (at least) two models:-

• Linear - species increase or decrease along the environmental gradient

• Unimodal - species rise to a peak somewhere along the environmental gradient and then fall again

A Theoretical Model

Environmental Gradient

Abundance

80706050403020100

100

80

60

40

20

0

Linear

-0.4 +0.4

+0.0

+7.0

Unimodal

-2.5 +3.5

+0.0

+250.0

Alpha and Beta Diversity

alpha diversity is the diversity of a community (either measured in terms of a diversity index or species richness)

beta diversity (also known as ‘species turnover’ or ‘differentiation diversity’) is the rate of change in species composition from one community to another along gradients; gamma diversity is the diversity of a region or a landscape.

A Short Coenocline

-0.5 +0.7

+0.0

+8.0

Ach m il

Agr s to

Air pra

Jun a rt

Pot pa l

Ra n fla

A Long Coenocline

Inferring Gradients from Species (or Attribute) Data

Indirect Gradient Analysis

• Environmental gradients are inferred from species data alone

• Three methods:– Principal Component Analysis - linear model– Correspondence Analysis - unimodal model– Detrended CA - modified unimodal model

Terschelling Dune Data

PCA gradient - site plot

PCA 1

PCA

2

2.01.51.00.50.0-0.5-1.0-1.5

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

Managmentbiodynamichobbynaturestandard

PCA Plot for Dune Species Data

PCA gradient - site/species biplot

Axis 1

Axi

s 2

210-1-2

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

Ach mil

Agr sto

Alo gen

Ant odo

Bel perBro hor

Ele pal

Ely rep

Jun art

J un buf

Leo aut

Lol per

Pla lan

Poa pra

Poa tri

Ran flaRum ace

Sag pro

Tri rep

Bra rut

Biplot for Dune Species Data

standard

nature

biodynamic& hobby

Arches - Artifact or Feature?

The Arch Effect

• What is it?

• Why does it happen?

• What should we do about it?

From Alexandria to Suez

CA - with arch effect (sites)

-3.0 +4.5

-3.5

+4.5

ALEX 07

ALEX 05

ALEX 06ALEX 08

CSRA 23

CSRA 16CSRA 17

CSRA 22CSRA 20CSRA 21

CSRA 12

ALEX 03

ALEX 02

CSRA 13

CSRA 15

ALEX 04

CSRA 11

CSRA 14

CSRA 18CSRA 19

ALEX 01

CSRA 30CSRA 32

CSRA 24CSRA 31

CSRA 25ALEX 10ALEX 09

CSRA 33

CSRA 35

CSRA 26

CSRA 29CSRA 34

CSRA 27CSRA 28

CA - with arch effect (species)

-3.0 +4.5

-3.5

+4.5

HAL SAL

ECH SERASP MIC

THY HIR

HAM ELEACH SANLAU SPIZYG DEC

CRO AEG

ART JUD

VER OFF

LAS HIR

LYC SHA

OCH BAC

PUL UND

IPH MUC

ZYG COC

LAU NUD

PAN TUR

KIC AEG

LYG RAE

AST GRA

ART MON

FAR AEG

ECH SPI

SAL LAN

ATR CAR

MOL CIL

EUP RET

CON LAN

ANA ART

PIT TOR

FAG ARA

SAL AEGZIL SPI

PER TOM

CAL COM

STI LAN

GYP CAP

AST SPI

HYO MUT

CLE DRO

RUT TUB

HEL ARA

SPH AUC

Long Gradients

A B C D

Gradient End Compression

CA - with arch effect (species)

-3.0 +4.5

-3.5

+4.5

HAL SAL

ECH SERASP MIC

THY HIR


CRO AEG

ART JUD

VER OFF

LAS HIR

LYC SHA

OCH BAC

PUL UND

IPH MUC

ZYG COC

LAU NUD

PAN TUR

KIC AEG

LYG RAE

AST GRA

ART MON

FAR AEG

ECH SPI

SAL LAN

ATR CAR

MOL CIL

EUP RET

CON LAN

ANA ART

PIT TOR

FAG ARA

SAL AEGZIL SPI

PER TOM

CAL COM

STI LAN

GYP CAP

AST SPI

HYO MUT

CLE DRO

RUT TUB

HEL ARA

SPH AUC

CA - with arch effect (sites)

-3.0 +4.5

-3.5

+4.5

ALEX 07

ALEX 05

ALEX 06ALEX 08

CSRA 23

CSRA 16CSRA 17

CSRA 22CSRA 20CSRA 21

CSRA 12

ALEX 03

ALEX 02

CSRA 13

CSRA 15

ALEX 04

CSRA 11

CSRA 14

CSRA 18CSRA 19

ALEX 01

CSRA 30CSRA 32

CSRA 24CSRA 31

CSRA 25ALEX 10ALEX 09

CSRA 33

CSRA 35

CSRA 26

CSRA 29CSRA 34

CSRA 27CSRA 28

Detrending by Segments

-3.0 +4.5

-3.5

+4.5

HAL SAL

ECH SERASP MIC

THY HIR


CRO AEG

ART JUD

VER OFF

LAS HIR

LYC SHA

OCH BAC

PUL UND

IPH MUC

ZYG COC

LAU NUD

PAN TUR

KIC AEG

LYG RAE

AST GRA

ART MON

FAR AEG

ECH SPI

SAL LAN

ATR CAR

MOL CIL

EUP RET

CON LAN

ANA ART

PIT TOR

FAG ARA

SAL AEGZIL SPI

PER TOM

CAL COM

STI LAN

GYP CAP

AST SPI

HYO MUT

CLE DRO

RUT TUB

HEL ARA

SPH AUC

DCA - modified unimodal

-1.0 +5.5

-1.5

+4.5

CSRA 23

CSRA 16

CSRA 17

CSRA 22

CSRA 12

CSRA 21CSRA 20

CSRA 13

CSRA 15

CSRA 11CSRA 14

CSRA 18CSRA 19

CSRA 29

CSRA 26

CSRA 34

CSRA 27CSRA 28

CSRA 35CSRA 33

ALEX 09

CSRA 25CSRA 31

ALEX 10

CSRA 24

CSRA 32CSRA 30

ALEX 01

ALEX 04ALEX 02ALEX 03

ALEX 08ALEX 05ALEX 06ALEX 07

HAM ELEACH SANLAU SPIZYG DECCRO AEG.ART JUD

VER OFF.

LAS HIR

LYC SHA

OCH BAC

PUL UND

IPH MUC.

ZYG COC

LAU NUD

PAN TUR

KIC AEG

LYG RAE

AST GRA

FAR AEG

ECH SPI

ATR CAREUP RETPIT TOR

FAG ARA

SAL AEG

ZIL SPI

CAL COM

STI LAN

AST SPI

HYO MUTCLE DRO

RUT TUB

HEL ARA

GYP CAPPER TOM

ANA ART

CON LAN

MOL CIL

SAL LAN.

ART MON

HAL SALECH SER.

THY HIR

ASP MIC

SPH AUC

Making Effective Use of Environmental Variables

Approaches

• Use single responses in linear models of environmental variables

• Use axes of a multivariate dimension reduction technique as responses in linear models of environmental variables

• Constrain the multivariate dimension reduction into the factor space defined by the environmental variables

Unconstrained/Constrained

• Unconstrained ordination axes correspond to the directions of the greatest variability within the data set.

• Constrained ordination axes correspond to the directions of the greatest variability of the data set that can be explained by the environmental variables.

Direct Gradient Analysis

• Environmental gradients are constructed from the relationship between species environmental variables

• Three methods:– Redundancy Analysis - linear model– Canonical (or Constrained) Correspondence

Analysis - unimodal model– Detrended CCA - modified unimodal model

Direct Gradient Analysis• Basic PCA

yik = b0k + b1kxi + eik

– xi - the sample scores on the ordination axis

– b1k - the regression coefficients for each species (the species scores on the ordination axis)

• In RDA there is a further constraint on xi

xi = c1zi1 + c2zi2

• Makingyik = b0k + b1kc1zi1 + b1kc2zi2 + eik

Direct Gradient Analysis

cca(species_data ~ e1 + e2 + ... + en + Condition(e5), data=environmental_data)

cca(varespec ~ Al + P*(K + Baresoil) + Condition(pH), data=varechem)

Lake Nasser - Egypt

CCA - site/species joint plot

-1.0 +1.0

-1.0

+1.0

MaEs10

AlEd6AlEd5

ZaEs20

KrEs14

KrWs12

MrWd3MrWd2

TuW23TuW24TuW22

IbEd16

MrWs4

MaEd9

IbEs15

MaW11

AlEs7AlEs8

IbWd18

MrEd1

Annelida

Protozoa

Turbellaria

Tardigrada

Nematoda

Cladocera

Insecta

Ostracoda

Copepoda

Rotifera

CCA - species/environment biplot

-1.0 +1.0

-1.0

+1.0

TDSMg

lgNO2

EC

DO

NO3

Ca

pH

WD

PO4TH

lgTSS

Annelida

Protozoa

Turbellaria

Tardigrada

Nematoda

Cladocera

Insecta

Ostracoda

Copepoda

Rotifera

Removing the Effect of Nuisance Variables

Partial Analyses

• Remove the effect of covariates – variables that we can measure but which are of

no interest– e.g. block effects, start values, etc.

• Carry out the gradient analysis on what is left of the variation after removing the effect of the covariates.

Cluster Analysis

Different types of data

example

Continuous data : height

Categorical data

ordered (nominal) : growth rate very slow, slow, medium, fast, very

fast

not ordered : fruit colour yellow, green, purple, red, orange

Binary data : fruit / no fruit

Similarity matrixWe define a similarity between units – like the correlation between continuous variables.

(also can be a dissimilarity or distance matrix)

A similarity can be constructed as an average of the similarities between the units on each variable.

(can use weighted average)

This provides a way of combining different types of variables.

relevant for continuous variables:

Euclidean

city block or Manhattan

Distance metrics

A

B

A

B

(also many other variations)

Similarity coefficients for binary data

simple matching

count if both units 0 or both units 1

Jaccard

count only if both units 1

(also many other variants)

simple matching can be extended to categorical data

0,1 1,1

0,0 1,0

0,1 1,1

0,0 1,0

hierarchical

divisive

put everything together and split

monothetic / polythetic

agglomerative

keep everything separate and join the most similar points (classical cluster analysis)

non-hierarchical

k-means clustering

Clustering methods

Agglomerative hierarchical

Single linkage or nearest neighbour

finds the minimum spanning tree: shortest tree that connects all points

chaining can be a problem


Complete linkage or furthest neighbour

compact clusters of approximately equal size.(makes compact groups even when none exist)


Average linkage methods

between single and complete linkage

Testing Significance in Ordination

Randomisation Tests

Lake Species Richness Area Fertilised

1 32 2.0 yes

2 29 0.9 yes

3 35 3.1 yes

4 36 3.0 yes

5 41 1.0 no

6 62 2.0 no

7 88 4.0 no

8 77 3.5 no

Randomisation Tests

0.5950 0.0894 0.0259 0.0047 0.2879 0.1839 0.0493 0.0166 0.1810 0.0001 0.0028 0.0838 0.0016 0.4809 0.0072 0.0094 0.0084 0.0315 0.0807 0.1322 0.1649 0.0068 0.4786 0.0842 0.0066 0.3674 0.1496 0.0501 0.0434 0.0544 0.0643 0.0107 0.0101 0.3152 0.0015 0.3450 0.0004 0.1151 0.0125 0.0635

Randomisation Example

Model: cca(formula = dune ~ Moisture + A1 + Management, data = dune.env)

Df Chisq F N.Perm Pr(>F)

Model 7 1.1392 2.0007 200 < 0.005 ***

Residual 12 0.9761

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05

Multivariate Description. What Technique? Response variable(s)... Predictors(s) No Predictors(s) Yes... is one distribution summary regression models...

Documents

linear regression slide

raw data slide

scree plot slide

biplot slide

variable space slide

indirect gradient analysis

linear discriminant

gulls variables