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Ordination Methods
Correspondence Analysis (CA)
Detrended Correspondence Analysis (DCA)
Factor Analysis (FA)Principles of Canonical Analysis
Redundancy Analysis (RA)
Canonical Correspondence Analysis (CCA)
Canonical Correlation Analysis (CCorA)
Discriminant Analysis (DA)
1. Direct Gradient Analysis................................2
2. Few species...........................................4
4. Monotonic responses to gradients (low beta).......Linear regression
4. Nonmonotonic responses to gradients.(high beta)......Generalized linear models
2. Many species..........................................5
5. Monotonic responses .......... ........... .......RDA
5. Nonmonotonic responses.............................6
6. concerned about arch effect..................DCCA
6. not concerned about arch effect...............CCA
1. Indirect Gradient Analysis..............................3
3. Only distance values are available....................7
7. Monotonic responses ............................PCoA
7. Nonmonotonic responses..........................NMDS
3. Raw data available....................................8
8. Monotonic responses ...............................9
9. Variables noncommensurate......PCA - corr. matrix
9. Variables commensurate..........PCA - cov. matrix
8. Nonmonotonic responses............................10
10. Feel OK about prespecifying number of dimensions,
not worried about local optima, not interested in
species scores..............NMDS
10. Not as above, but willing to accept either ar ch
effect or detrending/rescaling................11
11. Don't like arch, detrending OK ..........DCA
11. Arch OK, or only interested in axis 1.....CA
Dichotomous Key forOrdination Methods
Not 100% accurate, but agood place to start.
(see Palmer 1998)
Correspondence Analysis (CA)
Correspondence was developed independently by severalauthors over a period of ca. 30 years and given many differentnames in the literature:
Contingency table analysis
RQ-techniqueReciprocal averagingCorrespondence analysisReciprocal ordering
Dual scalingHomogeneity analysis
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Correspondence Analysis (CA)
Correspondence analysis was first proposed for analyzingtwo-
way contingency tables. In such tables, the states of the firstdescriptor (rows) are compared to the states of the seconddescriptor (columns). Data in each cell of the table are
frequencies. These frequencies are positive integers orzeroes.
In EEB, the most common application of CA is for the analysisof species data (0/1, or abundance) at different sampling sites.A species-site table essentially contains frequencies.
Correspondence Analysis (CA)
In general, CA may be applied to any data table that isdimensionally homogeneous (i.e., the physical dimensions ofall variables are the same) and only contains positive
integers or zeroes.
The !2distance (D16), which is a coefficient that excludes
double-zeroes, is used to quantify the relationship amongrows and columns. NB: Some authors have questioned the
efficacy of !2distance for certain types of data.
Correspondence Analysis (CA)
CA is primarily a method of ordination. As such, it is similar to PCA; it
preserves in the space of the principal axes (i.e., after rotation), theEuclidean distance between profiles of weighted conditional probabilities.
In other words, CA preserves the !2distance between the rows and the
columns of the contingency table.
Correspondence analysis proceeds along three steps:
(1) the contingency table is transformed into a table of contributions to the
Pearson chi-square statistic after fitting a null model to the table.
(2) Singular value decomposition (SVD) to that table and the eigenvalues
and eigenvectors are computed (as in PCA).
(3) Further matrix manipulations lead to the tables required for plotting in
ordination space.
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Let's use an example other than the stand "species situation wehave been looking at (although we could do this here too) and
consider the relative abundance (0, +, ++) of a particular speciesobserved at 100 sites. The temperature at each site wasrecorded and coded (1,2,3):
Temp.
(Descr.-1)
(Descr.-2)
Sp. is:Rare
(0)
Abund.
(+)
Very Abund.
(++)
Row
Sums
Cold (1) 10 10 20 40
Med. (2) 10 15 10 35
Warm (3) 15 5 5 25
Col. Sums 35 30 35 100
CA- Example -
MatrixQcontains the proportions pijand the marginal totals pi+
and p+jof the rows and columns, respectively. Identifiers of therows and columns are given outside the matrix brackets inparentheses:
CA- Example -
The eigenvalues of Q'Qare: #1= 0.09613 (70.1%) and#2 = 0.04094 (29.9%) and
#3= 0 (because of centering)
CA- Example -
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The normalized eigenvectors of Q'Q are then:
And the normalized eigenvectors of QQ' are then:
CA- Example -
In Scaling Type-1, Fand Vare determined to produce a CA joint plot:
Now, to put the rows (matrix F) at the centroids of the columns, the
position of each row along an ordination axis is computed as the mean ofthe column positions, weighted by the relative frequencies of the
observations in the various columns of that row...
CA- Example -
Consider the first row of the original data. The relative frequenciesof that row are 0.25, 0.25, 0.50. Multiplying matrix Vby that vector
provides the coordinates of the first row of the ordination diagram:
continuing:
CA- Example -
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CA- Example -
Now, using Fand V,we can construct the
ordination plot:
CA using R
CA.csv
CA using R
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CA using R
CA using R
Data Tables
Correspondence analysis has been applied to many types ofdata tables other than contingency tables.
However, as a caveat, recognize again that in order for CA towork correctly, the data table must be dimensionally
homogeneous(i.e., in the same physical units) and non-negative($0).
If the data do not meet these assumptions, they may be
transformed or recoded. This is a critical step in CA.
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Arch Effect
Let's return to notion of coenocline distortion that we first
considered in PCA. Recall that most of the these proceduresrequire linear(or at least monotonic) responses. Speciesdata, in particular, is usually unimodallydistributed across a
gradient.
Recall that this problem usually manifests itself in the form of
an archor horseshoein the data projection.
Some ecologists are willing to tolerate this distortion while
others feel that an attempt should be made to recover theoriginal gradient via detrending.
Arch Effect
The most extreme form of the arch effect usually occurs
while attempting to apply a Euclidean distance measure tospecies abundance data. A horseshoeis formed because the ends actually
contract and fold inwards at the endsof Axis-1 and bend along Axis-3. Thisis because ED considers the extreme sites
to be very near each other.
In most instances, CA does not exhibit such a dramatic
folding towards the terminal portions, but rather just bendsalong Axis-1 to form an arch.
Detrended Correspondence
Analysis (DCA)
When a single axis is is enough to order the sites and speciescorrectly, a second axis, which is independent of the first, can be
obtained by folding the first axis in the middle and bringing theends together.
Subsequent independent axes can be obtained by folding the firstaxis in three parts, four parts, etc.
This process is referred to as Detrended Correspondence Analysis(DCA).
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Detrended Correspondence
Analysis (DCA)
Recall this data set used to evaluate PCA & PCO where 3species were unimodally distributed across a coenocline:
-1
0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Axis2
Axis 1
-0.1
-0.2
-0.4
-0.5
-0.6
0.1
0.2
0.4
0.5
0.6
-0.1-0.2-0.4-0.5-0.6 0.1 0.2 0.4 0.5 0.6
PCA w/Euclidean Distance
vs.
DCA via quadratic polynomial
Detrended Correspondence
Analysis (DCA)
Two main approaches have been proposed to remove the archeffect:
detrending by polynomials(previous example), anddetrending by segments.
Both methods lead to detrended correspondence analysis.
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Detrending by Segments
When detrending by segments (Hill and Gauch 1980), axis-I is
divided in to a number of "segments" and, within each one, themean of the scores along axis II is made equal to zero; in otherwords, data points in each segment are moved along axis-II to
make their mean coincide with the abcissa.
Proximities among points should in no casebe interpreted as
meaningful! Segments can generate large differences in scores forpoints that are near each other in the original ordination but happento be on either side of a segment division.
The number of segments is arbitrary. Different numbers ofsegments lead to different ordinations.
Detrending by Segments
Various software packages use 10 as a minimum number of segmentsand 46 as a maximum; 26 being a recommended starting or default
value. This of course necessitates data sets with considerably moreobservations than 26. There are no empirical rules for the "correct"number of detrending segments.
After detrending by segments, the DCA ordination has the interestingproperty that the axes are scaled in units of the average standard
deviation (SD) of species turnover (Gauch 1982). Along a regulargradient, a species typically appears, rises to a modal value, anddisappears in 4 SD; similarly, a complete turnover in species
composition often occurs over 4 SD. Thus, the length of axis-I is oftenused as a measure of the length of the ecological gradient.
Detrending by Polynomials
Detrending bypolynomials(Hill and Gauch 1980) directly follows fromthe fact that an arch is produced when a gradient of sufficient length ispresent in the data. When a sufficient number of species are present
and replace each other along the gradient, the second axis of the CAapproaches a quadratic functionof the first one (i.e., a second degreepolynomial).
When detrending is sought, detrending by polynomials is an attractivemethod because it results in a continuous functionof the previous
axes, without the discontinuities generated by detrending-by-segments. However, the downside of this method is that it imposes avery specific polynomial modelthat the data must correspond to. It
also does not solve terminal gradient compressions at the ends of theordination axes.
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1
2
3
4
567
8
9
10
DCA Axis 1
DCAAxis2
DCA of WI Forest Data
R2Axis-I = 0.866R2Axis-II = 0.07610 Segments
NB: Wedge Effect
To Detrend or Not To Detrend
That is the Question...
The controversy over detrending has raged inthe literature for well over 20 years now.
Wartenburg et al. (1987) argue that the arch is an important andinherent attribute of the distances among sites, not simply a
mathematical artifact. The only effect of DCA is to flatten thedistribution of points onto axis-I. They also argue that detrending bysegments is completely arbitrary and has no theoretical justification.
Peet et al. (1988) still support DCA on the grounds that detrendingand rescaling may facilitate ecological interpretation & called for
improved algorithms.
To Detrend or Not To Detrend
That is the Question...
Minchin (1987) produced a nice comparison of several
ordination techniques and found DCA to perform poorly onmost accounts.
He found that DCA actually removed real pattern from the data
and produced significant distortion which he referred to as a"tongue" or subsequently a "wedge" in the data and this was a
simple artifact of the algorithm.
However, Palmer (2010) argues for the viability of DCA,
especially in certain applications. He highlights thecharacteristics of both DCA and NMDS at his Ordinationwebsite
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Factor Analysis
In the social sciences, analysis of the relationships among thedescriptors of a multidimensional data matrix is frequently carried out
via Factor Analysis (FA).
Recall that the goal of PCA is to account for a maximum amount of
the variancein the data, whereas the goal of factor analysis is toaccount for the covarianceamong descriptors.
Put another way, PCA is directed towards reducing the diagonalelements of R. Factor analysis is directed more towards reducing theoff-diagonal elements of R. Since reducing the diagonal elements
reduces the off-diagonal elements and vice versa, both methodsachieve much the same thing.
Factor Analysis
To do this, FA assumes that the observed descriptors are linearcombinations of hypothetical underlying variables (i.e., the factors).
Originally FA was used to evaluate such things as intelligence. Manyvariables could be measured such as age, parental education, family
income, etc. Multiple variables might play out to show that Factor-1
was determined by all variables related to education and Factor-2 tosocio-economic conditions (for example).
There are few applications of FA in EEB, so I will not cover it in
depth. An excellent treatment can be found in Tabachnick and Fidell(1996).
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Canonical Analysis
Canonical analysis is the simultaneous analysis of two, or
eventually several data tables. It permits biologists to do a directcomparisonof two data matrices. Hence, canonical analysis andits derivatives are known as direct ordination methods.
Often, in ecology, one is interested in the relationship between afirst table describing species composition and a second table of
environmental descriptors, observed at the same locations(i.e.,objects or samples).
Canonical Analysis
Previous to this, we have considered indirect ordinationmethods
(PCA, PCO, NMDS, CA, DCA) in that we would ordinate a species "stand matrix and then conduct some form of correlation or regressionanalysis on the ordination vectors to relate objects or descriptors to
externally obtained environmental information. This procedure isperformed a posteriori.
In canonical analysis, with two matrices (XandY), one is constrainedby the other, and both are examined simultaneously. This permits oneto directly test a priorihypotheses by bringing all of the variance ofY
that is directly related to Xand allowing formal tests of thehypotheses.
Canonical Form
In mathematics, a canonical formis the simplest and mostcomprehensive form to which certain functions, relations, orexpressions can be reduced without loss of generality.
For example, the canonical form of a covariance matrix is its matrix ofeigenvalues.
In general, most methods of canonical analysis employ eigenanalysis(some extensions have been described using NMDS).
Canonical analysis combines the concepts of ordination andregression. It involves a response matrixYand an explanatory
matrix X. (See next slide.)
Like previous ordination methods, canonical analysis produces
orthogonal axes from which scatter diagrams may be plotted.
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Variables y1...yp
Objects1t
on
Y
Objects1
ton
Variablesx1...xp
Xy
Var. y
Variables y1...yp
Objects1ton
Y
Variablesx1...xp
X
Simple
ordinationof matrix
Y: PCA,CA, etc.
Ordination
of y(single
axis) undertheconstraint of
X: aka
multipleregression
Ordination ofYunder
the constraint of X:Redundancy Analysis
(RDA) or CanonicalCorrespondence
Analysis (CCA)
Problems of canonical analysis can be r epresented via a partitionedcovariance matrix resulting from the fusion ofYand Xdata sets and
producing a joint dispersion matrix SY+X...
SubmatricesSYY(orderp !p) and SXX(m !m) concern each oftwo sets of descriptors, respectively, where SYX(p !m) and itstranspose S'YX= SXY(m !p) account for the covariances among
the descriptors of the two groups.
Redundancy Analysis
In redundancy analysis (RDA), each canonical ordinationaxis corresponds to a direction, in the multivariate scatter of
objects (Y), which is maximally related to a linearcombination of the explanatory variables X. A canonical axis
is thus similar to a principal component.
Two ordinations of the objects are obtained: (1) linearcombinations of theYvariables (matrix Fin PCA), (2) linearcombinations of the fittedY-hat variables (matrix Z), which
are thus also linear combinations of the Xvariables.
RDA preserves the Euclidean distance among objects in
matrixY-hat containing values ofYfitted by regression tothe explanatory variables X.
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Canonical Correspondence Analysis
Canonical correspondence analysis (CCA) is similar to
RDA. The difference is that it preserves the !2distance
(as in CA), instead of the Euclidean distance amongobjects.
Calculations are a bit more complex sinceYcontains
fitted values obtained by weighted linear regression of
matrix Qof correspondence analysis on the explanatoryvariablesX. As in RDA, two ordinations of the objects
are obtained.
^
Canonical Correlation Analysis
In canonical correlation analysis (CCorA), the
canonical axes maximize the correlation between
linear combinations of the two sets of variablesYand X.
This is obtained by maximizing the among-variable-
group covariance (or correlation) with respect to the
within-variable-group covariance.
Two ordinations of the objects are again obtained.
Canonical Discriminant Analysis
In canonical discriminant analysis, the objects are
divided in to kgroups, described by a qualitative
descriptor.
The method maximizes the dispersion of the
centroids of the kgroups. This is obtained by
maximizing the ratio of the among-object-group
dispersion over the pooled within-object-groupdispersion.
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Canonical Analysis
Unfortunately, we do not have the time to develop
the details of the algebra of each of the 4 methods
of canonical analysis previously described. But,you have now gained all of the necessary skills
necessary to interpret the details on your own
should you need to pursue one of these analyses.
Two excellent sources of of information on thesemethods can be found in Legendre and Legendre
(1998), ter Braak and !milauer (1998), and Lep"
and !milauer (2003).
Canonical Analysis
As an alternative to a detailed treatment of
mathematics behind each method, I would like to
develop a worked example.
Let's develop a data set using the number of fish
observed at 10 sites along a transect running from
the beach of a Caribbean island, with water depths
going from 1 to 10 m. The first three sites are onsand and the others alternate between coral and
"other substrate" (coded as 0/1).
Tropical Fish Data Set
Site
No.
Sp-
1
Sp-
2
Sp-
3
Sp-
4
Sp-
5
Sp-
6
Sp-
7
Sp-
8
Sp-
9
Depth
(m)
Coral Sand Other
1 1 0 0 0 0 0 2 4 4 1 0 1 0
2 0 0 0 0 0 0 5 6 1 2 0 1 0
3 0 1 0 0 0 0 0 2 3 3 0 1 0
4 1 4 0 0 8 1 6 2 0 4 0 0 1
5 1 5 17 7 0 0 6 6 2 5 1 0 06 9 6 0 0 6 2 10 1 4 6 0 0 1
7 9 7 13 10 0 0 4 5 4 7 1 0 0
8 7 8 0 0 4 3 6 6 4 8 0 0 10
9 7 9 10 13 0 0 6 2 0 9 1 0 1
10 5 10 0 0 2 4 0 1 3 10 0 0 0
% 60 50 40 30 20 10 45 35 25
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Tropical Fish Data Set
Because we wish to conduct a direct gradient analysis (i.e., wehave both species data and environmental data from the same
samples), and we have numerous species (9), with roughlymonotonic responses (although one may be unimodal; e.g., 7) weselect RDA as the method of choice.
RDA is particularly appropriate when the gradients are short andspecies distributions are linear (or generally monotonic).
The software of choice for this type of analysis has for the lastdecade been CANOCO. Mathematically, this software is excellent
but its ease of use is not the best and graphics are poor. R nowhas applications to handle most of these ordination procedures(i.e., RDA and CCA).
RDA using R- Tropical Fish Data Set -
RDA using R- Tropical Fish Data Set -
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RDA using R- Tropical Fish Data Set -
RDA using R- Tropical Fish Data Set -
RDA using R- Tropical Fish Data Set -
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RDA using R- Tropical Fish Data Set
Discriminant Analysis
A common situation arises in EEB applications where
one starts with an already known grouping of objects,
and one wishes to assess how well a group ofquantitative descriptors can explain the object
groups. Thus, the problem is no longer how to define
or delineate groups, but rather how to interpret them.
This is the realm of discriminant analysis.
Discriminant analysis is a method of linear modeling,
like analysis of variance, multiple regression, and
canonical correlation analysis. DA is frequently used
in systematics.
Discriminant Analysis
DA proceeds in two steps:
(1) It first tests for the differences in the explanatoryvariables (X), among the predefined groups. This part
of the analysis is identical to the overall test performed
in the MANOVA.
(2) If the test supports the alternative hypothesis ofsignificant differences among groups in the Xvariables,
the analysis proceeds to f ind the linear combinations
(called discriminant functions) of the Xvariables that
best discriminate the groups.
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Discriminant Analysis
Like one-way ANOVA, discriminant analysis considers asingle classification criterion (i.e., division of the objects into
groups) and allows one to test whether the explanatoryvariables can discriminate among the groups. Testing for
differences among group means in DA is identical toANOVA for a single explanatory variable and to MANOVAfor multiple explanatory variables.
When it comes to modeling, i.e., finding the linear
combinations of the variables (X) that best discriminateamong the groups, DA is a form of "inverse analysis" wherethe classification criterion is considered to be the response
variable (y) whereas the quantitative variables areexplanatory (matrix X).
Discriminant Analysis
Note that discriminant analysis (DA) is also called
canonical variates analysis(CVA). This method was
first proposed by Fisher (1936) where he publishedthe now famous data set where he described the
morphology of 150 specimens of irises (Iridaceae)
using 4 measured flower characters (lengths and
widths of sepals and petals) belonging to three
species.
Again, in the interest of time, we will bypass the
mathematical treatment of DA and work through the
iris data set using a software application (NCSS).
Discriminant Analysis
Shown here are Fisher'sdata for the 150 plants
(first 31 shown), fourvariables, and three
species (coded 1,2,3:1 = Iris setosa, 2 = Irisversicolor, and 3 = Iris
virginica . Note that I.versicoloris actually a
polyploid hybrid of theother two species.Datawere originally collected
by the botanist EdgarAnderson of the Missouri
Botanical garden andused with permission.
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This chart plots the valuesof the first and seconddiscriminant function
scores. By looking at thisplot you can see what theclassification rule would be.The first function appearsto be the most important in
separating the threespecies.