Vegetatio, Vol. 34, 3: 191-197, 1977

ON THE INTERPRETATION OF PRINCIPAL COMPONENTS ANALYSIS IN ECOLOGICAL CONTEXTS

Scott NICHOLS*

Section of Ecology & Systematics, Cornell University, jIthaca, New York 14853, USA

Keywords: Indirect gradient analysis, Ordination, Principal components analysis

Introduction

The application of principal components analysis (PCA) to ecological problems has led to disagreements over proper interpretation. Controversy about PCA as an ordi- nation technique is related to the evolution of two separate definitions for ordination. Goodall (1954) proposed 'ordination' as an antonym to 'classification', indicating a display of relationships among items using a continuous scale, as opposed to assigning items to discrete classes. Consideration of the traditional goals of plant ecology led to a restricted definition for ordination as an arrange- ment of samples and species on axes relating to environ- mental gradients. The need for a term complementing classification led to a broader definition for ordination, as any arrangement summarizing relationships among items. Both definitions are currently in use (Orl6ci 1966, Goodall 1970, Dale 1975, Noy-Meir & Whittaker !977).

PCA 'ordination' under the broader definition can be an effective tool for examining community structure. In this paper, I will develop some of the considerations neces- sary for using PCA effectively, stressing the meaning of the extracted axes, and the implicit, differential weightings of species in any analysis. PCA is not often appropriate for ordination under the restricted definition. Gauch & Whit- taker (1972) and Kessel & Whittaker (1976) provide empirical evidence of PCA's inappropriateness. Here, I can present the analytical evidence. In sequence, this paper considers PCA's structure (independent of user purpose or data transformation), the overwhelming effects of im- plicit weightings and the art of data transformation, the poor compatibility of PCA with indirect gradient analysis

* I would like to thank R. H. Whittaker, H. G. Gauch, R. E. Moeller, and S. R. Searle for their guidance and assistance.

(i.e. restricted-definition ordination), and a general treat- ment of applications of PCA.

PCA structure

In the most common use of PCA in ecology (and the only use considered here), the investigator has collected a set of abundance or importance values for many species over many samples, and has organized the data into a matrix. The elements of the data matrix may be centered, rescaled, or transformed as appropriate. A cross-products matrix is then calculated. I will limit my discussion to a matrix of cross-products between species, summed over samples. The cross-products are used as measures of similarity between species, similarity implying similar distribution patterns in the samples collected. The eigenvectors and eigenvalues of the cross-products matrix are then extracted. This involves finding vectors u and scalars 2 that solve the matrix equation

A u ~ ,~u

where A is the cross-products matrix. There are as many 2's and u's as there are rows (species) in A. (Some 2's may be zero.)

The properties of the extracted eigenvectors and eigen- values are most easily discussed in geometric terms. I re- commend the geometric explanations of PCA presented by Cassie (1963), Morrison (1967), and Pielou (1969) as starting points. Consider just what is being analyzed: the sums of cross-products between species, only. Geometri- cally, this is equivalent to looking at scatter plots of one species vs. another, only using as many axes as there are species. PCA will return scatter plots on a new set of axes, established by a rigid rotation of the original species axes.

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The eigenvector terms are the directional cosines relating the (scatter plot) species axes to the component axes. Each sample has a score on each axis, calculated as the sum (over all species) of the products of each species's importance value for that sample times that species's eigenvector coefficient. As such, the principal component scores are linear combinations of species importance scores, and will behave much like species importance scores. The principal component axes, then, represent 'combined' or 'derived' species. This is easily seen in cases where most of the eigen- vector terms for an axis have the same sign ('unipolar' axis). If there are eigenvector terms of opposite sign ('bipolar'), the axis is then a contrast between negatively related distributions. The 'strength' of a species's contribution to a component axis is indicated by the magnitude of the eigenvector coefficient. The primary effort in any PCA should be the examination of the eigenvector coefficients, which define the extracted axes. The investigator must determine which species combine to define which axes, and why.

Determining the number of axes that are 'significant' or 'interesting' is not necessarily straightforward. In any real data set, every PCA axis will express some real variation. Because PCA axes are required to be orthogona], successive axes deal with the residual variation present after extrac- tion of preceding axes. Often, a similarity among several species will be extracted on one axis, and the next axis will be associated with the residual differences among those same species. Beyond that exception lies a key point: the number of important axes often approximates the number of 'resolvable' species distribution patterns in the data.

'Resolvable' has a pragmatic meaning here. Species not tied to one of the first few components usually have most of their variation 'bled away' by virtue of casual, partial similarity to the several species linked with the first few axes. This is the source of some of the reduction power of PCA, and the reason why the axes beyond the first 3 or 4 are not readily interpretable. In practice, the extent of reduction, i.e. the fraction of the total multivariate varia- tion captured, as expressed in the eigenvalues, is over- whelmingly governed by the particular species weighting structure present in any given data set. This indicates why species weightings and transformations will be so impor- tant: we must have a procedure to guarantee a meaningful weighting structure if the order of axis extraction, and the PCA itself, is to be meaningful. Only secondarily does the //-diversity influence the number of components with large eigenvalues. The number of important axes increases with increasing //-diversity, because more resolvable patterns

will exist with higher//-diversity. The actual decision about how many axes to present must be considered case by case, and usually must be based on the interpretation of the first several axes via their eigenvector coefficients, rather than on the 'variance captured' expressed in the eigenvalues.

Most presentations of PCA results include axis 2 vs 1 (or sometimes, 3 dimensional) scatter diagrams. Such plots are of the same genre as plots of one species vs. another, and should be handled much the same way. Points in different regions of a plot represent samples in regions rich or poor in the species establishing each axis, species that are readily identified by their eigenvector coefficients.

Some workers plot the eigenvector coefficients on 2 axes. Such plots display the degree of similarity between each species and each of the first 2 components. The information in these plots is usually readily apparent upon inspection of the eigenvector coefficients directly. I would caution against attaching abstract meanings to the axes of these plots - consider the definition of the components to which each species is now being compared.

This description of PCA's structure suggests how PCA might aid the examination of complex data. Three points stand out. First, the axis vs. axis scatter diagrams are an ordination under the broader definition - a display of relationships among samples, based entirely on species composition, usually in only 1,2, or 3 dimensions. Without PCA, there were as many dimensions as species to be considered.

Second, in the eigenvectors, PCA can summarize a large similarity matrix. Species combine via their eigenvector coefficients-to define axes that indicate the important points in the cross-products matrix, beyond the original tabulation of pairwise relationships. Plots of eigenvector coefficients are also ordinations under the broader defini- tion. These plots might be used as plexus diagrams of species relationships.

Third, there are cases where the investigator might re- place species scores with component scores for subsequent analyses, with a great reduction in number of variables.

Data transformations and weightings

Different weights are implicitly given to every species in any PCA procedure, in that species with large terms in the crossproducts matrix will be important in the PCA. PCA will 'collect' similar species, but the order of the axis extraction will be determined by the large terms, while the

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species with small terms will have little influence. All variations of PCA can be obtained by transforming or re- scaling the species scores prior to computing a cross- products matrix. If the choice of transformation leads to certain terms being large or small for undesirable reasons, the analysis will be of little value.

Species with large cross-product terms will be those with large total variation under the transformation used for the analysis. In any real system, this total variation will have a vectorial component (variation potentially explain- able with reference to environmental and biological factors) and random components (associated primarily with sampling error and with patchiness). Unfortunately, the proportions of the vectorial and random components within the total variation are unknown. There are, however, some systematic trends that can be anticipated, the most important being the relationship between variation and abundance.

Most raw abundance data show increasing variances with increasing mean values. This will, a priori, weight the more abundant species. This can be desirable: the abundant species often show a higher proportion of vectorial varia- tion than rare species, and usually have more impact on the community as well. In many instances, however, the single most abundant species so dominates the analysis that the only meaningful axis returned is established by that species, only. This reveals nothing about interspecific relationships, rendering the analysis trivial.

Several transformations are designed to remove the dependence of variance on the mean. Logarithmic trans- formation will remove the dependence if variance is pro- portional to the mean, and sometimes will yield normally distributed errors. For many data sets, however, the log transformation is too 'strong', in the sense that variance then increases with decreasing abundance. PCA can then be disastrous: rare species, showing perhaps mostly random variation, will dominate the analysis by virtue of their high total variation on a log scale. Square, cube, and fourth root transformations are also used to reduce variance-mean dependency. For any data set, there will be some power transformation that will eliminate the syste- matic relationship of variances and means. Such a trans- formation should not necessarily be considered optimal, because the vectorial fraction of the variation will usually be lower in less abundant species.

An altemative procedure that removes the dependence of the variance on the mean is the use of the correlation matrix. This is the cross-products matrix calculated with the species scores rescaled

X i j - - "Xi

(n - 1)~sl

(i &j subscripts refer to species and sample, respectively; and s are the mean and standard deviation over all n samples). The total variance for each species is thus set equal to unity. If, as is usually the case, the proportion of random variation increases with decreasing abundance, then the random variation of rare species will have an undesirably important influence on the analysis. Addi- tionally, given a case where, say, 5 species exhibit one pattern and 2 exhibit another, the 5-species pattern will automatically draw the first axis. Given the equality of the total variances and the residual nature of successive axes, this situation may not always be desirable. In general, rare species should be omitted from PCA when using correla- tions.

Noy-Meir, Walker & Williams (1975) have considered other varieties of standardizing transformations for several purposes. Careful consideration of the total, random, and vectorial variations is especially required before applying any of the more specialized transformations. The strategy is to prevent or allow for PCA results that are purely consequences of the numerical weighting structure, and only then examine the results for ecological structure. Clearly, no generally 'optimal' procedure exists; the most meaningful procedures for any data set are set specific. Whether a transformation is weak, moderate, or strong depends not on the transformation, but on the data.

PCA and indirect gradient analysis

Many investigators have studied or used PCA as a technique for indirect gradient analysis. They sought a technique which, using only species importance data, could produce an ordering of samples interpretable with respect to environmental gradients. From this, investiga- tors hoped to perceive the arrangement of species in rela- tion to the underlying environmental variables. The ideal technique might even 'predict' a pattern for the underlying variables, which could be compared with the patterns of measured variables in hopes of identifying the variable or variables most impor tant to community organization. Pioneers in indirect gradient techniques tentatively accepted PCA as a procedure that could guide them toward gradient interpretations. Some subsequent users accepted PCA un- critically as a one-step technique for indirect gradient analysis, without considering the actual structure of PCA.

One proposition that has often been accepted without

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reference to PCA structure is that the principal component axes should represent underlying ecoclines in a data set. The mathematical structure of PCA prevents this from being correct, whatever the form of the species-environ- ment relations. Each principal component axis has a precise definition as a certain linear combination of species. Interpreting these linear combinations directly as ecoclines builds in a level of abstraction that simply does not exist. Treating the higher axes as spurious, or as distortions of an ecocline is likewise misleading, because the higher axes do have a legitimate meaning having nothing to do with ecoclines. I have already discussed factors controlling the number of important component axes. The dependence of fl-diversity on the number of environmental variables is probably unpredictable; the dependence of the number of important PCA axes on fl-diversity is definitely over- whelmed by purely numerical effects related to weightings of the data. The number of important component axes therefore has little or nothing to do with the number of important environmental gradients.

A corrected proposition can be stated, which is consistent both with the structure of PCA and with the goals of in- direct gradient analysis: under certain situations, there may be a nearly linear relation between the first principal com- ponent axis and an underlying environmental gradient. This statement is completely distinct from 'axis = coeno- cline', and is in fact the basis of past successful use of PCA as an indirect gradient technique.

There are three situations that could lead to a nearly linear relation between PCA axes and environmental gradients, thus allowing PCA to be used as a single-step indirect gradient technique. First, there could be a linear relation between each species's abundance scores and an environmental variable. Except over short segments of a gradient, this would rarely be encountered. Second, there may be some transformation of the species abundance scores that will be linearly related to an underlying gradient. This is possible in cases where species-to-gradient relations were monotonic in the raw data; but with sufficient fl- diversity for a sequence of bell-shaped abundance curves to be expressed, no transformation can linearize all the relations. Third, a quasi-linear axis-to-gradient relation may result if a transformation is found which inflates the weightings of the species at the extremes of an underlying gradient. A candidate transformation has been offered (Hill 1973), so this third situation is worth examining in more detail.

Reciprocal averaging, which is an eigenvector procedure employing a double-standardizing transformation, has in

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some instances (see Gauch, Whittaker & Wentworth 1977) yielded a nearly linear axis-to-gradient relation based on the inflation of the weightings for ecologically distant species. The RA transformation leads to a non-symmetric association matrix, whose elements can be viewed as sums of cross products between one set of species scores trans- formed to

~xij~xij i = 1 j = l

times an untransformed set. Hill (1973) introduced a computation procedure using principal components, in which the species scores are transformed to

Xij n m '

( Y, x,j)~( Zx,j)~ i ~ l j = l

and centered about the species means in the transformed scale. Cross-products are then calculated, followed by eigenextraction. To return to the reciprocal averaging scaling, each species's eigenvector coefficients are divided by

n

( 2 Xij) ½" i = l

These rescaled vectors are the eigenvectors of the non- symmetric association matrix. A sample's score on an RA axis is the sum (over species) of products of these eigen- vector coefficients times the untransformed species scores, divided by the total of the species scores for that sample. Hill also introduced a 0 to 100 scaling for both eigenvector coefficients and sample scores. This does not affect inter- pretation, except that the 0 to 100 scaling of the coefficients obscures whether an axis is a comparison (unipolar) or a contrast (bipolar).

The RA transformation cannot guarantee that the extreme species on an unknown gradient will be the most heavily weighted. This would require that the extreme samples on the gradient show greater dominance by their most abundant species than the interior samples, and that the abundance-variance relationships in the samples be such that the transformation always promotes the weight- ings for the ex t r eme species over those of perhaps more abundant species not at the extremes. Neither requirement can be checked without reference to a known gradient,

and there is no necessary reason that either requirement should be met in any given data set.

One should never omit examination of the weightings structure with RA. The non-symmetric association matrix can be examined directly, but it is probably easier to follow the computational procedure by examining the PCA of the symmetric matrix, and by evaluating the rescaling to RA, in sequence. Assessment of the number of important RA axes follows the same rules applicable to all PCA's. One may or may not be able to decide if the data structure was compatible with the requirements for RA to perform as a one-step indirect gradient technique.

Applications

Applications of PCA for analysis of ecological data follow almost directly from the consideration of PCA structure. Here, I will merely sketch how PCA might function in ~tudies of interspecific and species-environment relations. The capabilities of PCA fall roughly under three headings: reduction of number of variables, summarization of simi- larity matrices, and 'ordination' under the broader defini- tion. I have cited some examples which develop each of these capabilities in a clear manner. Of course, only rarely would a study be concerned with one function only.

Simply reducing the number of variables is often the main purpose of PCA in psychology and anthropology, but this aspect has limited use in ecology. Reduction of variable number in ecology has been used with environmental data to produce a small, orthogonal set of variables describing environmental variation. (e.g. Austin 1968, James 1971). Austin's (1968) PCA of species scores shows some purely reductive aspects. Nichols (1977) has used PCA to express the temporal patterns of groups of similarly distributed phytoplankton species as single variables. With variable reduction, the investigator might replace species (or environmental) scores by principal component scores, and look for systematic patterns in the component scores that were worthy of further study.

Summarizing large species-association matrices should be a major function of PCA in ecology. Surprisingly, this aspect has not been as well developed as the (broad- definition) ordination aspect to follow. Important points in an association matrix beyond the tabulation of pairwise relationships are called to the investigator's attention by PCA. Goodall (1970) suggested using PCA to identify 'ecological groups'. Noy-Meir (1971) suggested a similar function for non-centered PCA. If the groups themselves

become entities of interest, then the summarization aspect of PCA converges with the variable reduction aspect, and PCA functions as a classification technique. If the species that combine to form a particular component axis include species with known sensitivities to some variable, the investigator could consider that the other species in the combination might have similar requirements (e.g. Goodall 1954). This could serve as a working assumption, or become a testable hypothesis.

Graphical presentations of eigenvector coefficients (e.g. Goff 1975, Yarranton 1967) usually have purposes similar to those of the preceding paragraph. This type of 'ordina- tion' of species can also be obtained from eigenanalysis of a matrix of sample similarities (see Pielou 1969), but operating with species similarities is usually more intui- tively sensible to most workers.

Plots of samples as points on PCA axis 2 vs. 1 plots are probably the most commonly published feature from PCA. The investigator may see if his samples fall into single or multiple clusters (e.g. Goodall 1954, Cassie 1972). Some- times single, non-linear orderings ('manifolds') can be seen in 3 PCA dimensions (Gauch pers. comm.). Dale (1975) has discussed the problems of deciding between clustered vs. continuous alternatives: the decision is usually based on the presence or absence of very few points in very limited regions of axis 2 vs. 1 plots. Presence or absence of crucial 'intermediate' samples will depend on the efficiency of sampling - the likelihood of encountering such samples if they do exist. Even if no decision about data 'structure' can be made, these (broad-definition) ordina- tions can suggest further analyses which might be approp- riate for the data.

Environmental variables often show systematic pattems when mapped on axis 2 vs. 1 plots if the requirement that the pattern be linear is dropped. Environmental gradients will meander through axis vs. axis plots very much as they will through species vs. species plots (the preservation of interstand distance, and controllable type B distortion of Orl6ci 1974). Often one can find patterns that, though obscure in the raw data, now make intuitive sense, and are worthy of further study. Mapping principal component scores in space (e.g. Goodall 1954, Cassie & Michael 1968), or plotting them against known gradients (e.g. Nichols 1977) may be a more efficient way of displaying the same information, but both approaches are worth trying. With either type of plot, the purpose isto expose possible rela- tionships between community patterns and the environ- ment. This converges with the purposes of (limited- definition) indirect ordination, but does not include the

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limitations of linearity required for single-step indirect ordination.

Conclusions

Investigators using PCA on similarity matrices must operate with the structure described in the first section of this paper. That structure is unaffected lay the desire of an investigator for a one-step indirect gradient technique. Failure to reconcile assumed structure with actual struc- ture, particularly regarding axis definition, has led to the recent controversies over PCA interpretation. I fear that some disappointed ecologists might now unnecessarily dismiss PCA from their arsenal of techniques, without examining the usefulness of the reduction and summariza- tion aspects. These aspects are not trivial: PCA gives unique and objective reductions that are both predictable and comprehensible. Ecological interpretations must often be subjective, which suggests a greater emphasis on using PCA for hypothesis generation than for final demonstra- tion of ecological principles. The key points are that the maximum variance and orthogonal axis structure basic to PCA has proven to be a most useful summarization tech- nique over a wide variety of conditions, and that the objec- tivity of the eigen-extractions allows investigators to deter- mine precisely why any PCA behaved as it did. I hope my discussion will suggest further useful applications of PCA for data reduction and summarization, hypothesis genera- tion, and, in limited situations, indirect gradient analysis.

Whatever the intended application, users must respect the purely numerical effects of their data on any PCA. Numerical effects arising from the different abundances and variances of each species will overwhelm any analysis unless special care is taken. Choice of a transformation is that special care: the user tries to adjust for the numerical considerations, so that the ecological relationships might be revealed. In view of the importance of weightings and transformations, it is surprising that they have been ignored or only casually considered until recently.

Happily, it is not difficult to evaluate transformations and species weightings. In truth, a new set of species im- portance scores is analyzed with every transformation. The investigator should plot species distributions using these transformed scores against the (spatial or temporal) sampling pattern, or against known environmental varia- bles, to observe the actual data he is analyzing. (Include the centering step, if applicable. Similarities in absence pattern, with zero abundances as departures from the

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mean, are as real to the analysis as any others. Directions of departures, if internally consistent, are arbitrary to the analysis: try turning centered scores vs. gradient plots upside down !) The investigator can evaluate subjectively the pattern he sees, looking for anything that might destroy the ecological sensibility of the analysis. The cross-products matrix can be examined for structure prior to analysis to determine which species have the largest weights, and to decide if the weightings are meaningfully consistent with what is known about the total, random, and vectorial variations for the data. A plot of total (and, if known, error) variance vs. mean, including all species, will usually display the systematic relationships between variance and abundance.

It is best ifa transformation can be chosen apriori. Often one cannot be, and the investigator must evaluate several choices for appropriateness, based on the numerical con- siderations described. (This is qualitatively different from a blind application of many transformations, selecting the one which gives the most ecologically intriguing result!) Only occasionally will data be encountered for which no transformation will allow a meaningful analysis. With the properly selected transformations, PCA can be a useful technique for a wide variety of situations and purposes.

Summary

Principal components analysis is well suited for many data analysis problems in ecology, particularly for data reduc- tion and hypothesis generation; but the structure of PCA is poorly suited for indirect gradient analysis. Whatever the intended application of PCA, the user must exercise special care in selecting data transformations to prevent the analysis from being overwhelmed by the purely numerical effects in the variance structure of the data.

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Accepted 15 February 1977

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