HUGH G. GAUCH, JR.
Ecology and Systematics, Cornell University, Ithaca, New York 14850
GENE B. CHASE2
Education, Cornell University, Ithaca, New York 14850 AND
ROBERT H. WHITTAKER
Ecology and Systematics, Cornell University, Ithaca, New York 14850
Abstract. Direct gradient analysis has shown that along many environmental gradients species distributions show bell-shaped forms, overlap broadly, and have their centers and limits scattered. This finding is the basis of a technique called "Gaussian ordination" presented here, which arranges samples by maximizing the fit of Gaussian curves to the individual species' distributions. Ordination proceeds by (1) a first-guess arrangement of samples, (2) sorting of species for relative usefulness in ordination, and (3) fitting of least-squares Gaussian curves to the individual species, followed by (4) iterative fitting that changes the ordination values to produce an optimal Gaussian fit for all species together. Gaussian ordination is appropriate for sample sets with a major axis of environmental and community variation. The technique has produced successful ordinations of simulated and field data. Because Gaussian ordination is based on an explicit model of vegetation structure and computes the species and sample parameters for this model, the technique represents a convergence of research in direct and indirect ordination.
Key words: Catenation; Gaussian distribution; gradient analysis; ordination; vegetation models.
INTRODUCTION
Given a matrix of species scores or importance values P for species i and samples j, here denoted P(i,j), we may desire to arrange the samples (or the species) in a manner that reveals their ecological relationships. When environmental variables are also measured, the samples may be presented along these environmental gradients; this is direct ordination. Alternatively, various algorithms may be used to arrange the samples according to their species compositions; this is indirect ordination. In general, much can be said for studying sample relationships both along environmental gradients and by species composition; both approaches are productive and widely used. They are complementary rather than competitive.
Direct and indirect ordinations have each produced a major result of particular relevance here. First, direct gradient analysis has shown that "species distributions show a rounded or bell-shaped form in most cases, overlap broadly, and have their centers and limits scattered along the gradient" (Whittaker 1956; also Mclntosh 1967, Whittaker 1967, Noy-Meir 197 la, Gauch and Whittaker 1972a). In multivariate environmental space, species importances form Gaussian solids. Figure 1 shows typical examples of species distributions along an environmental gradient. Second, indirect ordinations, such as the Wisconsin polar or Bray and Curtis (1957) technique and principal components analysis (Orloci 1966, 1973), give curvilinear distortions in the or-dinated positions of samples representing a linear community gradient (Noy-Meir and Austin 1970, Swan 1970, Jeglum et al. 1971, Gauch and Whit-taker 1972a, Beals 1973, Whittaker and Gauch 1973). Distortion results from the assumption of linear species distributions in the algorithms for these ordinations, and the distortion increases with increasing beta (between-habitat) diversity (Gauch and Whittaker 1972b). Apparently some of the techniques of indirect ordination have been used with insufficient concern for the results of direct ordination. The distortion in Bray-Curtis ordination can be corrected for if beta diversity is within a few half-changes (Gauch 1973a).
FIG. 1. Species populations of forest trees along a moisture gradient in two typical cases (Whittaker 1967). Above, with low beta (between-habitat) diversity, at 460-470 m elevation in the Siskyou Mountains, Oregon; below, with high beta diversity, at 1,830-2,140 m elevation in the Santa Catalina Mountains, Arizona.
The distortion in indirect gradient analysis is a consequence of assumptions in conflict with knowledge of species distributions. However, there is the possibility of using those known distributions as the basis for indirect ordination. That is, given a matrix of P(i,j) values, we may seek to obtain those ordination values x(j) which best arrange the species importance values into Gaussian curves. This is the central concept of the ordination procedure presented here that we call "Gaussian ordination."
Gaussian ordination has several interesting prospects. High or even modest beta diversity is the poison of other indirect ordinations (Gauch and Whittaker 1972b), but it is the fuel of Gaussian ordination. This may indicate a fundamental step toward realism. Second, Gaussian ordination offers an integrated approach in that it encourages comparison with results from direct gradient analysis. In particular, Gaussian ordination is based on a definite, explicit, and testable model of vegetation structure (as in Gauch and Whittaker 1972a). As a consequence, results from Gaussian ordinations may be used, like those from direct ordinations, to evaluate conceptions of the population structure of vegetation (Whittaker 1970). Gaussian ordination is thus a logical convergence of direct and indirect ordination studies. And, third, in addition to providing ordination values for the samples, this ordination technique computes species parameters such as position and height of the mode, and dispersions. This may well permit quantitative research into such problems as habitat space and overlap, and species widths and packing along habitat gradients (cf. May and MacArthur 1972).
PROCEDURE
General comments
Gaussian ordination depends on three assumptions.
First, the samples must represent the Gaussian model. Second, the data for a
Gaussian ordination must contain some reasonable range of beta diversity to
give the algorithm working leverage with those data. And third, there must be
a predominating complex-gradient. These requirements are met by many sets of
data. Gaussian ordination is not appropriate,
however, for sets of samples affected by several major and almost equally important directions of community and environmental variation.
The algorithm of Gaussian ordination is complex. We have already stated that the inherent complexity of vegetation structure itself makes inevitable the need for complex analyses of such data (Gauch and Whittaker 1972b). Therefore, it is probably well to present the algorithm of Gaussian ordination in basic outline before presenting it in detail.
As noted, we are given a P(i,j) matrix of species importances for I species in J samples, and desire to obtain ordination values x(j) which reveal the ecological relationships among the samples. We shall assume the model of vegetation structure, and seek those ordination values x(j) which best arrange the species importance values into Gaussian curves; that x(j) vector is sought which minimizes the sum of squared deviations (SSD) of observed species importance values from those expected by the Gaussian curves fitted to the species. In other words, given a whole matrix with numerous Gaussian distributions in its individual vectors (species records), we seek a single (ordination) vector that simultaneously maximizes the arrangement of all the vectors in the data matrix into Gaussian curves. The algorithm must attempt to fit a number of Gaussian curves all at once with a single x(j) vector.
The computation of this x(j) vector may be discussed in three steps, as presented in the following three sections. First, one must obtain or provide a first guess of the x(j) ordination values. This is a necessary prelude to a later iterative procedure. Second, the species in the data matrix are sorted according to informativeness. This is necessary to obtain rapid, economical convergence to an answer. And third is the actual fitting of species with Gaussian curves and production of ordination values.
These steps are done by computer programs in the Cornell Ecology Programs series in IBM's FORTRAN IV (Gauch 1973b). Information on the series is available upon request; documentations including listings, and program decks, are available at cost.
Although here applied to phytosociological data, these algorithms and programs may be useful in other applications or sciences using multivariate data with Gaussian distributions. Principal components analysis is the appropriate analog with linear distributions and, we believe, has often been used with Gaussian data for lack of a more appropriate algorithm.
First guesses
Gaussian ordination begins by obtaining first guesses for the ordination values x(j). These may be supplied from the investigator's knowledge of his data. For example, if samples were taken going up a mountain, the elevation of the samples might serve as good first guesses. Also, a weighted-average estimate of moisture conditions might serve. Alternatively, another indirect ordination may be used to supply first guesses, such as Bray-Curtis or principal components ordination. The better the first guess, the more rapid convergence will be to an answer in Gaussian ordination.
Sorting species
As will be apparent in the following section, Gaussian ordination involves nested iterations and a large amount of computation. Therefore, it is desirable first to sort out the species in the data matrix according to informativeness, so the next step may begin with a few most informative species, and then go on to incorporate less informative ones. The computer program that does this sorting is "Cornell Ecology Program 8A. Data Screening for Species Importances Matrices."
Input to the data screening program is the species importance matrix, (first-guess) ordination values x(j), and certain parameters. Species may be eliminated according to the following four tests, applied in the order given; a species eliminated by one test is then ignored by any following tests. (1) The number of positive entries is less than the specified input parameter. In particular, since the Gaussian equation has three parameters, it requires at least three positive datum points to solve for the equation. Hence this test should use at least 3 as a minimum number. (2) The maximum importance value that a species has is below an input threshold value. This can be used, if desired, to eliminate minor species whose occurrence may be largely a matter of chance. (3) The species is not required to account for a given percentage of the variance of the original matrix. The species are ranked according to variance, and those with least variance may be deleted until the remaining variance is within that specified by an input parameter. For example, one may request to keep 85% of the original variance. The reason for this test is that, in general, species of too small a variance are those which will turn out not to be producing clearly defined Gaussian curves. And, (4) species may be further deleted, in order of least variances, until only an input parameter number of species remain. This can be used to obtain, say, the 30 most informative species. Thus it is possible to reduce data matrices automatically to a given size.
A statistical profile is printed out for the data matrix including for each species the number of positive entries, its average and maximum values, variance and percentage variance, relativized height and position of the mode on the first-guess ordination axis, and reason for deletion if the species is deleted.
If requested as an input option, the species importances (for those species retained) are graphed on the ordinate with the first-guess ordination values on the abscissa. These graphs give more detail than just relative height and position of the mode provide, and may be scanned to see which species are forming fairly good Gaussian curves.
The species retained are punched out in a new deck in order according to variance. This is almost what we want for the next step. It is important for the next step that the first group of species be well scattered along the ordination axis so that each sample (or with very few exceptions) includes at least one of the species in the first group. Normally this is the case; if not, one may want to move certain species (occurring along portions of the axis as yet poorly represented) from lower in the matrix up to the top group which is given first analysis.
Besides deleting insignificant data and organizing the remaining species according to informativeness, the output of this program provides the investigator with information that may be studied to decide whether the data at hand are appropriate for Gaussian ordination. If there are not at least a few species showing reasonable Gaussian distributions, one may try again with a different approach to get a good set of first guesses for the ordination values x(j). If this does not succeed, the chances are that the data do not fulfill the requirements or constraints of Gaussian ordination (as discussed earlier), and probably would fail to produce results in the third step.
The sorting, and so on, done by this preliminary program makes all later steps of the analysis more economical, and provides a logical point for the ecologist to scan his data in order to decide whether or not to carry out the third and final step. Noy-Meir (1971b) has a more general discussion of thresholds and information content, with quantitative bases for deciding how much information is carried by the various species; also see Orloci and Mukkattu (1973).
Gaussian fitting
We enter this third step given a matrix of species importance values (with most or many of the species presumed to have Gaussian distributions) as the ordinate values; a first guess of ordination values x(j) which are the abscissa values; and a data matrix arranged according to species informativeness. We now seek an iterative procedure for improving the x(j) values to arrange simultaneously all the species' data into the best possible Gaussian curves. The computer program that does this is "Cornell Ecology Program 8B. Gaussian ordination."
For the sake of efficiency, this procedure begins with only the most informative species, the number of which is supplied as an input parameter. Use of these few species produces rapid improvement while operating on a small matrix. When the rate of improvement drops (as discussed later), the entire data matrix is brought in; and when for this entire data matrix the rate of improvement drops, the final results are presented.
The essential task of Gaussian ordination may be stated in succinct formal terms as follows. Let x be the vector of ordination values, and yi the data vector of species importance values for species i, and Si the least-squares Gaussian function fitting the data of species i, where Si is
where Ai is the maximum, 
i
the mode, and 
i the standard deviation
of the above Gaussian equation for species i, and x is the independent
variable. The problem is then to accomplish the following minimization :
The inner minimization of the square of the L2 norm is identically the SSDi from a variation of parameters of the Gaussian equation, and has been worked out by Gauch and Chase (1974).
The outer minimization presents a complex problem, and is discussed in Appendix I.
In summary, each iteration has two basic parts. First the species are fitted to least-squares Gaussian curves, and then the x(j) values are moved to form better Gaussian curves. Roughly speaking, this means reducing the residual SSD of the system by alternating adjustments of the species' parameters and then the samples' ordination values, until the two-dimensional configuration (as in Fig. 1) maximizes the formation of Gaussian curves. Only x is an independent variable; that is, the goal of minimizing the SSD is to be accomplished only by changes (improvements) in the ordination values x. (The parameters of the Gaussian equations are dependent on the outer minimization.)
After an iteration is completed, the new x vector is compared with the previous x vector, and a decision is made. The options are to terminate unsuccessfully, continue iterating with the same number of species, continue iterating but add in more species, or terminate successfully. This decision is based essentially on the amount of change in the x vector and the amount of change in variance accounted for; the details are explained in the computer program and its accompanying documentation.
The program prints out the Gaussian parameters and sum of squared deviations for each species, and the current ordination values. Upon successful termination, it also punches out results if requested. The output for the final iteration includes the option of computing and punching out the residual matrix of observed minus expected values (with the values of the residual matrix rescaled into 0 to 100 for sake of convenience).
RESULTS
To evaluate the performance of Gaussian ordination, one may ask questions on three levels. First, what mathematical accuracy does the method have? Second, how does the accuracy of the results and the computer expense of Gaussian ordination compare with other ordination techniques? And third, how flexible are input data requirements and how clear and useful are ordination results, in applications to field data? To investigate these three questions, we have used both simulated and field data. Vegetation samples for which the location and composition of samples, as well as the parameters of the species, are known precisely were simulated by Cornell Ecology Program 1 (Gauch and Whittaker 1972a). Such simulated data are necessary for us to evaluate the first two questions; otherwise there is no exactly known null hypothesis against which to test or measure. Four sets of field data were used in evaluating the third question.
Certain arbitrary constants and limits on iterations were set in our program to have it terminate when approximately 99.9% to 99.99% of the possible variance is accounted for. This corresponds to an accuracy in ordination values of approximately a few tenths of one percent error, and the same for species parameters (unless the mode is outside the sampling interval, as discussed by Gauch and Chase 1974). This accurary is arbitrary, but we shall hereafter refer to results of this accuracy as "perfect recovery" of the structure of the input data.
The first question regarding mathematical accuracy was investigated by means of a number of simulated data matrices. Let us call errors in species importance values "y-scatter," and errors in ordination values "x-scatter." Using data with no x-scatter and no y-scatter, ordination values and species parameters are recovered perfectly. Using data with only x-scatter of typical amounts (e.g., 15% random noise), results are recovered perfectly after several iterations. Given y-scatter typical of field data (e.g., 15% random noise), results are altered because the real structure of the data in their independent variables has been altered. However, there is an averaging effect because each species has a number of datum points, so the error in species parameters is less than that in the individual importance datum values. Furthermore, in computing the ordination values there is a second and compounding averaging effect, in that the results from a number of species are averaged to obtain ordination values. Hence y-scatter alters the real, independent structure of the data, but Gaussian ordination has two averaging effects that enable it to hold up against sampling errors rather well. Given both x-scatter and y-scatter, the outcome is the same as if only the y-scatter were present; the program will recover perfectly from the x-scatter part of the error, and errors will be largely averaged out from the y-scatter part. Moreover, Gaussian ordination has recovered results perfectly even from random x vectors as first-guesses, but approximately twice as many iterations are needed as in the case of a fairly reasonable first-guess to reach the solution.
There is one exception to these statements. In some cases, with high beta diversity (above several half-changes) or very high noise, the program is dependent upon a reasonable first-guess to converge to an answer. We consider this limitation to result from our current algorithm for the outer minimization, and expect that this problem can be reduced by other means (Appendix I).
The second question concerns the accuracy and cost of Gaussian ordination as compared to other ordinations. For our simulated data, Gaussian ordination recovered perfectly as much structure as possible. Principal components analysis and Bray-Curtis ordinations have several percent error in ordination values for the same data; also, they are more vulnerable to sampling errors.
Computing times for Bray-Curtis and Gaussian ordinations are roughly proportional to the number of species times the number of samples; therefore, computing time increases linearly with both sample and species numbers. However, the computing time for Gaussian ordination is larger than that for Bray-Curtis ordination by a factor of about 25. By contrast, the computing time for principal components analysis increases roughly with the third power of the size of the real, symmetric matrix whose eigenvectors must be found (and may be taken to be the number either of species or of samples, whichever is smaller); an increase with the third power can rapidly become expensive. Concerning computer memory requirements, principal components analysis is most demanding (since a secondary variance-covariance matrix must also be stored), Gaussian ordination is intermediate, and Bray-Curtis ordination is very modest.
FIG. 2. Ordination positions of samples after Gaussian ordination (on the ordinate) compared with that in a weighted-average compositional ordination by Whittaker and Niering (1965) (on the abscissa). In the composite transect illustrated in Fig. 1, based on the weighted-average ordination, samples 1-4 became composite sample 1, and all other composite samples included five samples in the sequence as numbered (5-9, 10-14 . . . 45-49).
The third question concerns performance with field data, and has been evaluated by means of four sets of data: (1) The Santa Catalina Mountains trees shown in Fig. 1, lower panel, using 10 composite samples along the moisture gradient, (2) the original 49, 0.1-ha samples, as summarized in the first example, (3) ravine forests in the Finger Lakes Region of New York (Lewin 1973, 1974), extending from Acer saccharum-Fagus grandifolia bottom forests through Tsuga canadensis slopes to uplands with Quercus borealis and Q. alba, 33 tree species in 30 0.1-ha samples, and (4) a coastal terrace sequence in Mendocino County, California (West-man 1971, 1975) extending from Sequoia semper-virens forest through mixed evergreen and Pinus muricata forest to Cupressus pygmaea pygmy forest, 31 species of all strata by coverages in 61 samples. The ordination axis in the first two cases is the topographic moisture gradient with 10 values given by Whittaker; for the ravines, Lewin used six elevation positions; for Westman's data, a Bray-Curtis ordination and the soil pH were used. These values were used as the first-guesses for Gaussian ordination.
For Whittaker's summarized data, Gaussian ordination increased the variance accounted for from 86.7% to 94.3%, at the same time on the average halving the displacements of observed data values from the fitted Gaussian curves. The observed and expected species scores are given in Table 1.
TABLE 1. Distribution of species populations along a topographic moisture gradient as observed (roman numbers) in a composite transect, and as fitted (italic numbers) by a Gaussian ordination. The data are densities per hectare of trees in a transect for the 1,830-2,140 m elevation belt in the Santa Catalina Mountains, Arizona (Fig. 1, lower panel).
For the original samples from the Santa Catalina Mountains (set 2), Gaussian ordination accounts for 84.8% of the variance using Gaussian curves, whereas with the original (weighted average) moisture index values 48.0% of the variance was accounted for by Gaussian curves. A marked improvement in sample arrangement has thus resulted from the Gaussian ordination, even though the overall sequence of samples is similar (Fig. 2) and the relative positions of species are much the same in the two ordinations. Figure 3 illustrates the fits after Gaussian ordination for three species, two of which were well fitted and one (with apparently more irregular numbers in the samples) poorly fitted. The variance accounted for was higher, after a smaller number of iterations, for the 10 composite samples of set (1) than for the 49 original samples of set (2). Grouping and averaging samples into composite samples thus facilitates Gaussian ordination.
For Lewin's data, the increase in variance accounted for by using Gaussian ordination was only 2.6%, and most of the species' data look like the poor fit, rather than the good fits, in Fig. 3. The Gaussian ordination was, however, realistic in the central position of Tsuga along the gradient and the arrangement of other species as either more mesic or more xeric than Tsuga. The ravine forests have low beta diversity, and some species have bimodal distributions because of the strong predominance of Tsuga in the middle of this gradient. Lewin's data were chosen to be a difficult case; it is encouraging that Gaussian ordination of these data is feasible.
For Westman's data, soil pH arranges the samples in a manner allowing 22.1% of the variance to be accounted for by Gaussian curves; his Bray-Curtis ordination does better with 46.5%. Gaussian ordination beginning with the pH values as a first-guess achieves 65.8% variance accounted for; from the Bray-Curtis first-guess, 63.5% is achieved. In both cases improvement is trivial after about 15 iterations. Clearly, the pH values are a very crude first-guess, but are adequate as such for Gaussian ordination. The Gaussian ordinations represent the sequence of communities and major species correctly (as interpreted in the field by Westman). However, certain samples that were "deviant" in possessing high coverages of species present in very few other samples were thrown off to the extremes of the gradient, though these extremes do not seem (considering other species present) the best ordination positions for those samples. This, and the 64%-66% of variance accounted for, reflect the fact that the Mendocino data also are a relatively hard case for Gaussian ordination. Although there is a strong primary gradient from Sequoia to Cupressus forest, the samples are low in alpha diversity and are fire-affected. The "deviant" composition of some samples may result from either fire disturbance or other effects of environment that depart from the primary gradient.
FIG. 3. Fits of Gaussian curves to species distributions after Gaussian ordination, from the transect data used also in Fig. 2 and the lower panel of Fig. 1. The figure reproduces the computer graphs for three species: Quercus hypoleucoides (top), Pinus chihuahuana (middle) and Quercus ariwnica (bottom). The datum points are species importance values; the Gaussian curves were fitted by the Gaussian ordination computer program. The abscissa is in each case the 0-100 axis of Gaussian ordination; the ordinates are differently scaled for the three species, from zero to 186, 34, and 45 tree stems per 0.1 ha, respectively.
In all four cases, the variance accounted for by the model is good to high, suggesting the appropriate-ness of the Gaussian model for the vegetation patterns sampled. Gaussian ordination has computed all the parameters of the system as modeled, as well as given statistics to evaluate the fit of this model. Individual species exhibiting bimodality, skew, or generally poor fit to the model, may be noted in the output and studied to seek an ecological explanation for their departure from the usual case. The full models fitted to these sets of data could be used for other purposes, such as computation of various synthetic properties of the system (e.g., beta diversity and species relative importances), and prediction of stand compositions at unsampled points along the ordination axis.
DISCUSSION
From the tests using both simulated data (with and without noise), and field data, we judge that Gaussian ordination is workable and gives informative results. From a practical viewpoint, its ability to recover the structure of vegetation samples is rewarding. From a philosophical viewpoint, it combines the understanding of vegetation structure derived from direct gradient analysis, with objective determination of the major direction of variation in a set of community samples that is desired in indirect ordination. We offer it as one solution to the problems of distortion of sample positions that affect principal components and other linear models.
Alternative techniques for ordination without assuming linearity are discussed by Noy-Meir (1971a) as "catenations," and are based on local monoton-icities. He gives the definition: "catenation is suggested here as a collective name for all methods which are designed to order elements (e.g. sites, species) in continuous sequences, catenae, or on multidimensional surfaces defined by several such catenae, in a way which optimally accounts for local similarities." He traces the history of such methods, showing that several have been proposed since 1951; but their development has been meager and their theoretical significance unappreciated. His algorithm appears to escape most of the curvilinear distortion of sample positions characteristic of ordinations by principal components analysis; also it works on data with several axes of variation, provided that one axis predominates. Catenation is a fundamental concept for dealing with the non-linearity typical of vegeta-tional data (Noy-Meir 1971a, 1974).
Several comments may be made concerning future directions in ordination studies. First, the variety of data and landscapes, sampling design, amount of data collected, availability of computers, general viewpoints and prejudices, purposes of the analysis, and so on, may be expected to maintain a diversity of techniques of ordination (as well as of classification). Second, this diversity will make objective evaluation and comparison of these techniques important. Third, development of complex non-linear, multivariate algorithms must be expected. Fourth, simpler ordination techniques will have continued importance, especially for large data sets for which complex analyses, even by computers, may be too expensive. This should justify study of the performance and limitations of simple ordinations (as in Gauch 1973a). Finally, and most important, awareness should grow of the need to base or test ordination techniques against explicit models of vegetation structure. The formation of such models should draw heavily upon direct gradient analysis (Gauch and Whittaker 1972a). Of the complex algorithms, we suggest that Gaussian ordination and catenation show promise, and are worthy of further development and testing.
1This research was supported by a grant from the National Science Foundation. Manuscript received April 11, 1973; accepted April 22, 1974.
2Present address: Mathematics Department, Messiah College, Grantham, PA 17027.
ACKNOWLEDGMENTS
We appreciate helpful suggestions from J. E. Dennis, S. A. Levin, and T. R. Wentworth, and preparation of the manuscript by Beth Lanyon. We dedicate this work to the first-named author's father, Hugh Gauch, who retired recently from a distinguished career in plant physiology.
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