Showing papers on "Principal component analysis published in 1987"

An evaluation of the relative robustness of techniques for ecological ordination

Abstract: Simulated vegetation data were used to assess the relative robustness of ordination techniques to variations in the model of community variation in relation to environment. The methods compared were local non-metric multidimensional scaling (LNMDS), detrended correspondence analysis (DCA), Gaussian ordination (GO), principal components analysis (PCA) and principal co-ordinates analysis (PCoA). Both LNMDS and PCoA were applied to a matrix of Bray-Curtis coefficients. The results clearly demonstrated the ineffectiveness of the linear techniques (PCA, PCoA), due to curvilinear distortion. Gaussian ordination proved very sensitive to noise and was not robust to marked departures from a symmetric, unimodal response model. The currently popular method of DCA displayed a lack of robustness to variations in the response model and the sampling pattern. Furthermore, DCA ordinations of two-dimensional models often exhibited marked distortions, even when response surfaces were unimodal and symmetric. LNMDS is recommended as a robust technique for indirect gradient analysis, which deserves more widespread use by community ecologists.

Multi‐way principal components‐and PLS‐analysis

Abstract: The Lohmoller–Wold decomposition of multi-way (three-way, four-way, etc.) data arrays is combined with the non-linear partial least squares (NIPALS) algorithms to provide multi-way solutions of principal components analysis (PCA) and partial least squares modelling in latent variables (PLS). The decomposition of a multi-way array is developed as the product of a score vector and a loading array, where the score vectors have the same properties as those of ordinary two-way PCA and PLS. In image analysis, the array would instead be decomposed as the product of a loading vector and an image score matrix. The resulting methods are equivalent to the method of unfolding a multi-way array to a two-way matrix followed by ordinary PCA or PLS analysis. This automatically proves the eigenvector and least squares properties of the multi-way PCA and PLS methods. The methodology is presented; the algorithms are outlined and illustrated with a small chemical example.

Temporal encoding of two-dimensional patterns by single units in primate inferior temporal cortex. III: Information theoretic analysis

Abstract: Ablation and single-unit studies in primates have shown that inferior temporal (IT) cortex is important for pattern discrimination. The first paper in this series suggested that single units in IT cortex of alert monkeys respond to a set of two-dimensional patterns with complex temporal modulation of their spike trains. The second paper quantified the waveform of the modulated responses of IT neurons with principal components and demonstrated that the coefficients of two to four of the principal components were stimulus dependent. Although the coefficients of the principal components are uncorrelated, it is possible that they are not statistically independent. That is, several coefficients could be determined by the same feature of the stimulus, and thus could be conveying the same information. The final part of this study examined this issue by comparing the amount of information about the stimulus that can be conveyed by two codes: a temporal waveform code derived from the coefficients of the first three principal components and a mean rate code derived from the spike count. We considered the neuron to be an information channel conveying messages about stimulus parameters. Previous applications of information theory to neurophysiology have dealt either with the theoretical capacity of neuronal channels or the temporal distribution of information within the spike train. This previous work usually used a general binary code to represent the spike train of a neuron's response. Such a general approach yields no indication of the nature of the neuron's intrinsic coding scheme because it depends only on the timing of spikes in the response. In particular, it is independent of any statistical properties of the responses. Our approach uses the principal components of the response waveform to derive a code for representing information about the stimuli. We regard this code as an indication of the neuron's intrinsic coding scheme, because it is based on the statistical properties of the neuronal responses. We measured how much information about the stimulus was present in the neuron's responses. This transmitted information was calculated for codes based on either the spike count or on the first three principal components of the response waveform. The information transmitted by each of the first three principal components was largely independent of that transmitted by the others. It was found that the average amount of information transmitted by the principal components was about twice as large as that transmitted by the spike count.(ABSTRACT TRUNCATED AT 400 WORDS)

Multivariate Analysis of Ecological Communities

Abstract: 1 Ecological data.- 1.1 Types of data.- 1.2 Forms of data.- 1.3 Standardization and transformation of data.- 1.4 Constructing association data.- 2 Preliminary inspection of data.- 2.1 Displaying data values.- 2.2 Mapping.- 2.3 Displaying distributions of variables.- 2.4 Bivariate and multivariate displays.- 3 Ordination.- 3.1 Direct gradient analysis.- 3.2 Principal components analysis.- 3.3 Correspondence analysis.- 3.4 Ordination methods when rows or columns are grouped.- 3.5 Principal coordinates analysis.- 3.6 The horseshoe effect.- 3.7 Non-metric ordination.- 3.8 Case studies.- 4 Methods for comparing ordinations.- 4.1 Procrustes rotation.- 4.2 Generalized Procrustes analysis.- 4.3 Comparing ordination methods by multiple Procrustes analysis.- 5 Classification.- 5.1 Agglomerative hierarchical methods.- 5.2 Divisive hierarchical methods.- 5.3 Non-hierarchical classification.- 5.4 Visual displays for classification.- 5.5 Case study.- 5.6 Methods for comparing classifications.- 6 Analysis of asymmetry.- 6.1 Row and column plots.- 6.2 Skew-symmetry analysis.- 6.3 Case studies.- 6.4 A proof of the triangle-area theorem.- 7 Computing.- 7.1 Computing options.- 7.2 Examples of Genstat programs.- 7.3 Handling missing values.- 7.4 Conclusion.- 7.5 List of software.- References.- Appendix Matrix algebra.- A.1 Matrices and vectors.- A.2 Particular forms of matrices.- A.3 Simple matrix operations.- A.4 Simple geometry and some special matrices.- A.5 Matrix inversion.- A.6 Scalar functions of matrices.- A.7 Orthogonal matrices.- A.8 Matrix decompositions.- A.9 Conclusion.

Application of principal components analysis to change detection

Abstract: Principal components analysis (PCA) has been applied for land-cover change detection with multitemporal Landsat Multispectral Scanner (MSS) data. Previous work found that the higher order principal components were able to account for land-cover changes. In the computation of principal components, the eigenvectors used for transformation can be derived from a covariance matrix (with non-standardized data) or a correlation matrix (with standardized data); and from the total area or a subset area of specific land-cover types. In this paper, we examine the effect of using different types of matrices for principal components transformation with special emphasis on its application in landcover change detection in the Kitchener-Waterloo-Guelph area, Ontario, Canada. It is found that standardized principal components are more accurate than the non-standardized components because of their better alignment along landcover changes in the multiemporal data structure. Statistics extracted from the total study area are also better and more reliable than those extracted from the subset area. It is concluded that principal components analysis is scene dependent, and the use of this technique requires a careful appraisal of the eigenstructures and images of the principal components. Y = A\"'X • computation of the eigenvectors, and • linear transformation of the data set. 1 \" C = '\" (X -M)(X-M)·' \" K-l i~ I ,

A Comment on Shearing as a Method for “Size Correction”

Abstract: The "shear" method of Humphries et al. (1981) is based on a path model intended to explain differences in form by multiple factors: one for size and one or more for shape differences. Its adaptation for "removing" the effects of a within-population size-factor from between-group morphometric analyses is presented in compact matrix form, simplified, and compared to the method of orthogonal projection proposed by Burnaby (1966). While the size- correction methods give similar results for most real data sets, Burnaby's procedure with k = 1 (i.e., using a single composite size variable) is recommended for this purpose owing to its geometrical and computational simplicity. An example based on artificial data demonstrates that sheared principal components are not necessarily uncorrelated with size. Path modeling of size and shape together is a different purpose than size-correction, and is better served by a different procedure. (Allometric growth; morphometrics; size; shape; principal components analysis.)

Temporal encoding of two-dimensional patterns by single units in primate inferior temporal cortex. II: Quantification of response waveform

Abstract: The purpose of this study was to describe how the responses of neurons in inferior temporal (IT) cortex represent visual stimuli. In the preceding paper we described the responses of IT neurons to a large set of two-dimensional black and white patterns. The responses to different stimuli showed temporal modulation of the spike trains. This paper develops a method for quantifying temporal modulation and shows that the stimulus determines the distribution over time, as well as the number, of spikes in a response. The responses were quantified using an orthogonal set of temporal waveforms called principal components. The principal components related to each neuron were extracted from all the responses of that neuron to all of the stimuli, regardless of which stimulus elicited which response. Each response was then projected onto the set of principal components to obtain a set of coefficients that quantified its temporal modulation. This decomposition produces coefficients that are uncorrelated with each other. Thus each coefficient could be tested individually, with univariate statistics, to determine whether its relation to the stimulus was nonrandom. The waveforms of the principal components are unconstrained and depend only on the responses from which they are derived; hence, they can assume any shape. Nonetheless, the 21 neurons we analyzed all had principal components that belonged to only one of two sets. The two sets could be characterized by their first principal component, which was either phasic or tonic. This suggests that these neurons may use as few as two different mechanisms in generating responses. The first principal component was highly correlated with spike count, and both were driven by the stimulus. Higher principal components were uncorrelated with spike count, yet some of them were also driven by the stimulus. Thus the principal components form a richer description of the stimulus-dependent aspects of a neuronal response than does spike count. Bootstrap tests showed that several principal components (usually 3 or 4) were determined by the stimulus. Since higher principal components were not correlated with the spike count, the stimulus must have determined the distribution of spikes in the response as well as their number. However, it is possible that the number and distribution of spikes are both determined by the same characteristics of the stimulus. In this case, the temporal modulation would be redundant, and a simple univariate measure would be sufficient to characterize the stimulus-response relationship.(ABSTRACT TRUNCATED AT 400 WORDS)

Cross-Validation in Principal Component Analysis

Abstract: SUMMARY This paper describes a form of cross-validation, in the context of principal component analysis, which has a number of useful aspects as regards multivariate data inspection and description. Topics covered include choice of dimensionality, identification of influential observations, and selection of important variables. The methods are motivated by and illustrated on a well-known data set. 1. Data Set and Objectives Jeffers (1967) described two detailed multivariate case studies, one of which concerned 19 variables measured on each of 40 winged aphids alatee adelges) that had been caught in a light trap. The 19 variables are listed in Table 1. Principal component analysis (PCA) was used to examine the structure in the data, and if possible to answer the following questions: (i) How many dimensions of the individuals are being measured? (ii) How many distinct taxa are present in the habitat? (iii) Which variables among the 19 are redundant for distinguishing between taxa, and which must be retained in future work? Of the 19 variables, 14 are length or width measurements, four are counts, and one (anal fold) is a presence/absence variable scored 0 or 1. In view of this disparity in variable type, Jeffers elected to standardise the data and thus effect the PCA by finding the latent roots and vectors of the correlation (rather than covariance) matrix of the data. The elements of each latent vector provide the coefficients of one of 19 linear combinations of the standardised original variables that successively maximise sample variance subject to being orthogonal with each other, and the corresponding latent root is the sample variance of that linear combination. The 19 observations for each aphid were subjected to each of these 19 linear transformations to form the 19 principal component scores for that aphid. The above questions were then answered as follows: (i) The latent roots of the correlation matrix were as given in Table 1. The four largest comprise 73.0%, 12.5%, 3.9%, and 2.6%, respectively, of the total variance (19.0) of the standardised variables; the dimensionality of the data was therefore taken to be 2. (ii) When the scores of the first two principal components for the 40 aphids were plotted against orthogonal axes, the resulting 40 points divided into four groups as shown in Figure 1. Hence, four distinct species were identified for the aphids. (iii) From consideration of the size of coefficients in the first three principal components, it was concluded that only the four variables length of tibia, number of ovipositor

Statistical analysis for the angular central Gaussian distribution on the sphere

Abstract: SUMMARY The angular central Gaussian distribution is an alternative to the Bingham distribution for modeling antipodal symmetric directional data. In this paper the statistical theory for the angular central Gaussian model is presented. Some topics treated are maximum likelihood estimation of the parameters, testing for uniformity and circularity, and principal components analysis. Comparisons to methods based upon the sample second moments are made via an example.

Theory of the distribution of error eigenvalues resulting from principal component analysis with applications to spectroscopic data

Abstract: The distribution of error eigenvalues resulting from principal component analysis is deduced by considering the decomposition of an error matrix in which the errors are uniformly distributed. The derived probability function is Where λ0j is the jth error eigenvalue, r and c are the numbers of rows and columns in the data matrix, and N is the normalization constant. This expression is tested and validated by investigations involving model data. The distribution function is used to determine the number of factors responsible for various sets of spectroscopic data taken from the chemical literature (including nuclear magnetic resonance, infrared and mass spectra).

Two generalizations of the common principal component model

Abstract: SUMMARY Under the common principal component model the covariance matrices Ti of k populations are assumed to have identical eigenvectors, that is, the same orthogonal matrix diagonalizes all Ti simultaneously. This paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all Ti,, while the remaining p - q eigenvectors are specific in each group. This is called a partial common principal component model. A related modification assumes that q eigenvectors of each matrix span the same subspace, a problem that was first considered by Krzanowski (1979). For both modifications this paper derives the normal theory maximum likelihood estimators. It is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available. The methods are illustrated by numerical examples.

Rotation of principal components: a reply

Abstract: Jolliffe's comments raise some interesting and important points concerning both unrotated principal component analysis and rotated principal component analysis that deserve further consideration. These are examined herein, with particular attention to my original review of rotation.

Rotation of principal components: Some comments

Abstract: A recent paper by Richman (1986) provides a very useful review of the ideas of rotation in principal component analysis, in the context of climatology. However, the paper also raises a number of points which are open to discussion. The present note comments on a number of these points, and aims to give a balanced view of the usefulness of principal component analysis, both with and without rotation, in climatology.

A regionalization of Austria's precipitation climate using principal component analysis

Abstract: S-mode principal component analysis (PCA) is performed on correlation matrices of precipitation data of Austria for summer and winter half-year totals for the period 1951–1980. Application of the dominant-variance selection Rule N (Overland and Preisendorfer, 1982) proves two or three eigenvalues to be significant, where three principal components (PCs) on average account for 68.3 and 79.4 per cent of total variance in the summer and winter half-years, respectively. The Varimax-rotated PC loadings allow for a subdivision of Austria into three homogeneous (with respect to the underlying processes) regions. Intercomparison of the PC primary patterns, variances and time series of the principal components of three different networks confirms the spatial stability of these regions and turns attention towards seasonal differences. The three PCs are identified by assignment of large-scale weather types to them which are known to be precipitation producing in the region wherein the respective PC is dominant. The results of this study and their wide variety of applicability reveal PCA and Rule N as useful tools in identifying homogeneous groups of variables which can be ascribed a physical meaning. The paper contains a short account of the analysis of empirical orthogonal functions, the theory of PCA and its connection to factor analysis.

An analysis of international exchange rates using multivariate DLM's

Abstract: New models for multiple time series are introduced and illustrated in an application to international currency exchange rate data. The models, based on matrix-variate normal extensions of the dynamic linear model (DLM), provide a tractable, sequential procedure for estimation of unknown covariance structure between series. A principal components analysis is carried out providing a basis for easy model assessment. A practically important elaboration of the model incorporates time- variation in covariance matrices.

Component analysis of spatial and spectral patterns in multispectral images. I: Basis

Abstract: A new (to our knowledge) theory of component pattern analysis in multispectral images is developed by using the methods of principal component analysis and nonlinear optimization with a nonnegativity constraint. Given images of a scene in different color bands, we estimate both the spectral curves of components included in the image and the spatial pattern corresponding to each spectral curve. In this method, neither spatial nor spectral features of the components are necessary, but the physical rule of nonnegative absorptivity and density nonnegativity is used for any material of any optical frequency at any position in the image. Experimental results of component analysis with real microscopic image data are shown to demonstrate the effectiveness of the proposed method.

An Evaluation of the Effects of Variable Sampling On Component, Image, and Factor Analysis.

Abstract: An attribution of a latent variable procedure, such as factor analysis, that is sometimes cited is superior generalization from the variables sampled to the population of variables. Principal component analysis, image component analysis, and maximum likelihood factor analysis were compared to assess the effects of variable sampling. Three other independent variables were manipulated: (a) the size of the loadings or saturation, (b) the average number of variables per factor, and (c) the number of variables sampled. Five different correlation matrices were generated for each condition. Ten different variable samples were randomly selected. The sample pattern was then compared to the appropriate population pattern. The effects of method of analysis were relatively minor and very complex. The results with respect to degree of saturation and average number of variables per factor were clear and dramatic. Differential effects on boundary cases and nonconvergence problems were also found.

A software defoliant for geological analysis of band ratios

Abstract: Vegetation impedes the geological analysis of band ratio images, because it is both widely distributed in the surficial environment and can be spectrally similar to ferric oxides and clays when sampled by broad-band imaging devices. We address this problem by a technique we call ‘directed principal component analysis’ (DPCA) that involves calculating principal components on two input band ratio images. One ratio is a geological discriminant, conTused by the presence of vegetation; the second ratio is chosen for its suitability as a vegetation index. Once computed, the second DPC has the properties of a geological discriminant, but is less influenced by vegetation. The effects of vegetation, which are strongly correlated between the two input ratios, contribute chiefly to DPC#1. This simple method, applied selectively to airborne thematic mapper data, substantially reduces the effects of vegetation.

On the Interface between Cluster Analysis, Principal Component Analysis, and Multidimensional Scaling

Abstract: This paper shows how methods of cluster analysis, principal component analysis, and multidimensional scaling may be combined in order to obtain an optimal fit between a classification underlying some set of objects 1,…,n and its visual representation in a low-dimensional euclidean space ℝs. We propose several clustering criteria and corresponding k-means-like algorithms which are based either on a probabilistic model or on geometrical considerations leading to matrix approximation problems. In particular, a MDS-clustering strategy is presented for-displaying not only the n objects using their pairwise dissimilarities, but also the detected clusters and their average distances.

Some additional results on principal components analysis of three-mode data by means of alternating least squares algorithms

Abstract: Kroonenberg and de Leeuw (1980) have developed an alternating least-squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. The present paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC model.

A multivariate solution for cyclic data, applied in modelling locomotor forces

Abstract: The variability in dependent biological data (measured as forces at the human foot during pedalling) was reduced, by principal components analysis, to two major components which, combined, accounted for 46% of the variability observed in sets of 26 observations per cycle. On the basis of weighting over the cycle, the components were interpreted as due to Power Production and to Phase Switch from powergeneration to recovery. Force measurements were made for three frequencies of leg pedalling (1.00, 1.66, and 2.33 Hz at 10 N ergometer resistance). For each principal component, the mean (or summed) deviation over 75 consecutive cycles was found to increase linearly (p<0.05) with velocity of leg movement, by 7% and 85% respectively. Analysis of autocorrelations over cycles of movement showed that, in contrast to the strong interconnections of force measures within a cycle of movement, between-cycle dependence was very low. The statisticaltechnique described provides a useful descriptive and inferential method for analyzing dependent cyclic data. The resulting model of the locomotor forces generated at the foot implies that control of power output is substantially for one cycle of a limb and that variability of force increases at the point when the leg switches from power-generation to recovery.

Application of principal component analysis to the prediction of lucerne forage protein content and in vitro dry matter digestibility by NIR spectroscopy

Abstract: Application of principal component regression (PCR) was proposed for the development of a prediction equation of forage composition by near infra-red spectroscopy. PCR involves two steps: (a) the creation of new synthetic variables by principal component analysis (PCA) of spectral data, and (b) multiple linear regression with these new variables. Results obtained by this procedure have been compared with those generated by the conventional application of multiple linear regression (MLR) on spectral data. The comparison used the determination of protein content and in vitro dry matter digestibility (IVDMD) in 345 samples of lucerne forages. For protein determination, results of both procedures were quite similar (correlation coefficients: 0.978 and 0.980; standard errors of calibration: 0.86 and 0.84% DM; standard errors of prediction: 0.81 and 0.80% DM respectively for MLR and PCR prediction equations). The same was observed for IVDMD determination (correlation coefficients: 0.942 and 0.951; standard errors of calibration: 1.89 and 1.71% DM; standard errors of prediction: 2.22 and 2.22% DM, respectively). A large number of PCA variables were necessary for an accurate prediction of both constituents. The influence of the number of regression terms introduced in the PCR equation has been studied. The criterion for stopping the introduction of new terms in PCR did not seem as critical as in MLR.

A Cautionary Note on the Use of Principal Components Analysis: Supportive Empirical Evidence

Abstract: In a recent edition of this journal, Borgatta et al. (1986), using hypothetical data, illustrated how the results produced by principal components analysis can be substantially different from those of common factor analysis. The present article, using seven well-known data sets, extends their work into the empirical domain, and also compares the results of the maximum likelihood factor analysis model with those of the principal components model. The results strongly support those of Borgatta et al. Indeed, the discrepancies in the empirical results reported here are often larger than their hypothetical example suggests. It was found that, when comparing the performance of the principal components model with the common factor and maximum likelihood models, differences can be expected to occur in (1) the magnitudes of the factor loadings, (2) the signs attached to the factor loadings, and, most important, (3) the interpretation of the factors themselves.