Showing papers on "Principal component analysis published in 1984"

Multivariate Analysis: Methods and Applications

Abstract: Selected Aspects of Multivariate Analysis. Principal Components Analysis. Factor Analysis. Multidimensional Scaling. Cluster Analysis. Multiple Regression. Some Practical Considerations: Data Analysis Problems. Cross-Classified Frequency Data. Canonical Correlation Analysis. Discriminant Analysis: The Two-Group Problem. Multiple Discriminant Analysis and Related Topics. Linear Structural Relations (LISREL). Latent Structure Analysis. Appendixes. References. Index.

Multivariate Data Analysis in Chemistry

Abstract: Any data table produced in a chemical investigation can be analysed by bilinear projection methods, i. e. principal components and factor analysis and their extensions. Representing the table rows (objects) as points in a p-dimensional space, these methods project the point swarm of the data set or parts of it down on a F-dimensional subspace (plane or hyperplane). Different questions put to the data table correspond to different projections.

The use of biplots in interpreting variety by environment interactions

Abstract: Gabriel (1971) proposed a technique for displaying the rows and columns of a twoway table as a two-dimensional biplot so that any element of the table can be approximated by the inner product of vectors corresponding to the appropriate row and column. The technique is useful for investigating the pattern of response of varieties over different environments, and substantially increases the information available from the more familiar methods of regression and principal component analysis without need for additional computation.

Complex Principal Component Analysis: Theory and Examples

Abstract: Complex principal component (CPC) analysis is shown to be a useful method for identifying traveling and standing waves in geophysical data sets. Combinations of simple progressive and standing oscillations are used to examine the properties of this technique. These examples illustrate that although CPC analysis allows for the identification of traveling waves, many of the drawbacks associated with conventional principal component analysis remain, and sometimes become worse; e.g. the interpretation of CPC solutions is more difficult since both amplitude and phase relationships must be considered. A method for linearly transforming complex principal components was devised in order to identify regional relationships within large geophysical data sets. The errors in CPC analysis resulting from limited sample sizes are discussed.

Common Principal Components in k Groups

Abstract: This article generalizes the method of principal components to so-called “common principal components” as follows: Consider the hypothesis that the covariance matrices Σ i for k populations are simultaneously diagonalizable. That is, there is an orthogonal matrix β such that β' Σ i β is diagonal for i = 1, …, k. I derive the normal-theory maximum likelihood estimates of the common component Σ i matrices and the log-likelihood-ratio statistics for testing this hypothesis. The solution has some favorable properties that do not depend on normality assumptions. Numerical examples illustrate the method. Applications to data reduction, multiple regression, and nonlinear discriminant analysis are sketched.

Principal Curves and Surfaces

Abstract: : Principal curves are smooth one dimensional curves that pass through the middle of a p dimensional data set. They minimize the distance from the points, and provide a non-linear summary of the data. The curves are non- parametric and their shape is suggested by the data. Similarly, principal surfaces are two dimensional surfaces that pass through the middle of the data. The curves and surfaces are found using an iterative procedure which starts with a liner summary such as the usual principal component line or plate. Each successive iteration is a smooth or local average of the p dimensional points, where local is based on the projections of the points onto the curve or surface of the previous iteration. A number of linear techniques, such as factor analysis and errors in variables regression, end up using the principal components as their estimates (after a suitable scaling of the co-ordinates). Principal curves and surfaces can be viewed as the estimates of non-linear generalizations of these procedures. Principal Curves (or surfaces) have a theortical definition for distributions: they are the Self Consistent curves. A curve is self consistent if each point on the curve is the conditional mean of the points that project there. The main theorem proves that principal curves are critical values of the expected squared distance between the points and the curve. Linear principal components have this property as well; in fact, we prove that if a principal curve is straight, then it is a principal component. These results generalize the usual duality between conditional expectation and distance minimization. We also examine two sources of bias in the procedures, which have the satisfactory property of partially cancelling each other.

Analysis of Variance Models in Orthogonal Designs

Abstract: Summary This paper presents an approach to analysis of variance modelling in designs where all factors are orthogonal, based on formal mathematical definitions of concepts related to factors and experimental designs. The structure of an orthogonal design is described by a factor structure diagram containing the information about nestedness relations between the factors. An orthogonal design determines a unique decomposition of the observation space as a direct sum of orthogonal subspaces, one for each factor of the design. A class of well-behaved variance component models, stated in terms of fixed and random effects of factors from a given design, is characterized, and the solutions to problems of estimation and hypothesis testing within this class are given in terms of the factor structure diagram and the analysis of variance table induced by the decomposition.

Bird numbers and habitat characteristics in farmland hedgerows

Abstract: A statistical analysis is presented of the relationship between bird numbers and the charecteristics of forty-two hedges on a Dorset farm. Multiple regression, principal components analysis and regression of actors, and canonical correlation analysis were used and compared.

Principal component analysis of flexible systems--Open-loop case

Abstract: A generic class of flexible systems, characterized by finitely many lightly damped harmonic oscillators, is analyzed by means of the "open-loop principal component analysis," that is, singular value analysis and Gramian balancing. As the main result, it is shown that, as the damping ratio goes to zero, the balanced state coordinates are decoupled and coincide with the modal coordinates. Further, simple formulas expressing the "asymptotic singular values" as functions of the modal parameters are derived.

Reducing the dimensionality of compositional data sets

Abstract: The high-dimensionality of many compositional data sets has caused geologists to look for insights into the observed patterns of variability through two dimension-reducing procedures: (i)the selection of a few subcompositions for particular study, and (ii)principal component analysis. After a brief critical review of the unsatisfactory state of current statistical methodology for these two procedures, this paper takes as a starting point for the resolution of persisting difficulties a recent approach to principal component analysis through a new definition of the covariance structure of a composition. This approach is first applied for expository purposes to a small illustrative compositional data set and then to a number of larger published geochemical data sets. The new approach then leads naturally to a method of measuring the extent to which a subcomposition retains the pattern of variability of the whole composition and so provides a criterion for the selection of suitable subcompositions. Such a selection process is illustrated by application to geochemical data sets.

Principal Component Analysis in the Presence of Group Structure

Abstract: SUMMARY A nested series of hypotheses on dispersion structure is identified when observations are grouped in a multivariate sample. A simple method of estimation is suggested for one of these hypotheses, and results using this method are compared with those previously obtained by maximum likelihood methods. Using these hypotheses, an analogy may be drawn between comparison of principal components between groups and comparison of regressions between groups.

Morphometric Variation in Introduced Populations of the Common Myna (Acridotheres tristis): An Application of the Jackknife to Principal Component Analysis

Abstract: -Morphometric variation in samples of common mynas from 11 localities in Australia, New Zealand, Fiji and Hawaii was analyzed with principal component analysis. We attempt to avoid two shortcomings in many previous applications of principal component analysis in morphometrics by analyzing separately variation within and among populations, and by applying the jackknife procedure to reduce subjectivity in interpretation of the principal components. Within populations, only principal component I appears to have a stable orientation and this orientation is common to all localities. Component II may be a simple vector, differing across localities, but similarity in the second and third eigenvalues warns that the associated components could be largely arbitrary. The variance along component I does differ across localities. Among populations, again only the first component displays convincing stability, although component II may be a stable but extremely simple vector. Both within and among populations, component I appears to be a general factor representing size and size-related shape variation. The apparently simple patterns of covariation displayed by the introduced populations may be attributable to bottlenecks in small founding populations and the short time since the introductions were made. Future studies should incorporate some form of testing to confirm putative patterns of character covariation, and doing so probably will require sample sizes much larger than has been the custom. [Skeletal morphometrics; principal component analysis; jackknife; population variation; mynas.] A general justification for a morphometric approach to evolutionary studies arises from the common belief that morphology is a primary and direct means by which organisms interact with their environment; variation in size and shape can have physiological and mechanical consequences (Gould, 1966; Alexander, 1968, 1971; McMahon, 1973, 1975; Pedley, 1977). Support for this belief tends to come from large-scale phenomena. Morphological differences among species, for example, are related convincingly to functional differences (e.g., Lack, 1947; Bock, 1970; Liem, 1973; Abbott et al., 1977). It then is assumed generally that such correspondences apply continuously through finer scales of variation. Thus, we arrive at an assumption of morphometricians that measured differences among individuals have functional consequences to the organisms involved, though empirical support for this assumption is rather sparse (but see Grant et al. [1976] and references therein; Herrera, 1978). By further extension, morphological differences among geographically separated populations are believed not only to affect function but also to correlate with environmental differences, leading to generalizations such as the ecogeographic rules. Finally, it is assumed that both withinand amongpopulation variation is adaptive and, therefore, has been crafted by natural selection. Correlation of these two sources of variation is a central prediction of the synthetic theory of evolution (Sokal, 1978). Our goal herein is to learn how skeletal variation is organized within populations of birds, and whether the pattern of this variation is involved in among-population differentiation. Additionally, we wish to describe the pattern of among-population variation and to determine its relationship to within-population structure. Principal component analysis is a common analytical approach here, but many applications have incorporated two methodological

Morphometric variation in introduced populations of the common myna (acridotheres tristis): an application of the jackknife to

Abstract: Morphometric variation in samples of common mynas from 11 localities in Austra- lia, New Zealand, Fiji and Hawaii was analyzed with principal component analysis. We attempt to avoid two shortcomings in many previous applications of principal component analysis in morphometrics by analyzing separately variation within and among populations, and by ap- plying the jackknife procedure to reduce subjectivity in interpretation of the principal compo- nents. Within populations, only principal component I appears to have a stable orientation and this orientation is common to all localities. Component II may be a simple vector, differing across localities, but similarity in the second and third eigenvalues warns that the associated components could be largely arbitrary. The variance along component I does differ across lo- calities. Among populations, again only the first component displays convincing stability, al- though component II may be a stable but extremely simple vector. Both within and among populations, component I appears to be a general factor representing size and size-related shape variation. The apparently simple patterns of covariation displayed by the introduced popula- tions may be attributable to bottlenecks in small founding populations and the short time since the introductions were made. Future studies should incorporate some form of testing to confirm putative patterns of character covariation, and doing so probably will require sample sizes much larger than has been the custom. (Skeletal morphometrics; principal component analysis; jack- knife; population variation; mynas.) A general justification for a morpho- metric approach to evolutionary studies arises from the common belief that mor- phology is a primary and direct means by which organisms interact with their en- vironment; variation in size and shape can

Factorial Kriging Analysis: A Substitute to Spectral Analysis of Magnetic Data

Abstract: Factorial kriging analysis is a new method which combines kriging analysis and principal component analysis into the framework of geostatistics. The variables are split into principal components corresponding to different frequency ranges.

Principal components, factor analysis, and multivariate allometry: a small-sample direction test

Abstract: A small-sample procedure is proposed for testing hypotheses about the direction of multivariate normal principal axes. Any specified direction is acceptable as a principal axis if its sample multiple correlation with the other unspecified principal axes does not reach statistical significance. Unlike the asymptotic criterion of Anderson (1963, Annals of Mathematical Statistics 34, 122-148), the present criterion does not associate the specified vector with any particular characteristic root of the sample covariance matrix, but the closeness of hypothetical and observed vectors can be estimated by evaluation of angles. The direction test can also be applied approximately to a single general factor in factor analysis. The present procedure is illustrated and compared with Anderson's procedure in an analysis of trivariate allometry in the skeletal dimensions of 68 male martens.

Exploring Multivariate Data using the Minor Principal Components

Abstract: The minor principal components of a random vector x are the orthonormal linear combinations of the variables in x which have minimum variance. This fact may be used to detect interrelations amongst the variables as well as to identify those variables which can be predicted with high accuracy from the others. Application of these ideas to the variable selection problem in multiple regression and to the identification of outliers in multivariate data are discussed.

Mathematical models of the event-related potential.

Abstract: : In electrophysiology, the Event Related Potential is assumed to be composed of several underlying component wave forms. Principal Component Analysis is a statistical technique that has been used to uncover the components by analysis of the observed wave form. The mathematical assumptions behind Principal Component Analysis are examined, and their plausibility is questioned. It is pointed out that under certain conditions the component forms may not accurately be recovered by Principal Component Analysis. Under other circumstances violations of some of the mathematical assumptions does not appear to affect the accuracy of recovery of component waveforms. The points made are illustrated by an analysis of simulated wave forms constructed from known components.

Eigenvalue Shrinkage in Principal Components Based Factor Analysis

Abstract: The concept of shrinkage, as (1) a statistical phe nomenon of estimator bias, and (2) a reduction in ex plained variance resulting from cross-validation, is ex plored for statistics based on sample eigenvalues. Analytic solutions and previous research imply that the magnitude of eigenvalue shrinkage is a function of the type of shrinkage, sample size, the number of vari ables in the correlation matrix, the ordinal root posi tion, the population eigenstructure, and the choice of principal components analysis or principal factors analysis. Hypotheses relating these specific indepen dent variables to the magnitude of shrinkage were tested by means of a monte carlo simulation. In par ticular, the independent variable of population eigen structure is shown to have an important effect on shrinkage. Finally, regression equations are derived that describe the linear relation of population and cross-validated eigenvalues to the original eigenvalues, sample size, ordinal position, and the number of vari ables facto...

Data transformation as a scaling operation in ordination of plankton

Abstract: Data transformation is seen here as an aspect of scaling such that we are less interested in the quirks and properties of each transformation but are more concerned with the general scaling properties and trends of suites of transformations. Over two years of daily phytoplankton abundance data for 30 species from a temperate lake (Llyn Maelog, North Wales) were subjected to a series of scale-ordered transformations. Two major classes of transformation were systematically varied: binary and smoothing. Binary transformation scaled the cutoff threshold between ‘presence’ and ‘absence’ of a species to various levels of abundance. With successively smaller universes and smoothing windows and successive species exclusion, ordinations of sample dates revealed smaller scaled structures in the order: annual cycles of species turnover, seasonal areas of attraction and uniqueness of individual sample dates. Gradual increases in the length of the smoothing window resulted in gradual shifts in the positions of points in sample data ordination, but not necessarily in the species ordinations. Thus sample data structures are more stable with change in scale than are species data structures. These differences in stability are discussed in the context of filtering characteristics of data collection and data analysis. Transformations producing similar species statistics (means, variances and skews) did not generally give similar ordination results, while some transformations giving similar ordinations differed in species statistical parameters.

Three-mode principal component analysis illustrated with an example from attachment theory

Abstract: The three-mode principal component model—here referred to as the Tucker3 model—was first formulated within the context of the behavioral sciences by Ledyard Tucker (1963). In subsequent articles, Tucker extended the mathematical description and its programming aspects (1964, 1966). In the context of multidimensional scaling, references to his model occur frequently (Carroll and Chang 1972; Takane, Young, and de Leeuw 1977; Jennrich 1972), as the Tucker3 is the general model comprising many other individual differences models. A discussion of the relation between multidimensional scaling and three-mode principal component analysis can be found in Tucker (1972), Carroll and Wish (1974), Takane, Young, and de Leeuw (1977), and Carroll and Arabie (1980). Other approaches to three-mode analysis include threemode common factor analysis within the context of linear structural equation models (Bloxom 1968; Bentler and Lee 1978, 1979; Law and Snyder 1981). Sands and Young (1980) presented a restricted form of three-mode principal component analysis in the spirit of Harshman's PARAFAC2 model (1972), but they included an optimal scaling phase in their algorithm to accommodate data with lower measurement levels, missing data, and different data conditionalities (see also Young 1981). In this chapter, we first present the three-mode principal component model. on a conceptual level by providing various informal ways of looking at it. Secondly, we provide an outline of some technical aspects connected with analyzing this type of model. Finally, an example treating data from attachment theory

The climatic regions of New Brunswick: a multivariate analysis of meteorological data

Abstract: New Brunswick was divided into 11 climatic regions by means of three multivariate statistical analyses (principal component analysis, R and Q type, and cluster analysis) of data on precipitation, v...

The use of projections for dimensionality reduction of flow cytometric data

Abstract: In order to visualize multiparameter flow cytometric measurements it is desirable to reduce the dimensionality of the data to two while preserving important features of each data pattern. We describe a method that projects two-parameter data onto a straight line that forms an angle theta with one of the original parameter axis. The angle is selected interactively or by principal component analysis. The procedure is applied on-line or off-line to pairs of parameters of the original data set that has been collected in LIST mode.

A state space approach for the 2-D harmonic retrieval problem

Abstract: In this paper, a State Space approach to the 2-D harmonic retrieval problem is presented. Under certain assumptions, it is shown that the data and covariance matrices have finite rank and also possess a desirable algebraic structure. Then methods employing the Principal components algorithm are developed to estimate the state space parameters and the sinusoid parameters directly from the data and from the covariance information. Simulation results to support the methods are also provided.