Showing papers on "Principal component analysis published in 1973"

Discarding Variables in a Principal Component Analysis. Ii: Real Data

Abstract: In this paper it is shown for four sets of real data, all published examples of principal component analysis, that the number of variables used can be greatly reduced with little effect on the results obtained. Five methods for discarding variables, which have previously been successfully tested on artificial data (Jolliffe, 1972), are used. The methods are compared and all are shown to be satisfactory for real, as well as artificial, data, although none is shown to be overwhelmingly superior to the others.

Information Extraction, SNR Improvement, and Data Compression in Multispectral Imagery

Abstract: The Karhunen-Loeve transformation is applied to multispectral data for information extraction, SNR improvement, and data compression. When applied in the spectral dimension, the transform provides a set of uncorrelated principal component images very useful in automatic classification and human interpretation. Significant improvements in SNR and estimates of the noise variance are also shown to be possible in the spectral dimension. Data compression results using the transform on one-, two-, and three-dimensional blocks over three general types of terrain are presented.

On the Investigation of Alternative Regressions by Principal Component Analysis

Abstract: SUMMARY In a multiple regression problem, let the p x 1 vector x consist of the dependent variable and p -1 predictor variables. The correlation matrix of x is reduced to principal components. The components corresponding to low eigenvalues may be useful in suggesting possible alternative subregressions. This possibility is analysed, and formulae derived for the derivation of subregressions from the principal components.

On the investigation of alternative regressions by principal component analysis.

Abstract: SUMMARY In a multiple regression problem, let the p x 1 vector x consist of the dependent variable and p -1 predictor variables. The correlation matrix of x is reduced to principal components. The components corresponding to low eigenvalues may be useful in suggesting possible alternative subregressions. This possibility is analysed, and formulae derived for the derivation of subregressions from the principal components.

Standard errors for rotated factor loadings

Abstract: Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.

The analysis of variation between and within genotypes and environments.

Abstract: Methods of analysing variation between and within genotypes and environments are discussed. Principal component analysis, or an equivalent technique proposed by Mandel (1971) for examining interactions in two-way tables, is suggested as an appropriate method in many circumstances, followed by analysis of variance on these principal components for replicated data. Various techniques are applied to yields of carrots from a trial in which eight varieties were grown in 34 environments representing a set of 17 site/year combinations at two densities. The largest source of variation within genotypes is found to be that between environments but not conversely. Two other sources of variation are identified within genotypes and environments, one representing the interaction of varieties with site/year effects and the other their interaction with densities. Analysis of variance indicates the varieties and environments contributing to these interactions. The general implications of the use of principal component analysis are discussed, particularly in situations such as that with the carrot data where the method of joint regression analysis fails because the genotype-environment interaction contains more than one independent component.

Identification of the Structure of Multivariable Stochastic Systems

Abstract: Publisher Summary This chapter discusses the problem of identifying the structure of a multivariable linear system when observations on the input and output are given. The approach presented in the chapter involves the reduction of the dimensions of the input and output vectors, and it is based on the application of principal components analysis to the frequency domain representations of the input and output processes, taking into account the loss function associated with the control of the system under study. Some results on tests of significance for the eigenvalues of a spectral density matrix are also included in the chapter. The chapter discusses the applications of principal components analysis to the problems of reducing the dimension of the input variables and reducing the dimension of the output variables. The appropriate criteria for both problems are quite different.

Principal components of random variables with values in a seperable hilbert space

Abstract: we extend the theory of principal components to random variabies X with values in a separable HILBERT space and prove optima! properties well-known for finite-dimen-sional spaces, Further we give a...

Region-Building — a Comparison of Methods

Abstract: This study deals with the use of quantitative methods to build up uniform or formal regions, the construction of farming-type regions in eastern Norway being used as an example. To achieve reasonable uniformity in the size of the basic areal units, some of the original administrative areas are first combined; the resulting areal units are then described by four orthogonal variables extracted by principal component analysis. Six different methods of aggregating the basic units to form regions are discussed, each method using the same measure of similarity (D2) between the basic units. The homogeneity of the regions produced by these methods is compared, and Ward's method (Ward 1963) shown to give the best (though not necessarily the optimal) solution.

Relationships Among Coefficients of Vectors Used In Principal Components

Abstract: This paper discusses, in terms of correlations, the relation between coefficients of the characteristic vectors used in principal component analysis, both within each vector and among vectors. These relationships will be illllstrated with examples (1) where the variates represent a battery of physical tests related to rocket testing and (2) where they represent equally spaced observations on a continuous curve, a case related to film measurements.

The use of principal components analysis in physical anthropology

Abstract: A principal components analysis of eighty fossil and modern hominid skulls employing a total of 25 measurements is presented The results are discussed briefly, but the central concern of this paper is to examine how the multivariate functions are arrived at and what they show in terms of biological function A number of sources of error are discussed deriving from the method of calculating the cross-products matrix, the incomplete representation of sample variance, the use of data sets involving more than one biological function, and the use of samples that are unevenly constructed It is concluded that more than one kind of cross-product matrix should be used in principal components analysis so that distortions that arise from the method of normalizing the cross-product matrix and the position of the origin of the new set of axes counter-balance each other; that data sets should never encompass more than one functional complex; and that constituent samples in the total sample matrix should be of equal size as far as possible It is also concluded that the number of variables that can be usefully employed in one data set is limited, by analogy with the law of diminishing returns

Thoughts on use of principal component analysis in petrogenetic problems

Abstract: The use of principal component analysis in studying chemical trends in volcanic rock suites is described. It is suggested that eigenvectors generated from a correlation matrix, rather than a covariance matrix, could be used in this context. In the latter situation many elements are swamped by silicon's numerical size and range. In the former situation the alkalies and titanium begin to show their true importance.

Likelihood ratio tests of certain hypotheses about principal components

Abstract: μ and covariance matrix.Σ. Let δ1≧ δ2≧ ... ≧ δp≧0 be the latent roots of Σ and γ1, γ2, ..., γp be a set of the corresponding orthonormal latent vectors of Σ. Then Σ=rar', where r=(γ1,...,γp)and ⊿=diag(δ1,...,δp)-The i-th prillcipal component of x is defined by Si=γ 乞'(x一 μ) The hypotheses considered here are H,k:γ1=γ10,..., rk=γko and δ1=δ10,... ,δk=δ κ0, Ha:γ1=γ10, Ha: γ1=γlo and δ2=...=δP=δ, whereγ10,...,γko are given orthonormal vectors and atio's are given values such thatδ10≧ ... ≧Ok0>0. In each case the alternatives are K1:Σ>0. We also consider to test the hypothesis Ha against the alternatives K2:δ2=...=δp=δ. Anderson [2] has given an asymptotic test for testing Ha against Ki under the assumption that r10 corresponds to a latent root of multiplicity one. Kshirsagar[4] considered a goodness of fit test for a hypothetical principal component in certain situations and gave exact tests for testing H3 against K1 whenδis known and for testing Ha against Ka. The procedures hitherto proposed for these hypotheses are different from ones by the likelihood ratio(=LR)method. The purpose of this

Principal Components Analysis for Virtually Unlimited Data Matrices

Abstract: THE object of this note is to point out that principal component and factor analyses can be performed on larger numbers of variables than is generally realized. Furthermore, this can be done with computers and subroutines that are currently available. Given a data set consisting of a large number of measurements on each subject, it is common to devise a small number of ad hoc scales in order to reduce the number of measurements to a more reliable

Principal component maps instead of trend surface maps for modal and chemical data of granitoid rock bodies

Abstract: Trend surface analysis may be an unsuitable method for the design of contour maps and of petrogenetic models for granitoid rock bodies. In granitoid bodies we are often dealing with several successive and superimposed petrogenetic processes, which lead to very complex regional patterns of modal or chemical data. Before drawing the contour map of a rock body it is of advantage, therefore, to separate the superimposed components of the complex pattern. The trend surface analysis allows only for the separation of the geometrically simple parts from those showing a more complex, or even random, distribution. Principal component analysis permits us to break multivariate data down into those components which are independent and result from different petrogenetic processes. For each of the most important eigenvectors which are calculated from the covariance matrix of data one contour map is drawn. The number of contour maps is therefore low and their interpretation relatively easy. The advantage of the principal comoponent analysis is demonstrated by its application to twelve granitoid rock bodies.