A generalization of extendedQ-mode factor analysis to data matrices with variable row sums

The extended method of Q-mode factor analysis developed by Miesch for data matrices with constant row sums is generalized to data matrices with variable row sums. With the algorithm provided it is possible to compute factor scores in the metric of the original data and compute goodness-of-fit statistics and model geological systems unconstrained by constancy of row sums of data points.     

Dual extraction ofR-mode andQ-mode factor solutions

It is mathematically possible to extract both R-mode and Q-mode factors simultaneously (RQ-mode factor analysis)by invoking the Eckhart-Young theorem. The resulting factors will be expressed in measures determined by the form of the scalings that have been applied to the original data matrix. Unless the measures for both solutions are meaningful for the problem at hand, the factor results may be misleading or uninterpretable. Correspondence analysis uses a symmetrical scaling of both rows and columns to achieve measures of proportional similarity between objects and variables. In the literature, the resulting similarity is a χ ² distance appropriate for analysis of enumerated data, the original application of correspondence analysis. Justification for the use of this measure with interval or ratio data is unconvincing, but a minor modification of the scaling procedure yields the profile similarity, which is an appropriate measure. Symmetrical scaling of rows and columns is unnecessary for RQ-mode factor analysis. If the data are scaled so the minor product W'Wis the correlation matrix, the major product WW'is expressed in the Euclidean distances between objects. Therefore, RQ-mode factor analysis can be performed so that the Rmode is a principal components solution and the Qmode is a principal coordinates solution. For applications where the magnitudes of differences are important, this approach will yield more interpretable results than will correspondence analysis.     

Principal components analysis using the hypothetical closed array

The application of R-mode principal components analysis to a set of closed chemical data is described using previously published chemical analyses of rocks from Gough Island. Different measures of similarity have been used and the results compared by calculating the correlation coefficients between each of the elements of the extracted eigenvectors and each of the original variables. These correlations provide a convenient measure of the contribution of each variable to each of the principal components. The choice of similarity measure (variance-covariance or correlation coefficient)should reflect the nature of the data and the view of the investigator as to which is the proper weighting of the variables—according to their sample variance or equally. If the data are appropriate for principal components analysis, then the Chayes and Kruskal concept of the hypothetical open and closed arrays and the expected closure correlations would seem to be useful in defining the structure to be expected in the absence of significant departures from randomness. If the data are not multivariate normally distributed, then it is possible that the principal components will not be independent. This may result in significant nonzero covariances between various pairs of principal components.     

Comparison of automatic classification methods applied to lake geochemical samples

Geochemical samples from part of Lake Geneva were analyzed for 29oxides and trace elements. The variables and samples were subjected to R- and Q-mode analyses. The following techniques were applied in sequence: data transformation (normalization and standardization), data reduction (principal component and factor analysis), and automatic classification (dendrograph). The data were treated using various combinations of these techniques, and the resulting classifications evaluated by means of several criteria. The best classification of the samples is given by a cluster analysis performed on four principal components computed from standardized variables. The discriminatory power of the variables also was measured and determined to depend on their degree of intercorrelation. As a final result, the 29original variables were reduced to four components and the sediment samples classified into four facies, leading to easily interpretable geochemical maps.     

The use of correspondence analysis in geochemistry

When factor analysis is used in geochemistry, it may be useful for factors to be transformed by rotations in order to be identified either to the end members of a mixing model (Miesch, 1976a),or to known chemical equilibriums. It requires that the formula for recomputing data from the factors may be written in a factored manner, which is generally not the case in correspondence analysis. The present paper shows that this becomes possible with data having constant row sums. As an example, the method is tested on the lavas of Paricutin Volcano, already examined by using an extended Q-mode factor analysis (Miesch, 1979).Recomputation of the data after projection gives simular results for both methods. Otherwise, the fact that correspondence analysis provides centered factors makes it well suited to the study of chemical reactions leading to constant mass transformations.     

An effect of closure on the structure of principal components

The principal components transformation generates, from any data array, a new set of variables—the scores of the components—characterized by a total variance exactly equal to that of the initial set. It is in this sense that the transformed variables are said to contain, preserve, or account for, the variance of the original set. The scores, however, are uncorrelated. In the course of the transformation, what becomes of the strong interdependence of variance and covariance so characteristic of closed arrays? The question seems to have attracted little attention; we are aware of no study of it in the earth sciences. Experimental work reported here shows quite clearly that the overall equivalence of variance and covariance imposed by closure, though absent from the component scores,may emerge in relations between the coefficientsof each of the lower-order components; if the raw data are complete rock analyses, the sum of all the covariances of the coefficients of such a component is negative, and is very nearly equal to the sum of all the variances in absolute value. (In all cases so far examined, the absolute value of the first sum is a little less than that of the second.) The principal components transformation provides an elegant escape from closure correlation if a petrographic problem can be restated entirely in terms of component scores, but not if a physical interpretation of the component vectors is required.     

Estimation of a multidimensional covariance function in case of anisotropy

A model of a multivariate covariance function with an ellipsoidal directional correlation scale has been developed. The axes of the ellipsoidal scale are related to the eigenvalues and eigenvectors of a matrix B which characterizes the ellipsoid of the range of influence. The matrix B is found to be related to a matrix T which can be estimated directly from sparse sampling data and can be used to determine estimates of the matrix B. The method has been applied to both two-dimensional and three-dimensional cases. The numerical results show that the satisfactory accuracy is obtained with sparse sampling data from an anisotropic random function.     

Sampling variation of principal components

The convenience of reducing the dimension of a data matrix by principal component analysis invites substantive interpretation of the coefficients of the components. To test the consistency of component coefficients, 10 samples of approximately 25, 50, 100, and 200 items each were randomly drawn, with replacement, from a source sample consisting of 2086 subalkaline asalt analyses. From each sample principal components were calculated using 9 major oxides as variables. Although the eigenvalues are remarkably consistent, both across and within sample size groups, the coefficients of the eigenvectors are subject to considerable sample variance. It is sometimes assumed that the coefficients of the components calculated from small samples are well enough known to be used in detailed petrological interpretation. Our results indicate that the validity of this assumption should be tested in each specific research even when rather large samples are used. The testing procedure used here is suitable of a sufficiently large reservoir of sample items is available; in the absence of such a reservoir complete simulation could be used.     

A generalization of extendedQ-mode factor analysis to data matrices with variable row sums

The extended method of Q-mode factor analysis developed by Miesch for data matrices with constant row sums is generalized to data matrices with variable row sums. With the algorithm provided it is possible to compute factor scores in the metric of the original data and compute goodness-of-fit statistics and model geological systems unconstrained by constancy of row sums of data points.     

Thoughts on use of principal component analysis in petrogenetic problems

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.     

Multifractal Distribution of Eigenvalues and Eigenvectors from 2D Multiplicative Cascade Multifractal Fields

Two-dimensional fields (maps) generated by isotropic and anisotropic multiplicative cascade multifractal processes are common in many fields including oceans, atmosphere, the climate and solid earth geophysics. Modeling the anisotropic scaling property and heterogeneity of these types of fields are essential for understanding the underlying processes. The paper explicitly derives the eigenvalues and eigenvectors from these types of fields and proves that the eigenvalues and eigenvectors are described by non-conservative multifractal distributions. This results in a new multifractal model implemented in eigen domain to characterize 2D fields not only with respect to overall heterogeneity and singularity as characterized by the ordinary multifractal model applied to the field itself, but also with respect to orientational heterogeneity of the field. It may also result in a new way to characterize the distribution of extreme large eigenvalues involved in other studies such as principal component analysis. A newly defined operator and its properties as derived in this paper may be useful for studying other types of multifractal cascade processes.     

Estimation of regionalized compositions: A comparison of three methods

A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable.     

Some properties of principal component scores

Commonly used methods for calculating component scores are reviewed. Means, variances, and the covariance structures of the resulting sets of scores are examined both by calculations based on a large set of electron microprobe analyses of melilite (supplied by D. Velde)and by a survey of recent geological applications of principal component analysis. Most of the procedures used to project raw data into the new vector space yield uncorrelated scores. In exceptions so far encountered, correlations between scores seem to have been occasioned by the use of unstandardized variables with components calculated from a correlation matrix. In a number of cases substantive interpretations of such correlations have been proposed. A different set of correlations results for the same data if scores are computed from standardized variables and components based on the covariance matrix. If unscaled components are rotated by the varimax procedure, the result is a return to the original space. In the work reported here, nevertheless, scores calculated from varimax-rotated scaled vectors are uncorrelated.     

Empirical behavior of Chayes'' null model

Chayes' t-test for closure correlations is developed from various approximations and assumptions. The empirical behavior of this test is observed through the use of random-sampling number matrices of order to thirteen. The experiments demonstrate a lower limit for the reliability of Chayes' model of about 91/100. That is, about one chance in ten remains that an error due to the statistical testing is committed on some product-moment correlations in a given matrix of correlation, given a confidence level of 90 percent. With increase in the number of variables, the test for departure from zero correlation can be used provided the sample size remains small.     

A type curve approach to estimatingp for ap-normal transformation

The p-normal transformation plays an important role in reservoir characterization for data sets that are neither normally nor log-normally distributed. The key step in the transformation is to estimate the value of pfor a given data set. Even though there are several ways to determine p,these are more inconvenient than the quicker and easier type curve approach to estimate pwe present in this paper. In addition, the method provides the p-normal transformation with a visual interpretation. We demonstrate the technique by analyzing reservoir permeability and porosity data from the East Velma West Block Sims Sand Unit, Oklahoma.     

Projeto RADAM: a better look at the Brazilian tropics

Fifty-four % of Brazil's national territory, most of it in the Amazon, has been covered by Projeto RADAM. The original images are difficult to obtain and the examples used in this article were taken from secondary, but still adequate mosaics of the radar imagery. These illustrate the principal physical units of the Amazon, thevárzea, theterra firme, and several types of highland, as well as the principal kinds of rural settlement: native clearings, ranches and new colonies. Two illustrations are included from the extension of this imagery into northeastern Brazil, one showing the coastal mountain range and the other an oasis-like settlement in the semi-arid interior.     

Remote sensing of atmospheric water vapour from Bhaskara IISAMIR data and its comparison withNOAA-7 water vapour data

Applying the method of ‘statistical linear regression’, atomspheric water vapour over oceanic areas has been estimated from the 19GHz and 22 GHz data of the satellite microwave radiometer (SAMIR) system onboard the Bhaskara II satellite. In the absence of any simultaneousin situ measurements on water vapour over ocean, theSAMIR-derived water vapour data have been compared with like data from theNOAA-7 satellite. It is suggested that a positive bias seen in theSAMIR data could be due to calibration errors in the basic data. In view of the observed bias, the original regression equation is modified and then used to obtain water vapour distributions over ocean for winter and south-west monsoon seasons usingSAMIR data of several orbits.     

A method for determining the reversibility of a Markov sequence

This paper describes, given a tally matrix with strictly positive entries, a method to determine whether the associated Markov process is reversible, and (for reversible Markov processes) methods to compute the reversibility matrix from the tally matrix. If the tally matrixN is symmetric, then it is shown that the Markov process must be reversible and the reversibility matrixC equalss (R ^–1NR^–1), whereR is the diagonal matrix whosei ^th diagonal entry is the sum of the entries of thei ^th row ofN (for everyi) ands denotes the sum of all the entries ofN. Because a symmetric tally matrix is of special importance in applications, a ² test is proposed for determining, in the presence of experimental errors, whether such a matrix is symmetric.     

Study of horizontal scales of motion observed over Kharagpur during MONTBLEX-1990

Characteristic wavelengths for theu andv components of wind are studied using the Monsoon Trough Boundary Layer Experiment (MONTBLEX) data obtained from a Doppler Sonic Detection and Ranging System (sodar) over the land station Kharagpur (near sea-coast). The principal stability parameter (Z _i/L_o) is used to infer the behaviour of the non-dimensional form of the characteristic wavelength (L _H) within the entire stability range occurring during the sounding periods. This is compared with GATE - 1974 results (over the sea surface) published by Fitzjarrald (1978).     

Principal components analysis of three-way tables

A three-mode principal components method allows visualization of the structural or taxonomic relationships within three-way data tables. The fundamental model includes three sets of eigenvectors and a core matrix relating the principal components of each mode. Formal relationships between the method and the usual principal components formulation allow calculation of loadings and scores for each mode; taken with the core matrix, these provide a number of points of view in graphical analysis of three-mode data. The model compares favorably with alternative formulations in terms of simplicity of computation, generality, and symmetry of operation among the modes. An organic geochemical example illustrates the method.