Showing papers on "Principal component analysis published in 1975"

Eigenvector Analysis of Monthly Mean Surface Data

Abstract: Eigenvector or principal component analysis has been carried out on monthly means of surface pressure, temperature, and rainfall defined on grids extending over both hemispheres and the tropical belt. In each case, 10 out of a possible 120 components were sufficient to account for more than 80% of the observed variance regardless of whether the annual cycle was first removed. The major pressure components can be interpreted in terms of meridional and land-sea temperature gradients, and reflect a basic difference in the seasonal cycle in the Northern and Southern Hemispheres. While both pressure and temperature show coherent departures on a hemispheric scale, rainfall components are regional in character, and the variance reduction is more evenly distributed over the major components. The use of principal components as climatological indices merits greater attention than has so far been given.

On objectives of methods of ordination

Abstract: The objectives of a number of methods of ordination are examined and a major distinction made between two approaches. The first of these has as a primary objective the efficient redescription of data, and is typified by principal components analysis. However the linear additive model implied in component analysis and the predominance of unique variance, together with lack of scale invariance suggests that other methods of dimensionality reduction might be more appropriate ecologically—either the non-metric methods of multidimensional scaling or the methods of factor analysis. The second approach, typified by Curtis and McIntosh continuum analysis, seeks to order the stands so that the resulting data matrix has a particular form, and is not directly concerned with dimensionality reduction. Continuum analysis is not the only such pathseeking method, and the objectives of several others are briefly examined. Finally the methods of Hill for seriation and the intrinsic dimensionality approach of Trunk seem to provide methods close to those required for the examination of ecological data. Concluding comments are made on problems of interpretation and the effects of sampling and description on the value of the results, especially in the light of the present tendency to employ simulated data to test the efficacy of methods of analysis.

Minimum Variance Properties of Principal Component Regression

Abstract: In the context of the usual regression model, , this article shows that the coefficient obtained when Y is regressed on the first principal component of the independent variables may be interpreted as an estimator of that , which can be estimated with least variance. Analogous results hold for succeeding components, and the property can be expressed in terms of volumes of confidence ellipsoids. To estimate using r principal components, choice of those components associated with the r largest characteristic roots minimizes the trace of a variance matrix. The results are illustrated with data on French imports.

Analysis of Interactions in Incomplete Two‐Way Tables

Abstract: SUMMARY In incomplete two-way tables, analysis of variance can be used only to find adjusted means and sums of squares. It is proposed that, even in very incomplete tables, missing plot values can sometimes be fitted and the interaction sum of squares partitioned into principal components in the same way as with complete data. An example is given in detail in which this method of analysis gives a useful guide to the combination of results from many different trials.

Introduction to uses and interpretation of principal component analysis in forest biology.

Abstract: The application of principal component analysis for interpretation of multivariate data sets is reviewed with emphasis on (1) reduction of the number of variables, (2) ordination of variables, and (3) applications in conjunction with multiple regression.

Repeatability of principal components in samples: normal and non-normal data sets compared

Abstract: The asymptotic theory of the distribution of the latent roots and vector of principal components analysis (PCA) of samples has hitherto been tied t multivariate normal (MVN) distributions. However, much real behavior dat are not normal. The results of the present study show, using arguments base on Monte Carlo methods, that if there is enough meaningful structure in th population, then PCA of samples of data with multivariate non-normal (NN distribution may provide answers relative to the "true" population principal components (PC) of comparable reliability to PCA of samples of MVAT data.

An assessment of the relationship between the surface properties of chalk soils and slope form using principal components analysis

Abstract: Summary Five line transects were sited over grass-covered, convex-concave slopes in the Berkshire and Wiltshire chalk downs, with the object of describing and interpreting the variation in selected surface soil properties. Previous work had concentrated upon the correlation between single variables and slope gradient and slope length, whereas the emphasis of the present paper lies in examining the interrelationships amongst the variables through the use of principal components analysis. A consistent pattern of variation was established between the five sites. The first component derived from the analysis was interpreted as a combined organic matter and soluble constituents factor, which accounted for between 50 and 60 per cent of the total variance. The second component was identified as particle-size variation and accounted for a further 13 to 18 per cent of the total variance. Whereas the distribution of the component scores for the second component supported the original idea of a sharp junction effect in the pattern of soil properties on the slope, the first and most important component indicated a more diffuse transition from net erosion to net deposition conditions.

Comparison of automatic classification methods applied to lake geochemical samples

Abstract: 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.

Component Analysis: an Alternative System

Abstract: Summary The main purpose of the paper is to introduce a system of components as a substitute for the principal components. Some difficulties involved in the use of the principal components are pointed out and an alternative system, regression components, is described. It is shown that the use of the latter components may lead to conclusions different from those obtained by means of the former. The results of analyses for a number of examples are reported, and comparisons with principal components have been drawn. The effect of changing the sign of variates is also dealt with, and results for two examples are reported. The use of components as regressors in regression analysis and the methods of studying relationship between groups of variates are discussed. It is suggested that the use of canonical variates should be supplemented or substituted by other methods. It is emphasized that the variates must be chosen and used so as to obtain meaningful results.

Eigenvector analysis as an aid to airmass recognition

Abstract: At each of the eighteen Australian Radiosonde stations with a sufficient length of record, the standard upper air variables are converted to potential temperatures, mixing ratio, and equivalent potential temperature for individual soundings. Only data for the winter month of July and data from the atmosphere below 800 mbs is considered. A Principal Component or Eigenvector Analysis reduces these variables into coefficients or weightings on the significant eigenvectors of the data set. It is shown that the weightings on the first eigenvector are representative of the airmass present at the time of the sounding. A partial collective analysis assesses the degree to which the weightings group together, i.e. have similar characteristics in the vertical. Finally the areal continuity of the airmasses, so defined, is discussed.

The applicability of principal component analysis to the separation of multiple plane-wave signals

Abstract: It is shown that principal component analysis (eigenvector-eigenvalue decomposition of array spectral matrices) can be used for the approximate separation of superposed plane-wave signals only if at least one of the following conditions is satisfied: 1. (a) The signals are grossly unequal in amplitude. 2. (b) The directions and the velocities of the various signals are such that if the array is beamed at a given signal the other signals do not fall on prominent sidelobes of the array response. Application of the method for such cases is demonstrated on composited long-period seismograms using recordings from the D ring and center element of ASA.

Large sample and jackknife procedures for small sample orthogonal least squares inference

Abstract: This paper concerns orthogonal least squares estimation and (approximate) distribution theory for the particualr estimators considered. A small Monte Carlo study is included.

The Application of Pattern Recognition Techniques to the Characterization of Atmospheric Aerosols.

Abstract: : Measurements were made of the amounts of 24 components in 24-hour samples of atmospheric particulates collected over a 12-month period in the greater Tucson area. Techniques of pattern recognition were used to examine the data base, which also included meteorological information collected daily over the same period of time. Clustering methods grouped similar components together. Principal Component Analysis showed that most of the variance was contained in only a few dimensions.

A program for rotation to smooth functions

Abstract: Deacription. In an application of principal components analysis to a set of learning curves, Tucker (1966) advanced the following criterion for identifying meaningful components. The variables were consecutive trials, so the loadings of any meaningful component should form a smooth curve when plotted against trial number. Any particularly jagged curve would, according to this line of reasoning, probably be associated with random error. Similar considerations would lead us generally to expect that any meaningful component would be smooth in appearance when the variables consisted of successive measurements taken from some presumably continuous process. Other examples might be found in the principal components analysis or factor analysis of electrophysiological responses sampled over time, of economic time series, or of data from studies of growth or development. Tucker used the smoothness criterion in evaluating the meaningfulness of components already chosen according to some other criterion (contribution to total sum of squares). The present program performs a transformation of an arbitrary set of components into a new set that are as smooth as possible. The measure of smoothness associated with a vector y' =(y1 , Y2 , ., Yro) is taken to be:

Generalised partial correlation and principal components

Abstract: of Australasian PhD thesis Generalised partial correlation and principal components Neville Robert Bartlett Basically, the thesis consists of three main parts that can tie briefly summarised as (i) introduction and preliminaries, (ii) general partial correlation and applications, and (iii) optimality of principal components. Chapters 1 and 2 contain the introduction and the preliminary theory concerning random variables that take values in a separable Hilbert space H . A general space L (S) of such random variables is defined. Examples of the space H are (i) the real line, (ii) the complex plane, and (iii) R\" , the euclidean q-space. In Chapter 3 we define a bivariate correlation coefficient, a multiple correlation coefficient and a partial correlation coefficient for H-valued random variables. The properties of these coefficients are given and their relationships with the existing theory are discussed. The partial correlation coefficient is of particular interest since it can be used in two ways; Received lU January 197*. Thesis submitted to the Flinders University of South Australia, December 1973. Degree approved, November 197U. Supervisor: Professor J.N. Darroch. 155 156 Neville Robert Bartlett (i) to compare a regression type model and a regression type hypothesis; and (ii) to measure the improvement in predicting a random variable when more variables or information are added to an existing predictor. Because the general partial correlation coefficient is the correlation between two particular random variables we can plot these variables to obtain a scatter diagram. The uses of such scatter diagrams are considered and examples given. In Chapter k we use the theory of Chapter 3 to define correlation coefficients for non-negative random variables whose sum is bounded by a constant. These variables are called bounded-sum random variables. Scatter diagrams are given and correlation coefficients are calculated and compared with those obtained by Darroch and Rate I iff [2]. The existing optimal properties of principal components are generalised, in Chapter 5, to H-valued random variables and then further generalisations are made. Okamoto [3] shows that the principal components of X , , X are optimal, in some senses, over all random variables in o L (S) . By considering only those random variables in $2, a closed linear subspace of L (5) , we give general results that become those of Okamoto when fi_ = L (S) . These generalised results are used to answer a conjecture of Rao [4] concerning optimality of principal components of the o instrumental variables Z. , , Z € L (S) . 1 m An optimal property of principal component time-series (see Brill inger, [/]) is generalised in Chapter 6. The optimality of the principal component time-series is extended to a larger class of timeseries. In Chapter 7 we consider ways in which particular \"linear\" combinations of vectors X , . . . , X , taking values in fr , can be defined so that they have optimal properties analogous to those for principal components of X^ , . . . , X P a r t i a l c o r r e l a t i o n 157 References [J ] D.R. BriI linger, \"The generalisation of the techniques of factor analysis , canonical correlation and principal components to stationary time se r i e s\" , Roy. Statist. Soo. Conf. Cardiff, Wales, 196U. [2] John N. Darroch and Douglas Ratcliff, \"Null correlation for proportions I I \" , J. Internat. kseoc. Math. Geol. 2 (1970), 307-312. [3] Masashi Okamo+o, \"Optimality of principal components\", MuLtivariate analysis II, 673-685 (Proc. Second Internat . Sympos. Multivariate Analysis, Wright State University, Ohio, 1968. Academic Press, New York and London, 1969). [4] C.R. Rao, \"The use and interpretat ion of principal component analysis in applied research\", Sankhya, Ser. A 26 (196U), 329-358.

A Multivariate Methodology for Controlling Industrial Processes

Abstract: The combination of tree multivariate techniques (Principal Component Analysis, Cluster Analysis and Discriminant Analysis) may be used to isolate and determine bounds for the critical control variable in a ``black box'' type industrial process. If input samples can be grouped into homogeneous quality categories, then it may be possibie to use input and operating data to estimate discriminant functions which will differentiate between various quality categories. These functions could then be used to predict quality levels based upon input and operating characteristics. In addition, the coefficents of te discriminant functions would provide information as to the critical control variables and their bounds.