Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data

The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.     

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.     

多维标度法在矿产预测中的应用

在矿产预测中常涉及一些定性变量,对这些变量的分析和应用,必须将定性描述的地质特征转化为用数值表示的变量,这就需要处理此问题的方法——多维标度法。笔者介绍了计量性的Ｔｏｒｇｅｒｓｏｎ法、准计量性的林知已夫数量化理论和非计量性多维标度法,并列举了其在矿产预测中的应用实例。     

Torgerson方法在某斑岩铜钼矿蚀变与矿化分析中的应用

Torgerson方法是一种多维标度法。用于研究变量或样本对象，它从一组对象的两两之间的相似性度量或非相似性度量出发，求出这组对象在某低维空间中的标度，从而发现这组对象的整体关系。通过它在某斑岩铜钼矿的蚀变与矿化分析的实例应用，说明该方法在矿产资源评价方面具有一定的实效性。     

Assessment of water quality and identification of pollution sources of three lakes in Kashmir, India, using multivariate analysis

Multivariate statistical techniques, such as cluster analysis, principal component analysis (PCA) and factor analysis (FA) were applied to evaluate and interpret the water quality data set for 13 parameters at 10 different sites of the three lakes in Kashmir, India. Physicochemical parameters varied significantly (p?<?0.05) among the sampling sites. Hierarchical cluster analysis grouped 10 sampling sites into three clusters of less polluted, moderately polluted and highly polluted sites, based on similarity of water quality characteristics. FA/PCA applied to data sets resulted in three principal components accounting for a cumulative variance of 69.84, 65.05 and 71.76% for Anchar Lake, Khushalsar Lake and Dal Lake, respectively. Factor analysis obtained from principal components (PCs) indicated that factors responsible for accelerated eutrophication of the three lakes are domestic waste waters, agricultural runoff and to some extent catchment geology. This study assesses water quality of three lakes through multivariate statistical analysis of data sets for effective management of these lakes.     

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.     

An application of factor analysis for the study of the hydrogeological conditions in Plio-Pleistocene aquifers of NW Achaia (NW peloponnesus, Greece)

R- andQ-mode factor analysis is applied to 51 groundwater samples collected from wells drilled in the Plio-Pleistocene aquifers of NW Achaia, Greece. The purpose ofR- andQ-mode factor analysis application is to identify (i) the regional groundwater flow pattern, and (ii) the deterioration of groundwater quality. Sixteen hydrogeological parameters are used in order to examine their importance and to provide significant insight into their correlations. In theR-mode factor analysis, a six-factor model is suggested which can explain more than 77.5% of the total variance. The contribution of each factor at every site (factor scores) also is computed. Maps are constructed showing the geographical distribution of the factor scores. From these maps, the high salinity areas are delineated (seawater intrusion, possible appearance of halite layers) and the areas with elevated contribution of karastic-water are defined. Using theQ-mode correspondence analysis the meaning of the electrical conductivity as the most important variable in groundwater quality characterization is demonstrated.     

Spatial declustering weights

Because of autocorrelation and spatial clustering, all data within a given dataset have not the same statistical weight for estimation of global statistics such mean, variance, or quantiles of the population distribution. A measure of redundancy (or nonredundancy) of any given regionalized random variable Z(u_α)within any given set (of size N) of random variables is proposed. It is defined as the ratio of the determinant of the N X Ncorrelation matrix to the determinant of the (N - 1) X (N - 1)correlation matrix excluding random variable Z(u_α).This ratio measures the increase in redundancy when adding the random variable Z(u_α)to the (N - 1 )remainder. It can be used as declustering weight for any outcome (datum) z(u_α). When the redundancy matrix is a kriging covariance matrix, the proposed ratio is the crossvalidation simple kriging variance. The covariance of the uniform scores of the clustered data is proposed as a redundancy measure robust with respect to data clustering.     

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.     

Correspondence factor analysis: An outline of its method

Correspondence factor analysis is a multivariate technique that may be applied to any type of data and to any number of data points. It detects associations and oppositions existing between subjects and objects, measuring their contribution to the total inertia for each factor. The probabilistic character of the data matrix is taken into consideration, and together with the principle of distributional equivalence, results in stability. The projection of the subjects and the objects onto the same set of factorial axes enables two-dimensional graphs to be drawn which offer aid in the interpretation of the results.     

Composition and similarity of global anomodont-bearing tetrapod faunas

Anomodont synapsids represent the dominant herbivores of Permian and Triassic terrestrial vertebrate ecosystems. Their taxonomic diversity and morphological disparity in combination with their cosmopolitan distribution makes them an ideal study object for macroevolutionary patterns across the most devastating extinction event in earth history. This study provides a thorough review of anomodont-bearing tetrapod faunas to form the basis for a faunal similarity analysis and future studies of anomodont diversity. The stratigraphic correlation and composition of all known anomodont assemblages is revisited, including a discussion of the validity of the globally distributed anomodont species. The similarity analysis of anomodont faunas is performed on the basis of presence–absence data of anomodont taxa, using explorative methods such as cluster analysis (UPGMA) and non-metric multidimensional scaling (NMDS). The recovered faunal groupings indicate a common biostratigraphic age and furthermore reflect biogeographic patterns. Even though endemism and faunal provinciality was a constant element in anomodont faunas of the Permian and Triassic, the available evidence indicates that the end-Permian extinction resulted in a distinct uniformity that was unique to Early Triassic anomodont faunas. This is in particular characterized by the global distribution and overwhelming abundance of the disaster taxon Lystrosaurus. In contrast, cosmopolitan anomodonts also existed in the Late Permian (e.g., Diictodon) and Middle Triassic (e.g., Shansiodon), but those taxa coexisted with endemic faunal elements rather than dominated the fauna as Lystrosaurus did.     

To the infrared spectroscopy of natural uranyl phosphates

An interpretation based on site and factor symmetry analysis has been given of infrared spectra of the natural layered uranyl phosphates metaautunite, metatorbernite, metauranocircite I and II, saléeite and sabugalite, of the deuteroanalogues of metaautunite, metatorbernite and metauranocircite II and of the Raman spectrum of metatorbernite. The principal correlation relationships for the analysis of virbrational spectra G _M (symmetry of free ion) ? G _S (site symmetry) ? G _F (factor symmetry) have been described. The results have been correlated with data from X-ray structure analysis.     

Failure modes and bearing capacity of strip footings on soft ground reinforced by floating stone columns

This study evaluates the failure modes and the bearing capacity of soft ground reinforced by a group of floating stone columns. A finite difference method was adopted to analyze the performance of reinforced ground under strip footings subjected to a vertical load. The investigation was carried out by varying the aspect ratio of the reinforced zone, the area replacement ratio, and the surface surcharge. General shear failure of the reinforced ground was investigated numerically without the surcharge. The results show the existence of an effective length of the columns for the bearing capacity factors N _c and N _γ. When certain surcharge was applied, the failure mode of the reinforced ground changed from the general shear failure to the block failure. The aspect ratio of the reinforced zone and the area replacement ratio also contributed to this failure mode transition. A counterintuitive trend of the bearing capacity factor N _q can be justified with a shift in the critical failure mode. An upper-bound limit method based on the general shear failure mode was presented, and the results agree well with those of the previous studies of reinforced ground. Equivalent properties based on the area-weighted average of the stone columns and clay parameters were used to convert the individual column model to an equivalent area model. The numerical model produced reasonable equivalent properties. Finally, a theoretical method based on the comparison of the analytical equations for different failure modes was developed for engineering design. Good agreement was found between the theoretical and numerical results for the critical failure mode and its corresponding bearing capacity factors.     

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.     

A new multidimensional scaling technique based upon associations of triple objects—Pijk and its application to the analysis of geochemical data

A new multidimensional scaling (MS) technique, referred to as the Pijk model, is formulated on the basis of associations among triple objects (samples or variables), instead of pairs of objects as used in the usual MS methods, such as factor analysis. The computational scheme provided for this method is the reduction of an original problem to a standard eigenvalue-eigenvector problem. The major goal of the technique is simplification and reduction of data structures and the rescaling of original objects into a new and reduced space, so that patterns and relations of the original objects can be conventiently examined in two-dimensional “factor” plots. The Pïjk method is illustrated and tested by using a set of geochemical data related to the epithermal gold and silver vein deposits in the Walker Lake quadrangle of Nevada and California. The characteristics of element associations suggested in the Pijk analysis are consistent with field observations. A preliminary comparison between the new method and the ordinary factor analysis also is made on the basis of the same data set. Results are encouraging in that analysis by the Pijk model captures triple-object associations that might be missed by the ordinary factor analysis which considers only pair-variable correlations     

Amphipod predation by the infaunal polychaete,Nephtys incisa

Survival of the tubicolous amphipodAmpelisca abdita in a sediment toxicity test was negatively correlated with the presence of indigenousNephtys incisa. The gut contents ofN. incisa held in uncontaminated sediment included identifiable portions of the amphipods.A. abdita andMicrodeutopus gryllotalpa. Although most nephtyid polychaete worms are active predators,Nephtys incisa has been considered a nonselective deposit-feeder. We coclude thatN. incisa will prey upon amphipods under laboratory conditions. This predation may be a factor in the exclusion of surface-dwelling amphipods from theNephtys-Nucula community common in southern New England.     

Scaling properties of corner frequencies of Kamchatka earthquakes

Scaling properties of earthquake populations bear the major information on the physics of the source process of an earthquake. To determine scaling properties, source spectra of more than 400 earthquakes of Kamchatka were determined in a frequency range 0.1–30 Hz using materials of digital registration of PET station, and characteristic frequencies of earthquakes were estimated. The range of magnitudes is 4–6.5, the range of distances is 80–220 km. To enable reduction of a spectrum to the source, attenuation properties of the medium around PET were determined beforehand. It is revealed that source spectra show several corner (characteristic) frequencies: f _c1, f _c2 and f _c3; where the spectral trend changes: from f ⁰ to f ^?1, from f ^?1 to f ^?2, and from f ^?2 to f ^?3, respectively. Although in some cases f _c1 ≈ f _c2 in agreement with the usual ω^?2 spectral model, the main part of spectra has more complicated character. For a large part of the studied earthquakes a source-controlled upper cutoff of acceleration spectrum, or corner frequency f _c3, is observed. This is an important fact, as the existence of f _c3 (source-controlled f _max) is not recognized in the bulk of the seismological literature. For f _c1, the observed scaling agrees with the usual hypothesis of similarity of the earthquake sources of different size (magnitude); it is close to f _c1 ～ M ₀ ^?1/3 , where M ₀ is seismic moment. For f _c2, scaling is close to f _c2 ～ M ₀ ^?0.17 ～ f _c1 ^0.5 , that indicates an expressed violation of similarity. For f _c3, scaling is close to f _c2 ～ M ₀ ^?0.08 ～ f _c1 ^0.25 , so that similarity is broken even sharper in this case. Hypotheses about possible causes of the observed scaling are discussed.     

Efficient upscaling of hydraulic conductivity in heterogeneous alluvial aquifers

An efficient method to upscale hydraulic conductivity (K) from detailed three-dimensional geostatistical models of hydrofacies heterogeneity to a coarser model grid is presented. Geologic heterogeneity of an alluvial fan system was characterized using transition-probability-based geostatistical simulations of hydrofacies distributions. For comparison of different hydrofacies architecture, two alternative models with different hydrofacies structures and geometries and a multi-Gaussian model, all with the same mean and variance in K, were created. Upscaling was performed on five realizations of each of the geostatistical models using the arithmetic and harmonic means of the K-values within vertical grid columns. The effects of upscaling on model domain equivalent K were investigated by means of steady-state flow simulations. A logarithmic increase in model domain equivalent K with increasing upscaling, was found for all fields. The shape of that upscaling function depended on the structure and geometry of the hydrofacies bodies. For different realizations of one geostatistical model, however, the upscaling function was the same. From the upscaling function a factor could be calculated to correct the upscaled K-fields for the local effects of upscaling.