Effect of trends on correlation in open and closed three-component systems

It is well known among geologists that closure of an open-number system, as when stratigraphic rock thicknesses are converted to percentages, introduces correlations among the components even in the absence of correlations in the open system. In closed three-component systems the covariances are single-valued functions of the closed variances and are exactly predictable. If the open system has inherent correlation (point correlations) among its components the corresponding closed covariances reflect their presence in a predictable manner. If areal trends are present in the open system, the open covariances are themselves affected, but this trend effect can be completely removed to recover the initial point correlations among the components. Areal trends in open systems strongly influence the structure of the closed variance-covariance matrices, and the situation becomes increasingly complicated if the open system has both point correlations and areal trends. The paper considers the problems involved, and includes Monte Carlo runs to compare computed and predicted variances and covariances as data sets are followed from open systems with correlation but no trend to the closed equivalent of open systems with point correlations and trends.     

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.     

Correlations between global features of terrestrial fields

Studies of correlation coefficients between different sets of global geophysical data may lead to useful inferences concerning their relationship or independence. If one data set is allowed to rotate with respect to another, the statistical theory is complicated and extra care is required before one can conclude that there is any statistical significance to a maximized correlation coefficient. If, for some relative rotation, two spherical harmonic fields are significantly correlated, then their individual degree component harmonics of dominant power must also be significantly correlated. Rotations can be found that result in high correlations between the dominant low-degree spherical harmonics of the geomagnetic and tertestrial gravity field potentials, but rotations can also be found that result in equally high, yet meaningless, correlations if the lunar gravity field is substituted for the geomagnetic field. To explain such high correlations, the theoretical correlation distribution function between normally distributed component harmonics is derived and then verified for lowdegree harmonics by using a Monte Carlo technique which takes into account the three-dimensional rotation group. Some curious properties surface: (1)the correlation distribution function for all possible relative orientations is almost the same between identical and uncorrelated fields; and (2)a system for determining the correlation distribution function from randomly selected fields or from randomly rotated fields is almost ergodic.     

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.     

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.     

Multivariate techniques as alternative statistical tools applied to magnetic proxies for pollution: a case study from Argentina and Antarctica

Several statistical analyses—as alternative tools—were applied to magnetic monitoring studies. Magnetic and chemical data from two environments have been gathered from previous papers and studied separately. Univariate and multivariate analyses were first examined, revealing a link between magnetic and chemical variables. The latter analyses, in particular, canonical correlation analysis, showed very good canonical correlations: R = 0.950 (Antarctica) and R = 0.891 (Argentina). On the other hand, in order to classify the data according to the degree of contamination, principal coordinates and discriminant analyses, as well as the comparison of several multivariate means were performed. Three groups were distinguished in both case studies, which were well classified at a low margin of error and quite different from each other at a significant level: 0.01 (Antarctica) and 0.05 (Argentina). The joint use of these statistical analyses also showed, in agreement with previous studies, that the relevant variables in order to identify atmospheric pollution are: magnetic susceptibility, saturation of isothermal remanent magnetisation, anhysteric susceptibility/magnetic susceptibility, remanent coercivity, and copper, lead, zinc and chromium contents.     

Seasonal and interannual variations in pigments in the Adriatic Sea

Spatial and temporal variability of pigments was studied from the CZCS satellite data and fromin situ chlorophyll and transparency for the period 1979-1985. The three Adriatic sites, Northern, Middle, and Southern Adriatic are differently influenced by meteorological, hydrological and oceanographic parameters. The differences between seasonalin situ chlorophyll and remotely sensed pigment concentrations (from CZCS satellite data) from the Adriatic are large in winter. Through the correlation analysis, pigments were compared to meteo-oceanographic and hydrological parameters from different Adriatic sites. The PCA (principal component analysis) was applied to the pigment data series and significant components were compared. Different correlations are obtained for warm and cold periods of the year pointing to seasonal differences in the underlying mechanism of pigment variability. The first PC is influenced mainly by temperature. In the warm period more parameters seem to influence the pigment field, than in the cold period. The pigments in the Adriatic are in good correlation to a number of hydrologic and meteo-oceanographic factors.     

BLU Estimators and Compositional Data

One of the principal objections to the logratio approach for the statistical analysis of compositional data has been the absence of unbiasedness and minimum variance properties of some estimators: they seem not to be BLU estimator. Using a geometric approach, we introduce the concept of metric variance and of a compositional unbiased estimator, and we show that the closed geometric mean is a c-BLU estimator (compositional best linear unbiased estimator with respect to the geometry of the simplex) of the center of the distribution of a random composition. Thus, it satisfies analogous properties to the arithmetic mean as a BLU estimator of the expected value in real space. The geometric approach used gives real meaning to the concepts of measure of central tendency and measure of dispersion and opens up a new way of understanding the statistical analysis of compositional data.     

Examination of the Chayes-Kruskal procedure for testing correlations between proportions

The Chayes-Kruskal procedure for testing correlations between proportions uses a linear approximation to the actual closure transformation to provide a null value,p _ij , against which an observed closed correlation coefficient,r _ij , can be tested. It has been suggested that a significant difference betweenp _ij andr _ij would indicate a nonzero covariance relationship between theith andjth open variables. In this paper, the linear approximation to the closure transformation is described in terms of a matrix equation. Examination of the solution set of this equation shows that estimation of, or even the identification of, significant nonzero open correlations is essentially impossible even if the number of variables and the sample size are large. The method of solving the matrix equation is described in the appendix.     

Development of a Multimetric Index for Integrated Assessment of Salt Marsh Ecosystem Condition

Tools for assessing and communicating salt marsh condition are essential to guide decisions aimed at maintaining or restoring ecosystem integrity and services. Multimetric indices (MMIs) are increasingly used to provide integrated assessments of ecosystem condition. We employed a theory-based approach that considers the multivariate relationship of metrics with human disturbance to construct a salt marsh MMI for five National Parks in the northeastern USA. We quantified the degree of human disturbance for each marsh using the first principal component score from a principal components analysis of physical, chemical, and land use stressors. We then applied a metric selection algorithm to different combinations of about 45 vegetation and nekton metrics (e.g., species abundance, species richness, and ecological and functional classifications) derived from multi-year monitoring data. While MMIs derived from nekton or vegetation metrics alone were strongly correlated with human disturbance (r values from ?0.80 to ?0.93), an MMI derived from both vegetation and nekton metrics yielded an exceptionally strong correlation with disturbance (r = ?0.96). Individual MMIs included from one to five metrics. The metric-assembly algorithm yielded parsimonious MMIs that exhibit the greatest possible correlations with disturbance in a way that is objective, efficient, and reproducible.     

Variation of chemical composition of alkaline granitic rocks in the central part of the Kola Peninsula

Chemical composition and origin of alkaline granitic rocks in the Keivy area on the Kola Peninsula were investigated. Linear correlation analysis and principal-component analysis were used to determine the interrelation of major petrogenetic elements in alkaline granite and surrounding alkaline metasomatites. Estimates of linear correlation coefficients turned out to be different, and principal-component analysis of the chemical data revealed that there were three main components influencing variation of chemical composition. These factors can be interpreted in terms of petrological processes, which are different for alkaline granite and for the surrounding metasomatites, indicating a different origin of the rocks.     

Model-based remote sensing algorithms for particulate organic carbon (POC) in the Northeastern Gulf of Mexico

Hydrographic data, including particulate organic carbon (POC) from the Northeastern Gulf of Mexico (NEGOM) study, were combined with remotely-sensed SeaWiFS data to estimate POC concentration using principal component analysis (PCA). The spectral radiance was extracted at each NEGOM station, digitized, and averaged. The mean value and spurious trends were removed from each spectrum. De-trended data included six wavelengths at 58 stations. The correlation between the weighting factors of the first six eigenvectors and POC concentration were applied using multiple linear regression. PCA algorithms based on the first three, four, and five modes accounted for 90, 95, and 98% of total variance and yielded significant correlations with POC with R ² = 0.89, 0.92, and 0.93. These full waveband approaches provided robust estimates of POC in various water types. Three different analyses (root mean square error, mean ratio and standard deviation) showed similar error estimates, and suggest that spectral variations in the modes defined by just the first four characteristic vectors are closely correlated with POC concentration, resulting in only negligible loss of spectral information from additional modes. The use of POC algorithms greatly increases the spatial and temporal resolution for interpreting POC cycling and can be extrapolated throughout and perhaps beyond the area of shipboard sampling.     

The relationship between foliation and strain: an experimental investigation

A multilayered salt/mica specimen with embedded strain markers was shortened to produce a fold and the distribution of strain was subsequently mapped out over the profile plane. On a fine scale the initial foliation, which is parallel to the undeformed layers, is folded by tight kinks to produce two new foliations; one is defined by the preferred orientation of kink boundaries and the other by the preferred orientation of (001) of mica. In the hinge region of the fold the first of these new foliations is parallel to the local λ₁λ₂-principal plane of strain whereas the preferred orientation of mica is bimodal and is symmetrical about the λ₁λ₂-plane. Elsewhere the two new foliations are not parallel to the principal plane of strain and angular divergencies of up to 30–35° are measured. If a March model with initial random mica orientation is assumed for the development of mica preferred orientation then the correct value of strain is predicted but the orientation of the principal plane of strain can be grossly in error. A theoretical analysis of the angular relationships to be expected between kink boundaries and the λ₁λ₂-plane of strain confirms that for the type of geometries experimentally developed, large divergences of up to 35° should be common. In rocks where the foliation has developed by processes similar to those recorded here, large angular divergencies between the foliation and the λ₁λ₂-principal plane of strain should be expected as the rule.     

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.     

全风化花岗岩化学及矿物成份在全土和粘粒中的不同表征

对全风化花岗岩化学和矿物成份的测试常用全土和粘粒部分(粒径<2μm)分别进行。业已发现两者由于物质组成和结构特征的差异,对土工程地质性质所起作用不同。本文给出了6个全风化花岗岩样品的全土和粘粒部分筛分法和移液管法测定粒度成份,用X荧光光谱仪做全量化学分析,及XRD矿物学分析的结果。对比这两部分数据发现,属于含砾土的这些样品化学成份的变化同矿物一致,尤其是同粘土矿物成份含量变化很一致。同全岩相比,粘粒化学成份中的SiO₂减少了近50%,K₂O含量也降低,而Al₂O₃、Fe₂O₃和H₂O⁺都明显升高。粘粒中氧化物相对含量升高者居多;全岩矿物成份以石英、粘土矿物和长石为主,粘粒中埃洛石和高岭石占大多数,其次为伊利石;粒度成份、化学成份和矿物含量相关性比较明显的是:石英和角砾正相关,和砂粒负相关,长石和砂粒正相关,粘土矿物含量和Al₂O₃、烧失量(LOI)和埃洛石含量正相关。     

Groundwater depletion and quality deterioration due to environmental impacts in Maheshwaram watershed of R.R. district,AP (India)

Maheshwaram watershed is situated in Ranga Reddy district of Andhra Pradesh at a distance of about 30 km south of Hyderabad, capital of Andhra Pradesh. The watershed has an area of 60 km² and has hard rock aquifers with semi-arid climate. The study area has been expanding at a fast pace and now has the distinction of being one of the fastest growing urban centers facing the problem of groundwater depletion and quality deterioration due to the absence of perennial source of surface water and also due to over exploitation. Human activities involving industrial and agricultural development and the inadequate management of land and water resources have, directly or indirectly resulted in the degradation of environment viz. water and soil. In the present study chemical analysis of groundwater samples of the study area, collected during pre- and post-monsoon seasons of 2007–2008 has been carried out. The analyzed data are utilized to characterize the hydro chemical process dominant in the area. Various classification methods such as Piper, Back and Hanshaw, Wilcox, USA. Salinity Laboratory are employed to critically study the geochemical characteristics of groundwater of the study area. Finally, principal component analysis (PCA) is also employed to the chemical variables of groundwater to characterize the hydro chemical process that is dominant in the area. In the analysis four principal components emerged as significant contributors to the groundwater quality. The total contribution of these four components is about 85–87%. The contribution of the first component is about 49–50% and has significant positive loadings of Ca²⁺, Mg²⁺, Na⁺, and Cl⁻ ions. The second, third, and fourth principal components have significant positive loadings of F⁻, NO₃ ⁻, SO₄ ²⁺, and HCO₃ ⁻ ions.     

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.     

Ordination and character correlations of selected recent British ostracoda

Forty-six characters were measured on each of 14 Recent ostracode specimens representing 13 species collected along the British coast. Results obtained from ordination using principal components analysis agreed closely with results from cluster analysis, but ordination gave a better representation of taxonomic distances computed from the original data. Cophenetic correlations were 0.857 for the cluster analysis and 0.935 for distances computed from projections used in ordination. Characters showed considerable intercorrelation, and the first principal component was reified as a general-size factor correlated highly with height. The remaining principal components could not be reified precisely.     

Influence of initial state and intermediate principal stress on undrained behavior of soft clay during pure principal stress rotation

It is important to be fully aware of the dynamic characteristics of saturated soft clays under complex loading conditions in practice. In this paper, a series of undrained tests for soft clay consolidated with different initial major principal stress direction ξ were conducted by a hollow cylinder apparatus (HCA). The clay samples were subjected to pure principal stress rotation as the magnitudes of the mean total stress p, intermediate principal stress coefficient b, and deviator stress q were all maintained constant. The influences of intermediate principal stress coefficient and initial major principal stress direction on the variation of strain components, generation of pore water pressure, cyclic degradation and non-coaxiality were investigated. The experimental observations indicated that the strain components of specimen were affected by both intermediate principal stress coefficient and initial major principal stress direction. The generation of the pore water pressure was significantly influenced by intermediate principal stress coefficient. However, the generation of pore water pressure was merely influenced by initial major principal stress direction when b?=?0.5. It was also noted that the torsional stress–strain relationships were affected by the number of cycles, and the effect of intermediate principal stress coefficient and initial major principal stress direction on the torsional stress–strain loops were also significant. Stiffness degradation occur under pure principal stress rotation. Anisotropic behavior resulting from the process of inclined consolidation have considerable effects on the strain components and non-coaxial behavior of soft clay.

     

Tests of Significance for Structural Correlations in the Linear Model of Coregionalization

In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.