Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.     

On the analysis of pollen-climate canonical transfer functions

Canonical correlation analysis, as described by Webb and Bryson, Quaternary Research 1972, provides a means of reconstructing past climates quantitatively from fossil pollen using a pollen-climate transfer function. This paper presents a method for analysis of variance of the transfer function model. This method is used to identify ecological relationships among the pollen and climate variables, to select climatically sensitive taxa, and to investigate the importance of site factors. Several criteria are presented, in addition to those used by Webb and Bryson, for choosing canonical variate pairs to include in the transfer function model, namely: the variate pair relationships should be ecologically meaningful; the transfer function model should yield stable paleoclimatic estimates; and, the variate pair relationships should be statistically meaningful. The application of these criteria to the set of variate pairs used in the transfer function model of Webb and Bryson is described and modifications are suggested.     

Multigroup discrimination of river, beach, and dune sands using axial-ratio sample statistics of quartz grains

Optimal discrimination among several groups can be achieved by simultaneous diagonalization of pooled within-group, W, and among-group, A, sums of squares and cross-product matrices formed by utilizing axial-ratio sample statistics of quartz grains belonging to different sieve grades. This method maximizes the ratio of among-group cross products to within-group cross product quadratic forms (V'AV/V'WV)and simultaneously yields discriminant scores whose correlation coefficients are zero for group means as well as for within each group. This procedure enables a simple Euclidean distance measure for partitioning the discriminant space for assignment. Although W^–1 and Amatrices are symmetric, the W^–1 Amatrix needed for multigroup discrimination is asymmetric and hence the eigenstructure of W^–1 Ais obtained by simultaneous diagonalization of Wand Amatrices. The first four sample statistics (mean, standard deviation, skewness, kurtosis) of normalized axial-ratios are required for discrimination, although the mean and standard deviation are the most important discriminators.     

Asymmetric measures of association, closed data, and multivariate analysis

The association between constant-sum variables X_i and X_j expressed as percentages can be calculated as a product-moment correlation between X_i and X_j/(100 – X_i ) and a correlation between X_j and X_i/(100 – X_j ). An asymmetric, square matrix may be formed from these coefficients, and multivariate analysis performed by two methods: singular value decomposition and canonical decomposition. Either analysis avoids problems in the interpretation of correlation coefficients determined from closed arrays, and provides information about dependencies among the variables beyond that obtained from the usual correlation coefficient between X_i and X_j.Two examples show the canonical decomposition to have the greater usefulness.     

Compressed History Matching: Exploiting Transform-Domain Sparsity for Regularization of Nonlinear Dynamic Data Integration Problems

In this paper, we present a new approach for estimating spatially-distributed reservoir properties from scattered nonlinear dynamic well measurements by promoting sparsity in an appropriate transform domain where the unknown properties are believed to have a sparse approximation. The method is inspired by recent advances in sparse signal reconstruction that is formalized under the celebrated compressed sensing paradigm. Here, we use a truncated low-frequency discrete cosine transform (DCT) is redundant to approximate the spatial parameters with a sparse set of coefficients that are identified and estimated using available observations while imposing sparsity on the solution. The intrinsic continuity in geological features lends itself to sparse representations using selected low frequency DCT basis elements. By recasting the inversion in the DCT domain, the problem is transformed into identification of significant basis elements and estimation of the values of their corresponding coefficients. To find these significant DCT coefficients, a relatively large number of DCT basis vectors (without any preferred orientation) are initially included in the approximation. Available measurements are combined with a sparsity-promoting penalty on the DCT coefficients to identify coefficients with significant contribution and eliminate the insignificant ones. Specifically, minimization of a least-squares objective function augmented by an l ₁-norm of DCT coefficients is used to implement this scheme. The sparsity regularization approach using the l ₁-norm minimization leads to a better-posed inverse problem that improves the non-uniqueness of the history matching solutions and promotes solutions that are, according to the prior belief, sparse in the transform domain. The approach is related to basis pursuit (BP) and least absolute selection and shrinkage operator (LASSO) methods, and it extends the application of compressed sensing to inverse modeling with nonlinear dynamic observations. While the method appears to be generally applicable for solving dynamic inverse problems involving spatially-distributed parameters with sparse representation in any linear complementary basis, in this paper its suitability is demonstrated using low frequency DCT basis and synthetic waterflooding experiments.     

Modern pollen assemblages and vegetational history of the moraines of the Klutlan Glacier and its surroundings,Yukon Territory,Canada

Modern pollen assemblages have been studied from surficial lake muds and moss polsters collected from five vegetated ice-cored moraines of the Klutlan Glacier. The youngest vegetated moraine (K-II) is characterized by high pollen values for Salix and Hedysarum, K-III by high Salix and Shepherdia canadensis and low Hedysarum and Picea, K-IV by high Betula, Salix, and Shepherdia, and K-V and the Harris Creek moraine (HCM) by high Picea. Variations are summarized by canonical variates analysis. A percentage pollen diagram from Gull Lake on the upland east of the glacier records vegetational development since the deposition of the White River volcanic ash 1220 ¹⁴C yr ago. An initial species-rich treeless vegetation was replaced by birch-alder-willow shrub-tundra, and this by open Picea glauca forest similar to present vegetation around the lake. Sites on HCM show two basic stratigraphies. Triangle Lake reflects vegetational succession from Salix-Shepherdia canadensis scrub similar to that on K-III today, through open Picea woodland of K-IV type, to closed Picea forests of K-V and HCM. Heart Lake and Cotton Pond reflect vegetational development following melting of ice underlying the spruce forests of HCM. These two types are summarized by positioning the fossil spectra on the first two canonical variate axes of the modern surface spectra.     

Scanning electron microscope evidence for offshore-onshore sand transport at Fire Island, New York, USA

Balancing the sediment budget of the Atlantic shoreline of Long Island has proved to be problematic because more sediment has been found to be moving westward in littoral drift than was estimated to be supplied by wave erosion of the Ronkonkoma terminal moraine at Montauk Point. There is a need for the existence of an additional, possibly offshore source. As part of a larger Atlantic shore study, 11 beach samples from Fire Island were compared with samples representing the known glacial source at Montauk Point, and also with four offshore samples to the south. Comparisons were based upon quartz sand grain surface texture patterns derived by scanning electron microscope checklist analysis. Initial evidence of an offshore-onshore link was provided by qualitative results examining surface texture variability between samples, illustrated by photographic evidence and written notes. This evidence was confirmed consistently during quantitative analysis. In particular, canonical variate analysis grouped two offshore samples representing deposits of buried glacial to fluvioglacial lobes with beach samples from Fire Island in adjacent onshore locations or slightly downdrift. In addition, canonical variate analysis and factor analysis using the spssx package, together with cluster analysis using the ARTHUR package, depicted the main factors and texture variables largely responsible for sample discrimination. The significance of both the offshore-onshore link and the associated factors influencing quartz grain surface texture was revealed by the fact that they are strongly related to each other and to the regional geomorphology.     

Relationship between stratigraphical variation in the morphology of Gabonella elongata and geochemical composition of the host sediments

One hundred and sixty-two microspheric and 110 megalospheric specimens of the Late Cretaceous benthic foraminifer Gabonella elongata de Klasz and Meijer were picked from 13 borehole samples and analyzed for stratigraphical variation in four variables. Principal component and canonical variate analyses indicate a larger inter-level variation in the relative size of the final chamber in the microspheric generation than in the megalospheric. The morphometrical measurements were weighed against the concentration of 13 chemical elements determined at each level. The analyses indicate that G. elongata developed large tests in arenaceous sediments, whereas carbonaceous, argillaceous sediments seem to have resulted in small tests. Especially the microspheric generation tends to show greater reduction in growth in the ultimate chamber in carbonaceous, argillaceous sediments than in more arenaceous. This indicates that the genetically controlled growth pattern of the last chamber may be influenced by environmental factors such as agitation and clearness of the water.     

Environmental discrimination based on quartz grain surface textures: a statistical investigation

ABSTRACT The relative abundance of thirty-two surface features identified on the surfaces of quartz sand grains from eight coded samples was noted independently by five scanning electron microscopists. These data (reduced to binary form) were the subject of canonical variate analyses which discriminated clearly between all samples. Thirteen variables were important in distinguishing between the eight samples. This indicates that there are no single key surface features which, on their own, would allow rapid environmental discrimination. The contention that a multiplicity of surface features should be recorded for the purpose of environmental discrimination is, therefore, upheld.
Operator variance, although considerable in the recognition of individual surface features, is negligible in discrimination of samples based on binary data. The SEM technique of analysis of quartz grain surface textures is a reliable and statistically valid means of discriminating between samples from different environments.     

用微生物对单一碳源利用方法探讨重金属在城市土壤中积累的环境效应

Biolog研究表明，英国阿伯丁市城市土壤的微生物群落结构显著有别于农村土壤，并使微生物对碳源的消耗量增加，消耗速度加快。城市土壤中不仅重金属Cu,Pb,Zn,Ni的含量明显高于农村土壤，而且其化学形态的主成分分析表明，有效态Pb,Zn,Cu及有机态Ni和Cu是导致城市土壤区分于农村土壤的主要因子。相关性分析表明，Biolog的这种变化规律与重金属的上述化学形态密切相关；典型变量分析表明，重金属对城市土壤微生物群落结果的这种损伤具有长期性效应及不可恢复性。     

Salt marsh litter and detritivores: A closer look at redundancy

Most primary production of angiosperms in coastal salt marshes enters the detritivore food web; studies of this link have predominantly focused on one plant species (Spartina alterniflora) and one detritivore species (Littoraria irrorata). In mesocosm experiments, we studied the rates and pattern of decomposition of litter derived from four plant species common in southeastern United States coastal salt marshes and marsh-fringing terrestrial habitats. Crustanceans and gastropods were selected as detritivores feeding on, and affecting degradation of, the litter of two monocotyledons and two dicotyledons. Despite interspecific similarities in consumption, detritivores exhibited species-specific effects on litter chemistry and on the activity of litter-colonizing microbiota. The chemical composition of feces depended upon both the litter type and the detritivores’ species-specific digestive capabilities. Growth rates and survival of detritivores differed among litter species. Different salt marsh detritivores are likely to have different effects on decomposition processes in the salt marsh and cannot be regarded as functionally redundant nor can the litter of different plant species be regard ed as redundant as food for marsh detritivores.     

Quantifying bioirrigation using ecological parameters: a stochastic approach†

Irrigation by benthic macrofauna has a major influence on the biogeochemistry and microbial community structure of sediments. Existing quantitative models of bioirrigation rely primarily on chemical, rather than ecological, information and the depth-dependence of bioirrigation intensity is either imposed or constrained through a data fitting procedure. In this study, stochastic simulations of 3D burrow networks are used to calculate mean densities, volumes and wall surface areas of burrows, as well as their variabilities, as a function of sediment depth. Burrow networks of the following model organisms are considered: the polychaete worms Nereis diversicolor and Schizocardium sp., the shrimp Callianassa subterranea, the echiuran worm Maxmuelleria lankesteri, the fiddler crabs Uca minax, U. pugnax and U. pugilator, and the mud crabs Sesarma reticulatum and Eurytium limosum. Consortia of these model organisms are then used to predict burrow networks in a shallow water carbonate sediment at Dry Tortugas, FL, and in two intertidal saltmarsh sites at Sapelo Island, GA. Solute-specific nonlocal bioirrigation coefficients are calculated from the depth-dependent burrow surface areas and the radial diffusive length scale around the burrows. Bioirrigation coefficients for sulfate obtained from network simulations, with the diffusive length scales constrained by sulfate reduction rate profiles, agree with independent estimates of bioirrigation coefficients based on pore water chemistry. Bioirrigation coefficients for O₂ derived from the stochastic model, with the diffusion length scales constrained by O₂ microprofiles measured at the sediment/water interface, are larger than irrigation coefficients based on vertical pore water chemical profiles. This reflects, in part, the rapid attenuation with depth of the O₂ concentration within the burrows, which reduces the driving force for chemical transfer across the burrow walls. Correction for the depletion of O₂ in the burrows results in closer agreement between stochastically-derived and chemically-derived irrigation coefficient profiles.     

Probability of earthquake occurrence estimated from results of rock fracture experiments

The hazard rate, the number of fracture occurrences per unit time, which has been obtained from laboratory experiments of rock fracture, is obtained for the earth's crust by analysing the statistical distribution of geodetically-observed ultimate strain. The associated hazard function has two coefficients, A and B, to be determined. Comparison of the coefficients obtained by the results of rock-fracture experiments with the geodetically determined ones discloses that B is independent of the size-effect. It is therefore concluded that, if A is estimated from the statistics of the geodetically observed ultimate strain and B is obtained from fracture experiments of rock forming a local part of the crust, the probability of a local large-scale earthquake occurrence can be estimated.     

Within-taxon morphological diversity in late-Quaternary Neotoma as a paleoenvironmental indicator, Bonneville Basin, Northwestern Utah, USA

Ecological data indicate that as the amount of precipitation in an arid areas increases, so too does mammalian taxonomic richness. This correspondence has been found in two late-Quaternary mammalian faunas from Utah, one from Homestead Cave in the Bonneville Basin. We use the remains of two species of woodrat (Neotoma cinerea and Neotoma lepida) from Homestead Cave to test the hypothesis that as the amount of precipitation in an arid area increases, so too does morphological diversity within individual mammalian taxa. Morphological diversity is measured as corrected coefficients of variation and as richness of size classes of mandibular alveolar lengths. Coefficients of variation for N. cinerea are few and coincide with moisture history if temporally successive small samples are lumped together. More abundant coefficients of variation for N. lepida coincide only loosely with moisture history, likely because such coefficients measure dispersion but not necessarily other aspects of variation. Richness of size classes of N. lepida is high during the early and late Holocene when moisture was high, and lowest during the middle Holocene when climate was most arid.     

Mg tracer diffusion in aluminosilicate garnets at 750-850° C, 1 atm. and 1300° C, 8.5 GPa

 Tracer diffusion coefficients of Mg in natural aluminosilicate garnets of composition Alm₃₈Pyr₅₀Gr₁₀Sp₂ and Alm₇₃Pyr₂₁Gr₅Sp₁ have been measured at 1 bar, 750-850° C and at 8.5 GPa, 1300° C by chemically depositing a salt layer enriched in ²⁶Mg on the specially prepared surface of a garnet single crystal. Diffusion anneals at 1 atmosphere (101325 Pa) were carried out at a controlled f _{O 2} of ∼10^−17.5 bars maintained by a flowing gas mix of CO-CO₂. Annealing conditions were carefully chosen to avoid decomposition of garnet by redox reactions. High pressure anneals were carried out in a multianvil apparatus. Induced diffusion profiles (0.1–0.6 μm) were measured by an ion-microprobe with SIMS attachment. Diffusion coefficients at 1 atmosphere are in excellent agreement with extrapolation of data from high P-T experiments (Loomis et al. 1985; Chakraborty and Ganguly 1992) and also with the low temperature (750–900° C) dataset of Cygan and Lasaga (1985) if the diffusion coefficients are assumed to be proportional to f _{O 2} ^1/6. Such an f _{O 2} dependence, however, makes this dataset inconsistent with the recent dataset of Schwandt et al. (1995) on garnets of composition (Alm₁₅Pyr₇₂Gr₁₃Sp₀) unless a strong compositional dependence of Mg tracer diffusivity for Mg-rich garnets is invoked. The present experimental results show that such a compositional dependence is weak to non-existent for garnets with >38 mole percent almandine component. It is emphasized that the temperature dependence of diffusion coefficients at constant oxygen fugacities (activation energy ≈54 kcal/mol) are different from that along an oxygen fugacity buffer (activation energy ≈64.5 kcal/mol), as already pointed out by Chakraborty and Ganguly (1991). This distinction is of importance for modelling natural processes. The measurements at low temperatures either eliminate the need for, or greatly reduce the uncertainty of, extrapolation of laboratory data for modelling metamorphic processes. The high pressure results combined with those from Chakraborty and Ganguly (1992) and Loomis et al. (1985) indicate that pressure dependence of Mg tracer diffusivity in garnets is much stronger than that in forsterite (Chakraborty et al. 1994). This difference in pressure dependence of diffusivity may be caused by the difference in compressibility of the coordination polyhedra of Mg between olivines and garnets. Activation volumes of Mg tracer diffusion as high as 8 cm³/mol may be estimated using the present data in combination with earlier results. These data suggest that at a temperature of 1300° C, Mg tracer diffusion rates in garnets will decrease by an order of magnitude for every 100 km depth. The pressure effect will be stronger at lower temperatures. For calculations involving diffusion coefficients of garnets at high pressures (e.g. mantle xenoliths, eclogites) the pressure dependence of diffusivity must be taken into account. Received: 21 December 1994 / Accepted: 22 September 1995     

Allometry in the pliocene bivalveAstarte omalii: A new look at an old data set

The relationships between ontogeny and phylogeny have been studied for the bivalve Astarte omaliifrom the Lower, Middle, and Upper Pliocene of Belgium using canonical variates analysis, principal components in the form of multivariate allometry, and bivariate power functions or allometric equations. The results indicate that the three samples represent segments of a single evolutionary lineage which is characterized mainly by paedomorphosis where adult descendants are similar to juvenile ancestors. Recapitulation or peramorphosis is only observed for a single parameter. The Lower and Middle Pliocene samples are considered as Astarte omalii omalii,whereas the Upper Pliocene specimens are placed in A. omalii basteroti.The evolutionary trends most likely reflect changes in the living positions of the bivalves.     

Maris Gullying: canonical analysis of erosion and sedimentation sequences (Baronnies - France)

Abstract

The gullying action on Jurassic and Cretaceous marls in the Baronnies is observed, warm season after cold season, thanks to a network of measuring stakes arranged in 12 gullies. The climate, characterized by summer heath, rather cold but luminous winters, and quite heavy rain, can be classified as mountainous-mediterranean. Seasonal ablation/sedimentation measures have been carried out upon 119 stakes five times within two years (768 days): two in automn, two in early spring (april), and one at end-of-spring.

Ablation and sedimentation sequences are described among the stakes population by Correspondance Factor Analysis (CFA), and their covariations in space and time are explained by Canonical Correspondance Analysis (CCA) according to biophysical environment variables.

Axis 1 of the CFA opposes ground level variations according to their amplitude, high or moderate, regardless of the direction of variation and of measurement date. Axis 2 presents a temporal variation, opposing, on one side, sedimentation sequences at post-winter terms and ablation sequences at postsummer terms, to their converse on the other side.

The CCA indicates that talwegs in their medium and lower parts are more likely to undergo hard gullying or strong accumulation, especially on steep, or conversely flat, vegetationfree slopes of oxfordian marls; on the contrary, interfluve ridges and Gonson moderate gentle to moderate slopes on cenomanian marls, possibly with some vegetation are less dynamic (canonical axis 1). This result allows to extend former observations on rythmic, contrasted variations of gullies bottoms, well different of interfluves and gonson, progressive, regular, ones.

Owing to their varied reactions to seasonal erosive agents (cryoclasty, concentrated run-off), upper parts of talwegs and lower parts of gully slopes and interfluves (canonical axis 2) originate seasonal contrasts revealed by CFA’s axis 2. However, to insure this point of view, more sampling is needed.     

Uphill diffusion,zero-flux planes and transient chemical solitary waves in garnet

Diffusion profiles in minerals are increasingly used to determine the duration of geological events. For this purpose, the distinction between growth and diffusion zoning is critical; it requires the understanding of complex features associated with multicomponent diffusion. Seed-overgrowth interdiffusion experiments carried out in the range 1,050–1,250°C at 1.3 GPa have been designed to quantify and better understand Fe–Mg–Ca interdiffusion in garnet. Some of the diffusion profiles measured by analytical transmission electron microscope show characteristic features of multicomponent diffusion such as uphill diffusion, chemical solitary waves, zero-flux planes and complex diffusion paths. We implemented three different methods to calculate the interdiffusion coefficients of the D matrix from the experimental penetration curves and determined that with Ca as the dependent component, the crossed coefficients of the D matrix are negative. Experiments and numerical simulations indicate that: (1) uphill diffusion in garnet can be observed indifferently on the three components Fe, Mg and Ca, (2) it takes the form of complementary depletion/repletion waves and (3) chemical waves occur preferentially on initially flat concentration profiles. Derived D matrices are used to simulate the fate of chemical waves in time, in finite crystals. These examples show that the flow of atoms in multicomponent systems is not necessarily unidirectional for all components; it can change both in space along the diffusion profile and in time. Moving zero-flux planes in finite crystals are transitory features that allow flux reversals of atoms in the diffusion zone. Interdiffusion coefficients of the D matrices are also analyzed in terms of eigenvalues and eigenvectors. This analysis and the experimental results show that depending on the composition of the diffusion couple, (1) the shape of chemical waves and diffusion paths changes; (2) the width of the diffusion zone for each component may or may not be identical; and (3) the width of diffusion calculated at a given D and duration may greatly vary. D matrices were retrieved from thirteen sets of diffusion profiles. Data were cast in Arrhenius relations. Linear regressions of the data yield activation energies equal to 368, 148, 394, 152 kJ mol⁻¹ at 1 bar and frequency factors D_o equal to 2.37 × 10⁻⁶, −4.46 × 10⁻¹⁶, −1.31 × 10⁻⁵, 9.85 × 10⁻¹⁵ m² s⁻¹ for [(D)\tilde]_FeFe^Ca \tilde{D}_{FeFe}^{Ca} , [(D)\tilde]_FeMg^Ca \tilde{D}_{FeMg}^{Ca} , [(D)\tilde]_MgFe^Ca \tilde{D}_{MgFe}^{Ca} , [(D)\tilde]_MgMg^Ca \tilde{D}_{MgMg}^{Ca} , respectively. These values can be used to calculate interdiffusion coefficients in Fe–Mg–Ca garnets and determine the duration of geological events in high temperature metamorphic or magmatic garnets.     

Solution of Kepler’s equation for rectilinear motion

The properties of the solution of the kinematic equation (Kepler’s equation) presented as a series in powers of a function of time are considered for the case of unperturbed, rectilinear elliptical and rectilinear hyperbolic motion. Kepler’s equation for unperturbed, rectilinear elliptical motion has the form E − sinE = z ³/6, where E is the eccentric anomaly determining the position in the orbit, z ³/6 is the mean anomaly, which is proportional to the time measured from an encounter, and its solution can be represented as a series in powers of z. It is established that the coefficients of the series are positive. The asymptotic for the coefficients in the region of convergence of the series is found, which covers the entire orbit. The series continues to converge over the entire boundary of the circle of convergence. The kinematic equation for unperturbed, rectilinear hyperbolic motion has the form sinhH − H = ζ ³/6, where, as before, ζ ³ is proportional to time. The substitution E = iH, z = iζ reduces one equation to the other. The series for the solution in the hyperbolic case differs from the series for the elliptical solution only in its alternating-sign coefficients. However, the region of convergence covers only part of the orbit in the hyperbolic case.     

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.