Qualitative interpretation of physico-chemical and isotopic parameters in the Krka River (Croatia) assessed by multivariate statistical analysis

Multivariate statistical analyses were applied on the measured physico-chemical (Cd, Pb, Cu, Zn, Mg, Ca, O₂, alkalinity, temperature, pH, SAS, DOC and DIC) and isotopic parameters (δ¹³C and δ¹⁸O) to estimate and distinguish anthropogenic from natural influences to the water system of the Krka River. Analyses were conducted on the data collected during six years from twelve sampling sites. On the basis of orientation, positioning and grouping of parameters arranged by biplots, four main hypotheses were defined and finally statistically confirmed. Thereof, two main and distinct processes occurring in the Krka River could be highlighted: (i) upstream pollution, caused by the inflow of untreated waste-waters of city of Knin and (ii) downstream self-purification, caused by the sedimentation and/or co-precipitation of pollutants coupled by the inflow of clean subterranean water (groundwater recharge). Grouping of (i) hydrological and carbon cycle connected parameters, and (ii) anthropogenically influenced correlated parameters were proposed as a result of statistical analysis. Regarding the pH, it is shown that a stream section influenced by the subterranean inflow of Zrmanja River is statistically significantly different for all sampling campaigns during six years, being lower for about 0.5 pH unit.     

Multi-sample cluster analysis using Akaike's Information Criterion

Summary Multi-sample cluster analysis, the problem of grouping samples, is studied from an information-theoretic viewpoint via Akaike's Information Criterion (AIC). This criterion combines the maximum value of the likelihood with the number of parameters used in achieving that value. The multi-sample cluster problem is defined, and AIC is developed for this problem. The form of AIC is derived in both the multivariate analysis of variance (MANOVA) model and in the multivariate model with varying mean vectors and variance-covariance matrices. Numerical examples are presented for AIC and another criterion calledw-square. The results demonstrate the utility of AIC in identifying the best clustering alternatives. This research was supported by Office of Naval Research Contract N00014-80-C-0408, Task NR042-443 and Army Research Office Contract DAAG 29-82-K-0155, at the University of Illinois at Chicago.     

On the monotonicity of the power functions of tests based on traces of multivariate beta matrices

It is shown that for the MANOVA problem the power function of the test based on the trace of a multivariate beta matrix is monotonically increasing in each noncentrality parameter provided that the cutoff point is not too large. This result is also true for the problem of testing independence of two sets of variates.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero.     

On some distribution problems in Manova and discriminant analysis

Asymptotic expansions, valid for large error degrees of freedom, are given for the multivariate noncentral F distribution and for the distribution of latent roots in MANOVA and discriminant analysis. The asymptotic results are expressed in terms of elementary functions which are easy to compute and the results of some numerical work are included. The Bartlett test of the null hypothesis that some of the noncentrality parameters in discriminant analysis are zero is also briefly discussed.     

HE Plots for Multivariate Linear Models

Multivariate analysis of variance (MANOVA) extends the ideas and methods of univariate ANOVA in simple and straightforward ways. But the familiar graphical methods typically used for univariate ANOVA are inadequate for showing how measures in a multivariate response vary with each other, and how their means vary with explanatory factors. Similarly, the graphical methods commonly used in multiple regression are not widely available or used in multivariate multiple regression (MMRA). We describe a variety of graphical methods for multiple-response (MANOVA and MMRA) data aimed at understanding what is being tested in a multivariate test, and how factor/predictor effects are expressed across multiple response measures.

In particular, we describe and illustrate: (a) Data ellipses and biplots for multivariate data; (b) HE plots, showing the hypothesis and error covariance matrices for a given pair of responses, and a given effect; (c) HE plot matrices, showing all pairwise HE plots; and (d) reduced-rank analogs of HE plots, showing all observations, group means, and their relations to the response variables. All of these methods are implemented in a collection of easily used SAS macro programs.     

Generalized Hodges-Lehmann estimators for the analysis of variance

For X_i, …, X_n a random sample and K(·, ·) a symmetric kernel this paper considers large sample properties of location estimator satisfying , . Asymptotic normality of is obtained and two forms of interval estimators for parameter θ satisfying EK(X₁ − θ, X₂ − θ) = 0, are discussed. Consistent estimation of the variance parameters is obtained which permits the construction of asymptotically distribution free procedures. The p-variate and multigroup extension is accomplished to provide generalized one-way MANOVA. Monte Carlo results are included.     

Asymptotic joint distribution of the largest roots of several multivariate F matrices I. Quasi-independent case

An asymptotic expansion of the joint distribution of k largest characteristic roots C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is given, where S'_is and S₀ are independent Wishart matrices with common covariance matrix Σ. The modified second-approximation procedure to the upper percentage points of the maximum of C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is also considered. The evaluation of the expansion is based on the idea for studentization due to Welch and James with the use of differential operators and of the perturbation procedure.     

Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained.     

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.