Multivariate Situations Research Articles

Primeness of multivariate polynomial matrices

Different characterizations of primeness of polynomial matrices that are mutually equivalent in the one-dimensional case of univiriate polynomials lead to various primeness concepts in the multivariate situation. Applied to systems theory, this fact gives rise to a more refined view to observability and controllability of multidimensional systems. Algorithms for testing primeness properties are given based on computer algebraic techniques.

Application of the bootstrap to calibration experiments.

In calibration experiments, a number of samples of known concentration are used to establish the relationship between a measured response and sample concentration; this relationship is then used to estimate the unknown concentration of further samples from their measured responses. In addition to the estimates themselves, it is useful to have available some measure of their precision, usually given in the form of confidence limits. The standard method of inverting prediction limits is found to work well in simple situations, but in nonlinear multivariate calibration it becomes intractable. The bootstrap offers an alternative methodology, but in the calibration framework its application is not obvious. We describe some considerations in bootstrapping calibration data and compare our methods with a previous attempt and with the standard method in linear, nonlinear, and multivariate situations. The bootstrap is found to be a useful tool in those situations where the standard method is difficult to implement.

A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa

Abstract It seems difficult to find a formula in the literature that relates moments to cumulants (and vice versa) and is useful in computational work rather than in an algebraic approach. Hence I present four very simple recursive formulas that translate moments to cumulants and vice versa in the univariate and multivariate situations.

Selection of variables, and assessment of their performance, in mixed-variable discriminant analysis

Two problems frequently encountered by the practitioner when faced with multivariate situations are how to select the “best” variables for the objective in hand, and how to assess accurately subsequent performance of the selected variables. A typical framework in which these problems arise is discriminant analysis, and the variables available are often mixtures of qualitative and quantitative ones. Various methods have been proposed in recent years for selecting variables in such mixed-variable discrimination situations, and a brief review is first given of the possibilities. Some of the methods are simply variations on the same basic underlying model (the location model); some simulation studies comparing these variations are then described. Finally, the practical assessment of performance of a variable selection method is addressed. Analysis of some real data highlights the problems that can arise.

Modelling Capability

A survey is given of the concept of capability and the usual underlying assumption of Normality. The extension to non-Normal and multivariate situations is explored. The capability behaviour of systems is illustrated. A number of extensions to the idea of capability are suggested.

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.

VARIATIONAL PROBLEMS ON UNBOUNDED TWO-DIMENSIONAL DOMAINS

We consider the functional IΩ(u) = ∫Ω [ψ (u(x,y)) + ½K (∇ u)]dxdy defined for real valued functions u on ℝ2 and study its minimization over a certain class of functions u(·, ·). We look for a minimizer u⋆ which is universal in the sense that IΩ(u⋆)≤IΩ(u) for every bounded domain (in a certain class) and for every u(·, ·) which satisfies u|∂Ω=u⋆|∂Ω. This optimality notion is an extension to a multivariable situation of the overtaking optimality criterion used in control theory, and the minimal-energy-configuration concept employed in the study of certain chemical systems. The existence of such universal minimizers is established for a large class of variational problems. In the special case were K(∇ u) = ½ |∇ u|2 these minimizers are characterized as the functions u⋆(x, y)=ϕ(ax+by+c) for some explicitly computable ϕ:ℝ1→ℝ1 and constants a, b and c.

Generalization of normal discriminant analysis using fourier series density estimators

In this paper we examine the efficiency of a generalization of the traditional normal linear (LDA) or quadratic (QDA) discriminant analysis. This procedure (the generalized discriminant analysis, GDA) replaces each normal density used in the traditional classification rule by a Fourier series density estimator which 'adjusts' the normal density if the data deviate markedly from normality (for example, heavily skewed or multimodal). We derive the GDA in both the univariate and multivariate situations. In a simulation study for the univariate situation, we evaluate the relative efficiency of the GDA. In addition, we demonstrate the performance of the GDA through a series of multivariate applications. We conclude that if the distributions of the data do not deviate markedly from normality, the GDA is as efficient as the LDA or QDA. On the other hand, if either of the distributions deviates from normality, then the GDA, which performs as a semiparametric discriminant procedure, is more efficient than the LDA or QDA.

Testing Multivariate Partial, Semipartial and Bipartial Correlation Coefficients.

Stressing the close resemblance between univariate and multivariate situations, it is shown that a proper test for a multivariate semipartial correlation being zero is given by the test for the corresponding partial correlation. No legitimate test seems to be available for bipartial correlations. The procedures proposed by Cohen (1982) and Timm and Carlson (1976) are not well-founded and, in the case of multivariate semipartial correlations, unnecessarily conservative.

A method for testing whether model predictions fall within a prescribed factor of true values, with an application to pesticide leaching

A quantitative method is described for testing whether model predictions fall within a specified factor of true values. The technique is based on classical theory for confidence regions on unknown population parameters and can be related to hypothesis testing in both univariate and multivariate situations. A capability index is defined that can be used as a measure of predictive capability of a model, and its properties are discussed. The testing approach and the capability index should facilitate model validation efforts and permit comparisons among competing models. An example is given for a pesticide leaching model that predicts chemical concentrations in the soil profile.

Automatic tuning of commercial PID controllers for single-loop and multiloop applications

Certain continuous-time self-tuning algorithms are shown to be capable of generating tuning parameters for commercial proportional-integral-derivative (PID) controllers. They are also shown to be capable of generating a feedforward signal that decouples the disturbance from the interaction from adjacent loops in a multivariable situation. The approach is verified using a standard Atari personal computer supervising Eurotherm 820 PID controllers. Two systems are used to test the method: a laboratory water tank system under level control and an industrial multizone plastics extruder under temperature control. >

The exact distributions of the univariate and multivariate behrens-fisher statistics with a comparison of several solutions in the univariate case

The exact distributions of the univariate- and multivariate Behrens-Fisher statistics are derived. As is well known, these distributions are dependent on the unknown variances or covariance Matrices.The critical values of the exact distribution are computed and compared to the critical values of several approximations in the univariate case with the variances replaced by sample estimates.In the multivariate situation it is shown that this distribution is a special case of a generalization of a matrix Beta type II distribution as derived by van der Merwe and Nel (1987). Due to the computational intractability of the density function we could not compare it to an approximation.

Indices d'atypisme en paléontologie humaine

For a normally distributed variable, the index of atypicality associated with a measurement of that variable is defined as the probability of finding a result closer to the mean of the reference population than the one actually observed. The method extends straight forwardly to multivariate situations, thus providing a joint interpretation of multiple observations recorded on the same subject. In human palaeontology, atypicality indices allow to analyze and to compare the results obtained from variables closely associated with specific traits of the morphology. We briefly outline the principle of the method and give a computer program to carry out the calculations. Finally, we apply the use of univariate and multivariate atypicality indices to data recorded on 53 Upper Palaeolithic skulls.

Multivariate generalization of t'- statistic based on the mean square successive difference

The usual t-statistic is not useful if the successive observations have some kind of linear trend. This generally arises in the drug testing experiment and it is clearly pointed out by Shah (1988). He suggests to use t'-statistic which is defined by for testing μ = 0 where , and y1.,…,yk are independent observations from . We generalize this statistic to multivariate situation and define T'-statistic as where .The exact null distribution of T' and an approximate null distribution of T' are obtained. For p = 1, this approximate values are compared.

Experiences of reactor noise diagnostics applying parametric spectral analysis methods

The signal processing technology, like the choice of parametrization, methods of structure and parameter estimation, computation of special frequency functions, that are avilable in the reactor noise surveillance system of NPP Paks is reviewed. A method for decomposition of the spectral densities to study the effect of common source noises in multivariable situation is also proposed. The paper presents experiences using the above noise analysis methods using measurements on the 2nd and 3rd reactor units of NPP “Paks”.

On the Distribution of General Quadratic Functions in Normal Vectors

A representation in terms of independent standard normal variables tor the general quadratic form in normal variables in the univariate case, obtained by DIK and DE GUNST (1985), is extended to the multivariate situation. A representation for the quadratic function in normal vectors X'AX, where X is a random matrix with normally distributed elements and A a real symmetric matrix, is given in terms of independent and identically distributed central normal vectors. The representation is valid only when the covariance structure of X is of a special form, but all known results, especially necessary and sufficient conditions for X'AX to have a Wishart distribution, can easily be derived from it.

Standard Distance in Univariate and Multivariate Analysis

Abstract We propose to use the term standard distance for the quantity in univariate analysis and show that it can be easily generalized to the multivariate situation, where it coincides with the square root of the Mahalanobis distance between two samples.

Statistical model of changes in repeated multivariate measurements associated with the development of disease

This paper is concerned with the use of repeated multivariate measurements in risk factor analysis. Emphasis is placed upom the recognition of temporal variation patterns of the multivariate measurements prior to the development of disease. The proposed model is an extension of the Truett and Sorlie model [9] to a multivariate situation, but it includes a more unified approach on the identification of the most suitable model of multivariate pattern of temporal change. This model is applied to a cohort of individuals participating in the Wara Cardiovascular Study in Japan.

Computerized method for the graphic representation of multivariate periodontal data.

Appreciation of the multivariate features of data from periodontal research studies is sometimes difficult. This appears to be due to the form of tables and graphs in which information is made available in published studies. A computerized graphical technique, which appears well suited to the multivariable situation, is described. It uses structural analogy between the measurement scales and the variables measured and provides what can be a compelling iconic description of multifaceted data.

A bayesian approach to prediction in the presence of spurious observations for several models

The problem of spuriousity has been dealt with from a Bayesian perspective by, among others, Box and Taio (1968) and in several papers by Guttman with various co-authors, beginning with Guttman (1973), The main objective of these papers has been to obtain posterior distributions of parameters, and to base inference on these distributions. In the current paper, the Bayesian argument is carried one step further by deriving predictive distributions of future observations. Inferences are then based on these distributions. We will obtain predictive results for several models, First, we consider the univariate normal case with one spurious observation, This is then generalized to several spurious observations. The multivariate normal situation is studied next. Finally, we consider the general linear model with normal errors.