Multiple comparisons of several homoscedastic multivariate populations

The limiting joint distribution of correlated Hotelling’s T ² statistics associated with multiple comparisons with a control in multivariate one-way layout model is a multivariate central nonsingular chi-square distribution with one-factorial correlation matrix, which has the distribution function expressed in a closed form as an integral of a product of noncentral chi-square distribution functions with respect to a central chi-square density function. For pairwise comparisons, it is a multivariate central singular chi-square distribution whose distribution function is generally intricate. To overcome the complexity of the (exact or asymptotic) distribution theory of -type statistics appeared in simultaneous confidence intervals of mean vectors, improved Bonferroni-type inequalities are applied to construct asymptotically conservative simultaneous confidence intervals for pairwise comparisons as well as comparisons with a control.     

Improved variance estimation under sub-space restriction

For the linear regression model , we assume that for a given positive definite scale matrix , the error vector has a multivariate normal distribution and has the inverted Wishart distribution. For under an orthogonal sub-space restriction , we propose restricted unbiased, preliminary test and Stein-type estimators of variance of the error term, for when the scale of the inverse Wishart distribution is assumed to be unknown. We compare the weighted quadratic risks of the underlying estimators and propose dominance pictures for them.     

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.      

Bayesian linear regression and variable selection for spectroscopic calibration

This paper presents a Bayesian approach to the development of spectroscopic calibration models. By formulating the linear regression in a probabilistic framework, a Bayesian linear regression model is derived, and a specific optimization method, i.e. Bayesian evidence approximation, is utilized to estimate the model “hyper-parameters”. The relation of the proposed approach to the calibration models in the literature is discussed, including ridge regression and Gaussian process model. The Bayesian model may be modified for the calibration of multivariate response variables. Furthermore, a variable selection strategy is implemented within the Bayesian framework, the motivation being that the predictive performance may be improved by selecting a subset of the most informative spectral variables. The Bayesian calibration models are applied to two spectroscopic data sets, and they demonstrate improved prediction results in comparison with the benchmark method of partial least squares.     

Multivariate calibration analysis of colorimetric mercury sensing using a molecular probe

Selectivity is one of the main challenges of sensors, particularly those based on chemical interactions. Multivariate analytical models can determine the concentration of analytes even in the presence of other potential interferences. In this work, we have determined the presence of mercury ions in aqueous solutions in the ppm range (0-2 mg L⁻¹) using a ruthenium bis-thiocyanate complex as a chemical probe. Moreover, we have analyzed the mercury-containing solutions with the co-existence of higher concentrations (19.5 mg L⁻¹) of other potential competitors such as Cd²⁺, Pb²⁺, Cu²⁺ and Zn²⁺ ions. Our experimental model is based on partial least squares (PLS) method and other techniques as genetic algorithm and statistical feature selection (SFS) that have been used to refine, beforehand, the analytical data. In summary, we have demonstrated that the root mean square error of prediction without pre-treatment and with statistical feature selection can be reduced from 10.22% to 6.27%.     

Potential antioxidant compounds in Mallotus species fingerprints. Part I: Indication, using linear multivariate calibration techniques

Some Mallotus species are used in traditional medicine in Vietnam and China. Some also show interesting activities, such as antioxidant and cytotoxic ones. Combining fingerprint technology with data-handling techniques allows indicating the peaks potentially responsible for given activities. In this study it is aspired to indicate from chromatographic fingerprints the peaks potentially responsible for the antioxidant activity of several Mallotus species. Relevant information was extracted using linear multivariate calibration techniques, both before and after alignment of the fingerprints with correlation optimized warping (COW). From the studied techniques, Stepwise Multiple Linear Regression is least recommended as it made an inadequate variable selection. Principal Component Regression theoretically can take largely varying variables uncorrelated to the antioxidant activity into account. However, in practice in the actual case study this problem was limited. These problems in principle do not occur using Partial Least Squares (PLS) models. Of the tested PLS methods, Orthogonal Projections to Latent Structures was preferred because of its simplicity, reproducibility, reduced model complexity and improved interpretability of the regression coefficients, yielding a clearer view on the individual contribution of the compounds. Furthermore, reducing analysis times from 60 min to 35 and 22.5 min resulted in the same main compounds, indicated responsible for the antioxidant activity. Models built after alignment by COW did not result in additional information.     

Fractional calculus approach to the statistical characterization of random variables and vectors

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.     

General sum games with joint chance constraints

We consider an $n$-player non-cooperative game with continuous strategy sets. The strategy set of each player contains a set of stochastic linear constraints. We model the stochastic linear constraints of each player as a joint chance constraint. We assume that the row vectors of a matrix defining the stochastic constraints of each player are independent and each row vector follows a multivariate normal distribution. Under certain conditions, we show the existence of a Nash equilibrium for this game.     

Almost sure convergence for the maxima of strongly dependent stationary Gaussian vector sequences

In this paper, we prove the almost sure limit theorem of the maxima for a kind of strongly dependent stationary Gaussian vector sequences.