Extremes of multivariate ARMAX processes

We define a new multivariate time series model by generalizing the ARMAX process in a multivariate way. We give conditions on stationarity and analyze local dependence and domains of attraction. As a consequence of the obtained results, we derive new multivariate extreme value distributions. We characterize the extremal dependence by computing the multivariate extremal index and bivariate upper tail dependence coefficients. An estimation procedure for the multivariate extremal index is presented. We also address the marginal estimation and propose a new estimator for the ARMAX autoregressive parameter.     

On properties of probability-based multivariate process capability indices

Multivariate process capability indices (MPCIs) have been proposed to measure multivariate process capability in real-world application over the past three decades. For the practitioner's point of view, the intention of this paper is to examine the performances and distributional properties of probability-based MPCIs. Considering issues of construction of capability indices in multivariate setup and computation with performance, we found that probability-based MPCIs are a proper generalization of univariate basic process capability indices (PCIs). In the beginning of this decade, computation of probability-based indices was a difficult and time-consuming task, but in the computer age statistics, computation of probability-based MPCIs is simple and quick. Recent work on the performance of MPCI NMC_pm and distributional properties of its estimator reasonably recommended this index, for use in practical situations. To study distributional properties of natural estimators of probability-based MPCIs and recommended index estimator, we conducted simulation study. Though natural estimators of probability-based indices are negatively biased, they are better with respect to mean, relative bias, mean square error. Probability-based MPCI MC_pm is better as compared with NMC_pm with respect to performance and as its estimator quality. Hence, in real-world practice, we recommend probability-based MPCIs as a multivariate analogue of basic PCIs.     

Counting decomposable multivariate polynomials

A polynomial f (multivariate over a field) is decomposable if f=g °h{f=g \circ h} with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The relative error in our approximations is exponentially decaying in the input size.     

Economic design of the integrated multivariate EPC and multivariate SPC charts

The goal of engineering process control (EPC) is to minimize variability by adjusting some manipulative process variables. The goal of statistical process control (SPC) is to reduce variability by monitoring and eliminating assignable causes of variation. As suggested by Box and Kramer and others, it is possible to reduce both special cause and common cause variations by integrating EPC and SPC. In the integrated multivariate EPC (MEPC) and multivariate SPC (MSPC) charts, we propose some statistical and economic criteria, such as the average Euclidean distance from the target vector and the average quality cost (AQC) to evaluate the performance of the MEPC/MSPC charts. The traditional average run length (ARL), average Euclidean distance and AQC of three MSPC charts are investigated and compared. The results of the simulations show that the MEPC/MGWMA chart is more effective and more economical than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart in detecting small shifts of the mean vector. Copyright © 2006 John Wiley & Sons, Ltd.     

On extremal dependence: some contributions

The usual coefficients of tail dependence are based on exceedances of high values. These extremal events are useful and widely used in literature but an adverse situation may also occur with the upcrossing of a high level. In this context we define upcrossings-tail dependence coefficients and analyze all types of dependence coming out. We will prove that these coefficients are related to multivariate tail dependence coefficients already known in literature. We shall see that the upcrossings-tail dependence coefficients have the interesting feature of congregating both ??temporal?? and ??spatial?? dependence. The coefficients of tail dependence can also be applied to stationary sequences and hence measure the tail dependence in time. Results concerning connections with the extremal index and the upcrossings index as well as with local dependence conditions will be stated. Several illustrative examples will be exploited and a small note on inference will be given by presenting estimators derived from the stated results and respective properties.     

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

On toric codes and multivariate Vandermonde matrices

Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope P í \mathbbR^mP \subseteq {\mathbb{R}}^m . As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all m 3 2m \ge 2 . Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P ₁ to a second polytope P ₂, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.     

A Bayesian approach to the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices

Summary Two independent random samples of sizesN ₁ andN ₂ from multivariate normal populationsN _p (θ₁,∑₁) andN _p (θ₂,∑₂) are considered. Under the null hypothesisH ₀: θ₁=θ₂, a single θ is generated from aN _p(μ, Σ) prior distribution, while underH ₁: θ₁≠θ₂ two means are generated from the exchangeable priorN _p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.     

Fast and robust estimation of the multivariate errors in variables model

In the multivariate errors in variables models, one wishes to retrieve a linear relationship of the form y=β ^t x+α, where both x and y can be multivariate. The variables y and x are not directly measurable, but observed with measurement error. The classical approach to estimate the multivariate errors in variables model is based on an eigenvector analysis of the joint covariance matrix of the observations. In this paper, a projection-pursuit approach is proposed to estimate the unknown parameters. The focus is on projection indices based on half-samples. These lead to robust estimators which can be computed using fast algorithms. Fisher consistency of the procedure is shown, without the need to make distributional assumptions on the x-variables. A simulation study gives insight into the robustness and the efficiency of the procedure.     

On discrete multivariate distributions symmetric in frequencies

In this paper we describe a new class of discrete multivariate distributions which verify that their probability mass function is invariant when their univariate variables are permuted. These distributions may be generated by a multivariate extension of the Gauss function₂ F ₁ with matrix argument. A methodology that permits the fit of these distributions to real data is developed. A fit of a distribution for bivariate real data is shown and is compared with fits obtained by means of other usual bivariate distributions generated by extensions of the Gauss function.     

New capability indices for evaluating the performance of multivariate manufacturing processes

Generally, an industrial product has more than one quality characteristic. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several multivariate process capability indices have been developed in the past few years. Among them, Taam's MC_p and MC_pm indices have the drawback of overestimation and Hubele's three‐component capability vector lacks simplicity in practice. In this article, taking the correlation among multiple quality characteristics into account, we develop two novel indices; NMC_p and NMC_pm. Using two numerical examples we demonstrate that the true performance of multivariate processes are accurately reflected in our NMC_p and NMC_pm indices and in their associated interval estimates. Finally, simulation results show that our indices outperform both those of Taam and Hubele. Copyright © 2009 John Wiley & Sons, Ltd.     

Improving the performance of the multivariate exponentially weighted moving average control chart

Multivariate statistical process control (SPC) procedures are useful in cases where several process variables are monitored simultaneously. A significant disadvantage of these techniques is that the time required to detect a process shift increases with the number of variables being monitored. We show how the shift detection capability of one popular multivariate SPC scheme, the multivariate analogue of the exponentially weighted moving average control chart, can be significantly improved by transforming the original process variables to a lower‐dimensional subspace through the use of a U‐transformation. Copyright © 1999 John Wiley & Sons, Ltd.     

Select better suppliers based on manufacturing precision for processes with multivariate data

Process capability indices have been widely used in the manufacturing industry for measuring process reproduction capability according to manufacturing specifications. Processes with univariate data have been investigated extensively, but are comparatively neglected for processes with multivariate data. Chou (Chou, Y.M., 1994. Selecting a better supplier by testing process capability indices. Quality Engineering, 6, 427–438) developed a procedure using univariate C_p to determine whether or not two processes are equally capable, which allows one to select the supplier with better quality. However, for processes with multiple characteristics, no methods are available for comparing two processes with multivariate data. In this paper, we consider the supplier selection problem based on manufacturing precision in which the processes involve multiple quality characteristics. We derive the distribution of the corresponding test statistic, and calculate critical values required for the comparison purpose. A real-world application is presented for justification.     

Magnetic focusing of electrons and holes in bismuth

The electron and hole focusing effects in a magnetic field predicted by Pippard have been observed in bismuth. Two interlocking superconducting combs were evaporated onto the surface of a bismuth single crystal. The resistanceR(H) between the combs was measured with a superconducting voltmeter as a function of the magnetic fieldH applied parallel to the teeth of the combs. A local minimum in the resistance was observed when the magnetic field focused electrons or holes in extremal orbits between the teeth of different combs. A similar effect was observed when the combs were replaced by two parallel superconducting strips. In this case the dependence ofR(H) onH was in good qualitative agreement with the calculation of Gonçalves da Silva on a model whose Fermi surface was topologically equivalent to that of bismuth. From the values ofH at which the minima occurred we have deduced estimates forD _xe andD _ze , the diameters of the principal electron ellipsoid along the binary and trigonal axes. We findD _xe =(0.94±0.03)×10 ^–2 Å ^–1 andD _ze =(1.35±0.04)×10 ^–2 Å^–1. These values lie somewhat below other values in the literature. This discrepancy may occur because each minimum inR(H) occurs at a field slightly belowH ₀, whereH ₀ is the field for which the diameter of the extremal electron trajectory is equal to the separation of the superconducting strips.Work supported by the U.S. Atomic Energy Commission.Alfred P. Sloan Foundation Fellow.     

Optimal bandwidth selection for multivariate kernel deconvolution density estimation

Assume we have i.i.d. replications from the mismeasured random vector Y=X+ε, where X and ε are mutually independent. We consider a data-driven bandwidth, based on a cross-validation ideas, for multivariate kernel deconvolution estimator of the density of X. The proposed data-driven bandwidth selection method is shown to be asymptotically optimal. As a by-product of the proof of this result, we show that the average squared error, the integrated squared error, and the mean integrated squared error are asymptotically equivalent error measures.      

Bin width selection in multivariate histograms by the combinatorial method

We present several multivariate histogram density estimates that are universallyL ₁-optimal to within a constant factor and an additive term . The bin widths are chosen by the combinatorial method developed by the authors inCombinatorial Methods in Density Estimation (Springer-Verlag, 2001). The present paper solves a problem left open in that book.     

Adaptive multivariate double sampling and variable sampling interval Hotelling's T2 charts

In this article, two adaptive multivariate charts, which combine the double sampling (DS) and variable sampling interval (VSI) features, called the adaptive multivariate double sampling variable sampling interval T² (AMDSVSI T²) and the adaptive multivariate double sampling variable sampling interval combined T² (AMDSVSIC T²) charts, are proposed. The real purpose of using the proposed charts is to provide flexibility by enabling the sampling interval length of the DS T² chart to be varied so that the chart's sensitivity can be enhanced. The fundamental difference between the two proposed charts is that when a second sample is taken, the AMDSVSI T² chart uses the information of the combined sample mean vectors while the AMDSVSIC T² chart uses the information of the combined T² statistics, in deciding about the process status. This research is motivated by existing combined DS and VSI charts in the literature, which show convincing performance improvement over the standard DS chart. Consequently, it is believed that adopting this existing approach in the multivariate case will enable superior multivariate DS charts to be proposed. Numerical results show that the proposed charts outperform the existing standard T² and other adaptive multivariate charts, in detecting shifts in the mean vector, for the zero‐state and steady‐state cases. The performances of both charts when the shift sizes in the mean vector are unknown are also measured. The application of the AMDSVSI T² chart is illustrated with an example.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

Skewed multivariate models related to hidden truncation and/or selective reporting

The univariate skew-normal distribution was introduced by Azzalini in 1985 as a natural extension of the classical normal density to accommodate asymmetry. He extensively studied the properties of this distribution and in conjunction with coauthors, extended this class to include the multivariate analog of the skew-normal. Arnold et al. (1993) introduced a more general skew-normal distribution as the marginal distribution of a truncated bivariate normal distribution in whichX was retained only ifY satisfied certain constraints. Using this approach more general univariate and multivariate skewed distributions have been developed. A survey of such models is provided together with discussion of related inference questions.     

A boosting approach for understanding out-of-control signals in multivariate control charts

The most widely used tools in statistical quality control are control charts. However, the main problem of multivariate control charts, including Hotelling's T ² control chart, lies in that they indicate that a change in the process has happened, but do not show which variable or variables are the source of this shift. Although a number of methods have been proposed in the literature for tackling this problem, the most usual approach consists of decomposing the T ² statistic. In this paper, we propose an alternative method interpreting this task as a classification problem and solving it through the application of boosting with classification trees. The classifier is then used to determine which variable or variables caused the change in the process. The results prove this method to be a powerful tool for interpreting multivariate control charts.