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Modern Predictive Analytics and Big Data Systems Engineering
Published in Anna M. Doro-on, Handbook of Systems Engineering and Risk Management in Control Systems, Communication, Space Technology, Missile, Security and Defense Operations, 2023
Multivariate t-distribution is a generalization of the classical univariate Student’s t-distribution, which is of central importance in statistical inference (Nadarajah and Kotz 2005). The possible structures are numerous, and each one possesses special characteristics as far as potential and current applications are concerned (Nadarajah et al. 2005). Classical multivariate analysis is soundly tilted toward the multivariate normal distribution, while multivariate t-distributions offer a more viable alternative with respect to real-world data, particularly because its tails are more realistic (Nadarajah et al. 2005). We have seen recently some unexpected applications in novel areas such as cluster analysis, discriminant analysis multiple regression, robust projection indices, and missing data imputation (Nadarajah et al. 2005). These applications governed by multivariate t-distributions that are widely used in relation to big data being collected involving various homeland defense and national security operations. Here a p-dimensional random vector X = (X1, …, Xp)T is said to have the p-variate t-distribution with degrees of freedom n, mean vector μ, and correlate matrix R (and with denoting the corresponding covariance matrix) if its joint PDF is given by (Kotz and Nadarajah 2004): f(x)=[Γ([n+p]2)(πn)p2Γ|R|12][1+1n(x−μ)TR−1(x−μ)]−(n+p2)
Robust Bayesian hierarchical modeling and inference using scale mixtures of normal distributions
Published in IISE Transactions, 2022
Linhan Ouyang, Shichao Zhu, Keying Ye, Chanseok Park, Min Wang
In this section, several simulation experiments are conducted to illustrate the efficiency of the RBSUR. The underlying models in Equation (2) are used to generate experimental data. Specifically, we set the dimension of y as In the simulation experiments, we consider three predictors, which results in a total of 36 predictors to be analyzed, i.e., To further investigate the variable selection performance of the RBSUR, we consider several different proportions of the model sparsity in the SUR models. Without loss of generality, we assume that and The model parameters for significant predictors are set to be one. To obtain the heavy-tailed issue for the errors, we generate the errors using the following mixture distributions of the form where and determine the degrees of heavy-tailed behavior in the simulation data. The abbreviations ’’MVN’’ and ’’MVT’’ denote the multivariate normal distribution and the multivariate t-distribution, respectively. We consider which is calculated based on the experimental data in the case study in Section 4.2. As to the experiment designs, 50 runs are generated from a uniform distribution [0, 1], and there are 200 total data points collected for the four response variables.
Change detection in parametric multivariate dynamic data streams using the ARMAX-GARCH model
Published in Journal of Quality Technology, 2022
Miaomiao Yu, Chunjie Wu, Fugee Tsung
In this subsection, Monte Carlo (MC) method is applied to verify the asymptotic normality of QMLE, which is strictly proven in Theorem 2. Two multivariant models are considered, the low- and large-dimensional cases respectively, with the initial and where 0 and 1 are vectors, in which all the elements are 0 and 1 respectively. The results are calculated from N = 100 replicated simulations. In each model, follows from a multivariate normal distribution Other results, in which follows from a multivariate t distribution with mean vector 0, convariance matrix and 2 degrees of freedom can be found in Figure S3.13 of supplemental file.
Nonparametric monitoring of multivariate data via KNN learning
Published in International Journal of Production Research, 2021
Wendong Li, Chi Zhang, Fugee Tsung, Yajun Mei
the multivariate normal distribution ;the multivariate t distribution with ζ degrees of freedom;the multivariate mixed distribution, , i.e. a mixture of two multivariate normal distributions. In our simulation, we set r = 0.5, and the mean vector () is in the first component but is 0 for all other components.