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Stochastic Free Vibration Analysis of Pre-twisted Singly Curved Composite Shells
Published in Sudip Dey, Tanmoy Mukhopadhyay, Sondipon Adhikari, Uncertainty Quantification in Laminated Composites, 2018
Sudip Dey, Tanmoy Mukhopadhyay, Sondipon Adhikari
In general, a statistical measure of goodness of a model obtained by least squares regression analysis is the minimum generalized variance of the estimates of the model coefficients (Montgomery 1991). D-optimal design method is employed to provide a mathematical and statistical approach in portraying the input-output mapping by construction of meta-model with small representative number of samples. The problem of estimating the coefficients of a linear approximation is modelled by least squares regression analysis () Y=Xβ+ε
Statistical quality control
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
A plot of the ratio of the sample generalized variance to the population generalized variance can be used to monitor variability. Theoretical results are available for the sampling distribution but they are sensitive to an assumption of multivariate normality and Monte-Carlo simulation provides an easily implemented, if less elegant, alternative (Exercise-SQCgeneralizedvar). The trace of the sample variance-covariance matrix (that is the sum of the variances, which lie along the leading diagonal) might also be used to monitor variability but it ignores information from covariances (Exercise-SQCtrace).
Simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles with a new adaptive Shewhart-type control chart
Published in Quality Engineering, 2023
Hamed Sabahno, Amirhossein Amiri
Note that, as mentioned above, we assume that is a constant number, i.e., each element in the variance-covariance matrix shifts with the same multiplier (). In general, each element of may shift independently at different rates. In such cases, as and matrices have different dimensions (despite being interconnected), the multiplier () in is going to be different from the multiplier () in However, since Equation (27) only contains we only need to have the value. To compute we can use the concept of matrix generalization to change the matrices into numbers. According to Rencher (2002), the generalized form of a variance-covariance matrix (its determinant) is called the generalized variance, and it increases/decreases if the variances increase/decrease and/or the intercorrelations decrease/increase.
A review of dispersion control charts for multivariate individual observations
Published in Quality Engineering, 2021
Jimoh Olawale Ajadi, Zezhong Wang, Inez Maria Zwetsloot
Multivariate dispersion control charts monitor the process variability of multiple correlated quality characteristics. The most common method for monitoring the covariance matrix of the process is the generalized variance chart proposed by Alt (1985). This chart applies the determinant of the estimated covariance matrix as the monitoring statistic. Over the past decades, many methods have been developed for monitoring the covariance matrix. To obtain an estimate of the covariance matrix, usually a sample of observations are collected. Here, is the number of correlated quality characteristics to be monitored. We refer to this as grouped data. For details on multivariate dispersion charts for grouped data, see the review articles by Yeh, Lin, and McGrath (2006) and Bersimis, Psarakis, and Panaretos (2007).
Monitoring location and scale of multivariate processes subject to a multiplicity of assignable causes
Published in Quality Technology & Quantitative Management, 2020
Konstantinos A. Tasias, George Nenes
Recently, Liu, Zhong and Ma (2013) studied a multivariate synthetic control chart for monitoring the covariance matrix based on the combination of a conditional entropy chart and the conforming run-length chart. Li, Wang and Yeh (2013) and Mitra and Clark (2014), also, presented control charts for monitoring shifts in the covariance matrix of multivariate processes. Moreover, Lee and Khoo (2015) proposed a VSI multivariate synthetic control chart for shifts in the covariance matrix based on the generalized variance control chart.