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Multivariate Process Capability Indices Based on Proportion of Nonconformance
Published in Ashis Kumar Chakraborty, Moutushi Chatterjee, Handbook of Multivariate Process Capability Indices, 2021
Ashis Kumar Chakraborty, Moutushi Chatterjee
Wang and Chu [17] modified the proposed MPCI by dividing the variables into two broad groups: one set of variables which together follows multivariate normal distribution, and the other set of variables which does not follow multivariate normal distribution. For the first set of variables which jointly follow multivariate normal distribution Wang and Chu [17] suggested to use Mahalanobis distance (MD) as the distance measure between the observations. For the other group they suggested taking the Euclidian distance which Wang [15] referred to as the geometric distance (GD). The process yield for the GD variables can be defined as YieldGD=∫0MRDGDf(x)dx
Multivariate Methods
Published in Shayne C. Gad, Carrol S. Weil, Statistics and Experimental Design for Toxicologists, 1988
If Σ (and hence P) is of full rank n, then Σ (and hence P) will be positive definite. In this case, Var (aTX) = aT Σ a is strictly greater than zero for every a ≠ 0. But if rank (Σ) < p, then Σ (and hence P) will be singular, and this indicates a linear constraint on the components of X. This means that there exists a vector a ≠ 0 such that aTX is identically equal to a constant. The most commonly assumed and used multivariate distribution is the multivariate normal distribution.
Model Development
Published in Przemyslaw Biecek, Tomasz Burzykowski, Explanatory Model Analysis, 2021
Przemyslaw Biecek, Tomasz Burzykowski
For example, in linear regression we assume that the observed vector y follows a multivariate normal distribution: y¯~N(X¯'β¯,σ2I¯n), where θ̲’ = (β’, σ2) and In denotes the n x n identity matrix. In this case, equation (2.4T becomes .θ¯˜=argminθ¯∈Θ{1n‖y−X¯′β¯‖2+λ(β¯)}=θ¯˜=argminθ¯∈Θ{1n∑i=1n(yi−x¯′iβ¯)2+λ(β¯)}.
Modeling and optimization for multiple correlated responses with distribution variability
Published in IISE Transactions, 2023
Shijuan Yang, Jianjun Wang, Jiawei Wu, Yiliu Tu
Generally, three strategies can be used in Robust Parameter Design (RPD) to take into account the correlation among responses, i.e., considering correlations in the modeling process, correlations in the optimization process and correlations simultaneously in modeling and optimization processes. Principal Component Analysis (PCA) and Seemingly Uncorrelated Regression (SUR) approach are commonly used in the existing literature to consider correlations during the modeling process. The former solves the correlation problem by transforming multiple correlated responses into several linearly uncorrelated components, whereas the latter considers correlation using a covariance matrix under the assumption of a multivariate normal distribution. (De Paiva et al., 2014). Tensor analysis is also used by researchers to study the correlations of data with tensor structure in different dimensions (Yue et al., 2020). However, these approaches cannot solve the nonlinear correlation problem.
Solution Strategies for Three Problems in Empirical Fragility Curve Derivation Using the Maximum Likelihood Method
Published in Journal of Earthquake Engineering, 2018
Matthew R. Cutfield, Quincy T. M. Ma
Once and are known, the mean vector and covariance matrix defining the multivariate normal distribution may be readily calculated. After conditioning on the observed data, the intra-event errors at each site can be shown to follow the normal distribution [Park et al., 2007; Bradley and Hughes, 2012]:
A novel transfer learning model for predictive analytics using incomplete multimodality data
Published in IISE Transactions, 2021
Xiaonan Liu, Kewei Chen, David Weidman, Teresa Wu, Fleming Lure, Jing Li
The presentation of IMTL in Sections 3.1 to 3.3 is within the context of three modalities based on the consideration of notational simplicity. In this section, we provide the steps of extending IMTL to the general case of modalities:Given a multimodality dataset from an application, subjects (a.k.a. samples) are grouped into sub-cohorts with each sub-cohort having a different pattern of missing modalities.Depending on the type of the response variable, one can decide if the problem to be tackled should be formulated as regression or classification. For a regression problem, a multivariate normal distribution can be assumed for the modalities and the response. For classification problems, a multivariate normal distribution can be assumed for the modalities and a Bernoulli distribution can be assumed for the response. For most applications, there is at least one modality available across all the sub-cohorts. If this is the case, the aforementioned distributions can be modified into conditional distributions given the available modality. Next, one can write down the complete-data log-likelihood function under the distribution assumption.In the E-step of the EM algorithm, the key is to identify the sufficient statistics of the log-likelihood function, which include the missing modalities in each sub-cohort, the quadratic term of each missing modality, and pair-wise products between the missing modalities in that sub-cohort (if there is more than one missing modality). Then, one can derive expectations of the sufficient statistics given the observed data and parameter estimates from the previous iteration. Further, these expected sufficient statistics are plugged into the expected complete-data log-likelihood function, which will be maximized in the M-step.In the M-step, an intuitive, but mathematically-involved, approach is to equate the first-order partial derivative of each parameter to zero and solve the parameter-wise simultaneous equations to update the parameter set. Alternative approaches can be developed to solve the maximization problem easier, depending on the form of the log-likelihood function. For example, in the three-modality case, we use a notational trick, which allowed us to convert the maximization into least square estimations.