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Linear Transformations
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
The Ausdehnungslehre introduces two revolutionary concepts. First is the notion of a general n-dimensional vector space. Grassmann introduces the notion of linearly dependent vectors and develops the “elementary” theory of finite-dimensional vector spaces, as can be found in all of today’s books on linear algebra. The second idea is the introduction of multilinear algebra, which involves the product of vectors or “multivectors”. This leads to a complete theory of the “exterior algebra”, an area of mathematics whose importance could not have been anticipated at that time.
Multiple Tensor-on-Tensor Regression: An Approach for Modeling Processes With Heterogeneous Sources of Data
Published in Technometrics, 2021
Mostafa Reisi Gahrooei, Hao Yan, Kamran Paynabar, Jianjun Shi
In the past few years, multilinear algebra (and, in particular, tensor analysis) has shown promising results in many applications from network analysis to anomaly detection and process monitoring (Sun, Papadimitriou, and Philip 2006; Sapienza et al. 2015; Yan, Paynabar, and Shi 2015). Nevertheless, only a few works in the literature use tensor analysis for regression modeling. Zhou, Li, and Zhu (2013) has successfully employed tensor regression using PARAFAC/CANDECOMP (CP) decomposition to estimate a scalar variable based on an image input. The CP decomposition approximates a tensor as a sum of several rank-1 tensors (Kiers 2000). Zhou, Li, and Zhu (2013) further extended their approach to a generalized linear model for tensor regression in which the scalar output follows any exponential family distribution. Li, Zhou, and Li (2013) performed tensor regression with scalar output using Tucker decomposition. Tucker decomposition is a form of higher order PCA that decomposes a tensor into a core tensor multiplied by a matrix along each mode (Tucker 1963). Yan, Paynabar, and Pacella (2019) performed the opposite regression and estimated point cloud data as an output using a set of scalar process variables. Recently, convex and nonconvex optimization frameworks have been offered to deal with HD multi-response tensor regression problems (Chen, Raskutti, and Yuan 2019; Raskutti, Yuan, and Chen 2019).
Structured Point Cloud Data Analysis Via Regularized Tensor Regression for Process Modeling and Optimization
Published in Technometrics, 2019
Hao Yan, Kamran Paynabar, Massimo Pacella
The remainder of the article is organized as follows. Section 2 gives a brief literature review on functional and tensor regression models. Section 3 provides an overview of the basic tensor notations and multilinear algebra operations. Section 4 first introduces the general regression framework for tensor response data and then elaborates the two approaches for basis selection, that is, OTDR and RTR. Section 5 validates the proposed methodology by using simulated data with two different types of structured point clouds. In this section, the performance of the proposed method is compared with existing two-step methods in terms of the estimation accuracy and computational time. In Section 6, we illustrate a case study for process modeling and optimization in a turning process. Finally, we conclude the article with a short discussion and an outline of future work in Section 7.
Multilinear principal component analysis for statistical modeling of cylindrical surfaces: a case study
Published in Quality Technology & Quantitative Management, 2018
Massimo Pacella, Bianca M. Colosimo
In the targeted application the data set is summarized in a 3rd-order tensor with real entries. Let represent the tensor of data, and the centered data tensor obtained from by subtracting the mean from each sample (). Tensor is decomposed using the generalization of the SVD in Equation (1) called Higher Order Single Value Decomposition (HOSVD) in multilinear algebra (Lathauwer et al., 2000a). In particular, the centered 3rd-order tensor can be decomposed as follows.