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Session 2520

Linear and Neural-Network Methods for Condensing High-Dimensional Measurements

Michael L. Mavrovouniotis, Venkatramana N. Reddy Northwestern University (Evanston, IL)

Introduction Process data are the foundation of process monitoring, evaluation and control. Advancements in automation allow the collection of large volumes of process data. A process may be equipped with hundreds or even thousands of sensors with sampling intervals of seconds or minutes. As an important step towards process understanding, engineers need to uncover the significant patterns hidden in process data. Dimensionality reduction is a way of summarizing information carried by a large number of observed variables with a few latent variables. Through dimensionality reduction, we obtain not only a reduced data set but also a model that relates all observed variables to a few latent variables; such a model is valuable to many types of data screening tasks, such as data noise reduction, missing sensor replacement, gross error detection and correction, and fault detection. Given an m × n matrix representing m measurements made on n variables, n is the observed dimensionality of the data set. Reduction of data dimensionality aims to map the original data matrix to a much smaller matrix of dimension m × f (f << n), which is able to reproduce the original matrix with minimum distortion. The dimensionality reduction is useful when there exist correlations among the observed variables. The reduced matrix describes latent variables extracted from the original matrix. Ideally, the f latent variables should retain all nonrandom variations in the observed variables, exploiting correlations and redundancies in the measurements. Linear Approach The linear technique of Principal Component Analysis (PCA) is the most commonly used technique, and it has been shown to facilitate many types of data analysis in process engineering, including data validation and fault detection (Wise and Ricker, 1989), quality control (MacGregor, 1989), data visualization (Stephanopoulos and Guterman, 1989), and process monitoring (Raich and Cinar, 1994). Singular value decomposition (SVD) provides a computationally efficient method for PCA. Any m x n matrix A of rank r can be decomposed into the following form (Strang, 1988):

A = u1s1vT + u2s2 v T +... +ur sr vT 1 2 r ( s1 ≥ s 2 ≥...≥ sr > 0)

where si (i = 1, 2, ..., r) are positive scalars in descending order, ui (i = 1, 2, ..., r) are m x 1 orthonormal vectors and vi (i = 1, 2, ..., r) are n x 1 orthonormal vectors. The first f terms of the above decomposition provide the best approximation to A with f principal components. PCA is a linear technique in the sense that it uses linear functions to model relationships between observed variables and latent variables. Factor analysis (FA) is another linear technique for

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Reddy, V. N., & Mavrovouniotis, M. L. (1997, June), Linear And Neural Network Methods For Condensing High Dimensional Measurements Paper presented at 1997 Annual Conference, Milwaukee, Wisconsin. 10.18260/1-2--6666