Modelling and abnormal change detection in multivariate signals and systems using subspace projection techniques
The focus of the thesis is black-box modelling and the detection of abnormal events in multivariate systems. Subspace projection techniques have been widely applied for the modelling and monitoring of multivariate systems. The popularity of these techniques stems from the fact that these methods can address multicollinearity, a problem commonly encountered when modelling using ordinary least squares with strongly correlated input (process) variables. The subspace techniques of principal component analysis and partial least squares are the methodology of specific interest throughout the thesis. Several non-linear PLS algorithms have been proposed over the last decade. In this thesis analysis of existing non-linear PLS algorithms is undertaken. In particular, following a mathematical analysis of the non-linear PLS algorithm proposed by Baffi et al., (1999(a», it is proven that the algorithm is a non-linear extension of reduced rank regression. It is also argued that a 'true' non-linear generalization to linear PLS should be based on the maximization of a 'non-linear covariance' function if the spirit of linear PLS is to be preserved in its non-linear extension. A mathematical analysis of the algorithm of Wold et al., (1989) is undertaken and it is proven that this algorithm makes an attempt to maximize the non-linear covariance function but with certain limitations. The limitations of the algorithm of Wold et al., (1989) are addressed in two new non-linear PLS algorithms, NLPLS I and NLPLS2. Also following a critical analysis, all existing non-linear PLS algorithms are divided into three categories namely, quick and dirty, covariance based and error based depending on the underlying objective functions optimized by the algorithms. An application of PLS as a parameter estimator is explored and it is shown that when a subspace of dimension A ( < K, number of input variables) is correlated with the output variable and a PLS1 model is built using A latent variables then PLS1 gives an unbiased estimate of the parameters. One approach to extending PLS to take into consideration the dynamics of the process is to replace the inner static relationship between the t- and u-scores of conventional PLS by a dynamic relationship. An algorithm that integrates the dynamics of the data within a PLS framework is proposed. The performance of the algorithm is evaluated against alternative methodologies presented in the literature using an artificial data set and two simulations of chemical processes. The second aspect of the thesis is concerned with detecting abnormal changes in variancecovariance structure of variables. The conventional PCA based monitoring scheme is known to be insensitive to small changes in the variance-covariance structure of variables. A new monitoring scheme that derives a monitoring statistic from the PCA model identification procedure is proposed. The proposed scheme is compared with conventional PCA based monitoring scheme on two artificial data sets and a data set generated from a continuous stirred tank reactor system. A new monitoring scheme for detecting changes in the cross-covariance structure (between input and output variables) in a PLS based monitoring scheme is proposed. The derivation of monitoring statistic requires that a recursive algorithm exists for identifying the PLS model parameters. A new recursive PLS algorithm is derived and the statistic derived from it is used to detect change in parameters of an artificial system before applying to detect fouling in the heat exchanger of a CSTR system. The performance of the proposed scheme is also compared with conventional PLS based monitoring scheme.