Supervised Aggregative Feature Extraction for Big Data Time Series Regression

In many applications, and especially those where batch processes are involved, a target scalar output of interest is often dependent on one or more time series of data. With the exponential growth in data logging in modern industries, such time series are increasingly available for statistical modeling in soft sensing applications. In order to exploit time-series data for predictive modeling, it is necessary to summarize the information they contain as a set of features to use as model regressors. Typically this is done in an unsupervised fashion using simple techniques such as computing statistical moments, principal components or wavelet decompositions, often leading to significant information loss, and hence suboptimal predictive models. In this paper, a functional learning paradigm is exploited in a supervised fashion to derive continuous smooth estimates of time-series data (yielding aggregated local information), while simultaneously estimating a continuous shape function yielding optimal predictions. The proposed supervised aggregative feature extraction (SAFE) methodology can be extended to support nonlinear predictive models by embedding the functional learning framework in a reproducing kernel Hilbert spaces (RKHSs) setting. SAFE has a number of attractive features including closed-form solution and the ability to explicitly incorporate first- and second-order derivative information. Using simulation studies and a practical semiconductor manufacturing case study, we highlight the strengths of the new methodology with respect to standard unsupervised feature extraction approaches.