Parameter estimation in first-order autoregressive model for statistical process monitoring in the presence of data autocorrelation

Abstract

The present paper considers an application of the first-order autoregressive ( AR(1)) model to realizations, ( λ 1, …, λ n ) of an unobservable variable, λ, representing a quality characteristic of a process monitored at a sequence of 'time' intervals in mineral processing or manufacturing production. At time t i ( i=1, …, n) a set of m i observations x i T = ( x i 1 , ... , x i m i ) are made on the realization λ i . The unknown realizations are observed subject to errors, implying an errors-in-variables model, for the observed sequence of data. The model, referred to as an autocorrelative model, has a reasonably wide range of applications in process monitoring with autocorrelated data. Observed variograms of data in many production processes very often exhibit empirical forms such as a linear or an exponential model. Variation structure based on these models is consistent with an assumption of an autocorrelative model for the original sequence of observations and justifies a detailed study of efficient parameter estimation for the model. Application of such a model to process data requires both estimation of the unobservables, ( λ 1, …, λ n ), in constructing one-step-ahead predictions and also estimation of all the underlying model parameters. For given values of the underlying model parameters, estimation of the unobservables can be carried out most efficiently by Kalman-filter technique. Estimation of the model parameters can be handled by a number of techniques. Specific contributions of the present paper are: (i) a parametric approach comprising a comprehensive development of the full maximum likelihood technique for estimation of the model parameters in the presence of random effects, the number of which increases with the number of observations, (ii) a semi-parametric approach combining a direct or indirect fitting of a variogram combined with the method of moments, and minimum prediction error sum of squares techniques for estimation of model parameters and (iii) a modified procedure for developing an EWMA statistical control chart or process monitoring in the presence of data autocorrelation based on efficient parameter estimates together with its average-run-length properties.