Statistical properties of quadratic-type performance indices

Performance indices used in the analysis of performance of univariate and multivariate processes are often computed as ratios of stochastic variables. Consequently, they have an inherent variability. In order to provide a basis for assessing performance using these measures, it is important to understand the sampling distribution of these indices. In this paper the statistical properties of quadratic-type performance indices used in the analysis of performance of univariate and multivariate processes are derived. When the system parameters are known, or are treated as know uncertainty in the performance indices can be quantified through calculation of exact confidence intervals using results from mathematical statistics. Given the extensive computational requirements for computing exact confidence intervals, high quality approximations can in turn be used. The influence of data length and autocorrelation structure of the process on the width on these the confidence intervals is seen explicitly in the derived expressions. It is shown that in the case of non-normal driving forces, the resulting confidence intervals are theoretically justified using the central limit theorem. The results of Desborough and Harris [Can. J. Chem. Eng. 70 (1992) 1186] are shown to be special case of the approach derived in this paper. When the parameters of the time series are estimated from data, the uncertainty in the performance indices must be investigated using a fundamentally different approach. Several methods are outlined. The methodologies developed in this paper are applicable to a wide variety of quadratic performance measures encountered in process monitoring and analysis, controller design and filtering.