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A new approach to gross error detection provides unbiased estimates and 100( 1 -a ) % simultaneous confidence intervals of process variables when biased process measurements and process leaks exist. Presented in this article are estimation equations for process variables, as well as equations that help identify biased measurements and process leaks. These equations include the power function for a global test, and two types of a-level component tests and their power functions. Important strengths and weaknesses of this approach are compared to those of the serial compensation strategy, in particular, by varying the significance level ( a ) , the variance-covariance matrix ( C ) , the size of measurement bias (6), the number of biased variables, and the sample size ( N ) . Accuracy of 6 estimation and performance in detecting the presence of process leaks ( y ) are also evaluated and compared. The proposed approach has unique features that can provide a basis for improving the reconciliation of variables in process operations.

IntroductionŽ . Statistical process control SPC has been an active area of research for many decades. A broad spectrum of methods have been developed, including methods for univariate SPC Ž . such as Shewhart, moving-average MA , exponentially Ž . weighted moving-average EWMA , and cumulative-sum Ž . CUSUM charts. Methods for multivariate SPC include multivariate extensions of univariate methods, and methods that monitor latent variables obtained by combining the measured variables with a lower dimension of space. Popular methods for reducing the dimensionality of the measured data include Ž . principal-component analysis PCA and partial least-square Ž . regression PLS . Many extensions and applications of these Ž have been developed Kresta et al., 1991;Ku et al., 1995;. MacGregor, 1994 . Correspondence concerning this article should be addressed to B. R. Bakshi. Current addresses of: H. B. Aradhye, SRI International, Menlo Park, CA; R. A. Strauss, ExxonMobil, Fairfax, VA. J. F. Davis, University of California, Los Angeles, CA.Most existing univariate and multivariate SPC methods operate at a fixed scale, and are best for detecting changes at a single scale. For example, Shewhart charts analyze the raw measurements at the scale of the sampling interval or the finest scale, and are best for detecting large, localized changes. In contrast, MA, EWMA, and CUSUM charts inherently filter the data, and, therefore, process measurements at a coarser scale. They are best for detecting small shifts or features at coarse scales. Tuning parameters such as window length or filter constant determine the scale at which the measurements are represented.In contrast to the single-scale nature of SPC methods, data from most practical processes are inherently multiscale due to events occurring with different localizations in time, space, and frequency. A typical example of such data from a petrochemical process is shown in Figure 1. Figure 1a shows data during normal operation, while Figure 1b represents unusual operation due to a drier cooling event. In Figure 1b cess change at approximately 150 time units is at a very fine scale and localized in time, but spans a wide range of frequencies. The steady portions of the signal are at coarse scales and span a wide temporal range. Finally, the change between 425 and 675 time units consists of a small sharp change followed by a short steady section and a slow ramp at an intermediate scale. Ideally, techniques for detecting changes at different scales, such as those shown in Figure 1b, should adapt automatically to the scale of the features. In response to this need, many heuristic or ad hoc techniques have been proposed for overcoming the single-scale nature of SPC Ž charts. These include the Western Electric rules Western . Electric, 1956 , useful for identifying patterns in data, and Ž . combined Shewhart and CUSUM charts Lucas, 1982 for identifying large and small shifts. Other methods, such as Ž . CUSCORE charts Box and Ramirez, 1992 , may be specially designed to detect abnormal feat...