SPC techniques such as Shewhart, MA, EWMA and CUSUM control charts. ... Statistical Process Control (SPC) is the task of detecting abnormal process ...
Multiscale Statistical Process Monitoring and Diagnosis of Univariate and Multivariate Processes Bhavik R. Bakshi and Sermin Top Department of Chemical Engineering The Ohio State University Columbus, OH 43210 Abstract This paper presents a general framework for SPC based on representing the measurements at multiple scales by projection on a family of wavelets. The framework unifies several existing SPC techniques such as Shewhart, MA, EWMA and CUSUM control charts. These existing methods are special cases of the framework since they differ only in the scale at which they represent the measurements. Shewhart charts operate at the finest scale, CUSUM charts operate at the scale of all the measurements, which is the coarsest scale, while MA and EWMA charts operate at a scale determined by the value of the tuning parameters. The resulting multiscale SPC (MSSPC) method detects deterministic changes as large wavelet coefficients that violate detection limits determined separately at each scale. If the measurements are autocorrelated, the detection limits change according to the power spectrum of the normal measurements, and the wavelet coefficients will be approximately decorrelated. Consequently, MSSPC is able to monitor both uncorrelated and autocorrelated measurements without additional processing. The state of the process is confirmed by reconstructing the signal based on the large wavelet coefficients at each scale, and applying a detection limit determined from the scales at which the event was detected. Thus, MSSPC automatically adjusts the detection limits for each measurement, and extracts the signal feature representing the abnormal operation. Comparison of the average run length of MSSPC with that of other univariate methods indicates that MSSPC is a general approach that may be preferable for problems where it is necessary to detect all types of changes, from various types of process data. 1. Introduction Statistical Process Control (SPC) is the task of detecting abnormal process operation from the statistical behavior of variables. SPC determines the region of normal variation of a measured variable, and indicates abnormal operation if the measurements lie outside the normal region. A variety of control charts have been developed for SPC including, Shewhart, moving average (MA), exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts. These univariate methods have been extended for SPC of multivariate processes, but are not as practical or popular as multivariate SPC by empirical modeling methods such as principal component analysis (PCA) and partial least squares regression (PLS). Existing methods for univariate and multivariate SPC suffer from several limitations. Various control charts are best only for detecting certain types of changes. For example, a Shewhart chart can detect large changes quickly, but is slow in detecting small shifts in the mean, whereas CUSUM, MA and EWMA charts are better at detecting a small mean shift, but may be slow in detecting a large shift, and require tuning of their filter parameters (Montogomery, 1996). This
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limitation may be overcome by using heuristics such as the Western Electric rules, or Shewhart and CUSUM charts together (Lucas, 1982). Another limitation of existing methods is that they require the measurements to be uncorrelated, or white, whereas, in practice, autocorrelated measurements are extremely common. A common approach for decorrelating autocorrelated measurements is to approximate the measurements by a time series model, and monitor the residual error. Unfortunately, this approach is not practical, particularly for multivariate processes with hundreds of measured variables. Other approaches for decorrelating autocorrelated measurements without time-series modeling include taking the batch-means (Runger and Willemain, 1995), and finding the residuals between the measurements and their one-step ahead prediction by an EWMA model (Montgomery and Mastrangelo, 1995). Unfortunately, neither of these approaches are broadly applicable to a wide variety of stochastic processes, and lack multivariate generalizations. These limitations of existing methods are due to a mismatch between the nature of the measured data, and the nature of existing SPC methods. Measured data are inherently multiscale in nature due to contributions from deterministic or stochastic events occurring at different locations and with different localization in time and frequency. In contrast, existing SPC methods are inherently single-scale in nature. The filter used by existing SPC charts is at a fixed scale, as illustrated in Figure 1. Various methods differ only in the scale at which they represent the measurements. Thus, Shewhart charts represent data at the scale of the sampling interval, which is the finest scale, MA and EWMA charts represent data at a coarser scale, determined by the filter parameter, and CUSUM charts represent data at the scale of all the measurements, which is the coarsest scale. The disadvantages of existing methods may be overcome by developing a multiscale approach for SPC. Such an approach for multivariate SPC based on multiscale PCA was described by Bakshi (1998), and shown to perform better than multivariate SPC by conventional PCA and dynamic PCA.
a) Shewhart
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Figure 1. Filters for existing SPC methods This paper provides further insight into multiscale SPC (MSSPC) by studying its statistical properties, and comparing the performance with existing methods. The emphasis of this paper is on univariate SPC, since many more techniques are available for this task, and their statistical properties have been studied in detail. These results are also expected to be valid for multivariate SPC, since univariate MSSPC can be easily extended to multivariate MSSPC as described by Bakshi (1998). The remainder of this paper focuses on the properties of SPC techniques for monitoring uncorrelated and autocorrelated measurements. For uncorrelated measurements, the wavelet decomposition is carried out without downsampling, whereas for autocorrelated measurements, the wavelet decomposition includes downsampling to decorrelate the measurements. SPC methods are compared by their run length for detecting shifts of various sizes.
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2. Wavelets The family of wavelet basis functions may be represented as, ψ su (x) =
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where, s and u represent the dilation and translation parameters, respectively, and ψ(x) is the mother (b) wavelet. Any signal may be Figure 2. Wavelet decomposition (a) with decomposed to its contribution at downsampling, (b) without downsampling. multiple scales by convolution with the corresponding filters. If the translation parameter is discretized dyadically as, u = 2mk, the wavelet decomposition downsamples the coefficients at each scale, as shown in Figure 2a. This approach permits the use of orthonormal wavelets, which approximately decorrelate autocorrelated measurements, but may cause a time delay in detecting a change in the signal. Alternatively, the translation parameter may be discretized as, u = k, resulting in the decomposition shown in Figure 2b. The basis functions in this approach lack orthonormality, but can detect a shift with minimum time delay. Wavelet decomposition with downsampling is useful for SPC of autocorrelated measurements, whereas, wavelet decomposition without downsampling is useful for SPC of uncorrelated measurements. 8 4 0 -4 3 0
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Figure 3. MSSPC Methodology 3. Multiscale Statistical Process Control The methodology for MSSPC is depicted in Figure 3. The variable is decomposed to its multiscale representation by the selected wavelet, followed by applying a Shewhart control chart to the coefficients at each scale. Scales at which the current coefficient violates the detection
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limit are selected as being relevant at the current time. The signal and covariance at the selected scales are reconstructed by the inverse wavelet transform, and the state of the process is confirmed by checking if the current value of the reconstructed signal violates the corresponding detection limit. The signal reconstruction is essential for efficient detection of the return to normal operation. 4. MSSPC of Uncorrelated Measurements If the measurements are uncorrelated, there is no need to whiten the measurements by the wavelet decomposition. Consequently, the measurements are decomposed without downsampling. The resulting approach is equivalent to adapting the filter for each measurement to the scale that is best for detecting abnormal operation. Thus, MSSPC with Haar wavelets subsumes SPC by MA charts, while MSSPC with smoother, boundary corrected wavelets approximately subsumes SPC by EWMA. The performance of Shewhart, MA, and MSSPC with Haar wavelets is compared in Figure 4. Figure 4 compares the average number of samples required to detect a shift of different sizes. In each case, the parameters are adjusted so that the in-control run length, or the average number of samples before a measurement violates the detection limit in the absence of a 1000 shift, are equal. MSSPC-Haar These figures show Shewhart that if the objective MA of SPC is to detect only small shifts, it 100 is best to use a MA control chart, or if the objective is to detect only large 10 shifts, it is best to use a Shewhart chart. If the objective of SPC is to have a general 1 method that can 0 0.5 1 2 3 4 5 Mean shift detect both small and large shifts, and Figure 4. Average run length for SPC of uncorrelated data provide better performance on the average, it is best to use MSSPC. 5. MSSPC of Autocorrelated Measurements For SPC of autocorrelated measurements, since it is essential to decorrelate the data, MSSPC with dyadic downsampling is used. The nature of the wavelet filters, and the downsampling can decorrelate a wide variety of stochastic processes. Figure 5 depicts the ARL for an AR(1) process given by, x(t) = 0.9x(t − 1) + ε(t)
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Weighted batch 1000 means (Runger and Willemain, 1995) decorrelates the data by taking a weighted sum of 100 the measurements in a window of fixed size. The weights are 10 determined to decorrelate the MSSPC measurements. WBM This approach MCEWMA works best for 1 detecting small 0 0.5 1 2 3 4 5 shifts, and always Mean shift has a run length Figure 5. ARL of AR(1) process greater than the length of the window. Moving center line EWMA (MCEWMA) (Montgomery and Mastrangelo, 1995) fits an EWMA to the measurements to minimize the one-step ahead prediction error. The results in Figure 5 indicate that MSSPC 1000 performs well as a general method for detecting shifts of various sizes. Nonstationary stochastic processes present special challenges for SPC, since their mean tends to change over time. The ARL performance of MSSPC and MCEWMA are compared in Figure 6. In this case, the stochastic process is IMA(1,1) given by,
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Figure 6. ARL of IMA(1,1) process
x(t) = x(t − 1) + ε(t) − 0.5ε(t −1)
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which can be modeled optimally by EWMA. In this case, MCEWMA performs better than MSSPC for detecting large shifts due to better decorrelation by MCEWMA. 6. Conclusions The statistical properties of MSSPC studied in this paper indicate that MSSPC is an excellent general method for process monitoring. It can perform better than existing methods for processes where it is essential to detect all types of shifts in any type of measured data. If the objective of monitoring is to detect certain types of changes only, then an existing single-scale method can be tailored for this task to provide the best possible performance. For example, if the objective of monitoring is to detect only small shifts, an optimum MA, EWMA, or CUSUM method can be designed. Similarly, if the measurements can be modeled as an IMA(1,1) process, moving center line EWMA can result in excellent performance. In practice, since it is usually essential to detect all types of changes, and since the stochastic nature of the measurements need not follow an IMA(1,1) model, MSSPC is a method that is superior to existing methods on the average. References Bakshi, B. R., Multiscale PCA with Application to Multivariate Statistical Process Monitoring, AIChE Journal, 44, 7, 1596-1610 (1998) Runger, G. C., and T. R. Willemain, Model-Based and Model-Free Control of Autocorrelated Processes, J. Qual. Tech., 27, 4, 283-292 (1995) Lucas, J. M., Combined Shewhart-CUSUM Quality Control Schemes, J. Qual. Tech., 14, 2, 5159 (1982) Mastrangelo, C. M., and D. C. Montgomery, SPC with Correlated Observations for the Chemical and Process Industries, Qual. Reliab. Eng. Int., 11, 79-89 (1995) Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley and Sons, Inc., New York (1996)
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