To be presented at the American Control Conference, Denver, CO, June 4 6, Data Compression Issues with Pattern Matching in Historical Data

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1 To be presented at the American Control Conference, Denver, CO, June 4 6, 2003 Data Compression Issues with Pattern Matching in Historical Data Ashish Singhal Dale E. Seborg Department of Chemical Engineering University of California, Santa Barbara, CA Abstract It is a common practice in the process industries to compress plant data before it is archived. However, compression may alter the data in a manner that makes it difficult to extract useful information from it. In this paper we evaluate the effectiveness of a new pattern matching technique 1 for applications involving compressed historical data. We also compare several data compression methods with regard to efficiency, data reconstruction, and suitability for pattern matching applications. 1 Introduction Due to the advances in computer technology, large amounts of data produced by industrial plants are recorded as frequently as every second using commercially available data historians. 2, 3 Although storage media are inexpensive, the cost of building large bandwidth networks is still high. Thus, to minimize the cost of transmitting large amounts of data over company networks or the internet, data have to be compressed. One of the classic papers on compression of process data was published by Hale and Sellars. 2 They provided an excellent overview of the issues in the compression of process data and also described piecewise linear compression methods. Other researchers have developed several algorithms to compress time varying signals in efficient ways. Bristol 4 modified the piecewise linear compression methods of Hale and Sellars 2 to propose a swinging door data compression algorithm. Mah et al. 5 proposed a complex piecewise linear online trending (PLOT) algorithm that performed better than the classical box-car, backward slope and swinging door methods. Bakshi and Stephanopoulos 6 compressed process data using wavelet methods. Recently, Misra et al. 7 developed an online data compression method using wavelets where the algorithm computes and updates the wavelet decomposition tree before receiving the next data point. In this paper different data compression methods are evaluated not only on the basis of how accurately they represent process data, but also on how they affect pattern matching. In this paper, six data compression methods are evaluated for both accuracy and their effect on pattern matching. 2 Popular data compression and reconstruction methods for time-series data This section briefly describes some of the popular compression methods for time-series data. Because the accuracy of retrieved data depends not only on the method that was used for compression, but also on the method used for reconstruction, some simple reconstruction techniques that include zero-order hold and linear interpolation are also discussed briefly. 2.1 Data compression methods The box-car method is a simple piecewise linear compression method. This method records data when a value is significantly different from the last recorded value. 2 Because recording of a future value depends only on the value of the last recorded value and the recording limits, the box-car algorithm performs best when the process runs for long periods of steady-state operation. 8 The backward slope method is also a piecewise linear compression method and utilizes the trending nature of a process variable by projecting the recording limit into the future on the basis of the slope of the previously two recorded values. 2 The combination method combines the box-car and backward slope algorithms. 2 This algorithm handles cases when the system is at steady state as well as when process variables exhibit trends. Data averaging compression is a common compression techniques where the time-series data are simply averaged over a specified period of time. In this case, the compression is performed off-line, rather than online. Wavelet based compression. Wavelet transforms can be used to compress time-series data by thresholding the wavelet coefficients. 7, 8 Hard thresholding is a method by which only those wavelet coefficients that are greater than a specified threshold are retained. It is used for this research. For data compression, only the non-zero thresholded wavelet coefficients are stored. These thresholded coefficients can then be used to reconstruct data when Present address: Johnson Controls, Inc., 507 E. Michigan St., Milwaukee, WI Ashish.Singhal@jci.com Corresponding author. seborg@engineering.ucsb.edu 1

2 needed. In the present study, the recording limits on each of the process variables will be used as threshold values. Compression using commercial PI TM software (OSI Software, Because PI TM is widely used for data archiving, it is informative to compare the commercially available software with the classical techniques. In particular, the BatchFile Interface for the PI TM software was used in this research for data compression. 2.2 Data reconstruction methods All of the data compression methods described in the previous section produce lossy compression, i.e., it is not possible to reconstruct the compressed data to exactly match the original data. The accuracy by which compressed data can describe the original uncompressed data depends not only on the compression algorithm, but also on the method of data reconstruction. Many reconstruction methods are available such as the zero-order hold (ZOH) where the value of a variable is held at the last recorded value until the next recording. Linear interpolation (LIN) is a simple method that can overcome a part of this limitation by reconstructing data between recordings. It can provide more accurate reconstruction for situations where the process is at steady state, or situations where process variables show trends. More sophisticated methods such as spline interpolation, and expectation-maximization algorithm for data reconstruction have also been proposed. 9, 10 But these methods are sensitive to the amount of missing data, and do not perform well 9, 10 when a significant amount of data are missing. 3 Pattern matching approach In this article, the pattern matching methodology described by Singhal 11 and Singhal and Seborg 1 is used to compare historical and current snapshot datasets. First, the user defines the snapshot data that serves as a template for searching the historical database. The snapshot specifications consist of: (i) the relevant process variables, and (ii) duration of the abnormal situation. These specifications can be arbitrarily chosen by the user; no special plant tests or pre-imposed conditions are necessary. In order to find periods of operation in historical data that are similar to the snapshot data, a window of the same size as the snapshot data is moved through the historical data. The similarity between the snapshot and the historical data in the moving window is characterized by the S PCA and S dist 1, 12 similarity factors. The PCA similarity factor compares two datasets by comparing the angles between the subspaces spanned by the datasets, while the distance similarity factor compares datasets by calculating the Mahalanobis distance between their centers. 1 The historical data windows with the largest values of the similarity factors are collected in a candidate pool. The individual data windows in the candidate pool are called records. After the candidate pool has been formed, a person familiar with the process can then perform a more detailed examination of the records. The number of observations by which the window is moved through historical data is denoted as w, and is set equal to one-tenth to one-fifth of the length of the snapshot data window. 1 Detailed description of the similarity factors and the pattern matching methodology is provided by Singhal 11 and Singhal and Seborg Performance measures for pattern matching Two important metrics are used to quantify the effectiveness of a pattern matching technique. But first, several definitions are introduced: N P : The size of the candidate pool. N P is the number of historical data windows that have been labeled similar to the snapshot data by a pattern matching technique. The data windows collected in the candidate pool are called records. N 1 : The number of records in the candidate pool that are actually similar to the current snapshot, i.e., the number of correctly identified records. N 2 : The number of records in the candidate pool that are actually not similar to the current snapshot, i.e., the number of incorrectly identified records. By definition, N 1 + N 2 = N P. N DB : The total number of historical data windows that are actually similar to the current snapshot. In general, N DB N P. The first metric, the pool accuracy p, characterizes the accuracy of the candidate pool: p N 1 N P 100% (1) A second metric, the pattern matching efficiency η, characterizes how effective the pattern matching technique is in locating similar records in the historical database. It is defined as: η N 1 N DB 100% (2) Because an effective pattern matching technique should ideally produce large values of both p and η, an average of the two quantities (ξ) is used as a measure of the overall effectiveness of pattern matching.: ξ p + η 2 4 Simulation case study: continuous stirred tank reactor example In order to compare the effect of data compression on pattern matching, a case study was performed for a simulated chemical reactor. A nonlinear continuous stirred tank reactor (3) 2

3 (CSTR) with cooling jacket dynamics, variable liquid level and a first order irreversible reaction, A B was simulated. The dynamic model of Russo and Bequette 13 based on the assumptions of perfect mixing and constant physical parameters was used for the simulation. In the simulation study, white noise is added to several measurements and process variables in order to simulate the variability present in real world processes Generation of recording limits For the simulation study, 95% Shewhart chart limits were used calculate the recording limits. The chart limits were constructed using representative data that included small disturbances as described by Johannesmeyer et al. 14 The high and low limits for each variable were calculated using these data. 14 The recording limits for each variable were specified by calculating the Shewhart chart limits around the nominal value of each variable. The standard deviation for the i th process variable, σ i, was determined using the methodology described above. Then the recording limit for that variable was set equal to cσ i, where c is a scaling factor. The value of c was specified differently for each compression method as described later. The value of the standard deviation, σ, for each measured variable is reported by Singhal Results and discussion The data compression methods described in Section 2 were compared on the basis of the reconstruction error as well as the compression ratio. The compression ratio (CR) is defined as, CR No. of data points in original dataset No. of data points in compressed dataset and the mean squared error (MSE) of reconstruction is defined as, 1 n m MSE ɛi, 2 j (5) m n where m is the number of measurements in the original dataset; n is the number of variables; ɛ i, j = (x i, j ˆx i, j ), x i, j represents the j th measurement of the i th variable in the original data, and ˆx i, j is the corresponding reconstructed value. If the recording limit constant, c, is the same for all methods, then the resulting compression ratios will be different for each method. These type of results would indicate how effective each method is for compressing data. However, in order to compare the methods with respect to reconstruction accuracy, it is easier to analyze the results if all methods have the same compression ratio. A constant compression ratio requires adjusting the recording limits individually for each method. Because the accuracy of the data reconstruction is a key concern, the recording limits for each method were varied in order to achieve the the same compression ratio. As mentioned in the previous section, the recording limits i=1 j=1 (4) for a for given method and each process variable are proportional to their standard deviations. For example, the OSI PI TM recording limits were chosen as 3σ i, while the recording limits for the box-car method were adjusted to produce the same compression ratio as the PI TM method. Thus, the recording limits for the box-car method were 2.23σ i. The effectiveness of a compression-reconstruction method was characterized in two ways: (i) reconstruction error, and (ii) degree of similarity between the original data and the reconstructed data. The S PCA and S dist similarity factors were used to quantify the similarity between the original and reconstructed data. 5.1 Comparison of different methods with respect to compression and reconstruction Different data compression methods were first compared on the basis of reconstruction error. The recording limits for the OSI PI TM method were set to 3σ and data compression was performed using PI s proprietary algorithm. The compression ratio was calculated for each of the 28 datasets. The average compression ratio obtained for the 28 datasets was The recording limits for all other methods were then adjusted using numerical root finding techniques, such as the bisection method, to obtain an average compression ratio of approximately 14.8 for each method. The results presented in Table 1 show that the PI TM algorithm provides the best reconstruction of the compressed data, while wavelet-based compression is second best. Except for the box-car method, linear interpolation provided better reconstruction than zeroorder hold. The common practice of averaging data provides the worst reconstruction. 5.2 Effect of data compression on pattern matching Because the present research is concerned with pattern matching, it is interesting to investigate the effect of data compression on pattern matching. It is obvious that data compression affects pattern matching because the original and reconstructed data sets are not the same. In order to evaluate the effect of different compression methods on the effectiveness of the proposed pattern matching methodology, similarity factors between the original data and the reconstructed data were calculated to see how similar the reconstructed dataset was to the original one. For scaling purposes, the original dataset was considered to be the snapshot dataset while the reconstructed dataset was considered to be the historical dataset. The average values for S PCA, S dist and their combination, S F = 0.67 S PCA S dist, are presented in Table 2. Although the averaging compression method performed worst in terms of reconstruction error (cf. Table 1), it produced compressed datasets that show a high degree of similarity to the original ones, as indicated by high S PCA and S dist values. The wavelet compression method produces low MSE values as well as high S PCA and S dist values. These results demonstrate that wavelet-based compression is very 3

4 Table 1. Data compression and reconstruction results for the CSTR example for a constant compression ratio. Compression method Recording limit constant (c) Reconstruction method CR MSE Box-Car Linear Zero-order hold Backward-slope Linear Zero-order hold Combination Linear Zero-order hold Averaging Linear NA (over 1.25 min) Zero-order hold Wavelet Wavelet PI TM 3.0 PI TM accurate both in terms of reconstruction error and the similarity of the reconstructed and original datasets. Although the PI TM algorithm produces a very low MSE, it does not represent the data very well for pattern matching. The wavelet method produces both a low MSE and high similarity factor values. The wavelet transform preserves the essential dynamic features of the signal in the detail coefficients while retaining the correlation structure between the variables in the approximation coefficients. These two features of the wavelet transform produce low MSE and high S PCA values between the original and reconstructed data. These features also minimize mean shifts and result in high S dist values. By contrast the PI TM method, records data very accurately and produces very low MSE values, but its variable sampling rates disrupt the correlation structure between variables and produce low S PCA values. Variable sampling also affects the mean value of the reconstructed data and produces low S dist values. The detailed results for different operating conditions for the CSTR case study are reported by Singhal Pattern matching in compressed historical data The historical data for the CSTR example described in Section 4 were compressed using three different methods: wavelets, averaging, and a combination of the box-car and backward slope methods. The performance of the proposed pattern matching technique for compressed historical data was then evaluated. As described by Singhal, 11 and Singhal and Seborg, 1 a data window that was the same size as the snapshot data (S) was moved through the historical database, 100 observations at a time (i.e., w = 100). The i th moving window was denoted as H i. For pattern matching, the compressed data were reconstructed using the linear interpolation method. The same compression method was used for both the snapshot and historical data. The snapshot data were then scaled to zero mean and unit variance. The historical data were scaled using the scaling factors for the snapshot data. Similarity factors were then calculated for each H i. After the the entire database was analyzed for one set of snapshot data, the analysis was repeated for a new snapshot dataset. A total of 28 different snapshot datasets, one for each of the 28 operating conditions, were used for pattern matching. 11 Table 3 compares the pattern matching results for historical and snapshot data compressed using different methods. The best pattern matching results were obtained when the data were compressed using the wavelet method. The optimum N P values were determined by choosing the value of N P for which ξ had the largest value. Table 3 indicates that pattern matching is adversely affected by data compression when the data are compressed using either the averaging method or the combination of box-car and backward slope compression methods. By contrast wavelet-based compression has very little effect on pattern matching because similar results are obtained for both compressed and uncompressed data. Table 4 presents results for the situation when the snapshot data are not compressed while the historical data are compressed using the wavelet method. The p, η and ξ values in Table 4 are slightly lower compared to those in Table 3. Thus, if the historical data are compressed, it may be beneficial to compress the snapshot data as well to obtain better pattern matching. 6 Conclusions A variety of data compression methods have been compared and evaluated for pattern matching applications using a case study approach. Classical methods such as box-car, backward slope and data averaging compression methods do not accurately represent data either in terms of reconstruction error or similarity with the original dataset. Data compressed using the PI TM software very accurately represents the original data, but produces somewhat lower similarity factor values. Compression using the wavelet method produces reconstruction errors that are higher than those obtained with PI TM, but much lower than conventional compression methods such as box-car, etc. Data compressed using wavelets also show a high degree of similarity with the original data. For pattern matching applications, it is beneficial to compress the snapshot data prior to performing pattern matching. 4

5 Table 2. Effect of different data compression and reconstruction methods on pattern matching for the CSTR example. Compression method Recording limit constant (c) Reconstruction method S PCA S dist SF Box-Car Backward-slope Combination Linear Zero-order hold Linear Zero-order hold Linear Zero-order hold Averaging Linear NA (over 1.25 min) Zero-order hold Wavelet Wavelet 0.95 > PI TM 3.0 PI TM S F = 0.67 S PCA S dist Table 3. Effect of data compression on pattern matching for the CSTR example when both the snapshot and historical data are compressed using the same method. Compression Similarity Opt. N P p (%) η (%) η max (%) ξ (%) method factor S PCA only Original data S dist only S F S PCA only Combination S dist only S F S PCA only Averaging S dist only S F S PCA only Wavelet S dist only S F S F = 0.67 S PCA S dist For the simulated case study, data compression had only a minor effect on the effectiveness of a new pattern matching strategy. 11 Acknowledgements The authors thank OSI Software for providing financial support and the data archiving software PI TM, and Gregg LeBlanc at OSI for providing software support during the research. Financial support from ChevronTexaco Research and Technology Co. is also acknowledged. References (1) Singhal, A. and Seborg, D. E. Pattern Matching in Multivariate Time Series Databases Using a Moving Window Approach. Ind. Eng. Chem. Res., , (2) Hale, J. C. and Sellars, H. L. Historical Data Recording For Process Computers. Chemical Eng. Prog., (11), (3) Kennedy, J. P. Building an Industrial Desktop. Chemical Engr., (1), (4) Bristol, E. H. Swinging Door Trending: Adaptive Trend Recording? In Advances in Instrumentation and Control, volume 45. Instrument Society of America, Research Triangle Park, NC, (5) Mah, R. S. H.; Tamhane, A. C.; Tung, S. H. and Patel, A. N. Process Trending With Piecewise Linear Smoothing. Comput. Chem. Engr., ,

6 Table 4. Effect of data compression on pattern matching when snapshot data are not compressed and historical data are compressed. Compression Similarity Opt. N P p (%) η (%) η max (%) ξ (%) method factor S PCA only Original data S dist only S F S PCA only Combination S dist only S F S PCA only Averaging S dist only S F S PCA only Wavelet S dist only S F S F = 0.67 S PCA S dist (6) Bakshi, B. R. and Stephanopoulos, G. Compression of Chemical Process Data Through Functional Approximation and Feature Extraction. AIChE J., , (7) Misra, M.; Kumar, S.; Qin, S. J. and Seemann, D. Error Based Criterion for On-Line Wavelet Data Compression. J. Process Control, , (8) Watson, M. J.; Liakopoulos, A.; Brzakovic, D. and Georgakis, C. A Practical Assessment of Process Data Compression Techniques. Ind. Eng. Chem. Res., , (9) Nelson, P. R. C.; Taylor, P. A. and MacGregor, J. F. Missing Data Methods in PCA and PLS: Score Calculations with Incomplete Observations. Chemometrics and Intel. Lab. Syst., , (10) Roweis, S. EM Algorithms for PCA and SPCA. In Neural Information Processing Systems 11 (NIPS 98) (11) Singhal, A. Pattern Matching in Multivariate Time- Series Data. Ph.D. Dissertation, University of California, Santa Barbara, CA, (12) Krzanowski, W. J. Between-Groups Comparison of Principal Components. J. Amer. Stat. Assoc., (367), (13) Russo, L. P. and Bequette, B. W. Effect of Process Design on the Open-Loop Behavior of a Jacketed Exothermic CSTR. Comput. Chem. Eng., , (14) Johannesmeyer, M. C.; Singhal, A. and Seborg, D. E. Pattern Matching in Historical Data. AIChE J., ,