Contribution plots

Two different approaches for calculating variable contributions to T -statistic have been proposed. The first approach introduced by Miller et al. [198] and by MacGregor et al. [146, 177] calculates the contribution of each process variable to a separate score. can be written as 

Considering that variables with high levels of contribution that are of the same sign as the score are responsible for driving to higher values, only those variables are included in the analysis [146]. For example, only variables with negative contributions are selected if the score is negative. 

The overall contribution of each variable is computed by summing over all scores with high values. For each score with high values (using a threshold value of 2.5, for example) the variable contributions are calculated [146]. Then, the values over all the I high scores are summed for contributions that have the same sign as the score  

The second approach was proposed by Nomikos [217] and implemented on batch process data. This approach calculates contributions of each process variable to the T -statistic rather than contributions of separate 

Contribution to the SPE-statistic is calculated using the individual residuals. The contribution of variable j to the SPE at time k is 

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The signal contribution plot helps to understand the reasons for the spectral inconsistency linked to the individual image analysis, coming from the large difference in representativity of the image constituents in the different layers. Thus, three compounds have a very minor signal in the deepest layers and, as a consequence, the resolved spectra obtained from the individual analyses of these layers are of poor quality and differ significantly from the spectra obtained from the top... 

When or SPE charts exceed their control limits to signal abnormal process operation, variable contributions can be analyzed to determine which variable (s) caused the inflation of the monitoring statistic and initiated the alarm. The variables identified provide valuable information to plant personnel who are responsible for associating these process variables with process equipment or external disturbances that will influence these variables, and diagnosing the source causes for the abnormal plant behavior. The procedure and equations for developing the contribution plots was p-resented in Section 3.4. 

The decomposition technique given in [146] can be extended to the and SPEn values of state variables. The state variables are calculated by Eq. 4.67 in which the past data vector is used. When the or SPE chart of the state variables gives an out-of-control signal, contribution plots can be inspected to find the responsible variable for that signal. 

The computed values of the contributions of each process variable and its past values on all the state variables are plotted on a bar chart. The procedure is repeated for all process variables j = 1,..., p). Their contributions are plotted on the same bar plot to decide which variable(s) caused the out-of-control alarm in the multivariate chart of state variables. Use of state variables in SPM and their contribution plots are introduced and illustrated in [211] and [219], respectively. 

In the third fault at time 741 in steam valve fault, the contribution plots of SPEn showed the holding tube inlet temperature sensor as the cause of the alarms (Figure 7.28). In the fourth fault at time 961 in steam... 

Figure 7.24. Contribution plots of for the steam valve fault 3 (Table 5.1). Sampling time of the snapshot after the fault is introduced (a) 40, (b) 41. Reprinted from [143]. Copyright 2001 with permission from Elsevier.

Analysis of contribution plots can be automated and linked with fault... 

Contribution plots presented in Section 7.4 provide an indirect approach to fault diagnosis by first determining process variables that have inflated the detection statistics. These variables are then related to equipment and disturbances. A direct approach would associate the trends in process data to faults explicitly. HMMs discussed in the first three sections of this chapter is one way of implementing this approach. Use of statistical discriminant analysis and classification techniques discussed in this section and in Section 7.6 provides alternative methods for implementing direct fault diagnosis. 

P Miller and RE Swanson. Contribution plots The missing link in multivariate quality control. In 37th Annual Fall Technical Conf., ASQC, Rochester, NY, 1993. 

Examine now the temperature dependence of the specific heat of Cs3V202F7. The magnetic contribution, plotted in the range 2-30K in Fig. 14 Corresponds to corrected values for the lattice contribution determined from the isostructural diamagnetic compound CsjVjO Fj [48],... 

Miller P, Swanson RE, Heckler C, Contribution plots a missing link in multivariate quality control, Applied Mathematics and Computer Science, 1998, 8, 775-792. 

Westerhuis JA, Gurden SP, Smilde AK, Generalized contribution plots in multivariate statistical process monitoring, Chemometrics and Intelligent Laboratory Systems, 2000a, 51, 95-114. 

Statistical tests make possible the automatic detection of an outlier in both cases (they are defined as outliers in the first case and Q outliers in the second case). With these simple tests it will be possible to detect a fault in a process or to reject a bad product by checking just two plots, instead of as many plots as variables as in the case of the Sheward charts commonly used when the univariate approach is applied. Furthermore, the multivariate approach is much more robust, since it will lead to a lower number of false negatives and false positives, and much more sensitive, since it allows the detection of faults at an earlier stage. Finally, the contribution plots will easily outline which variables are responsible for the sample being an outlier. 

The performance of multivariate SPC by MSPCA is illustrated based on simulated data from a fluidized catalytic cracker unit. This simulation was provided by Honeywell to the abnormal situation management consortium. The data consist of 110 measured variables and several types of process faults. Only three components are enough to capture most of the variation in the data. The results of multivariate SPC by PCA and MSPCA are compared in Fig. 8 for a slow drift in the slurry pump around. This drift is present in variable numbers 55 and 97, and starts at 5 min and ends at 65 min. Conventional PCA is unable to detect the shift with more than 99% confidence, whereas MSPCA detects the shift consistently with 99 /o confidence after 24 min. The contribution plots for this fault at 20 min shown in Fig. 9 clearly indicate that MSPCA identifies the contributing variables, whereas PCA does not. Further theoretical comparison based on the average run length of steady-state PCA, dynamic PCA and MSPCA are also available [14],... 

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