PATH ANALYSIS FOR PROCESS TROUBLESHOOTING Biao
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1.2. Subtracting eqn 2 by eqn 6 yields y a l FTlh X lZe 1f X e 7 Taking mean square value on both sides of eqn 7 results in E y 8 a l FTl T E X1 X a l F lz 113 E X3 X lls 1g Bee lz Bee a h Fb a h Fr hs H3 la 12 Beet ly Ee gt Ee 9 II The equality is achieved if and only if lh a lz 0 and l3 0 The minimum of E y is achieved in the limit by least squares Therefore the least squares estimation can asymptotically converge to the true model 2 even though a number of redundant irrelevant variables have been included in the model The implication of this result is that if X is the source of the variability in y among all selected input variables then this source can be correctly identified by checking the estimated coefficients of all input variables The one which is statistically nonzero is likely to be the sources of variability Other input variables with zero coefficients although they are also correlated with y are in fact the response to X but not the source of y and can therefore be ruled out through this analysis One potential problem in the calculation is the collinearity of the input variables If two or more of input variables are highly correlated then the regression analysis may fail In this case PCA PLS based regression analysis may be applied Obviously the path analysis prop
3. exogenous input variables and those variables that are affected by others endogenous output variables With each of the latter output vari ables is associated a residual Certain conventions govern the drawing of a path diagram Directed arrows represent a path The path diagram is constructed as follows 1 A straight arrow is drawn to each output endogenous variable from each of its source x i Py Pye Pi2 Pis x fr y P23 Py3 X3 Fig 1 An example of path analysis 2 A straight arrow is also drawn to each output variable from its residual 3 A curved double headed arrow is drawn be tween each pair of input exogenous vari ables thought to have nonzero correlation The above procedure is illustrated in Fig 1 To calculate the coefficients for the path diagram we use standardized variable i e all variables have mean 0 and variance 1 If a regression model of original variables is given by Y bo b1X b2X2 BpXp E Then a multivariate regression model of the normalized variables can be constructed as Jou X1 M F222 X2 u2 Ta Be Zo Urr Ae ie Vle _ E Va ee fers Wee Y py S HI OYY VOIYY 6 or Ys pyiZi py2Z2 ERG pyrZ pye s 1 The coefficients PYk BaJ nk VOVY and PY VGee ovy are the path coefficients or the direct effects An example of the path diagram is shown in Fig 1 where py1 py2 Py3 Pye a
4. analysis tool The method of path analysis was developed by the geneticist to explain causal relations in population genetics Johnson and Wichern 1982 The goal of path analysis is to provide plausible explanations of observed correlations by constructing models of cause and effect relations variables In this study we will explore this method further and develop it for process troubleshooting applications 2 PATH DIAGRAM 2 1 What is path analysis The concept of path analysis is explained in this subsection according to Johnson and Wichern 1 Corresponding author biao huang ualberta ca 1982 For a more comprehensive discussion on the path analysis method readers are referred to Johnson and Wichern 1982 and references therein It is well known that a significant correlation between two variables does not imply a causal relationship For example the variation in both variables may be introduced by a third variable Or one of the two variables may affect the second variable through a third variable or many other variables When one variable X precedes another variable X in time it may be postulated that X causes X The relation can be represented in the path analysis as X X 2 Taking into account the error 2 the path diagram may be presented as X ZON X E2 The diagram may be written as a linear model X po 61X1 2 where X is considered to be a causal variable that is not infl
5. of Tag 34 can not be identified Step 2 Draw a control volume around Reformer see Fig 3 for control volume 2 Tag 19 and 20 as output Tag 29 30 31 34 as inputs In this step we will analyze Tag 19 only The analysis with Tag 20 as output will be performed in the next step Path analysis yields the following results 1 The two indices are calculated as ye 0 93 and ya 0 92 These results indicate that most of the variability in Tag 19 can be ex plained by the selected inputs and in addition the source can be easily identified 2 The direct effect table is summarized in Tab 6 This result clearly indicates that Tag 34 is the source of Tag 19 Table 6 Direct effect table Tag 29 Tag 30 Tag3l Tag 34 Tag 19 0 04 0 06 0 03 0 93 0 26 Step 8 Continuation of Step 2 with Tag 20 as output 1 The two indices are calculated as ye 0 94 and ya 0 97 These results indicate that most of the variability in Tag 20 can be explained by the selected inputs and the source can be easily identified 2 The direct effect table is summarized in Tab 7 This result clearly shows that Tag 34 is the source of Tag 20 Table 7 Direct effect table Tag 29 Tag 30 Tag3l Tag 34 e Tag 20 0 02 0 03 0 01 0 95 0 25 Comments Step 2 and 3 indicate that Tag 34 is actually the source of both Tag 19 and 20 This result explains why Tag 34 can not find its source from Tag 19 or 20 in Step 1 Step 4 Draw a control volume aroun
6. PATH ANALYSIS FOR PROCESS TROUBLESHOOTING Biao Huang Nina Thornhill Sirish Shah Dave Shook Dept of Chemical and Materials Engg University of Alberta Edmonton AB Canada T6G 2G6 Department of Electronic and Electrical Engg University of College London London U K WC1E 7JE Matrikon Inc Edmonton AB Canada T5J 38N4 Abstract In this paper a model free data driven approach to process troubleshooting is proposed The method is simple and can handle both univariate and multivariate processes The only information needed for such an analysis is the data The objective is to identify possible source of variability oscillation from all interacting variables To achieve this objective a model free method known as path analysis is used In this paper we will summarize the theory and algorithms developed for such an analysis An industrial case study is presented to demonstrate the feasibility of the proposed method Keywords Process monitoring troubleshooting data mining path analysis 1 INTRODUCTION If a control loop has no potential to improve performance by tuning the controller then one obvious choice is to trace the source of the upset and reduce the disturbances oscillations in the source One therefore has to search for among many loops which interact with the loop of con cern the source of the disturbances oscillations This can be a forbidding task for a large scale process without an appropriate
7. ag 29 is taken as output Tag 23 25 26 35 36 37 27 as inputs Path analysis yields ye 0 37 the selected input variables are not sufficient to explain the variability in Tag 29 Therefore the source of the Tag 29 oscillation can not be identi fied from the given tags 4 CONCLUSIONS In this paper the path analysis is proposed for process troubleshooting by tracing the source of variability oscillation Path analysis is similar to correlation analysis in terms of its simplicity but it provides a directional correlation information That is a correlation analysis reveals all possi ble correlation between two variables direct and indirect while path analysis reports the direct relation of two variables It is shown in this paper that path analysis can be used to trace the source of process variability The result has also been extended to tracing the oscillation by applying the path analysis to autocovariance data An in dustrial case study is presented to illustrate the effectiveness of the proposed algorithms 5 REFERENCES Johnson R A and D W Wichern 1982 Applied Multivariate Statistical Analysis Prentice Hall Thornhill N B Huang and H Zhang 20012 Detection of multiple oscillations in control loops To appear in Journal of Process Con trol Thornhill N F S L Shah and B Huang 20018 Detection of distributed oscillations and root cause diagnosis In Proceedings of CHEMAS Cheju Island Korea
8. ant to y Accordingly we partition X into X1 X2 and X3 X is the set of input variables that are directly affected by X and described by the following model Xg FX 3 where F is a coefficient matrix of an appropriate dimension and with Cov e 0 is a disturbance variable vector The posed condition Covu c 0 ensures X not to include any variable that is exactly the same as one of the variables in X or a linear combination of X1 Physically this tells us that we should not include any two or more input variables which are exactly the same or have exact linear relationship into the input variables Numerically this condition will avoid the collinearity problem in regression analysis X3 is a set of input variables that do not affect y and may be represented by the following model X3 v 4 where v is a disturbance variable vector and is independent of both X and X9 In addition e and v are mutually independent Now suppose we build a model of y by including all possible input variables as the input g X 1 X2 13 X3 5 where l l2 and l3 are model coefficients of ap propriate dimension All variables y X1 X2 and X3 have been normalized namely EX X7 I EX Xf I EX3X7 I Using model 5 one would like to know if the estimated model can converge to the true model 2 in the limit Substituting eqns 3 and 4 into eqn 5 yields gal X1 PX 17 X3 7F 7 F X ife X 6
9. d the whole process as shown in the flowchart see Fig 3 for control volume 3 Tag 11 is the output Tag 34 is a recycle stream and not an output Tag 23 25 26 27 35 36 37 30 31 are the inputs Some inputs such as light naphtha flow rate is not available and has not been included in the analysis Path analysis yields the following result shown in Table 8 Due to the space limit direct path coefficients with small values are omitted from the table The two indices ye 0 96 and ya 0 90 indicate that the selected inputs are sufficient to explain the output variability and the source of the variability can be easily identified The direct path coefficient from Tag 25 to Tag 11 clearly shows that Tag 25 is the source of the oscillation in Tag 11 Combining with the results obtained in previous steps now the question is which one of Tag 25 and Tag 34 is the source of the oscillation If there is no recycle from Tag 34 to Tag 25 then the result obtained in this step clearly shows that Tag 25 is the real source and Tag 34 is actually a response to Tag 25 However if there is a recycle from Tag 34 to Tag 25 then Tag 34 could be the source a result obtained in Thornhill et al 2001 Thornhill et al 2001b Table 8 Direct effect table Yr n Tae e Tag 11 0 96 0 90 1 0 0 20 Step 5 Draw a control volume around Feed unit Feed vaporizer superheater unit and Reformer feed pre heat unit see Fig 3 for control volume 4 T
10. ffect dominates the indirect effect and therefore it is fairly easy to isolate the source of the variability X l 0 97 E 0 33 0 76 0 15 0 73 0 16 X3 Fig 2 An example of path analysis If there are a large number of variables involved in the analysis the graphic representation may not be efficient The direct effect table can be con structed which lists the direct effect coefficients For example the direct effect of the path analysis figure shown in Fig 2 can be equally represented by Table 2 Another table is known as total effect table which shows the total effect from a input variable to the output variable by combining di rect effect and indirect effect For example the total effect from x to y according to Fig 2 can be calculated as 0 97 0 76 x 0 15 0 91 x 0 16 0 94 while the total effect from x2 to y can be calcu lated as 0 15 0 76 x 0 97 0 73 x 0 16 0 77 The total effect from this analysis is given in Ta ble 3 which is exactly the same as the correlation coefficients between the input variables and the output variable y as has been discussed above Table 2 Direct effect table 1 T2 x3 E y 0 97 0 15 0 16 0 33 Table 3 Total effect table ry T2 x3 E y 0 94 0 77 0 83 0 33 3 APPLICATION OF PATH ANALYSIS FOR OSCILLATION DETECTION One of the most important applications of the path analysis is for oscillation detection and trac ing the source of the oscillation Oscilla
11. iables while ye 1 indicates that the selected input variables are complete and explain all variability in the output variables ye 0 5 indicates that 50 variance of the output variables can be explained by the selected input variables Therefore ye 0 5 or Ye lt 0 5 is a typical in dication that additional input variables may need to be selected for a meaningful analysis e Significance index of the direct effect is de fined as ya 1 z ya 1 indicates that all effects are from the direct path the input variables are mutually independent and the source of variability can be identified easily Therefore ya lt 0 5 is typical indication that the source of the variability may not be iso lated even though the selected input vari ables are sufficient to explain the variability in the output 2 3 Asymptotic property of path analysis Consider a model given by y aTX e 2 where y is the variable of concern output vari able X is an input variable that directly affects y and e is a disturbance variable that is indepen dent of X The problem of interest is to isolate the source variables X from a group of input plausible source variables That is to isolate X4 from a set of input variables X Among this set of input variables some are also affected by the same source X and therefore a strong correlation apparently exists between these variables and y as well the remaining variables are irrelev
12. ol volume 1 Tag 11 and 34 as output variables Tag 19 and 20 as input variables Tag 3 is not an independent input as it is found to 3 aes SN Control 47 N x a7 Volume 4 A tp os 7 Feed E OA y tt 4 Y Control E see f t I i 1 i I l a 1 Feed i ey r A Vaporizer 1 en 4 1 Process gas J I i 1 A a os 1 1 Control Volume 2 I Y i y Reformer P sfeed preheat N 1 Le Control Volume 1 Fig 3 Schematic of process and control volumes Table 4 Summary of indices Ye Yd Tag 11 0 98 0 94 Tag 34 0 85 0 36 have an identical shape as Tag 20 Path analysis yields the following results 1 The completeness index and significance in dex are calculated and shown in Tab4 The first column of the table shows that Tag 19 and 20 can explain most variability of Tag 11 and 34 The second column shows that the source of Tag 11 s variability can be easily identified while the source of Tag 34 variabil ity may not be isolated easily 2 The direct effect table is shown in Tab 5 The Table 5 Direct effect table Tag 19 Tag 20 Tag 11 0 92 0 07 0 14 Tag 34 0 26 0 71 0 25 first column clearly indicates that Tag 19 is the source of Tag 11 The second column shows that Tag 20 is possibly the main con tributor to Tag 34 but Tag 19 also has a con siderable contribution Therefore a unique source
13. osed so far is limited to steady state analysis Process dynamics such as time delay may affect the result if the dis turbances are relatively fast Thus the algorithm discussed so far can only be applied to trace slow disturbances Extension to dynamic application such as for oscillation detection will be discussed in the next section 2 4 An example on path analysis The following example illustrates the path analy sis method Four variables x1 2 3 and y are used for analysis where y is the quality variable of the concern output variable All four variables are highly correlated with the correlation coeffi cients shown in Table 1 the last row of the table According to this simple correlation analysis all input variables seem to have a strong correlation with y with the minimum correlation coefficient 0 77 However the path analysis shown in Fig 2 clearly distinguishes x from others and indicates Table 1 Correlation coefficients T1 T2 T3 y arai 1 z 0 76 1 x3 0 91 0 72 1 Yy 0 94 0 77 0 83 1 that it is the real source of the variation in y The two indices can be calculated as Ye 0 89 ya 0 90 Both numbers are close to 1 indicating that the selected variables are able to explain most of the variability in y and the source of the variability can be easily identified ye 0 89 also indicates that 89 variability in y can be explained by the selected variables and yq 0 90 indicates that the direct e
14. re the path co efficients direct effect coefficients p is the cor relation coefficient between X and X 2 2 Path analysis It is interesting to see that the correlation coeffi cient between Y and X can be constructed from the path diagram This is shown below py x Corr Y Xi Cov s Zi Using 1 Cov Ys Zi Cov S gt py 325 Zi So py iis j l j l which is weighted sum of the path coefficients This correlation may be interpreted as the total effects from X to Y through all possible paths and therefore this total effect is nothing but the correlation coefficient between X and Y The difference between the direct effect and correlation coefficient is evident through this analysis Another interesting fact is the variance decompo sition Note that the following equation exists 1 Var Y Var pyiZi pyc i l r r gt pD PYiPikPYk Pye i 1 k 1 Tr F r 1 So py 2 5 5 PYiPikPYk Dye i 1 i 1 k i 1 Vg Vi Vu This equation may be interpreted as Total variance of the output Contribution from direct effects Contribution from indirect effects Contribution from unknown source Two useful indices can be defined e Completeness index of the selected variables is defined as ye va v which is bounded from 0 to 1 ye 0 indicates that the se lected input independent exogenous vari ables have no effect at all on the output de pendent endogenous var
15. tion is a dynamic behavior of the process and is determined by its amplitude frequency and phase While the amplitude and frequency can be captured by the static path analysis the phase lag or time delay clearly nullifies the static approach For oscillation detection or tracing the oscillation one is not interested in finding the phase infor mation of the oscillation as long as the frequency of the oscillation is captured Autocovariance or spectrum of a time series captures the oscillation characteristics including amplitude and frequency but is independent of the phase Therefore ap plying the path analysis to the autocovariance of data will circumvent the problem of phase lag or time delay Thornhill et al 2001 Thornhill et al 2001a have presented a MATLAB function to calculate filtered autocovariance and spectrum of time se ries The same algorithm is used here for dy namic path analysis A set of data courtesy of a SE Asian refinery has been used for oscillation detection in Thornhill et al 2001 Thornhill et al 2001b The same set of data is revisited by applying the path analysis to the autocovariance of the data The process diagram is shown in Fig 3 The question is to trace the source of the oscil lations The path analysis is applied to the au tocovariance of the data to search for the source which constitutes the following five steps Step 1 Draw a control volume around PSA unit see Fig 3 for contr
16. uenced by other variables The notion of a causal relation between X and X requires that all other possible causal factors be ruled out Statistically we specify that X and 2 be uncorrelated where 2 represents the collective effect of all unmeasured variables that could conceivably influence X and X2 To offset the influence of variable units the re gression equation is written in the standardized form as a ou X M Oee E2 ee a a y 722 022 y T11 022 Tce or written in a compact form Z2 p21 Z1 pre Note that all variables including the error 2 now have the same variance of 1 and mean of 0 The error also has a coefficient The parameters p in the standardized model are defined as path coefficients Mathematically it is equally logical to postulate that X causes X or to postulate a third model that includes a common factor In the latter case the correlation between X and X is spurious and not a cause effect correlation The path diagram is now E2 X2 X a Meee PS A E1 where we again allow for errors in the relation ship In terms of standardized variables the lin ear model implied by the path diagram above becomes Z pi3F3 Pie 1 Z2 p23 F3 Pie 2 where the standardized errors and 2 are un correlated with each other and with F3 A distinction is made between variables that are not influenced by other variables in the system
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