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Data Reconciliation & Gross Error Detection An Intelligent Use of Process Data
Shankar Narasimhan and Cornelius Jordache
Publishing Company Houston, Texas
Data Reconciliation & Gross Error Detection Copyright 02000 by Gulf Publishing Cornparty, Houston. Texas. All I-ightsresel-ved. This book, or parts thereof. may not be reproduced in any form without express written permission of the publisher.
Tr, our grtru Professor Richard S. H. Mak, who played
tlze roles of an initiator and a catalyst.
Gulf P ~ ~ b l ~ s hCompany i~ig Book Di\,ision PO. 13ox 260s 1Hou~lnn.l'esas 77252-260s "Since all measurements and observationb ar-e nothing more than approximations to the truth. the same must be t!-ue of 311 c:ilcuiations resting upon them, and thc: highest a m of all cn:npu!2iior!s matie COiICZi71iEs c0i:Cr::ie ~!:cllOnlsila i:lcsr t ? !o ~ approxirnatc, al; ncari)- YLL pr;?c!icab!c. t ~ i!i2 l tr:!ii~. R u t this can be accomplisi~cdin ;lo OCIICS \ri-;~te~:~:; ....................................... Coir!i>ii;a!ic)i-iA Si;lti'ci~\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pcr-fori!ianci. .'il~-a,u;-:.\ for Ij\.al~iatinsGross fir!-01-!ilcn!i!Ycativil '
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Appendix R: Gralth Theory Fundarr:enfal\. 3-3 Giaphh. PI-cicessGI-,i;;i~s.a:ici Su!;sra;lh\ . . . . . . . . . . . . . . . . . . . 37t. t'at!i.;. Cycles. aiitl C::nnccri~iry . . . . . . . . . . . . . . . . 1 Sp~niiinp'l'recs. I(~-ai?lhes. zrid Cti,?;.~!... . . . . . . . . . . . . . . . . . . . . . :;SO .Ll; Graph Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cutset\. I-undamcniais C~.lrsets.a i d F~ir-ida~ni.rltai C lcie.; . . . . . . . . . . . . ?" I [{efcrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3K3 ,.>:>
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Appendix C: Fundamentals of Probability and Statistics. 384 Randorn Variables 2nd PI-ohabilityDensity Functions . . . . . . . . . . . . . . . 383 Statistical Properties of IZandom Val-iables . . . . . . . . . . . . . . . . . . . . . . . . 589 Ffypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Acknowledaments
Thc authors are indebted to several people who have contributed to the preparation of this book. T h e m a i n contributions c a m e from Prof. N;lrasinlhan's students at the Indian Institute of Technology (1lT)hladras. 'f. Rcnganatllan and J . Prakash. currently doing their doctc?ra! prozrams. prepared the solutions for all examples lvith assislance fr-om Sreerari~Mriguluri. Mnrukuria Rajar~~ouli spent 110ui.s in :he riigtit psepal-ins tlic ~ a b i z sand iigurzs in dil.l'erent ch:i~::eu. Thc s~~ccessfiilccniltleiioii of ii-~i, book was clue to their eftoris. ii:--. S;icl-iln Pat\i.a!-dhan ;1.!1(! S. i'cslir;~\ an;im. faculty at f Jl' M d r a s . pro\~icli.c!critical in~)t!isii? ii:ipro\ c tile 'li!c.us and c!ari:y of- tni: text. Thanks are also due to Liii: RAGE softv.-are cie\~zl:)pment t a r n at E ~ g i ncers India Lirni!cd, R&D Center. c o n s i s i i ~ gi\f Dr. i\ladhukar Gasg. Dr. V. !?a\ ikuliiar. and XIIS. Sheoi-aJ Sin?i! fi-ti~;: ~,~hclir? PI- iiii. irlc1udc.d tl~atdescrihe the necessary b a i c co:iccpts fro111linex a!gsi?i:a. %rap11t!:co:v. atid prohabi1i;y ar;d :,ta:isiicai h\;po~hcci.;tc\tiil:.
The Importance of Data Reconciliafion and Gross Error Detection
PROCESS DATA CONDiflONLING METHODS I n aily modern chelnicni p!ant, pctrc~cheiiliiaiprocr3si or refiilel).. h~ndred5oi- eve:; tiiousai~dsof variabiei---silcfi 2s Eo\v r:;:c.s. terr?pei'atu!-cs, pres.;i!i.c;>, ie\.els. ~ : coiiij~o~i:io:~s---31-e d ~ . o L ~ : ~ Inieasured Iz!~ ail,! auio!na:icaliy I-ccoi-dedi'c o f proc, are expi:cted to be 111ol-eaccurare than the ~neasurements.Although the problerr, considered here is siiiiple, ir does have important indiistrial appiicatic~nsin accurate accoiliiting far- tlie material flows as. for example, in a lube blending piact. in t!le strtaln ; ~ n i i water distribution subsysleni of a p!ant, or ill a complete r e f i ~ i e ~ ~ .
Example 1-1 Let us consider a si~np!eprocess of a heat exchanger- witii a bypass as shown in Figurc 1-2. Let us also igrrorc the energy flows of t!~is process and focus only on the mass flows. It is assumed that the flo\vs of' all six strea~nsof this process are n~easur-etiand that thcse measurements con-
The Imnportance 1.f Dora Kcconciliu(ion und Gross Error I)etecriori Data Ke~~~nriliutinn and Gross Error Defection
Figure 1-2. Heat exchanger system with bypass.
tain random errors. If we denote the true value of the flow of stream i by the variable xi and the corresponding measured value by yi, then we can relate them by the following equations yi = xi + E ;
i = 1 ... 6
e
*
(I - 6)
-
xj = 0
6
,=,
(1 - 9)
where :he weight:, wi are chosen to reflcct the accuracy of the respective measurements. More accurate ~neasurernectsare given larger weights in order to force their aajustn~entsto be as small as possible. Gznerally, it is assumed that the error vxiances for all the measurements are known and that the weighs are choscn to be the invcrse of these variances. --. 1 nc re con cilia ti or^ problem is thus a constrained optimization problem
(1 - Saj
Q
The measured values (given in Table 1-1) d o r,ot satisfy the above equatiocz, sic:- t'.~;. rcrt-iy randerr? zrr2rs !r is desired to derive estim2tes of the flows that satisfy the above flow balances. Intuitively, we can impose the condition that the differences between the measured and estimated flows, also referred to as adjustments, should be as small as possible. As a first choice, we can represent this objective as
The above function is the familiar least-squares criterion used in regression. Since it is immaterial whether the adjustments are positive or negative, the square of the adjustment is minimized. Although other types of criteria may be used such as minimizing the sum of absolute adjustment, they do not have a statistical basis and also make the solution of the problem more difficult. The least-squares criterion is acceptable, if all measurements are equally accurate. The adjustment made to one measurement is given the same ~mportanceas any other. In practice, however, it is likely that some measurements are more accurate than others depending on the instrument being used and the process environment under which it operates. In order to acccunt for this, we can use a weighted least-squares objective as a more general criterion, given by ~ 4 i n z w , ( -~x i, ) '
where E, is the random error in measurement y,. The flow balances around the splitter, exchanger, valve. and miser can be written as
x i - x2
13
with the objective functioil given by Equation 1-9 and the constrain~s given by Equations 1-?a through 1-7d. The solution of this optirnizatior; prcblein can be obtained analytically for flow reconciliation. Table 1-1 shows the true, measured, and reconciled flows for the prGcess of Figure 1-2. The reconciled flows shown in column fm1r nf this t&!e are okclned by assu~ningthat all measurements are equally accurate (weights are all equal). It can be easily verified that while the measured values do not satisfy the flow balances, Equations 1-7a through 1-7d, the reconciled flows satisfy then?.
The Imnporlance of Duru Reconciliation ur:d Gloss Error Detecriorl
Table 1-1 Flow Reconciliation for a Completely Measured Process Stream Number
True Flow Values
Measured Flow Values
Min xl~x2>XS>x6
wl(yl -x112 +w2(y2 -x212 +WS(YS-xs12
Reconciled Flow Values
Since the unmeasured variables are present only in the constraint set. the simplest strategy for solving the problem is to eliminate them from the constraints. This will not affect the objective function since it does not involve unmeasured variables. Variable x3 can be eliminated by combining Equations I-7a and 1-7c, while variable x4 can be eliminated by combining Equations 1-7b and 1-7d. Thus, we obtain a reduced set of constraints which involves only measured variables. Systems With Unmeasured Variables
In the previous example, we have assumed that all variables are measured However, usually only a sub\et of the variables are measured The presence of unmeasured var~ablesnot only co~npl~cates the problem solutlon, but also ~ n t r o d u c enew ~ questions such as whether an unmeasured ariable can be ect~mated,or whether d meaxured varldble can be reconclled as Illustrated b> the following example
The reduced data reconciliation problem is now to minimize 1-11 subject to the constraints of Equations 1-12a and 1-12b. It can be observed that this reduced problem involving the variables x i , x2, x5, and x6 is simi!ar to the completely measured case, and an analytical solution can be used co obtain the reconciled values of the measured variables. Using {hi. same measured values for x,, x-,, x5, and x6 as given in Table 1-1, and assurni~gall measuremects to be equaily accurate, the reconc~ledi.a!uec whicn are obiained are shown in Tabie 1-2 In the col~imnmder Case !. Once the reconciied values f ~ the r measured variables ere okained, the estirxates of the uilmeasured variab!es can be calculated usil:g the original constraints. Thus the estimate of x4 is equal tc~that of x2, and the estirnate of xi is equal to that of xs. These values are also indicated in Table 1-2. By comparing with the resu!ts of Table 1-1, it can be observed that since There are fewer measured variables in this case, the estimates of some variables are less accurate than those derived for the completely measured system. The central idea that is gained from this case is that the reconciliation problem can be split or decomposed inro subproble~ns-the first being a reduced reconciliation problem involving only measured variables, followed by an estimation or coaptation problem for calculating the estiniates of anmeasured variables.
Let us consider the flow reconciliation problem of the simple piocess shown in Figure :-2. However, we will cot assume that all the flcws are rneasured as b e f ~ r r Instead, . we will assume that only selective fiu\vs are measured and in each case discltss the issues and prob!ems invclved in partia!ly measured systems. Case I . Flows of streams i , 2, 5, a n d 6 are measured, while the other two stream flows are unmeasureci.
The object~vein this case is to not only reconcile the measured flows. but also to estimate all ?he ~~nrr~r:icl~red flows as part of the reconciliation problem. As in Equation 1-6. we relate the measured and true stream flows.
'The constraints are still given by Equation 1-7. It should be noted that the constraints involve both measured and unnleasured flow variables. The objective function is the weighted sum of squares of adjustments made t c ~1ne:rsured variables. and is given by
. l
Tlre Irnpor-imzcr ofDu(a Rrro~rciliuriono~zdGI-osrError Detecr;on
Table 1-2 Flow Reconciliation of Partially Measured Process Reconciled Flow Values
Stream
Case 1-Streams 3 and 4 unmeasured
Case 2-Streams 3 , 4 , 5 , 6 unmeasured
Case 3-Streams 2,3,4,5 unmeasured
1 2 3 4 5 6
100.49 64.25 36.24 64.25 36.24 100.49
101.91 64.45 37.46 64.45 37.46 101.91
100.39 -
17
These unmeasured variables are also known as observable. A formal definition of the concepts of observability and redundancy is given in Chapter 3. It is sufficient at present to note that while the partially measured process in Case 1 gives a redundant and observable system, Case 2 gives rise to a nonredundant, observable system. Case 3. Only flows of streams I a n d 6 are measured.
The reduced reconciliation problem we obtain for this case is
-
.-
100.39
Case 2. Only flows of streams 7 a n d i 0, measured
In :his case, only Equations I-7a and 1-7b contain measured variables and are useful in the reconciliation problem. The objective f~inctionis set ~ 1 as 7 before to minimize the adjustment made to measured variables and is ~ i v e nby
As in Case I . we ti-y tc elin~inatethe un~rieasaiedvariables from the cc)nstr;,ints 1-7a arid 1-7b. Our attempt tc produce an equation involvirlg (mly ~neasuredvariables by suitably combining :he originai constraints ends in failfire. Thus, the reccncijiaiion problem we obtain is to minimize 1-1 -3 \vithout any constraints. It is immediately obvious that the best estirnarzs of x i anti x2 3ri given by their respective nleasured values which resclts in t.he least adjgstmcrtt of zero for 1-13. The estinlates of the unmsasurcc! variables can now be calcuiated using the constraints. The estimate of x6 is equal to x,. the eslimatc of x4 is equal to x2, and the estimates of x3 and xS are both equal to the difference between x, and x2. 'These values are a11 given in Table 1-2 under Case 2. Two irnpo17ant observations can be made in this case. First, no adjust111crttis made to the two measured variables x,, and x2. This is due to the fact that there is no additional information in the fonn of conssaints that relttte only the measured variables which can be exploited for adjusting their measurements. Such measured variables are also known as noni-eduildnnt val-inbles. Second, a unique estimate for every unmeasured variable is obtained using the constraints and estimates of meaxured variables.
Min wl(yl -x112 + ~ g ( y -6 ~ g ) ~ ,x6
(1- 14)
X1
such that:
Equation 1-15 is obtained by adding all the constraints 1-7a through 1-7d. Assuming that the measurements of xi and Q are equally accurate, their reconciled values obtained are given in Table 1-2 under Case 3. We now attempt to calculate the estimates of the remaining four variables. We wit1 not be successful, however, in obtaining unique estimates for these variables. In other words, :here are many soiutions-in fact, an infinite nurnher-which can satisfy the constraints. For example, one possible solution is to take the estimates of x; and x5 to he both equal to that of x,, and the esti~rkatesof x2 and x4 to be equal to zero. Alternatively, we can choose thc estimates of x2 and x4 :o 5e equal to that of x,,while the estimates of xa and x5 are chosen tc be zero. Without additional info-mation, there is no way of det-erminicg which of these myriad possible solutions is rnorc: accurate. The variables .x2. x3, x4. and x2 are denoted as unabservablc. in rhis case. An interesting feature of this case is thzt thoilgh there are some cnmeaslired variables which cannot be u~iquelyestimated, reconciliation of the variabies x, arid x6 can still be performed utilizing the available measurements. Therefore, Case 3 is a redundant, unobservable system. System Containing Gross Errors
In all the cases considered in Example 1, the measurements did not contain any systematic error or bias. In such cases, data reconciliation does reduce the error in measurements. We wi!l now examine the case when one of the measurements contains a systematic bias or gross error
and demonstrate the need to perform gross error detection along with data reconciliation.
Example 1-3 W e reconsider the flow process shown in Figure 1-2 for which the true stream flows are as given in Table 1-1. W e will assume that all flows are measured with measurements as given in Table 1- I, except that the measurement o f stream 2 contains a positive bias o f 4 units, so that its measured value is 68.45 instead o f 64.45. As before, we reconcile these measurements and obtain estimates which are shown in Table 1-3, in column 2, when all the measurements are used. A comparison o f these estimates with those listed in Table I - I , clearly shows that the accuracy o f the estimates has decreased due to the presence o f the gross error. Furthermore, although only the flow measurement o f stream 2 contains a gross error, the accuracy o f all the flow estimates has decreased. This is known as a smeariilg effect and it occurs due to reconciliation which exploits the spatial constraint rela'.'ions between differentvariables. In order for data reconciliation to be effective.it is therefore necessary to identify those measurements c~:itaininggross errors and either eliminate them or make appropriate con:l;enxation. The iast colurnn o f Table 1-3 shows the reconciled esti~natesobtained when the flow measurement o f strearn 2 is discarded acd not ussd ir: the reconci!iati~nprocess. Clearly. ?he accuracy o f the reconciled estimates has improved cocsiderably. even though Be recfundaricy has ciecrzased by discardins thz mzzsurement. Table 1-3
Flow Reconciliation When Stream 2 Flew Measurement Contains a Grass Error Recanciled Flow Values Stream
All measurements used
Stream 2 measurement elimii~ated
~-.
1 2
3 4 5 6 --
100.89 65.83 35.05 65.83 35.05 100.89
-
100.23 64.53 35.71 64.53 35.71 100.23 --
Thus far, we have not considered the important question o f how to identify the measurement contaiuing a gross error based only on the knowledge o f the measured values and constraint relations between variables. There are several ways o f tackling this problenl and in this example we illustrate one approach. Given a set o f measurements, we can initially reconcile them assuming that there are no gross errors in the data. In the flow process example considered here, the reconciled estimates obtained under this assumption have already been shown in the second colu~mlo f Table 1-3. From these reconciled estimates we can compute the differences between the measured and reconciled values (measurement adjustments) for all measured variables, and these are shown in Table 1-4. Table 1-4 Measurement Adjustments for Flow Process Stream
Measurement adjustments
I f the c~fistraintsare linear as in t!lis example, the expected variar;ce of the adjustments can be analytically derived which will be a function of the constraint matrix and !he measurement error variances. For the flow process example considered here, it cai; be shown :hat the standard drviation o f measurernent adjustments for every variable is 0.8165. A simple statistical test can be applied to determice i f the computed measurement aliustments fail within a confidence interval, say within a 220 interval. In this example, the 220 interval (95% confidence in~erval)is [-1.6 I .6]. From Table 1-4, we can observe that the measurement adjustments f ~ r the flows o f streams 2, 4, and 6 fall outside this interval and as a t~rstcut the measurements o f these streams can be suspected o f contai~~ing a gross error. Among these the measureme~itadjustment o f stream 2 has the largest magnitude and can be identified to contain a gross error. After discarding the measurement o f stream 2, we can again reconcile the data and compute the measurement adjustments to examine i f any more gross errors are present.
The procedure used above is a sequential procedure for gross error detection and makes use of the statistical test known as the rrreasurenzeizt test. A variety of statistical tests and methods for identifying one or more gross errors have been developed and are described in Chapters 7 and 8. Although in this example we have only considered a gross error in measurements, it is possible for a gross error to be present in the constraints due to an unaccounted leak or loss of material. Some of the methods described in Chapters 7 and 8 can also be used to identify such gross errors. The example also clearly demonstrates that data reconciliation and gross error detection have to be applied together for obtaining accurate estimates.
BENEFITS F R O M DATA RECONCILIATION A N D GROSS ERROR DETECTION Development of a data reconciliation and gross error detection package for a system and its practical implementation is a difficult and costly task and cannot be justified without its benefits for a particular industrial application. The justiiication for data reconciliation and gross error detection nlay come from the many iiiy~ortantapplications for improving Frocess performance shewn in Figure 1- 1 which requires accurate data for achieving expected benefits a:; outlir?ed below: i . A direct application of data reconciliation is in evaluating process yieids or i n assessing consumption of utilities in different process cnits. Keconciied values provide more accsrate estimates as compared to :he use of r2w nieastlremen!s. f;cr example, refinery-wide material balance reccnciliatior? aids in :t Getter estimate of overail refinery yields. Similarly. a plant-wide energy audit using reconciled flows and temperatures hetps in a better identification of energy inefficient processes and equipment. 2. Applications such as simulation ana opti~nizationof existicg process equipment rely on a model of the equipsilent. These models usually contain parameters which hatie to be estimated from plant data. This is also known as model tunillg, for which accurate data is essential. 'The use o f erroneous measurements in model tuning can give rise to incorrect model paranleters which can nullify the benefits achievable through optimization. There are two possible ways in which data reconciliation can be used for such applications which we illustrate using a simple example.
Let us consider the problem of optimizing the performance of an existing distillatior column. From the operating data, measurements of flows, temperaturzs and compositions of all inlet and outlet streams of the column can be obtained. One possible way is to reconcile these measurements using only overall material and energy balances around the column. The reconciled data can now be used along with a detailed tray to tray model of the column in order to estimate parameters such as tray efficiencies. The tuned model can then be used to optimize the performance of the column. Alternatively, a simultaneous data reconciliation and parameter estimation can be performed using the detailed tray-to-tray model of the column. In this case, if measurements of tray temperatures and/or compcsitions are available, they can also be used and reconciled as part of the proolem. Obviously, the second approach leads to a significant increase in effort and computation time. This approach is also referred to as rigorous on-line modeling and has been incorporated in many commercial steady-state simulators. 3. Data reconciliation can be very useful in scheduling maintenance of process equipment. Reconciled data can be used to accurately estimate key performance parameters of process equipment. For ewam~de.heat transfer coefficient of heat exchangers or the level of catalyst activity in reac!ors can be estimated and used to determine whether- ihe heat exchanper - should be cleaned or whcther the c a t a l y t sbculd be replacedJregenerated, respectively. 4. Many advanced control strategies such as rr,odel-based control or inferential contrci require accurate estimates of controlled vzriables. Dynamic data reconci!iatior, techciques can be used to derive accurate esGmates for better process coilttol. 5. Gross error detection not only improves the estimation accuracy of data reconciliation procedures but is also u s e f ~ lin identifj'ing instrw mentation problems which require special maintenance and correction. Incipient detection of gross errcrs can reduce maintenar,:, cost; a;;< provide a smoother plant operation. These methods can -.1~0 b? pxtended to detect faulty equipment.
A BRIEF HISTORY OF DATA RECONClLlATlON A N D GROSS ERROR DETECTION The problem of data reconciliati011was first introduced in 1961 and during the past four decades more than 200 research publications in the
22
Ilirra Kerations(edited by D.W.T. Kippin, J. C. Hale,
and J. F. Davis). Amsterdam: CACHE/Elscvier, 1994,429-436.
16. Simpson, D. E., V. R. Voller, and M. G. Everett. "An Efficient Algorithm for Mineral Processing Data Adjustment." Int. J. Miner. Proc. 31 (199 1): 73-96. 17. Heraud, N., D. Maquin, and J. Rago:. "Multilinear Balance Equilibration: Application to a Complex Metallurgical Process." Min. Metall. Proc. 1 1 (1991): 197-204. 18. Reid K. J, K. A. Smith, V. R. Voller, and M. Cross. "A Sunfey of Material Balance Computer Packages in the Mineral Industry." in 17rh Appiicnriot~sq/. L'onlputers and Operations Re.seurclz it1 the Miner01 Industry (edited by T . B. Johnson and R. J. Barnes). New York: AIME, 1982. 19. Stanley, G. M., and R.S.H. Mah. "Ertimation of Flows and Tempel-atures iil Process Networks." AIChE Journal 23 (1977): 6 4 2 4 5 0 . 20. Alrnasy. G. A. "Principles of Dynanlic Baiancin~."AICIIL' Joui-nul 36 (1991): 1321-1330. 21. Liehman, M. J., T. F. Edgar. and L. S. Lasdon. "Efficien: Data Reccnci!ration and Estirllation for Dynarnic Processes Using Nonlinear Prc>~ran~:nins Techniques." Cotnplrters Clzen~.Eilg~~g. 16 1992): S63-986.
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22. Ripps. D. L. "Adjustment of Expesinierita! Data." Chenr. Etz,~.PI-og;...i'w1[1 Scr. iZo. 53 61 (1965): 8-13.
and R. E. Carpani. "Application of Statis!ical Theory lo 23. Kei!ly, P . IM., Adjustment of klaierial Balances." presented at the 13th Can. Chzm. 5 2 Cod.. Mo~treal,Quebec, 1963. 54. Almasy, G. A, and T. Sztzno. "Checking and Conectior. of Measur-cmen[~ Basis of Linear Sys:enl Model." Proh. Co?7tro! l i f 0 ~ 1 17-!ieor-\. . 1 on (1975): 57-69. 25. Narajiillhsn, S., arid R.S.H. Mah. "Gerieraiized Likelihood Ratio kletttod fo: Grot.; F ~ n Identification." r AICizE Juut-rial 33 (1987)- 1514-1521. 26. Tarll!lane. ,\. C., C. Jordache, arid R.S.H. Mah. "A Bayesian Approach io Gross Error Detection in Chemical Process Data. Part I: Model Developand hztel. Lab. Sys. 4 (1 988): 33. ment." Clzc,mom~~rics 27. 'Tons. H., and C. M. Crowe. "Detection of Gross Errors in Data Reconciliation by Principal Component i2nalysis." AIClrL- Jourrzai 31 ( 1095!: 1712-1722.
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