Application of Cointegration Testing Method to Condition Monitoring and Fault Diagnosis of Non-Stationary Systems 1

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1 Applicaio of Coiegraio esig Mehod o Codiio Moiorig ad Faul Diagosis of No-Saioary Sysems Qia Che, Uwe Kruger, Yuyu Pa College of Aerospace Egieerig, Najig Uiversiy of Aeroauics ad Asroauics, 9 Yudao Sree, Najig 006, P.R. Chia Iellige Sysems ad Corol Research Group, Quee s Uiversiy Belfas, Belfas, B9 5AH, U.K. Absrac: his paper iroduces coiegraio esig mehod for moiorig mulivariae o-saioary processes. Compared wih Box-Jekis models, coiegraio esig mehod is able o model log-ru equilibrium relaioships ad circumve he effec of forecas recovery. Coiegraio has bee successfully uilised o describe such relaioships i ecoomics sysems, which are o-saioary i aure. Coiegraio mehod ca ideify a saioary process usig a liear combiaio of he o-saioary processes. his saioary process describes a log-ru equilibrium relaio bewee he o-saioary process variables, which provides a possibiliy o apply he coiegraio esig mehod o he modellig ad moiorig of egieerig sysems. A simulaio example from he lieraure shows he capabiliies ad beefis of coiegraio esig mehod for moiorig o-saioary processes. Copyrigh 006 USARH Keywords: Coiegraio esig, o-saioary sysems, saisical process corol, faul deecio.. INRODUCION No-saioary dyamic processes are a freque occurrece i egieerig sysems. As highlighed i he research lieraure, o-saioary sysems are difficul o moior, where exisig work predomialy relaes o he applicaio of Auo- Regressio Iegraed Movig Average (ARIMA model or he Wavele or Fourier aalysis. Box, e al, (994 showed ha (i o-saioary sequece is characerised by a varyig mea ad (ii differecig he origial o-saioary sequeces covers a o-saioary o a saioary siquece, i.e. a sequece ha has a cosa mea. Hece, osaioary sequece ca be modelled by a ARIMA(p,d,q model, where a oal of d differeaio have bee performed ad he saioary sequece is described by p lagged auo-regressive ad q lagged movig average erms. Berhouex ad Box (996 ad Wiksrom e al, (998 employed ARIMA models o geerae a se of saioary sigals for modellig a mulivariae sysem for process moiorig. However, as poied ou by Superville ad Adams (994 ad Apley ad Jiaju (999, ARIMA models affec he sigaure ad magiiude of faul codiios, a effec kow as forecas recovery. Backshi ad Sephaopoulos (996 proposed a sigal-based approach usig he wavele rasformaio ad moiored parameer chages of Haar waveles. Sigal-based echiques, however, may be dif- he auhors graefully ackowledge he suppor of K. C. Wog Educaio Foudaio, Hog Kog. Correspodig Auhor: qche@uaa.edu.c

2 ficul o impleme i a mulivariae coex if he process variables are ierrelaed, sice such relaios cao be modelled usig a sigle-variable aalysis. As summarised above, alhough o-saioary sequeces ca be described by mulivariae ARIMA models, he ihere differeiaio ca have a profoud impac upo he faul magiude ad sigaure. his paper uilises coiegraio esig mehod for modellig he relaioships bewee he process variables by ideifyig a coiegraig coefficies. hese coefficies are uilised o form a liear combiaio of he o-saioary process variables. his liear combiaio geeraes a saioary sequece or equilibrium residual sequece ha describes he log-ru equilibrium relaioship bewee he o-saioary sequeces. he residual sequeces ca he be uilised i a saisical process corol framework for process moiorig purpose. Sice coiegraio esig mehod does o rely o he saisical aalysis of differeiaed sequeces, he deficiecy of mulivariae ARIMA models, i.e. he isolaio of he correc faul magiude ad sigaure, are circumveed. he cocep of coiegraio esig was developed by Egle ad Grager (987 ad has bee successfully applied i ecoomerics ad, i paricular, o model he relaioships bewee power cosumpio ad ecoomic developme (Che ad Hurvich, 003; Hase, 003; Kaufma, 004; Galido, 005; Hassler, e al., 006. o verify he applicabiliy of he coiegraio esig mehod o egieerig sysems, [Pa ad Che, 006] employed i for a car egie codiio moiorig ad obaied some ecouragig resuls. I his paper a example of a simulaed osaioary Fluid Caalyic Crackig Ui (FCCU sysem was discussed. he resuls show ha he coiegraio esig mehod migh have a big poeial i osaioary process moiorig ad faul diagosis.. COINEGRAION PRELIMINARIES Accordig o Box e al. (994, if a o-saioary process variable η becomes saioary afer beig differeiaed d imes i is said o be iegraed of order d, deoed by η I(d. Egle ad Grager, (987 cosidered a se of such d-iegraed osaioary variables o possess i log-ru equilibrium if ad oly if: β y y + L y = ξ ( Here, he equilibrium residual is defied as ξ = β y ( where β = β, β L R, y = y, y y ( β ( L R. I geeral, y i are a iegraed o-saioary sochasic variables ad ξ is a saioary sequece ha is assumed o describe he deparure of y from is equilibrium. his implies ha here exiss a vecor β ha, accordig o Eq. (, produces a ξ ha is of order 0, i.e. ξ I(0. More precisely, he Egle-Grager mehod ideifies a specific coefficie vecor β such ha he residuals of he equilibrium relaioship are saioary ad sochasic.. Defiiio of coiegraio he defiiio of coiegraio is, accordig o Egle ad Grager (987, ha he o-saioary sequeces, sored i y, are coiegraed of order (d, b, deoed by y ~ CI(d,b if ad oly if:. All variables i y are iegraed of order d;.. here exiss a vecor β such ha he liear combiaio β y is iegraed of order I(d-b, where d b > 0 ad b beig he umber of reduced order iegraio. Here, β is kow as coiegraig vecor ad if b = d, he β y is, accordig o Eqs. ( ad ( saioary. If he elemes i y are iegraed of order d ad coiegraed of order (d,b, here may be as may as liearly idepede coiegraig vecors. If muliple coiegraig relaioships are foud, i may be difficul o ideify a behavioural relaioship. For saisical process moiorig, a coiegraig vecor correspodig o a mos saioary residual series should be seleced. his ca be accomplished usig he Dickey-Fuller (DF saisic (Dickey ad Fuller, 979. his hypohesis es relies o values of he DF saisic, which are compared wih a give crieria, usually wih a sigificace of 5%.. Coiegraio ad commo reds Coiegraed variables share commo reds. For simpliciy, cosider oly wo variables = + (3a y w ε y w + ε = (3b Each of he w s is a radom walk ad each of he ε s is a sochasic ad saioary variable. If variables y ad y are coiegraed of order (, here mus be ozero values of β ad β for which he liear combiaio β y y is saioary. hus βy y = β( w + ε ( w + ε (4 = ( βw w + ( βε ε is saioary if ad oly if β w w vaishes: ( β w w = 0, (5 Hece, if β w w = 0 he β w = -β w. Sice w ad w beig radom walks, i.e. osaioary sochasic reds, hey (i are equal up o a scalig facor -β /β, (ii sochasic o-saioary I( sequeces ha have he same sochasic red ad (iii coiegraed of order (,. he parameers of β mus cosequely suppress he o-saioary red ecapsulaed i he liear combiaio of y ad y. Pracically, more deailed ime series srucures, icludig a cosa shif ad/or deermiisic reds, ca be employed. For example, Waso (994 uilised AR( models of he form: y = ρ y + ρ y + ε (6a y = α + ρ y + ρ y + ε (6b y = α + µ + ρ y + ρ y + ε (6c For idusrial processes, he measured variables ofe show commo sochasic o-saioary ad/or

3 deermiisic reds. Uder he assumpio ha he recorded variables have he same order, I(d, ad he process ca be described by a liar ime ivaria model, hese reds, described by he coiegraig vecor, are compesaed such ha log-ru dyamic relaioships are esablished. hese he gover he uderlyig physical or chemical relaioships bewee he recorded variables. If, however, a umeasured disurbace acs upo he process, his dyamic equilibrium ca o loger correcly represe ha process. Cosequely, he presece of abormal process behaviour maifes iself by a chage i he sochasic saioary sequece ξ, which ca he be saisically evaluaed for faul deecio ad a subseque diagosis. Compared o previous work, sice o differeiaio is required o esablish moiorig saisics, faul magiudes ad sigaures ca be preserved for subseque aalyses. 3. UNI ROO ESS Before ideifyig a coiegraio model, i is esseial o verify ha each o-saioary sequece is iegraed of order I(d. he, i is required o ivesigae wheher he liear combiaio of he se of o-saioar variables is saioary, ha is o es wheher he residual sequece ξ is of order I(0. his process is referred o as he Ui Roo es (UR ad wo of he mos commoly applied echiques are discussed below. 3. Dickey-Fuller es (Dickey ad Fuller, 979 For simpliciy cosider a o-saioary ime series defied by Eq (6a which has o deermiisic red. he AR( model of he ime series is y = ρ y + ρ y + ε (7a or y = ( ρ + ρ y ρ y + ε (7b where, ε ~ Ɲ(0,σ is assumed o be a whie oise sequece ad he parameers ρ ad ρ are esimaed usig Ordiary Leas Square (OLS. Assumig ha ρ <, he hypohesis H 0 : ρ + ρ = 0 ca be esed o deemrie wheher y has a ui roo. If so, Eq (7b becomes y = ρ y + ε (8 sice ρ <, y is a saioary, zero mea sequece. More precisely, y is a I( o-saioary sequece. If H 0 is rejeced, carry o he es o he AR( model usig z = y ad cosrac Eqs (7a ad (7b usig z isead of y. If H 0 is ow acceped, he y is a I( osaioary sequece Coversially, if H 0 is rejeced agai repea his es usig higher order differeials of y uil H 0 is acceped. he hypohesis wheher ρ + ρ = ca be esed usig he -saisic: ˆ ρ ˆ + ρ ρ = (9 se ( ˆ ρ ˆ + ρ where se( refers o he sadard deviaio for he esimaed parameers. As discussed i Dickey ad Fuller (979, his saisic follows a DF disribuio. Based o Eq. (9, he hypohesis for he -saisic is esed wih a cofidece of %, 5% or 0% ad associaed cofidece limis ca be obaied from a DF able. 3. Augmeed Dickey-Fuller es Sice i ca o be guaraeed ha ε is a whie oise sequece, he parameer esimaio is biased. his was addressed by proposig a augmeed DF (ADF es. Cosider he deermiisic red described by he AR( model i Eq. (6c y = α + µ ρ y + ( ρ+ ρ y + ε (0 Afer esimaig he parameers of he model ad obaiig he -saisic, he ADF es ca be carried ou o es he hypoheses H 0 : ρ + ρ =, H 0 : µ = 0 ad H 0 : α = 0. Associaed cofidece limis o evaluae he deermiisic reds for ρ + ρ, µ ad α ca be obaied from a ADF crierio able. 4. COINEGRAION ESS 4. Egle-Grager es For a bivariae case, y ad y, Egle ad Grager (987 develop a wo-sep esig mehod o examie if hey are coiegraed. Sep: Esimae he parameer of he regressio of y o y usig OLS y = β y + e ( he predicio of y ad he esimaio of β are obaied as yˆ = βˆ y ( where he predicio error is give by eˆ = y yˆ = y ˆ β y (3 Sep: es ê ~ I(0. If H 0 is acceped, y ad y are coiegraed. 4. Johase es If he umber of variable exceeds wo, he above Egle-Grager es cao be applied. Johase, (988 ad Johase ad Juselius (990 iroduced a coiegraio es mehod based o Vecor Auo- Regressio (VAR model applicable i a geeral mulivariae coex. his es should be performed afer cofirmig ha he variables are o-saioary ad I(. Give a he daa vecor y = ( y, y, L, y R ~I(, i which he elemes y k are represeed by o-saioary sochasic variables, he Johase es is based o a uresriced VAR model. his implies ha a uresriced VAR model mus be cosruced prior o mulivariable coiegraio ess ad he ideificaio of he correspodig coiegraio coefficies. he workig of his es is bes moivaed usig he followig example. Whe dealig wih a rivarial ecoomic sysem based o he logarihm of omial wages, prices ad labour produciviy, wo coiegraig relaioships could iuiively be isolaed: deermiig (i he employme equaio ad (ii he wage equaio. I geeral, a coiegraio model of osaioary variables is give by

4 ξ = β y (4 where ξ is a saioary residual. A Vecor Error Correcio Model (VECM forms he basis for he Johase esig. Cosider he VAR(p model of a vecor of o-saioary variables (Egle ad Grager, 987: p y = Π y + ε (5 k k k = o derive he VECM, subrac y - from boh sides of Eq. (5 o produce: where p y = Πy + Φ y + ε (6 k k k = Π = I + Π p j= k+ p k = k = Π( (7a Φ = Π, k =,, L, p (7b k j Now, Π( is decomposed io a produc of hree marices, i.e. Π( = U( M( V ( (8 Here, M( is a diagoal marix of rak r. Furhermore, Π = -Π( which is also of rak r. Le β deoe a by r marix whose colums cosruc a basis of he row space for Π, ha is Π = δβ, where δ is a by r marix wih full colum rak, Eq. (6 becomes or p = + k k + k = y δβ y Φ y ε (9a p y = δξ + Φ y + ε (9b k k k = Here, ξ = β y ad ξ - is give by p ξ = δ δ δ y Φk y k ε (0 k = From Eq (9b, i ca be cocluded ha he elemes of ξ - are I(0 ad cosequely represe saioary sequeces. hus, he scaled sum of he I( elemes i y produce he saioary I(0 sequece ξ. Furhermore, he colum vecors of he marix β are defied as he coiegraig vecors. he VECM imposes k < ui roos i he VAR model by icludig firs differeials of y k ad r = k liear combiaios of he o-saioary variables. he zero hypoheses should be : ( or H0 Rak Π r Π = αβ ( he case β y = 0 could be ierpreed as he equilibrium of a dyamic sysem ad ξ as he equilibrium errors. I fac, Eq. (0 describes he self correcig mechaism of he sysem from is o-equilibrium sae. he ideificaio of δ, β ad Π is cosrai such ha δ ad β ca be idividually ideified. he marix β ca be ideified o he basis of he maximum likelihood (ML mehod usig he probabiliy desiy fucio of y a isace f( y = f ( y = i i i= ( π exp / / Ω ε Ω ε ( where ε ~ Ɲ(0,Ω. Ω is also a parameer marix o be ideified for he VAR model. Icludig a oal of samples Eq. ( becomes f = = = = i = = i= / / = ( π exp = ( y f ( y ( y ( f f y (3 f ( y Ω ε Ω ε Applyig he aural logarihm fucio of Eq. (3 produces L( Φ, L, Φp, Π, Ω = l( π (4 l Ω ε Ω ε = I should be oed ha he ML esimaio of Π is cumbersome. Compuaioally beeficial is a esimaio of Π by a oliear leas squares regressio of y usig lagged values. I coras, he maximisaio of Eq. (4 o ideify Φ, Φ,, Φ p- is a compuaioally iexpesive OLS regressio of y = αβ y -. his produces he residuals ê 0, followed by a regressio of y - o he lagged differeials of y geeraig he residuals ê. I ca be show ha aoher regressio model: e ˆ ˆ + ε (5 0 = Πeˆ ca be obaied based o he heorem of uiqueess of vecor orhogoal projecio. Subsiuig Eq. (5 io Eq. (4 leads o L( α,β,ω = l( π l Ω (6 ( e0 αβ e Ω ( e0 αβ e = Followig he precedig discussio, Johase (995 showed ha deermiig he rak of Πˆ is equivale o he es for he umber of caoical correlaios bewee ê 0 ad ê ha are differe from zero. his ca be coduced usig oe of he followig wo es saisics λ ( l( ˆ r r = λi (7a i= r+ λ ˆ max (, rr+ = l( λ r + (7b where ˆk λ are he eigevalues of he marix S0S00S 0 wih respec o he marix S, arraged i descedig order > ˆ λ ˆ > L > λ > 0, where S = ˆ ˆ ij eie j, i, j = 0,. (8 he eigevalues ca be obaied based o he deermia of he followig equaio λs S S S = ( he saisic i Eqs. (7, kow as he race saisic, ess he ull hypohesis ha he umber of

5 coiegraig vecors is less ha or equal o r agais a geeral aleraive. Kowig ha l( = 0 ad l(0, boh saisics are equal o zero if ˆi λ is zero, ad he larger he eigevalues he smaller l( ˆ λ i becomes. Likewise, he saisic i Eq. (6, kow as he maximum eigevalue saisic, ca be esed o obai r, H 0, agais he aleraive, H, of r + coiegraig vecors. As above, if ˆr λ + is close o zero, he saisic produces a small value. Furhermore, if he ull hypohesis is acceped, he r coiegraig vecors coaied i marix β ca be esimaed as he firs r colums of marix V ˆ = ( vˆ, L, vˆ which coais he eigevecors associaed o he eigevalues i Eq. (9: ( λ i S S0S00S0 vˆ i = 0, i =,, L, (30 Oce a esimae of β has bee obaied, he esimaes of δ, Π i ad ε i Eq. (9a ca be obaied by iserig his esimae i he correspodig OLS formulaio. However, if oly he coiegraio vecors are of ieres raher ha he complee VECM model, i is sufficie o compue he eigevecors of Eq. (30. As oulied above, oly oe coiegraig vecor, i.e. oe of he eigevecors of Eq. (30, is required o cosruc a moiorig model. o ideify he mos suiable cadidae, he ADF es ca be performed bes oe saisical propery of he saioary residuals. al. (995 is summarised below. his example icludes wo ipu ad wo oupu variables, u k = v k + e k ad y k = x k + f k, respecively. Here, u k ad y k are he measured ipu ad oupu variables, x k is calculaed by a s order ARX model usig x k- ad v k- ad e k ad f k are idepedely disribued saioary whie oise sequeces. Furhermore, he sequeces v k were o-saioary ad produced from a mulivariae ARIMA model. From he above process, wo daa ses of 000 samples each were geeraed, where he firs se served as a referece se for deermiig he coiegraio model ad he secod oe icluded a sep ype faul of magiude.8 superimposed o he s oupu variable afer 600 samples were simulaed. For faul deecio, a coiegraio model was esablished usig he seps summarised i Secio 4.3. he ideified coiegraio vecor was ( β ˆ = (3 Defiig he vecor z = (y y u u, he saioary residual variable, ξ, was give by ξ = z 0.7z z z 4. he residual vecor was ow deermied for boh daa ses ad Fig. shows ha he sep-ype faul could correcly be deeced. 4.3 Ideifyig coiegraio models he ideificaio of a coiegraio model, ha is he ideificaio of β, ivolves he followig seps:. Use ordiary leas squares (OLS o ideify a AR model for each variable o be esed,. Carry ou a ui roo es for each model o examie wheher hese are iegraed of order I(. 3. Use OLS o fi wo VAR models for y ad y o lagged values of y o obai wo residual vecors o form he eigevalue equaio. 4. Solve he eigevalue problem o obai he coiegraig vecors β j o geerae he equilibrium residuals ξ j ad examie he residuals o deermie he bes coiegraig vecor. Afer esablishig a coiegraio model usig he above seps, he applicaio of his model for faul deecio ad diagosis is as follows:. Apply he esablished model o ew daa ad moior he residuals o deec ay abormal variaios i he equilibrium residual ξ j.. If aomalous behaviour has bee deeced, diagose his eve by ivokig reduced coiegraio models as discussed ex. Fig. : Sequeces of ξ for fauly daa For faul diagosis, he followig 4 reduced models were obaied o isolae he impac of he faul o a sigle variable 5. APPLICAION SUDY o demosrae he workig of coiegraio esig mehod i a codiio moiorig coex, a applicaio sudy o he simulaio example i Ku e Fig. : Residuals from reduced coiegraio models

6 ξ = z z z ξ = z z z ξ = z 0.377z z ξ = z +.64z +.655z (3 Afer he faul was deeced, hese reduced models were applied i ur. Fig. shows he resula residual sequeces for each model ad clearly illusraes ha omiig y o faul iformaio is maifesed i he associaed residual sequece. I coras, each of he 3 remaiig reduced models, as well as he origial model, produced a offse of.8 afer 600 samples. 6. CONCLUSIONS his paper iroduces coiegraio esig mehod for codiio moiorig of mulivariae osaioary processes. his echique esablishes a liear combiaio of he osaioary variables such ha a saioary residual sequece is produced ha ca be iegraed io he saisical process corol framework for abormaliy deecio i process. A applicaio example o a simulaio from he lieraure showed ha coiegraio esig mehod ca deec a faul codiio ad correcly ideify he variable affeced by he faul. I coras o exisig work, forecas recovery ad he limiaios of sigalbased echiques are herefore overcome. I has bee foud ha differe coiegraio models based o he r coiegraig vecors have differe sesiiviies o he fauls. herefore, i should be ivesigaed how o uilise a oal of r coiegraio models for process moiorig ad diagose more complex faul scearios ha affec several variables. REFERENCES Apley, D.W. ad S. Jiaju (999. he glr for saisical process corol of auocorrelaed processes. IIE rasacios, 3(, Backshi, B.R ad G. Sephaopoulos (996. Compressio of chemical process daa by fucioal approximaio ad feaure exracio, AIChE Joural, 4(, Berhouex, P.M. ad G.E. Box (996. ime series models for forecasig wasewaer reame pla performace. Waer Research, 30 (8, Box, G.E.P, G.M. Jekis ad G.C. Reisel (994. ime series aalysis forecasig ad corol, 3 rd ed., Preice-Hall, Eglewood Cliffs, NJ. Che, W.W. ad C.M. Hurvich (003. Esimaig fracioal coiegraio i he presece of polyomial reds. Joural of Ecoomerics, 7(, 95-. Dickey, D.A. ad W.A. Fuller (979. Disribuio of he esimaors for auo-regressive ime series wih a ui roo. Joural of he America Saisi cal Associaio, 74, Egle, R.F. ad C.W. Grager (987. Co-iegraio ad error correcio: represeaio, esimaio ad esig. Ecoomerics, 55(, Galido, L.M. (005. Shor- ad log-ru demad for eergy i Mexico: a coiegraio approach. Eergy Policy, 33(9, Hase, P.R. (003. Srucural chages i he coiegraed vecor auoregressive model. Joural of Ecoomerics, 4(, Hassler, U., F. Marmol ad C. Velasco (006. Residual log-periodogram iferece for log-ru relaioships. Joural of Ecoomerics, 30(, Johase, S. (988. Saisical aalysis of coiegraio vecors, Joural of Ecoomic Dyamics ad Corol, (-3, Johase, S. ad K. Juselius (990. Maximum likelihood esimaio ad iferece o coiegraio wih applicaios o he demad for moey. Oxford Bullei of Ecoomics ad Saisics, 5, -6. Johase, S. (995. Likelihood-based iferece i coiegraed vecor auoregressive models. Oxford Uiversiy Press, Oxford. Kaufma, R.K. (004. he mechaism for auoomous eergy efficiecy: A coiegraio aalysis of he US eergy/gdp raio. Eergy Joural, 5(, Ku, W., R.H. Sorer ad Georgakis C. (995. Disurbace deecio ad isolaio by dyamic pricipal compoe aalysis. Chemomerics & Iellige Laboraory Sysems, 30, Pa, Y.Y.; Che, Q. (006; Moiorig ad Faul Diagosis of Dyamic Sysem usig Coiegraio esig Mehod, J of Compuer Measureme ad Corol, V.4 N.3, 8-84 (i Chiese wih Eglish absrac Superville, C.R. ad B.M. Adams (994. A evaluaio of forecas-based qualiy corol schemes. Commuicaios i Saisics: Simulaios ad Compuaios, 3(3, Waso, M.W. (994. Vecor auo-regressios ad coiegraio. I Hadbook of Ecoomerics, Vol. IV, (Egle, R.F. ad D.L. McFadde (Ed., Elsevier, Oxford. Wiksröm, C., C. Albao, L. Eriksso, H. Fridé, E. Johasso, A. Nordahl, S. Räar, M. Sadberg, N. Keaeh-Wold ad S. Wold (998. Mulivariae process ad qualiy moiorig applied o a elecrolysis process, par II mulivariae ime series aalysis of lagged lae variables. Chemomerics & Iellige Laboraory Sysems, 4,