autoregressive moving average (PARMA) processes is a prerequisite to their analysis. Means of ... representation of the lumped vector containing the periodic.
WATER
RESOURCES
RESEARCH,
VOL. 33, NO. 8, PAGES 1929-1934, AUGUST
1997
Periodic stationarity conditions for periodic autoregressive moving average processesas eigenvalue problems Taylan A. Ula Departmentof Statistics,Middle East TechnicalUniversity,Ankara, Turkey
Abdullah
A. Smadi
Departmentof Mathematics,A1 A1-BaytUniversity,A1-Mafraq,Jordan
Abstract. The determinationof periodicstationarityconditionsfor periodic autoregressive movingaverage(PARMA) processes is a prerequisiteto their analysis. Means of obtainingtheseconditionsin analyticallysimpleforms are sought.It is shown that periodicstationarityconditionsfor univariateand multivariatePARMA processes can alwaysbe reducedto eigenvalueproblems,which are computationallyand analytically easierto deal with. Two differentlumpingsof the periodicprocessare consideredalong this line. The first is the commonw-spanlumpingover all w periods.The secondlumping consideredis the p-span lumpingof the p th order periodicautoregressive processoverp periods,whichis basedon a recentlyintroducedlumpingtechnique.It is shownthatp-span lumpingmay yield the periodicstationarityconditionsin an analyticallysimplerform as comparedto w-spanlumpingwhenp < w. Introduction
Modeling of seasonalhydrologicprocesses, mainly thoseof streamflowseries,with periodicautoregressive movingaverage (PARMA) modelshas receivedconsiderableattention.The analysisof PARMA modelsrequiresperiodic(covariance)stationarity, that is, the dependenceof the autocovariancefunction on the period and the lag but not on the absolutetime. Gladysev[1961]gavea formal definitionof periodicstationarity for a generalperiodicprocessand showedthat a necessary and sufficientconditionfor suchstationarityis the stationarity of the so-called"lumped"vectorprocess,which containsthe periodicvariablesfor all w periodsas its elements(also referred to here as w-spanlumping). Tiao and Grupe [1980] showedthat the lumped process correspondingto a univariatePARMA processis a multivariate autoregressive moving average(ARMA) process.They gave formulas relating the parameters and orders of the PARMA processto those of the lumped process.We extend these results to the multivariate
PARMA
case with some no-
tational adjustmentsfor matrix parameters.Periodic stationarity conditionsfor univariateand multivariatePARMA processes canthen be obtainedfrom the stationarityconditionsof multivariateARMA processes, which are readily availablein termsof the rootsof a determinantalequation.I/ecchia[1985] and Ula [1990]employedthis approachto obtainexplicitlythe periodicstationarityconditionsfor someunivariateand multivariatePARMA processes, respectively. Tiao and Grupe [1980] employedwhat we call the forward representationof the lumped vector containingthe periodic variablesfrom the first to the last period in order. We employ the backwardrepresentationof the lumpedvector,whichcontains the periodic variablesin reverse order. This is more compatiblewith backwardrepresentations in time seriesmodCopyright1997 by the American GeophysicalUnion. Paper number 97WR01002. 0043-1397/97/97WR-01002509.00
els, and it analyticallysimplifiesobtainingthe periodicstationarity conditions. Sinceperiodicstationarityconditionsfor a PARMA process involveonly the autoregressive (AR) component,the analysis for periodic stationaritycan be restrictedto periodic autoregressive(PAR) processes for whichthe lumpedprocessis an AR process.Ula [1990]showedthat if the lumpedprocessis a first-orderAR process(AR(1)), then the roots of the determinantalequationleadingto the stationarityconditionsreduce to the eigenvaluesof the AR parameter matrix. Through this result and alsoby makinguse of severaltheoremsabout eigenvalues,he obtainedthe periodicstationarityconditionsfor a multivariatefirst-orderPAR process(PAR(l)) explicitlyin termsof the eigenvalues of the productof periodicAR parameters. Eigenvaluesare computationallyeasier to obtain (by hand or by commonlyavailablecomputerroutines) as compared to the roots of a determinantal equation, and more importantly,they are easier to deal with analyticallybecause there are a large number of theorems concerningthem. It shouldbe mentionedhere that our emphasisin thispaperis on analyticalsimplicityrather than on computationalsimplicity (which,practically,may not mean much becauseof the availabilityof all sortsof efficientcomputerroutines)becausetheoretical treatment of PARMA modelsalmost alwaysrequires an analyticalspecificationof periodicstationarityconditions. The stationarityconditionsfor a multivariate ARMA processcan be reducedto an eigenvalueproblemevenif the AR componentis not of firstorder [Fuller,1976,pp. 50-51; Barone and Roy, 1983;Barone,1987;Smadi, 1994]. This actuallyfollowsfrom the state-space representationof a higher-orderAR model by a first-orderAR model. We will employthis result here for expressing the periodicstationarityconditionsfor any PARMA processin terms of eigenvalues. Bentarziand Hallin [1994] studiedthe invertibilityof periodic movingaverage(PMA) processes and obtainedthe invertibilityconditionsfor a w period and fixed qth-order multivariate PMAw(q) processby consideringa lumping of the
1929
1930
ULA
AND
SMADI:
PERIODIC
STATIONARITY
processover a span of q rather than over a spanof w, which leadsto a PMAs(1) process,s being a functionof w and q. Sinceinvertibilityconditionsin termsof movingaverage(MA) parametersare analogousto stationarityconditionsin termsof
CONDITIONS
plicitform.Theseresultscanbe extended to the multivariate PARMA case with some notational adjustmentsfor matrix parameters.
Sincein time seriesmodelswe i•se a backwardrepresenta-
AR parameters, we adopttheirapproach to thedetermination tion(i.e.,relateXv,• in (1), for example,to Xv,•_• ratherthan of periodicstationarityconditionsfor a fixedp th-order multi- to Xv, •-+i), it is morenaturalto express the lumpedprocess variatePARw(p) processandcallitp-span lumping.Although vector Yv in the backwardform as Bentarzi and Hallin [1994] claimed the superiorityof their approachoverw-spanlumpingin termsof the degreesof the determinantalequationsinvolved,it will be shownhere that Yv = their approachmayyield the periodicstationarityconditionsin an analyticallysimpler form only when p < w.
(3)
X'v 1/
ratherthanin the forwardformYv = ( Xv, r 1, .. ', Xv,w) r r ßIt Lumped ARMA Representation of Multivariate PARMA
Processes
Definition
Consider the following m-dimensional,m = 1, 2, ..., and w-period, w = 2, 3, -.., PARMA processwith periodically varyingAR and MA ordersp • and q•, respectively,with p •, q,0, 1, 2,--., andz = 1,.-., w'
will be shownlater that the backwardform providesanalytical simplicityin obtainingthe periodicstationarityconditions.We therefore adopt this form of Yv in this paper. For anm-variatePARMAw(p•, q•) processthe mw-variate lumpedARMA(p*, q*) processcan be expressedas p*
q*
•0Yv- •'• •kY•,-k= O0a•,--•'• Okav-• k=l
pT
Xv,,- • q>,,,Xv,,-,: ev,,- •'• 0,,,e ....-i, t=l
(1) whereav= (eT..... '",ev, T 1)T, and•0and•aremw
i=1
v=0,
uncorrelated
x mw
AR parametermatricesdefined as _+1, _+2,...
whereXv,, is them-dimensional process vector(say,a streamflowvectorfor m stations) for, say,yearv andperiod are m-dimensional
(4)
k=l
qT
Im {l)0 •
• (•w,1 -- (•w,2
0m
Im
Om
Om
• •) W,W --1 ß
--(•w-l,1
.
o
• (])w-l,w-2
(5)
noise vectors with zero means
andperiodically varyingcovariance matrices •;,; andtb,,i and 0,,i are m x m periodically varyingAR and MA parameter matrices,respectively.The subscriptz - i in Xv,,_ i and ev,._i, denotingthe period,is restrictedto the range1, .-., w; thereforeXv,0 = Xv_•,w and Xv,_• = Xv_l,w_l, for
'l'w-
Om
Itn
'"
(6)
ß
ß
ß
example.To accountfor periodicallyvaryingnonzeroprocess
means•,, Xv,, is to be replacedby Xv,, - •,. We denotethe abovedefinedPARMA processas PARMAw(p,, q,) and as PARMA(p, q) if the ordersare fixed.The PARMA process includesthe periodicautoregressive PARw(p, ) and periodic moving averagePMAw(q,) processesas specialcaseswith q, = 0 and pc = 0, respectively;in which case the correspondingsummationterm in (1) is dropped. It followsfrom Gladysev[1961] that the m-dimensionaland
k= 1,...,p* in whichI m is the m X m identitymatrix, 0mis the m X m null
matrix,and tbi,• - 0m for j > Pi. The mw X mw MA parametermatricesO0 and Oa have the sameform as •0 and
ß a withtbi,•replaced by0•,;,and0go= 0mforj > qi. Using the forward form of Y interchangesthe rows and columnsin
w-periodperiodicprocess{Xv,1, ø'', Xv,w} is periodically (5) and (6); in partitionedform the first row is interchanged (covariance)stationaryif and only if the wm-dimensional with the last row, the secondrow is interchangedwith the T T, is second to the last row, and so on, and then the columns are Xv,w) interchangedin the same way. These forms of •0 and •a TiaoandGrupe [1980]showed thatif theperiodic process is reducefor the univariatecaseto thosegivenby Tiao and Grupe a univariatePARMAw(p,, q,) process,then the lumpedpro- [1980] and Vecchia[1985]. cessis a w-variateARMA(p*, q*) process,where In general,somefinalcolumns of the matrix•p. maybe zero,andthissimplifiesobtainingthe periodicstationarityconp* = max[(p•- z)/w] + 1 ditions, as to be shown later. It can be shown that the number (2) of zerofinalcolumns of •p., in partitioned form,is q* = max[(q,- z)/w] + 1 n = (wp* - 1) + min (•-PO (7)
lumpedprocess{Yv}, where Yv = ( X T v,•, '", (covarihnce)stationary.
in which Ix] is the integral part of x, i.e., the greatestinteger
lessthan or equalto x. Equation(2) is applicablealsofor the Using the forward form of Yv interchanges,these final zero m-variatePARMAw(p,, q,) procegsfor whichthe lumped column•with nonzerocolumns,andbecauseof that,obtaining processis an mw-variateARMA(p*, q*) process.Tiao and the periodicstationarityconditionsbecomesanalyticallymore Grupe[1980]alsogaveformulasrelatingthe parametersof the difficult. univariatePARMAw(p,, q,) processto thoseof the lumped Asanexample, consider them-variate PARMA2(3 , 1;1,1) process.Vecchia[1985] expressedtheseresultsin a more ex- process:
ULA AND
SMADI:
PERIODIC
STATIONARITY
(8a)
Imw
Omw
'''
Omw
Omw
Omw
Imw
'''
Omw
Omw
Omw
Omw
ß. .
Imw
Omw
are all 1, the sta- q• in (17) by premultiplyingand postmultiplying the latter by tionarityconditioncanstillbe reducedto an eigenvalueprob-
lemthrough thefollowing result [FUller, 1976, pp.50-51;Barone and Roy, 1983; Barone, 1987], which follows from the state-space representation of an AR model:the rootsof (12) are all < 1 in absolutevalueif andonlyif the eigenvalues of the m wp * x m wp * matrix
0m Im
Im
0m 0m
0m 0m 0m
C--,0m 0m0mIm 0m 0m Im
0m
(18)
1932
ULA AND SMADI: PERIODIC
STATIONARITY
CONDITIONS
whichis a symmetric orthogonal matrix,that is, C = Cr,
....
CrC = I, andthereforeC - C-1. The eigenvalues of a matrix do not changeby premultiplyingand postmultiplying it by a matrix and its inverse[Graybill,1969, p. 43]. Therefore the matricesin (14) and (17) havethe sameeigenvalues. The invertibilityof a PARMAw(p,, q,) processcan be
Om
Im
--•Tp-l,1 ....
•Tp,p-1 •Tp-l,p-2
Om
Om
defined as the invertibility of the correspondinglumped AT,1 ARMA(p*, q*) process[Tiao and Grupe,1980;Ula, 1993; (• Tp, p BentarziandHallin, 1994].The invertibilityconditionsfor the lumpedARMA(p*, q*) processin (4) canbe obtainedfrom (• Tp-l,p-1 = the stationarityconditions, basedon equations(12) and (13), by replacing•I,•,with O•, andp * with q*. Denoting•I, in (13) as O under thesesubstitutions, for the PARMA2(3, 1; 1, 1)
Om
Om
Om
'''
Om
(• Tp-l,p
Om
'''
Om
• ,
process,we have
(22)
2'101'1 0m) 01,10m
Lumped PAR(l) Representation PAR
Let sp (the productof s andp) be the leastcommonmultiple and w. It then followsthat
(19) ofp
It then follows from our previous discussionsthat the PARMA2(3, 1; 1, 1) processis invertibleif and only if the eigenvalues of 02,101.1 are all mp, equalityholdingonly forp - w, 2w, 3w, .... Therefore the dimensionof • is lessthan or equal to that of •I,. The numberof zerofinalcolumnsin •I, is givenby (7), which for the fixed-ordercasebecomesn = wp* - p. Followingthe discussion after (16), the eigenvaluesof •I, can be obtained
(XTTp, XTTp_i, ''' , XTTp_(p_l)) r andear= (eTTp, eTTp--1, ''' , from an rn(wp* - n) x m(wp* - n) lower dimensional errp-(p-1))r,T = 0, +_1,+2,"', andthemp x mp matrix, which for the fixed-ordercaseis an mp x mp matrix. matrices
Therefore, althoughthe dimensionmwp* x mwp* of •I, is
ULA
AND
SMADI:
PERIODIC
STATIONARITY
greater than or equal to the dimensionmp x mp of •, its eigenvaluescan be obtained from an mp x mp matrix that
CONDITIONS
1933
theycorrespondto the sameperiods,namely,periods1, 2, and 1 in order.
is of the same order as •. However, it should be noted that tp
The w-spanlumpingof the m-variatePAR2(3) processleads to a 2m-variateAR(2) process.The 4m x 4m matrix tp for 1, ..., p*, whereas• is constructedfrom mp x mp matrices this processcan be obtainedfrom (13) as 1Ak,1,wherek = 1, -.- , s. In thissense,the matrix• can A;,o •2,2 -+-•2,1•1,1 •2,3 q- •2,1•1,2 •2,1•1,30m be constructedmore easilyfor p < w. On the other hand, it shouldbe noted that • involvesan aggregationof matrices, whereas• involvesa productof matrices.This may introduce additional analyticaldifficultiesin the construction•. Also, 0m Im 0m 0m for the variable-orderprocessthe complexityof • is increased artificiallyto accommodatefor the maximumorder involved.It the nonzero eigenvaluesof which can be obtained from the may thereforebe concludedthat for w < p the w-spanlump- 3m x 3m submatrixoccupyingthe upper left-hand corner. ing is analyticallymore suitable for obtaining the periodic The periodicstationarityconditionis that the eigenvaluesof stationarityconditions,whereasfor p < w thep-span lumping thissubmatrixbe all < 1 in absolutevalue.It maybe notedthat may prove to be more suitable,dependingon the conditions. • is 3m x 3m, whereas tp is 4m x 4m. However, the As an example,considerthe m-variate PAR2(3) process nonzero eigenvaluesof tp can be obtainedfrom a 3m x 3m
is constructed frommw x mw matricestP•-ltPk,wherek --
(•1,1 Im
t[}=
Xv,•= q>•,•Xv-•,2 + q>•,2X•-•,•+ q>•,3X•-2,2 + e•,•
(26a)
Xv,2-- (•2,1Xv,1 q- (•2,2Xv-1,2 q- (D2,3Xv-l,1 + e•,2
(26b)
(•1,2 •}1,3 0m (32) 0m 0m 0m
matrix that is of the same order as •.
Furthermore, tp is
constructed easilywithtwo2m x 2m submatrices tp•-•tp•and (D•-i(D 2whereas • isconstructed bythemultiplication of four 3m x 3m matrices.Sincew = 2 < p = 3 here, tp is easier
For thep-span lumping(23) of this process,we have
to construct.
X3r
Zr= X3r-12
(27)
X3T-
e3T 1
•r= e3r-• / e3T-2] Im
At,0=
AT,1=
(28)
-- 4)3T, 1 -- 4)3T,2
0m
Im
0m
0m
The matrix tp for the PARMA2(3, 1; 1, 1) process,whichis alsoapplicablefor the PAR2(3, 1) process,was givenin (14). The matrix • for the PAR2(3, 1) processcannotbe obtained directlysincethe order of the processis not fixed.It may only be obtainedfrom the matrix• (equation(31)) of the PAR2(3) process,wherep = 3 = max (p• = 3, P2 = 1) by setting (•)2,2= (•)2,3= 0. For further comparison,considerthe m-variate PAR2(1) processfor whichp = 1 < w = 2 and the PAR2(2) process for whichp = w = 2. For the PAR2(1) process,
-- 4)3r-l,1
(29)
1'1 0m) tp= ((•2'1(• q>•,• 0m
Im
4)3T, 3 0m0m 4)3T-1,24)3T-1,30m
(30)
(33)
and • = ([}2,1([}1,1, the periodicstationarityconditionbeing that the eigenvalues of q>2,•q>•,• all be
