Lagodimos, AG

Verrijdt, JHCM Jos

Verrijdt, J.H.C.M.

Lagodimos, A.G.

NARCIS 

Moments of consecutive sums of ARMA(1,1) processes

Repository TU/e

Pure OAI Repository

 Moments of consecutive sums of ARMA(1,1) processesCitation for published version (APA):Verrijdt, J. H. C. M., & Lagodimos, A. G. (1993). Moments of consecutive sums of ARMA(1,1) processes. (TUEindhoven. Fac. TBDK, Vakgroep LBS : working paper series; Vol. 9306). Technische Universiteit Eindhoven.Document status and date:Published: 01/01/1993Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Department of Operations Planning and Control -- Working Paper Series Moments of Consecutive Sums of ARMA(l,l) Processes J.H.C.M. Verrijdt' and AG. Lagodimos" Research Report TUE/BDK/LBS/93-06 May, 1993 " Graduate School of Industrial Engineering and Management Science Eindhoven University of Technology P.O. Box 513, Paviljoen F16 NL-5600 MB Eindhoven The Netherlands Phone: +31.40.472681 Fax: +31.40.464596 KANTOR Management Consultants 4 Vasilissis Sophias Avenue 106 71 P.thens Greece This paper should not be quoted or refen-ed to without the prior written permission of the author Moments of Consecutive Sums of ARMA(l,l) Processes Abstract In this paper we develop expressions for the evaluation of the two central moments of sums of consecutive terms of ARMA(1,1) stochastic processes. We also present an expression for the evaluation of the correlation parameters between two consecutive sums of consecutive terms of such processes. These expressions subsequently form the basis for evaluating the corresponding parameters of AR(1) and MA( 1) processes, which can be viewed as special cases of the ARMA( 1, 1) process. J.H.C.M. Verrijdt Eindhoven University of Technology Graduate School of Industrial Engineering and Management Science Department of Operations Planning and Control P.O. Box 513 NL 5600 MB Eindhoven The Netherlands A.G. Lagodimos KANTOR Management Consultants 4 Vasilissis Sophias Avenue 106 71 Athens Greece Research Report: TUE/BDK/LBS/93-06 This working paper should not be quoted or referred to without written consent of the author(s). Comments are solicited and should be addressed to the author(s). 1. Introduction Problems in several areas of Management Science (queuing theory, inventory control, quality control and forecasting) necessitate the consideration of stochastic processes. As a result, models often involve sums of random variables (representing successive observations of a stochastic process) whose parameters need to be evaluated or estimated. Good examples are the models developed for the evaluation of the cumulative lead time demand parameters in inventory systems (see Eppen and Martin 1988, An et at. 1989 and Lagodimos et at. 1993) and for the determination of the action limits of quality control charts (see Wardell et al. 1992). One simplifying assumption usually underlying the analysis of such models is that successive terms of the stochastic processes involved are independent. This assumption, however, is not always realistic (see Wagner 1980), and so processes with serially correlated terms need to be considered. A general class of such processes is the Autoregressive Moving Average (ARMA) process, popularised by Box and Jenkins (1976). It is with the determination of the statistical parameters of consecutive sums of ARMA(l,l) processes that we are concerned here. The remainder of this paper is organised as follows. In section 2 we introduce the ARMA(l,l) model and deal with the problem of determining the model coefficients so as to obtain some desired mean, variance and autocorrelation characteristics. In section 3 we derive expressions for the first central moments (mean and variance) of consecutive terms of ARMA(l,l) processes. In section 4 we derive an expression for the autocorrelation coefficient between consecutive sums of consecutive terms of such processes. Finally, in section 5 we present some concluding remarks. 2. The ARMA(1.1) process Let {d t } be a discrete-time stationary stochastic process. In the following we use the notation jJ., dl and Ok to represent the (time invariant) mean, variance and autocorrelation coefficient between any two terms d t and dt+k of the process respectively. The ARMA(l,l) process is a particular member of the ARMA class of stochastic processes consisting of one autoregressive and one moving average term (see Box and Jenkins 1976). If {d t } is an ARMA(l,l) process, then: (1) where a, p and I' are real coefficients and {EI } is a stochastic error process representing a stream of independent identically distributed random variables having zero mean and constant variance V. In general, we need not make any assumption concerning the distribution of {EI }. However, when {EI} are normally distributed each term of the ARMA(l,l) process will be normally distributed as well. In order for {dl } to be stationary we need that I a I < 1 and I I' I < 1. There are also two particular cases: for a=O, {dl } becomes a Moving Average MA(1) process, while for 1'=0 it becomes an Autoregressive AR(l) process. It is useful to have an expression for representing d l +k as a function of d t• Using (1) we have: dt+k={3+adt+k_l +YE t+k-l +E I+k ={3+c'+k +yct+k-l +a{{3+ad,+k_2+Yc t+k-2+c,+k-l} ={3 (1 +a)+(c t+k +aE t+k-l)+Y(£ I+k-l +at t+k-2)+c?dt+k-2 1 After a number of iterative steps we obtain the following expression for dt+k: d,+k =Al +A2+A3+d'd/ k-l with Al :=PE d ;=0 k A2 :=E c/<-; et+; ;=1 k-l A ~ k-I-; 3 :=1' L..J a- e /+; ;=0 (2) We can now evaluate the statistical parameters of dt in terms of a, P and 1'_ First consider the mean,u_ Taking expectations of both terms of (1) we obtain: E[d,-ad,_d = E[p+ye/_l+et ] ,u-a,u = P ,u=p/(l-a) We can also evaluate the covariance of d t and dt+k- By definition: COV(dt,dt+k)=E[(dt -,u)(dt+k -,u)] =E[(dt -,u)(d'dt+A1 +A2+A3-,u)] =E[d'd~+d,AI +d,A2+d,A3-.ud,-,uddt -.uA1-.uA2-,uA3+,u2] =d'(E[d~-,u2)+E[d,A3] (3) =d'c?+yd'-IE[d,c/] (4) =d'c?+yd'-Iv where Using the relation 0k=Cov(dt,dt+k)/c?' the correlation coefficient Ok can be obtained directly_ We finally evaluate Var(dt). Using (1) we have: Var(dt -ad/_I) = Var(p +ye t-I +e ,) Var(d,) + Var(adt_I)-2Cov(d"adt_I)= Var(ye/_ 1)+ Var(e /) c?( 1 +a2) -2aCov(dt,d,_I)=(y2+ 1) V Finally, replacing Cov(dt,dt_I) above by expression (4) with k=l, we obtain: 2 if = (y2 + 2ay + 1 ) v 1-a2 which is a function of a and y and which can also be found in Box and Jenkins (1976). (5) One problem often encountered in practice is the determination of the process coefficients a, p and y and of the variance V of the error process {Et } in order to obtain some desired /J., cr and Ok. To deal with this problem, we simply need to solve the following algebraic system of equations: p = ,u(1-a) U2 = (y2+2ay+1)V 1-a2 Ok = ~+yd-l V if Since we are interested in stationary ARMA(1,1) processes we need only to consider values of a and y such that I a I < 1 and I y I < 1. In order to solve this system for a, p, " and V, we need to know the values of the statistical parameters Jl, if, °1 and °2 of {d t }. Notice that for the two special cases (the AR(1) and MA(1) process) we only need to know the values of/J., cr and °1, since we have only three unknown parameters. That is: a, p and V for the AR(1) process andP, " and V for the MA(1) process respectively. 3. Sums of consecutive terms In this section we develop expressions for the mean and variance of sums consisting of consecutive ARMA(1,1) terms of the form: L X:=Ed,+k k=l We start with E[X]. Since E[dt ] =Jl, it is straightforward that: (6) (7) The derivation of Var(dt) is more involving. It is convenient for this purpose to rewrite X in a more concise form. Introducing (2) in expression (6) we have: L X = Edl+k k=l L k-l L k L k-l L = EPEa;+E Ed-; El+;+yE E~-l-i Et+;+Edd, k=l ;=0 k=1 ;=1 k=1 ;=0 k=1 (8) 3 We can evaluate the sums in Xl' XZ• X3 and X4 as follows: = = = = = = = = = = = = = = = = = = L k-l EfiEd k=l i=O ~ l-ak fi L.J-k=l I-a L L-Ed] I-a k=l L L_a 1- aL ] I-a I-a L k E Ed- i €t+i k=li=1 L E (d- l€t+l +d-zE t+2+···+ a €t+k_l +€t+k) k=1 (€t+l)+(a€ t+l +Et+2)+···+(aL-1€ t+l +aL-2€t+z+···+€ t+L) L [L-k] ~ €t+k ~ ai L l_aL-k+1 ]; €t+k I-a 1 L . - E (1-a')et+L+l_j I-a j=l L k-l E E d- 1- ie t+i k=1 i=O L E (d-1e, +d-ze t+l + ... +a€t+k-2+€t+k-l) k=l (e,)+(ae ,+e '+l)+ ... +(~-le,+~-zet+l + ... +ae t+L-2+€,+L-l) e,&' d+ ~ e'+kr~~r d] l-aL 1 ~ I.-k --e,+ __ L.J (I-a- )et+k I-a I-a k=l _1_ [( l-aL) e, + Li (l-d)e t+L-j] I-a j=l L Eak k=l l-aL a __ I-a 4 Introducing these expressions in (8) we obtain: X = Xl +XZ +yX3 +X4dt {3 [ I-d-] I L , = - L-a -- + - E(I-d)Et+L+l_j I-a I-a I-a j=l I [ L-I,] I-d-+ y - (I-d-)E t + E (l-d)Et+L_j + a -- dt I-a j=l I-a Taking the variance of the above we have: Var(X) = Var(X2)+yZVar(X3 ) + Var(X4dt ) +2y Cov(X2,X3 )+2y COy (X3 ,X4dt ) (9) which is obtained by noticing that the random variables X2 and d, are independent. Each of the terms in the above expression can be evaluated as follows: = = [ I-~ 1-~] COY _-Et,a_-dt I-a 1-a 2 I-~ a -- Cov(Et,dt) I-a 2 I-~ = a V I-a [ 1 L, I-~ I L-I , ] = COy - E(I-d)Et+L+l_ ,,--Et+- E (l-d)Et+L_j I-a j=l J I-a I-a j=l = I {COV((1-~)Et+1'(1-~-1)Et+l)+COV((1_~-1)Et+2,(I-~-2)Et+J+ ... (1-a)2 +Cov( (l-a2)Et+L_1'(I-a)Et+L_l)} L-l = V E (l-d+1)(I-d) (l-a)2 j=l L-I = V E (l-d-d+1+a2i+I ) (l-a)2 j=l L-2 L-2 L-2 = V (L-I+aE d-a2 E d+a3 E (a2y (1-a)z j=O j=o j=o = V [L-1+ a(1-~-l) _ a2(1-~-1) + a 3(1_a2(L-l)] (1-a)2 I-a I-a 1-a2 5 Var(X:J = = = = = = = = = = = Introducing the above in (9) we finally obtain: Var(X) = V {L- 2a 1-<1- +az1-a2L +yZ [(I-d-/+L-I-2a 1-<1--1 +aZ1 -a2(L2-1)] (l-a)2 I-a l-a2 I-a I-a + (y2+ l)a2(1-<1-)z +2ay(1-<1-)z 1 +a2-2aOl +2y [L-l +a 1-<1--1 _a21 -c/--l +a3I -a2(L-l)] I (10) I-a I-a l-az 6 For the two special cases of the ARMA(I,l) process we have the following expressions: MA(I) process (a=O): Var(X) = V {L+y2(1+L-l)+2y(L-l)} = V {L+Ly2+ZyL-Zy} AR(I) process (y=O. 81 =a): Var(X) = = 4. Consecutive sums of consecutive terms In this section we evaluate the correlation between consecutive sums of consecutive ARMA(l,l) terms. Let X and Z be random variables defined as follows: L X:=E dl+k k=1 and L+l Z:= E dl+k k=L+l where L and I are integers representing the number of ARMA(l,l) terms included in X and Z. We are interested in evaluating Cov(X,Z) and the resulting correlation coefficient p. Define a new variable Y=X+Z. We can generally represent Cov(X,Z) as: Cov(X,Z)=Cov(X, Y-X)=Cov(X, Y)- Var(X) (11) Since Var(X) has already been evaluated in section 3, we only need Cov(X,Y). By definition we can write: Cov(X, Y)=E[XY]-E[X]E[Y] (12) and so we need to evaluate E[XY] and E[X]E[Y]. Using the results of section 3, it is convenient to rewrite X and Y as: 7 Hence: E[XY] = E[X]E[Y] = X:= _1_ [ Bl(L)+B2(L)+BiL)+BiL)+Bs(L)dt ] I-a Y:= _1_ [ Bl(l)+B2(l)+Bil )+B4(l)+Bs(l)dt ] I-a where A.:= L+l [ I-aP 1 B1(P) := f3 p-a I-a p . B2(P) := E (l-d)et+p+l_j j=l B3(P) := y(l-aP)et p-l B4(P) := y E (l-d)e,+p_j j=} Bs(P) := a(l-aP) 1 { B} (L )B} (A. )+B1 (L )Bs(l ),u+E[ B2(L )Bil)] +E[ B2(L)B il )]+E[ B3(L) Bil )] (l-a)2 +E[ BiL )Bs(l )dt] +E[ B iL )B2(l)] +E[ B iL)B iA.)] +Bs(L )Bl(l),u +E[Bs(L)B3(l)d,]+E[Bs(L)Bs(A.)d~ } 1 { B}(L)Bl(A.)+Bl(L)Bs(l),u+Bs(L)Bl(l),u+Bs(L)B5(l),u2 } (1-a)2 Introducing the above in (12) we have: Cov(X,y) = E[XY]-E[X]E(Y] = 1 { E[BiL)Bil)] +E[B2(L)B4(l)] +E[B3(L)Bil)] (1-a)2 (13) +E[ BiL )Bs(l )d,] +E[ BiL )B2(l)] +E[ B 4(L)B 4(l)] +E[ B5(L )B3(l )d,] +Bs(L )Bs(l){ E[ d;] _,u2} } We need to evaluate each of the terms in the above expression. We have: 8 L L+l ] E[BzCL)B2(l)] = E j~ (I-d) E t+L+1-j *j~ (I-d) E I+L+I+ 1-j = v t (1-d)(1-d+1) 1 = v it (1-d+1-d+a1i+1) 1 L-l L-1 L-1 = V L-d+1E d-aE d+d+2 E (a2y j=o j=o j=o = V L -d+ 1 1-£1- -a 1-~ +d+2 1-a2L 1-a 1-a 1-a2 E[B2(L)B4(l)] = E [t (1-d)EHL+1_j * /£1 (1-d)EHL+I_j 1 J=1 J=1 L = Y V E (1-d)(I-d+I- 1) j=1 L = Y V E (1-d+I- 1-d+a]J+I-l) j=1 = Y V [L-aI1 -£I- -a 1-~ +d+1I - a2L 1 I-a 1-a I-a2 E[B3(L)Bs(l)dll = E [Y (1-£l-)EI a (1-£l-+/)dl ] = y a (l-~)(l-£l-+I)E[ Eld,] = Y a V (1-aL)(1-aL+1) 9 = = = = = = = = = = = [ L-l L+I 1 EyE (l-d)EHL _j • E (l-d)Et+L+I+l_j j=1 j=1 L-1 I' V E (1-d)(1-d+1+l) j=1 I' V L-1- E d- E d+I+1+ E a'lJ+1+1 L-1 L-1 L-l 1 j=l j=l j=l I' V L-1-a - + 1-aL-1 d+Z 1-aL-1 d+3 1_a2(L-l) 1 1-a 1-a 1-aZ E [r ~ (l-d)eHL _i * r L~1 (l-d)eHL+I_i 1 L-l 1'2 V E (1-d)(1-d+/) j=1 L-l L-l L-1 1 y2V L-1-~d-~d+f+j~a'lJ+1 2 1-aL-1 1+1 1-aL-1 1+2 1_a2(L-1) 1 I' V L-1-a -a +a l-a 1-a 1-a2 E [ a(1-aL)dt Y(1-aL+f)Et ] I' a (1-aL)(1-aL+/)E[dtEtl I' a V (1-aL)(1-aL+f) Bs(L)Bs(A) Q2 a Z (1-aL)(1-aL+/) ~ l-a2 Replacing the above in expression (13) for Cov(X,Y) and using expressions (10) and (11), after some algebra, we finally obtain: 10 Cov(X,Z) Since = Cov(X, Y) - Var(X) = V {L_d+11-d- -a I-d- +al+21-a2L ] (l-a)2 I-a I-a l-a2 11-d- I-d- 1+1 l-a2L ] + I' L-a ---a--+a I-a I-a l-a2 + 1'2 (1-d-)(I-d-+/) + I' a (1-d-)(I-d-+ /) I-d--1 d+21-d--1 d+31 - a2(L-1) + I' L-1-a - + 1-a 1-a 1-a2 2 1-d--1 d+ 1 1-d--1 1+2 1- a2(L-1) + I' L-l-a +a I-a I-a 1-a2 + I' a (1-d-)(1-d-+ /) 2 + _a_ (l-d-)(l-<l-+/) 1-a2 - [L-2a 1-<1- +a21 -a2L ] I-a l-a2 2 [ 1. 2 1-<1--1 21- a2(L-1)] - I' (1-a-) +L-I-2a +a ---;:--1-a 1-a2 [ (1'2+ 1)a2(1-d-)2 +2ay (1-<1-)2] l+a2-2a(J1 [ 1-<1--1 21-<1-- 1 31- a2(L-1) 1 } - 21' L -1 +a -a +a ---;;--1-a 1-a 1-a2 Cov(X,Z) p = ---r.=;=.:::::;::;:;~:::::;::;;:::;::. JVar(X) Var(Z) the above can be used directly to evaluate p. It is interesting to consider the two special cases where the ARMA(1,1) process reduces to an AR( 1) or an MA( 1) process. 11 MA(1)-process (a=O): Hence: AR(1)-process (y=O): Cov(X,y) = Var(X) = Var(Z) = Cov(X,Y)-Var(X) = = Hence: Cov(X,y) = Var(X) = Var(Z) = Cov(X,Y)- Var(X) = v [ L+Lyz+2yL-y ] V [ L+Lyz+2yL-2y ] V [ 1+lyz+2yl-2y ] I' V p = = Cov(X,Y)- Var(X) JVar(X) * JVar(Z) y V [L_a1+11-d- -a I-d- +a1+zl -a2L +~(1-d-)(I-d-+/) ] (I-a)z I-a I-a l-a2 l-a2 V [ 2d-+1-Laz-2a+L ] (l-a)2(I-az) V [ 2a1+1-laz-2a+l ] (l-a)z(1-a2) V [ L(I-a2)-d+1(1-cf-)(1 +a)-a(I-d-)(1 +a)+d+2(1-a2L ) (l-a)2(I-a2) +a2(1-d-)(I-d-+1) - 2if-+ 1 + LaZ+ 2a-L) ] a V [(d--I)(d-I) ] (1-a)Z(1-a2) p = Cov(X,z)- Var(X) JVar(X) * JVar(Z) a(d--I)(d-I) = 5. Concluding remarks In this paper we derived expressions for the central moments (mean and variance) of sums of consecutive terms of ARMA(l,l) stochastic processes. We also derived an expression for the autocorrelation coefficient between consecutive sums of consecutive terms of ARMA(I,l) processes. The derivation of these expressions involves some lengthy and cumbersome algebra. Using a similar methodology, extension of these expressions to more general ARMA(p,q) processes (with p autoregressive and q moving average terms) is straightforward, although the algebra involved is much more involving. 12 References An, B.-G., Fotopoulos S.B., and Wang M.-C. (1989), "Estimating the lead-time demand distribution for an autocorrelated demand by the Pearson system and a normal approximation", Naval Research Logistics 36, 463-477. Box, G.E.P., and Jenkins, G.M. (1976), Time Series Analysis, Forecasting and Control, 2nd Edition, Holden-Day, San Francisco. Eppen, G.D., and Martin R.K. (1988), "Determining safety stock in the presence of stochastic lead time and demand", Management Science 34, 1380-1390. Lagodimos, AG., Kok AG. de, and Verrijdt J.H.C.M. (1993), "The robustness of multi-echelon service models under autocorrelated demands", Research Paper TUE/BDK/LBS/93-14, Department of Industrial Engineering and Management Science, Eindhoven University of Technology, The Netherlands. Wagner, H.M. (1980), "Research portfolio for inventory management and production planning systems", Operations Research 28, 445-475. Wardell, D.G., Moskowitz, H., and Plante, R.D. (1992), "Control charts in the presence of data correlation", Management Science 38, 1084-1105. 13 

Moments of consecutive sums of ARMA(1,1) processes

Abstract

Similar works

Full text

Available Versions

NARCIS

Repository TU/e

Pure OAI Repository