A Combinatorial Approach to PiecewiseLinear Time Series AnalysisMarcelo C. MEDEIROS , Alvaro VEIGA , and Mauricio G. C. RESENDEOver recent years, several nonlinear time series models have been proposed in the lit-erature.One model that has found a large number of successfulapplicationsis the thresholdautoregressivemodel (TAR). The TAR model is a piecewise linear process whose centralidea is to change the parameters of a linear autoregressive model according to the valueof an observable variable, called the threshold variable. If this variable is a lagged valueof the time series, the model is called a self-exciting threshold autoregressive (SETAR)model. In this article, we propose a heuristic to estimate a more general SETAR model,where the thresholds are multivariate.We formulate the task of Ž nding multivariate thresh-olds as a combinatorial optimization problem.We develop an algorithm based on a greedyrandomized adaptive search procedure (GRASP) to solve the problem. GRASP is an itera-tive randomized sampling technique that has been shown to quickly produce good qualitysolutions for a wide variety of optimization problems. The proposedmodel performs wellon both simulated and real data.Key Words: Combinatorial optimization;GRASP; Nonlinear time series analysis; Piece-wise linear models; Search heuristic.
1. INTRODUCTION AND PROBLEM DESCRIPTIONThe most frequently used approaches to time series model building assume that thedata under study are generated from a linearGaussian stochastic process (Box, Jenkins, andReinsel1994).One of the reasons for this popularity is that linearGaussianmodels provideanumberof appealingproperties, such as physical interpretations,frequencydomainanalysis,asymptotic results, statistical inference, and many others that nonlinear models still failto produce consistently. Despite those advantages, it is well known that real-life systemsare usually nonlinear, and certain features, such as limit-cycles, asymmetry, amplitude-dependent frequency responses, jump phenomena, and chaos cannot be correctly captured
Marcelo C. Medeiros is Assistant Professor, Department of Economics, Catholic University of Rio de Janeiro,Rio de Janeiro, RJ, Brazil (E-mail: mcm@econ.puc-rio.br). Alvaro Veiga is Assistant Professor, Department ofElectrical Engineering, Catholic University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil (E-mail: alvf@ele.puc-rio.br). Mauricio G. C. Resende is a Technology Consultant, Information Sciences Research Center, AT&T LabsResearch, Florham Park, NJ 07932 (E-mail: mgcr@research.att.com).c® 2002 American Statistical Association, Institute of Mathematical Statistics,and Interface Foundation of North AmericaJournal of Computational and Graphical Statistics, Volume 11, Number 1, Pages 236–258
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A COMBINATORIAL APPROACH TO PIECEWISE LINEAR TIME SERIES ANALYSIS 237
by linear statistical models.Over recent years, several nonlinear time series models have been proposed both inclassical econometrics (Tong 1990; Granger and Tera¨svirta 1993; van Dijk, Tera¨svirta, andFranses 2000) and in machine learning theory (Zhang, Patuwo, and Hu 1998; Kuan andWhite 1994; Leisch, Trapletti, and Hornik 1999). One model that has found a large numberof successful applications is the threshold autoregressive model (TAR), proposed by Tong(1978) and Tong and Lim (1980). The TAR model is a piecewise linear process whosecentral idea is to change the parameters of a linear autoregressive model according to thevalue of a single observable variable, called the threshold variable. If this variable is alagged value of the time series, the model is called a self-exciting threshold autoregressive(SETAR) model.In this article,we proposea heuristicto estimateSETARmodelswith thresholdsdeŽ nedby more than one variable. This is a generalization of the procedures described by Tongand Lim (1980) and Tsay (1989), where the switchingmechanism is controlled by a singlethreshold variable.Multivariate thresholds are useful in describing complex nonlinear behavior and allowfor different sources of nonlinearity. Several papers concerning multiple threshold vari-ables have appeared in the literature during the past years. However, they assumed thatthe threshold was controlled by known linear combination of individual variables. See, forexample, Tiao and Tsay (1994) where the thresholds are controlled by two lagged valuesof a transformed U.S. GNP series reecting the situation of the economy. In the presentframework, we adopt a less restrictive formulation, assuming that the linear combinationof variables is unknown and is jointly estimated with the other parameters of the model.An alternative approach is the adaptive spline autoregressive (ASTAR) model proposedby Lewis and Stevens (1991), which is based on multivariate adaptive regression splines(MARS) of Friedman (1991).We formulated the task of Ž ndingmultivariate thresholds as a combinatorial optimiza-tion problem. Combinatorial optimization is a Ž eld of applied mathematics that treats aspecial type of mathematical optimization problem where the set of feasible solutions isŽ nite.We developedan algorithmbased on a greedy randomized adaptive search procedure(GRASP), proposed by Feo and Resende (1989) (see also Feo and Resende (1995) andResende (1999)), to solve the problem.The article is organized as follows. Section 2 gives a general description of thresholdmodels.Section 3 presents the proposedprocedure.Section 4 dealswith the speciŽ cationofthemodel. Section 5 briey describes theGRASPmethodologyand presents its applicationto our particular problem. Section 6 presents some numerical examples illustrating theperformance of the proposed model. Section 7 shows an application with a real dataset.Concluding remarks are made in Section 8.
2. THRESHOLD AUTOREGRESSIVEMODELSThe threshold autoregressive (TAR) model was Ž rst proposed by Tong (1978) and wasfurther developed by Tong and Lim (1980) and Tong (1983). The main idea of the TARmodel is to describe a given stochastic process by a piecewise linear autoregressivemodel,