Seasonal Space‐Time ARIMA Modeling

Abstract

Parameter p is the autoregressive order of the model and X is a vector with components A, specifylng the spatial order of the kth autoregressive term.Likewise q is the moving average order of the model, and m is the vector of moving average spatial orders.Parameter d specifies the number of differences needed to induce stationarity in the original series.The space-time system of interest often exhibits a recurrent seasonal pattern across time of known period S. 'Although the general STARIMA model (1) may be adequate to accommodate some forms of seasonal cyclic behavior, there are alternative model forms that can express seasonal relationships in a parsimonious form.This paper extends previous work done by Pfeifer and Deutsch (198Ob) with respect to the STARIMA model family to introduce the general class of multiplicative seasonal STARIMA models as an appropriate model family for use with time histories of spatially located data that exhibit seasonal relationships.Section 2 presents and describes the multiplicative seasonal STARIMA class, after which section 3 describes the three iterative stages of the procedure for building these models with special emphasis on the considerations necessary with seasonal models.The paper concludes with section 4, an application of the seasonal STARIMA model to daily discharge data measured simultaneously at different locations in the Mississippi River.The resulting model is then applied to provide forecasts of the future occurrences of this space-time process. MULTIPLICATIVE SEASONAL STARIMA MODEL CLASSSuppose that ztr t = 1,2, . . .T, is a seasonal time series of monthly observations with S = 12.If the observation Z, is made in the month of January, we might expect that the relationship between this observation and the observation during January of all previous years could be modeled with the STARIMA model form.Specifically, this model is written as where vf = ( I -B y Furthermore, if this same relationship holds for every month of the year, (2) would be an appropriate representation of the seasonal relationship for all 2,.Now since it is also possible for the a, in (2) to exhibit space-time relationships within each 15384632, 1981, 2,

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