NCJ Number
60691
Date Published
1979
Length
17 pages
Annotation
THE CONDITIONS UNDER WHICH THE SPACE-TIME AUTOREGRESSIVE MOVING AVERAGE (STARMA) MODEL FAMILY PROCESS ARE STATIONARY AND STOCHASTIC ARE INVESTIGATED.
Abstract
THE STARMA MODEL FAMILY HAS PROVEN USEFUL IN MODELING TIME HISTORIES OF SPATIALLY LOCATED DATA. STARMA USES THE SPATIAL AUTOCORRELATION USUALLY EXHIBITED BY SPACE-TIME SERIES TO PROVIDE AN EFFICIENT REPRESENTATION OF THE SYSTEM. THE BASIC MECHANISM FOR THIS REPRESENTATION IS A HIERARCHICAL SPATIAL ORDERING OF THE NEIGHBORS OF EACH SITE AND A SEQUENCE OF WEIGHTING MATRICES. TWO SPECIAL SUBCLASSES OF THE STARMA MODEL ARE CONSIDERED: (1) THE STAR MODEL, USED WHEN ONLY AUTOREGRESSIVE TERMS ARE INVOLVED; AND (2) STMA MODELS, USED WHEN NO AUTOREGRESSIVE TERMS ARE INVOLVED. IN MODEL BUILDING, THE SELECTION OF A UNIQUE REPRESENTATION OF A GIVEN SPACE-TIME SYSTEM REQUIRES RESTRICTION OF THE PARAMETER SPACE TO BE EQUIVALENT TO THE STATIONARY DOMAIN. THE IMMOBILITY OF THE GENERAL MULTIVARIATE ARMA MODEL, OF WHICH THE STARMA IS A SPECIAL CASE, IS DEFINED, AND THAT OF THE STAR MODEL IS EXAMINED. STATIONARITY CONDITIONS OF THE FOUR PARAMETERS OF THE STAR MODEL ARE DEVELOPED FOR THE CASE OF A GENERAL SCALED MATRIX. WHEREAS ALL MOVING AVERAGE MODELS ARE STATIONARY (A FUNCTION ONLY OF THE AUTOREGRESSIVE TERMS), ONE USUALLY REQUIRES THAT THE MOVING AVERAGE TERMS BE SUCH THAT THE MODEL IS INVERTIBLE. THE INVERTIBILITY OF THE STMA MODEL IS DISCUSSED. CONSTRAINTS ON THE PARAMETER SPACE THAT ENSURE STATIONARITY ARE DEVELOPED FOR ALL STARMA MODELS OF AUTOREGRESSIVE TEMPORAL ORDER LESS THAN OR EQUAL TO TWO AND SPATIAL ORDER LESS THAN OR EQUAL TO ONE, WHEN THE MODEL FORM USES SCALED WEIGHTS. INVERTIBILITY FOR THESE SAME MODELS ARE ALSO GIVEN. MATHEMATICAL EQUATIONS FOR THE MODELS EXAMINED ARE PROVIDED, ALONG WITH GRAPHIC DATA AND REFERENCES. (RCB)