Orthogonal decompositions of 2-D nonhomogeneous discrete random fields

Abstract

Imposing atotal-order on a two-dimensional (2-D) discrete random field induces an orthogonal decomposition of the random field into two components: Apurely-indeterministic field and adeterministic one. The purely-indeterministic component is shown to have a 2-D white-innovations driven moving-average representation. The 2-D deterministic random field can be perfectly predicted from the field's “past” with respect to the imposed total-order definition. The deterministic field is further orthogonally decomposed into anevanescent field, and aremote past field. The evanescent field is generated by the columnto-column innovations of the deterministic field with respect to the imposed nonsymmetrical-half-plane total-ordering definition. The presented decomposition can be obtained with respect to any nonsymmetrical-half-plane total-ordering definition, for which the nonsymmetrical-half-plane boundary line has rational slope.