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Spatial statistics employ the microstructure function to rapidly compute an objective description of the material information provided by model(s) within a similar sample volume. The spatial statistics are computed by the following relationship  EQUATION  where EQUATION is the probability of finding local states h and h' separated by a vector t; h is a local state derived from signal i at the tail of t and h' is local state derived from signal i' is at the head . The complete set of statistics includes all the discrete set of statistics for all possible vectors within for the sampled region sampling pattern of the model. To better understand the definition above, it is useful to consider the numerator and denominator individually. The numerator is a cumulative sum of the positive outcomes where h and h' were observed to be separated by t. The denominator S_t^ii' S_t^ EQUATION  provides the total number of trials conducted with a vector t from the signal sources i corresponding to the local state indices h and h'. (Figure to illustrate statistics) The spatial statistics are computed for all vectors in the sample volume, L, that satisfy the Nyquist criteria, EQUATION for EQUATION.[ref] The correlation function of all vectors for states h and h' is defined asEQUATION. There are two types of correlations that are computed  Auto-correlation – occurs when h=h' and is represented as EQUATION.   Cross-correlation – occurs when h≠h'.