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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  f_t^(hh')=(∑_s▒〖m_s^h m_(s+t)^(h') 〗)/(S_t^(hhii'') ) EQUATION  where f_t^(hh') 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^ 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, |t_i |≤〖0.5 L〗_i EQUATION〗_i  for i=1…3.[ref] The correlation function of all vectors for states h and h' is defined as F_ ^(hh'). There are two types of correlations that are computed Auto-correlation – occurs when h=h' and is represented as F_ ^(hh).   Cross-correlation – occurs when h≠h'.   Both correlations functions are smooth and differentiable; they are readily amenable to interpolation methods to extract correlations of arbitrary vectors t. Auto-correlation functions maintain C_2^ , 2-fold symmetry, about the origin of the statistics, or the 〈0,0,0〉 vector; slightly more than half of the t vectors are unique. The cross-correlation functions F_ ^(hh') and F_ ^(h'h) are respectively anti-symmetric. (IS THIS TRUE FOR COMPLEX BASIS?) The complete set of spatial statistics is defined for all combinations of local state indices of the microstructure function in the following anti-symmetric block matrix