deletions | additions       diff --git a/Spatial Statistics.tex b/Spatial Statistics.tex  index b8e99a1..6a91776 100644  --- a/Spatial Statistics.tex  +++ b/Spatial Statistics.tex 
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\end{equation}  where $f^{hh'}_{t}$ 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 $S_{ii'}^{t}$ 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, $|t_i| \leq 0.5 L_{i}$ for $i = 1, .. 3$.[ref] The correlation function of all vectors for states $h$ and $h'$ is defined as EQUATION. There are two types of correlations that are computed computed:  \begin{enumerate}  \item  Auto-correlation – occurs when EQUATION $h = h'$  and is represented as EQUATION. F^{hh}.   \item  Cross-correlation – occurs when EQUATION. $h \neq h'$.   \end{enumerate}  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 EQUATION , 2-fold symmetry, about the origin of the statistics, or the EQUATION vector; slightly more than half of the $t$ vectors are unique. The cross-correlation functions EQUATION and EQUATION 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 EQUATION  with the auto-correlation functions on the diagonal and the cross-correlation functions on the upper and lower triangle. Previous literature reports the independent and dependent features of the complete statistics for an isolated, but common set of raw material information.[STEVEref]