this is for holding javascript data
tonyfast deleted file Spatial-Statistics.tex
almost 10 years ago
Commit id: 37d2b2e29b55f975d473fda792962535203f6331
deletions | additions
diff --git a/Spatial-Statistics.tex b/Spatial-Statistics.tex
deleted file mode 100644
index e0effb2..0000000
--- a/Spatial-Statistics.tex
+++ /dev/null
...
\subsection{Spatial Statistics}
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 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 EQUATION and is represented as EQUATION.
Cross-correlation – occurs when EQUATION.
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]
