Make sure you have as much content available for Wednesday in a folder.
m(x,h)dxdh - probablitily density function
descretize this --> insert image 1
this is based on binning - don't necessarily have to: could do linear interpolation
indicator funtions - bar graph
interpolation funtion triangle
The differences are 1 and 3 are orthonormal 2 is not 1 and 2 are local support basis but 3 is global support
create an ensemble of j microstructures that form our data set
therefore we want to look as statisics of the values
m is volume fraction of h~ at spacaial grid S
or: probability of finding h~ at S
if you use second interpretation 2 point statistics makes more sense as the joint probablity of events happening
continuous version of 2 point
there is no intergral over dh and dh' so dh and dh' are carried over
every cell in this microstructre is a different data point only possible because of assumption that it is independent of S adding the sum to look at spacial changes now say that there are H local states and S bins the size of f(hh't) is now H^2S (size of t and the size of s are the same) size fo fh is H and the size of f hh'h''tt' is H^3S^2
- exponentially increasing S is in millions H is in 10s(even if using euler angles) so going from 1 point to 2 point to 3 point is a huge change in size
Do we need all of them? - we definately need some of them Are there redundancies? Yes but that is for later
1 point all you see is volume fraction 2 point allows some information about morphology - information about spacial aspects THere is definately some information in 3 point but don't think that the huge increase in size is all important
How do we find out what is important? 1st data science problem
think of a simple 2D space - I have 2 point statisics - a very large vector
If i have micrographs and this is a large dimensional space then the entire vector I have J vectors each vector is just a point in 2D space the representation of this entire data set the dimensions of this space are on the order of 10^6 but the number of points is only on the order of 100 so it will not fill the entire space - extreemly sparse
Want to know what is the variance in the data
axis becomes the mean center - new origin is at the mean of the data set -simplifies computation
rotating the axis
x's are 2 point statisitic x' is a wieghted linear combination of 2 point statisitics looking for an orientation that maximizes the variance image
sticking with x1: Xi and Q are a vector and xk is a square matrix y = Xq (y = xi, q = Q and X = xk) the variance I am interested in is '(y)(y)^T' = q^T X^T X q q is the eigenvector of X^TX
This is the theory/logic behind it but there are smart algorithms to do this better. This is the general idea.