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Spatial Statistical Descriptors Tony Fast NIST Workshop
+How do we discuss the variety in materials science information?
Materials are hierarchical and multi-physics.
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Statistics are material descriptors
β-Titanium
REDUCED OUTPUT: Grain size Grain Faces Number of Grains Mean Curvature Nearest Grain Analysis
+First Order Statistics
n Effective statistics the describe a material volume n Volume Fraction, Phase Distribution, Mean’s, Standard Deviation’s n Often times the value is a single feature parameters, but the
information in spatial materials data contains information about the distribution. n The distribution increases the number of variables in the system,
but adds to the fidelity of the material feature description.
n Effective Statistics require: n Data processing
n Which could inject incorrect assumptions? n Limited return on the Time invested
n How do we get more information out of spatial datasets & faster?
+ Goals of today: Advanced Spatial Statistics and Signal Processing
n Practical manipulation of multidimensional and multimodal datasets.
n New statistics tools to quantify material structures.
n The variety of metadata and the uniformity of data.
n Advanced methods for extracting structure-property-processing connections.
n To start thinking differently about the data you generate, ingest, and manipulate.
+Focus on Scalability
n Datasets are getting larger, more channels can be extracted, and the features are less understood.
n Exploring the new space of data requires scalable parametric and statistical material feature descriptors.
+Types of Higher-Order Statistics
n Moving Window Average – Code demo of image processing filters
n Neighborhood Connectivity – Code demo of Delaunay tessellation and Voronoi Triangulation. n Shortest network path n GraphTehoryTest
n Chord Length Distribution -Probably a chord of length d will contiguously span a region containing some feature
n Pair Correlation Functions – In depth
n Vector-resolved spatial statistics – In depth
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Spatial Statistics
n Spatial statistics are a joint probability of material feature domain with a posterior probability relating to a spatial information.
Spatial statistics are the probability of finding <Feature A> and <Feature B> separated by a <Vector,Distance> of <d-Tuple>"
n Main Spatial Statistics to discuss n Pair Correlation Function
n Probability of two features two separated by a vector of magnitude r
n Vector resolved spatial statistics n Probability of two features two separated by a vector t n The pair correlation function is a reduced projection of the vector
resolved statistics
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Index or vector into a spatial condition
Numerator is occurrence of true conditions • Summation only occurs when
s + t is a valid vector
Denominator :Number of tests on the spatial condition • Number of valid s+t vectors
Joint Probability of two features i & j • If i=j, autocorrelation • otherwise, crosscorrelation
Index into features in the spatial materials signal • Direct or latent variables • Basis function representation
Digital Signals i & j • Gridded or Point Cloud • Experimental or Simulated • Periodic or non-periodic • Any scale
Spatial Statistics • Conditional, joint
probability
The Breakdown
+Vector Resolved Spatial Correlation Function of a Gridded Image
n Computing this relationship directly is costly.
n Since it is a convolution, we will use the Fourier transform again. n Used to compute the numerator and denominator separately.
Code that Animates the statistics
+ There is a Fourier Convolution Property
n Wikipedia
+First Consideration: Signal pattern n The input signals must be on an
even grid to use DFT methods.
n Work around
n Non-Uniform FFT’s ( Most accurate )
n Binning point cloud data ( Introduces uncertainty )
Pattern
Point
Boundaries
Gridded
+ The Fourier Transform introduces periodicity.
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Ø
Ø Ø
Source
Experiment
Boundary Conditions
Nonperiodic
Simulation
Boundary Conditions
Nonperiodic Periodic
Second Consideration: Periodicity Part 1
Group Discussion If the denominator is the number of counts, how will it change with t?
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The Denominator
n If any dimensions are nonperiodic then the denominator always varies with position. The number of times a variable can be tested.
when
n Convolution!
n Needs to be computed less frequently than the numerator.
n Partial Periodicity is possible.
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Ø
Ø Ø
1
1
Source
Experiment
Boundary Conditions
Nonperiodic
Simulation
Boundary Conditions
Nonperiodic Periodic
Second Consideration: Periodicity Part 2
+Pair Correlation Functions and Spatial Statistics
n Pair Correlation functions are a projection of the spatial statistics. Either the magnitudes of the vectors or an average of the vectors about their angle.
n Group exercise : design a workflow to compute pair correlation functions on periodic point cloud data.
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