An Slight Overview of the Critical Elements of Spatial Statistics

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+ Spatial Statistical Descriptors Tony Fast NIST Workshop

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Transcript of An Slight Overview of the Critical Elements of Spatial Statistics

Page 1: An Slight Overview of the Critical Elements of Spatial Statistics

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Spatial Statistical Descriptors Tony Fast NIST Workshop

Page 2: An Slight Overview of the Critical Elements of Spatial Statistics

+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

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+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?

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+ 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.

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+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.

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+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

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+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

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+ There is a Fourier Convolution Property

n  Wikipedia

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+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

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+ 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

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+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.