An Slight Overview of the Critical Elements of Spatial Statistics
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- 1. +Spatial Statistical Descriptors Tony Fast NIST Workshop
2. + How do we discuss the variety in materials science information?Materials are hierarchical and multi-physics. 3. + Statistics are material descriptors -Titanium REDUCED OUTPUT: Grain size Grain Faces Number of Grains Mean Curvature Nearest Grain Analysis 4. +First Order Statistics nEffective statistics the describe a material volume n nnEffective Statistics require: nnnVolume Fraction, Phase Distribution, Means, Standard Deviations 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.Data processing n Which could inject incorrect assumptions? Limited return on the Time investedHow do we get more information out of spatial datasets & faster? 5. +Goals of today: Advanced Spatial Statistics and Signal Processing nPractical manipulation of multidimensional and multimodal datasets.nNew statistics tools to quantify material structures.nThe variety of metadata and the uniformity of data.nAdvanced methods for extracting structure-propertyprocessing connections.nTo start thinking differently about the data you generate, ingest, and manipulate. 6. +Focus on Scalability nDatasets are getting larger, more channels can be extracted, and the features are less understood.nExploring the new space of data requires scalable parametric and statistical material feature descriptors. 7. +Types of Higher-Order Statistics nMoving Window Average Code demo of image processing filtersnNeighborhood Connectivity Code demo of Delaunay tessellation and Voronoi Triangulation. n nShortest network path GraphTehoryTestnChord Length Distribution -Probably a chord of length d will contiguously span a region containing some featurenPair Correlation Functions In depthnVector-resolved spatial statistics In depth 8. + Spatial Statistics nSpatial statistics are a joint probability of material feature domain with a posterior probability relating to a spatial information. Spatial statistics are the probability of nding and separated by a of "nMain Spatial Statistics to discuss nnPair Correlation Function n Probability of two features two separated by a vector of magnitude r 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 9. + The Breakdown Index into features in the spatial materials signal Direct or latent variables Basis function representationDigital Signals i & j Gridded or Point Cloud Experimental or Simulated Periodic or non-periodic Any scaleNumerator is occurrence of true conditions Summation only occurs when s + t is a valid vectorSpatial Statistics Conditional, joint probability Joint Probability of two features i & j If i=j, autocorrelation otherwise, crosscorrelation Index or vector into a spatial conditionDenominator :Number of tests on the spatial condition Number of valid s+t vectors 10. +Vector Resolved Spatial Correlation Function of a Gridded ImagenComputing this relationship directly is costly.nSince it is a convolution, we will use the Fourier transform again. nUsed to compute the numerator and denominator separately.Code that Animates the statistics 11. +There is a Fourier Convolution Property nWikipedia 12. +First Consideration: Signal pattern nThe input signals must be on an even grid to use DFT methods.PatternPoint BoundariesnWork around nNon-Uniform FFTs ( Most accurate )nBinning point cloud data ( Introduces uncertainty )Gridded 13. +The Fourier Transform introduces periodicity. 14. + Second Consideration: Periodicity Part 1 SourceExperimentSimulationBoundary ConditionsBoundary ConditionsNonperiodicNonperiodicPeriodic Group Discussion If the denominator is the number of counts, how will it change with t? 15. + The Denominator nIf any dimensions are nonperiodic then the denominator always varies with position. The number of times a variable can be tested.when nConvolution!nNeeds to be computed less frequently than the numerator.nPartial Periodicity is possible. 16. + Second Consideration: Periodicity Part 2 SourceExperimentSimulationBoundary ConditionsBoundary ConditionsNonperiodicNonperiodic1Periodic1 17. +Pair Correlation Functions and Spatial Statistics nPair Correlation functions are a projection of the spatial statistics. Either the magnitudes of the vectors or an average of the vectors about their angle.nGroup exercise : design a workflow to compute pair correlation functions on periodic point cloud data.