statistical learning methods for microstructures
DESCRIPTION
STATISTICAL LEARNING METHODS FOR MICROSTRUCTURES. Veera Sundararaghavan and Prof. Nicholas Zabaras. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 - PowerPoint PPT PresentationTRANSCRIPT
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Veera Sundararaghavan and Prof. Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: [email protected], [email protected] URL: http://mpdc.mae.cornell.edu/
STATISTICAL LEARNING METHODS FOR MICROSTRUCTURES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
WHAT IS STATISTICAL LEARNING
Statistical learning is all about automating the process of searching for patterns from large Statistical learning is all about automating the process of searching for patterns from large scale statistics.scale statistics.
Which patterns are interesting?Which patterns are interesting?
Mathematical techniques for associating input data with desired attributes and identifying Mathematical techniques for associating input data with desired attributes and identifying correlationscorrelations
A powerful tool for designing materialsA powerful tool for designing materials
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FOR MICROSTRUCTURES?FOR MICROSTRUCTURES?
Properties of a material are affected by the underlying microstructure
• Microstructural attributes related to specific properties
•Examples: Correlation functions -> Elastic moduli
Orientation distribution ->Yield stress in polycrystals
• Attributes evolve during processing (thermo mechanical, chemical, solidification etc.)
• Can we identify specific patterns in these relationships?
• Is it possible to probabilistically predict the best microstructure and the best processing paths for optimizing properties based on available structural attributes?
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TERMINOLOGYTERMINOLOGY
Microstructure can be represented in terms of typical attributesMicrostructure can be represented in terms of typical attributesExamples are volume fractions, probability functions, shape/size attributes, Examples are volume fractions, probability functions, shape/size attributes,
orientation of grains, cluster functions, lineal measures and so onorientation of grains, cluster functions, lineal measures and so on
All these attributes affect physical propertiesAll these attributes affect physical properties
Attributes evolve during processing of a microstructureAttributes evolve during processing of a microstructure
Attributes are represented in a discrete (vector) form as Attributes are represented in a discrete (vector) form as ‘features’‘features’
‘‘features’ are represented as a vector xfeatures’ are represented as a vector xkk, k = 1,…,n, k = 1,…,n where n is where n is
the dimensionality of the featurethe dimensionality of the feature
Every different feature is represented as Every different feature is represented as xxkk(i)(i) where superscript where superscript
denotes the denotes the iithth feature feature that we are interested in that we are interested in
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TERMINOLOGYTERMINOLOGY
Given a data set of computational or experimental Given a data set of computational or experimental microstructures, can we learn the functional differences microstructures, can we learn the functional differences
between them based on features?between them based on features?
We denote microstructures that are similar in attributes in We denote microstructures that are similar in attributes in terms of a class representation terms of a class representation ‘y’, y = 1..k‘y’, y = 1..k where k is where k is
number of classes. number of classes.
Classes are formed into hierarchies: Each level Classes are formed into hierarchies: Each level represented by feature represented by feature xx(i)(i)..
Structure based classes are affiliated with process and Structure based classes are affiliated with process and properties:properties: powerful tool for exploring complex powerful tool for exploring complex
microstructure design spacemicrostructure design space
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APPLICATIONSAPPLICATIONS
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quantification and mining associations
Input microstructure
Classifier
Feature Detection
MICROSTRUCTURE LIBRARIES FOR REPRESENTATION
Identify and add new classes
Employ lower-order features
Pre-processing
Sundararaghavan & Zabaras, Acta Materialia, 2004
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MICROSTRUCTURE RECONSTRUCTION
vision
Database
2D Imaging techniques
MicrostructureAnalysis
(FEM/Bounding theory)
Feature extraction
Pattern recognition Microstructure
evolution models
Process
Reverse engineerprocess parameters
3D realizations
Sundararaghavan and Zabaras, Computational Materials Sci, 2005
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Training samples
ODF
Image
Pole figures
STATISTICALLEARNING TOOLBOX
Functions:1. Classification
methods2. Identify new
classes
NUMERICAL SIMULATION OF
MATERIAL RESPONSE
1. Multi-length scale analysis
2. Polycrystalline plasticity
PROCESS DESIGN
ALGORITHMS
1. Exact methods(Sensitvities)
2. Heuristic methods
Update data
In the library
Associate datawith a class;
update classesProcesscontroller
STATISTICAL LEARNING TOOLBOXSTATISTICAL LEARNING TOOLBOX
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DESIGNING PROCESSES FOR MICROSTRUCTURES
Process sequence-1
Process parameters
ODF history
Reduced basis
Process sequence-2
New process parameters
ODF history
Reduced basis
Classifier Adaptive basis selection
Optimization
Reduced basisProcess
Probable Process
sequences & Initial parameters
Desired texture/propert
y
Stage - 1 Stage - 2
New dataset added
DATABASE
Optimum parameters
Sundararaghavan and Zabaras, Acta Materialia, 2005
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THIS LECTURE WILL COVER….THIS LECTURE WILL COVER….
•This lecture we will try to go into the math behind statistical learning and learn two really useful techniques – Support Vector Machines and Bayesian Clustering.
•Applications to microstructure representation, reconstruction and process design will be shown
• We will skim over the physics and some important computational tools behind these problems
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STATISTICAL LEARNING TECHNIQUESSTATISTICAL LEARNING TECHNIQUES
Regressor Prediction ofreal-valued output
InputAttributes
DensityEstimator
ProbabilityInput
Attributes
Classifier Prediction ofcategorical output
InputAttributes
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Regressor Prediction ofreal-valued output
InputAttributes
DensityEstimator
ProbabilityInput
Attributes
Classifier Prediction ofcategorical output
InputAttributes
This lecture
STATISTICAL LEARNING TECHNIQUESSTATISTICAL LEARNING TECHNIQUES
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Regressor Prediction ofreal-valued output
InputAttributes
DensityEstimator
ProbabilityInput
Attributes
Classifier Prediction ofcategorical output
InputAttributes
This lecture
Function approximation: Useful for prediction in regions that are computationally unreachable (not covered in this lecture)
STATISTICAL LEARNING TECHNIQUESSTATISTICAL LEARNING TECHNIQUES
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PRELIMINARIES OF SUPERVISED CLASSIFIERSPRELIMINARIES OF SUPERVISED CLASSIFIERS
Classifiery
x
Decision
Function
y = w.f(x)+bMicrostructure classes eg. based on a property
Microstructure features
denotes +1
denotes -1Two class problem: The classes for the test specimens are known apriori
Aim: To predict the strength of a new microstructureVolume fraction
Po
re d
ensi
ty
High strength
Low strength
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
f(x,w,b) = sign(w. x - b)
How would you classify this data?
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OCCAM’S RAZOROCCAM’S RAZOR
plurality should not be assumed without necessity William of Ockham, Surrey (England) 1285-1347 AD, theologian
•Simpler models are more likely to be correct than complex ones•Nature prefers simplicity. •principle of uncertainty maximization
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
f(x,w,b) = sign(w. x - b)
How would you classify this data?
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
f(x,w,b) = sign(w. x - b)
Any of these would be fine..
..but which is best?
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
f(x,w,b) = sign(w. x - b)
Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
f(x,w,b) = sign(w. x - b)
The maximum margin linear classifier is the linear classifier with the, um, maximum margin.
This is the simplest kind of SVM (Called an LSVM)Linear SVM
Support Vectors are those datapoints that the margin pushes up against
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
• Plus-plane = Plus-plane = { { xx : : ww . . xx + b = +1 } + b = +1 }
• Minus-plane = Minus-plane = { { xx : : ww . . xx + b = -1 } + b = -1 }
• The vector The vector ww is perpendicular to the Plus Plane. Why? is perpendicular to the Plus Plane. Why?
“Predict Class
= +1”
zone
“Predict Class
= -1”
zonewx+b=1
wx+b=0
wx+b=-
1
M = Margin Width
How do we compute M in terms of w and b?
Let u and v be two vectors on the Plus Plane. What is w . ( u – v ) ?
And so of course the vector w is also perpendicular to the Minus Plane
Claim: Claim: xx++ = = xx-- + + ww for for some value of some value of . Why?. Why?
x+
x-
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
• What we know:What we know:
• ww . . x+x+ + b = +1 + b = +1
• ww . . x-x- + b = -1 + b = -1
• x+x+ = = x-x- + + ww
• ||x+x+ - - x-x- | | = M= M
• It’s now easy to get It’s now easy to get MM in terms of in terms of ww and and bb
“Predict Class
= +1”
zone
“Predict Class
= -1”
zonewx+b=1
wx+b=0
wx+b=-
1
M = Margin Width
w . (x - + w) + b = 1
=>
w . x - + b + w .w = 1
=>
-1 + w .w = 1
=>
x-
x+Computing the margin widthComputing the margin width
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
Learning the Maximum Margin Learning the Maximum Margin ClassifierClassifier
“Predict Class
= +1”
zone
“Predict Class
= -1”
zonewx+b=1
wx+b=0
wx+b=-1
M =
MinimizeMinimize w.ww.wWhat are the constraints?What are the constraints?
ww . . xxkk + b >= 1 if y + b >= 1 if ykk = 1 = 1
ww . . xxkk + b <= -1 if y + b <= -1 if ykk = -1 = -1
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
denotes +1
denotes -1
This is going to be a problem!
What should we do?
Minimize w.w + C (distance of error points to their correct place)
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
wx+b=1
wx+b=0
wx+b=-1
M =
1
1.
2
R
kk
C ε
w w
7
11 2
Constraints?Constraints?w . xk + b >= 1-k if yk = 1
w . xk + b <= -1+k if yk = -1
k >= 0 for all k
MinimizeMinimize
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
Harder 1-dimensional datasetHarder 1-dimensional dataset
What can be done about this?
x=0
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SUPPORT VECTOR MACHINESSUPPORT VECTOR MACHINES
x=0
Quadratic Basis Quadratic Basis FunctionsFunctions
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SUPPORT VECTOR MACHINES WITH KERNELSSUPPORT VECTOR MACHINES WITH KERNELS
1
1.
2
R
kk
C ε
w w
Constraints?Constraints?w . (xk)+ b >= 1-k if yk = 1
w . (xk)+ b <= -1+k if yk = -1
k >= 0 for all k
MinimizeMinimize
Φ: x → φ(x)
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SUPPORT VECTOR MACHINES: QUADRATIC PROGRAMMINGSUPPORT VECTOR MACHINES: QUADRATIC PROGRAMMING
Datapoints with k > 0 will be the support vectors
Maximize
where
Subject to these constraints:
Then define:
Then classify with:
f(x,w,b) = sign(w. (x) - b)Φ
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MULTIPLE CLASSESMULTIPLE CLASSES
Class-AClass-B
Class-CA
CB
AB
C
Given a new microstructure with its ‘s’ features given by
find the class of 3D microstructure (y ) to which it is most likely to belong.
[1,2,3,..., ]p1 2
1 1 1 2 2 21 1 2 2 1 2 1 2{ , ,...., }, { , ,...., },..., { , ,...., }
s
T T T s s sm m s mx x x x x x x x x x x x
p = 3One Against One Method:
• Step 1: Pair-wise classification, for a p class problem
• Step 2: Given a data point, select class with maximum votes out of ( 1)
2
p p
( 1)
2
p p
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MULTIPLE FEATURESMULTIPLE FEATURES
Class - 1
3D Microstructures3D Microstructures
Class - 2
FEATURE – 1: GRAIN SHAPE
FEATURE – 2 GRAIN SIZES
Class - 1
Class - 2
Class - 3
Class - 4
Rose of intersectionsHeyn int. Histogram
100
200
30
210
60
240
90
270
120
300
150
330
180 0
20 40 60 80
30
210
60
240
90
270
120
300
150
330
180 0
1 2 3 4 5 6 7 8 9 10111213140
10
20
30
40
1 2 3 4 5 6 7 8 9 10111213140
5
10
15
1 2 3 4 5 6 7 8 9 10111213140
5
10
15
1 2 3 4 5 6 7 8 9 10111213140
5
10
15
HIERARCHICAL LIBRARIES – (a.k.a) DIVISIVE CLUSTERING
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DYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTSDYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTS
Space of all possible microstructures
New class
New class: partition
Expandable class partitions
(retraining)
Hierarchical sub-classes (eg. medium grains)
A class of microstructures (eg. equiaxial grains)
Dynamic Representation:
Axis for representation
New microstructure
added
Updated representation
distance measures
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QUANTIFICATION OF DIVERSE MICROSTRUCTURE
A Common Framework for Quantification of Diverse Microstructure
Representation space of all possible polyhedral microstructures
Equiaxial grain microstructure space
Qualitative representation
Lower order descriptor approach
Equiax grains
Grain size: small
Grain size distribution
Grain size number
No.
of
grai
ns
Quantitative approach
1.41.4 2.62.6 4.04.0 0.90.9 ……....
Microstructure represented by a set of numbers
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BENEFITSBENEFITS
1. A data-abstraction layer for describing microstructural information.
2. An unbiased representation for comparing simulations and experiments AND for evaluating correlation between microstructure and properties.
3. A self-organizing database of valuable microstructural information which can be associated with processes and properties.
• Data mining: Process sequence selection for obtaining desired properties
• Identification of multiple process paths leading to the same microstructure
• Adaptive selection of basis for reduced order microstructural simulations.
• Hierarchical libraries for 3D microstructure reconstruction in real-time by matching multiple lower order features.
• Quality control: Allows machine inspection and unambiguous quantitative specification of microstructures.
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PRINCIPAL COMPONENT ANALYSISPRINCIPAL COMPONENT ANALYSIS
Let be n images.
1. Vectorize input images2. Create an average image
3. Generate training images
1 2 n, ,.....
1
1=
n
iin
i i 4. Create correlation matrix (Lmn)
5. Find eigen basis (vi) of the correlation matrix
6. Eigen microstructures (ui) are generated from the basis (vi) as
7. Any new face image ( ) can be transformed to eigen face components through ‘n’ coefficients (wk) as,
Tmn m nL
i i iLv v
i ij ju v
( )Tk ku
Representation coefficients
Reduced basis
Data Points
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REQUIREMENTS OF A REPRESENTATION SCHEMEREQUIREMENTS OF A REPRESENTATION SCHEME
REPRESENTATION SPACE OF A PARTICULAR MICROSTRUCTURE
Need for a technique that is autonomous, applicable to a variety of microstructures, computationally feasible and provides complete
representation
A set of numbers which completely represents a microstructure within its class
2.72.7 3.63.6 1.21.2 0.10.1 ……....
8.48.4 2.12.1 5.75.7 1.91.9 ……....
Must differentiate other cases: (must be statistically representative)
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PCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLEPCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLE
Eigen-microstructures
Input Microstructures
Representation coefficients (x 0.001)
Image-1 quantified by 5 coefficients over the eigen-microstructures
0.0125 1.3142 -4.23 4.5429 -1.6396
-0.8406 0.8463 -3.0232 0.3424 2.6752
3.943 -4.2162 -0.6817 -9718 1.9268
1.17961.1796 -1.3354-1.3354 -2.8401-2.8401 6.20646.2064 -3.2106-3.2106
5.82945.8294 5.22875.2287 -3.7972-3.7972 -3.6095-3.6095 -3.6515-3.6515Basis 5
Basis 1
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EIGEN VALUES AND RECONSTRUCTION OVER THE BASISEIGEN VALUES AND RECONSTRUCTION OVER THE BASIS
1.Reconstruction with 100% basis
2. Reconstruction with 80% basis
3. Reconstruction with 60% basis
4. Reconstruction with 40% basis
4 23 1
Reconstruction of microstructures over fractions of the basis
Significant eigen values capture most of the image features
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INCREMENTAL PCA METHODINCREMENTAL PCA METHOD
• For updating the representation basis when new microstructures are added in real-time.
• Basis update is based on an error measure of the reconstructed microstructure over the existing basis and the original microstructure
IPCA :
Given the Eigen basis for 9 microstructures, the update in the basis for the 10th microstructure is based on a PCA of 10 x 1 coefficient vectors instead of a 16384 x 1 size microstructures.
Updated BasisNewly added data point
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ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, 1974)ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, 1974)
Identify intercepts of lines with grain boundaries plotted within a circular domain
Count the number of intercepts over several lines placed at various angles.
Total number of intercepts of lines at each angle is given as a polar plot called rose of intersections
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GRAIN SHAPE FEATURE: EXAMPLESGRAIN SHAPE FEATURE: EXAMPLES
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GRAIN SIZE PARAMETERGRAIN SIZE PARAMETER
Several lines are superimposed on the microstructure and the intercept length of the lines with the grain boundaries are recorded
(Vander Voort, 1993)
The intercept length (x-axis) versus number of lines (y-axis) histogram is used as the measure of grain size.
GRAIN SIZE FEATURE: EXAMPLESGRAIN SIZE FEATURE: EXAMPLES
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SVM TRAINING FORMAT
CLASSIFICATION SUCCESS %
Total Total imagesimages
Number of Number of classesclasses
Number of Number of Training imagesTraining images
Highest Highest success ratesuccess rate
Average Average success ratesuccess rate
375375 1111 4040 95.8295.82 92.5392.53
375375 1111 100100 98.5498.54 95.8095.80
ClassClass Feature Feature numbernumber
Feature Feature valuevalue
Feature Feature numbernumber
Feature Feature valuevalue
11 11 23.3223.32 22 21.5221.52
22 11 24.1224.12 22 31.5231.52
Data point
GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM
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CLASS HIERARCHYCLASS HIERARCHY
Class –2Class –1
Class 1(a) Class 1(b) Class 1(c) Class 2(a) Class 2(b) Class 2(c)
Level 1 : Grain shapes
Level 2 : Subclasses based on grain sizes
New classes:
Distance of image feature from the average feature vector of a class
IPCA QUANTIFICATION WITHIN CLASSESIPCA QUANTIFICATION WITHIN CLASSES
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Class-j Microstructures (Equiaxial grains, medium grain size)
Class-i Microstructures (Elongated 45 degrees, small grain size)
Representation Matrix
Image -1 Image-2 Image-3…
Component in basis vector 1
123 23 38
2 91 54 -85
3 -54 90 12
Average Image
21 23 24…
Eigen Basis
0.9 0.84 0.23..
0.54 0.21 0.74..
The Library – Quantification and image representation
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REPRESENTATION FORMAT FOR MICROSTRUCTUREREPRESENTATION FORMAT FOR MICROSTRUCTURE
Improvement of microstructure representation due to classificationImprovement of microstructure representation due to classification
Date: 1/12 02:23PM, Basis updated
Shape Class: 3, (Oriented 40 degrees, elongated)
Size Class : 1, (Large grains)
Coefficients in the basis:[2.42, 12.35, -4.14, 1.95, 1.96, -1.25]
Reconstruction with 6 coefficients (24% basis): A class with 25 images
Improvement in reconstruction: 6 coefficients (10 % of basis) Class of 60 images
Original image Reconstruction over 15 coefficients
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MICROSTRUCTURE REPRESENTATION USING SVM & PCAMICROSTRUCTURE REPRESENTATION USING SVM & PCA
COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION
Does not decay to zero
A DYNAMIC LIBRARY APPROACH
•Classify microstructures based on lower order descriptors.
•Create a common basis for representing images in each class at the last level in the class hierarchy.
•Represent 3D microstructures as coefficients over a reduced basis in the base classes.
•Dynamically update the basis and the representation for new microstructures
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PCA MICROSTRUCTURE RECONSTRUCTIONPCA MICROSTRUCTURE RECONSTRUCTION
Pixel value round-off
Basis Components
X 5.89
X 14.86
+
Project
onto basis
Reconstruct using two basis components
Representation using just 2 coefficients (5.89,14.86)
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MOTIVATIONMOTIVATION
1. Creation of 3D microstructure models from 2D images
2. 3D imaging requires time and effort. Need to address real–time methodologies for generating 3D realizations.
3. Make intelligent use of available information from computational models and experiments.
vision
Database
Pattern recognition
MicrostructureAnalysis
2D Imaging techniques
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LITERATURE: STOCHASTIC MICROSTRUCTURE RECONSTRUCTION
Methods available are optimization based: Features of 2D image are matched to that of a 3D microstructure by posing an optimization problem.
1) Does not make use of available information (experimental/simulated data)
2) Cannot perform reconstructions in real-time.
Need to take into account the processes that create these microstructure (Oren and Bakke, 2003) for correctly modeling the geometric connectivity.
Key assumptions employed for 3D image reconstruction from a single 2D image
Randomness Assumption (Ohser and Mucklich – 2000).
1. Grains in a polyhedral microstructure are assumed to be of the similar shapes but of different sizes.
2. Two phase microstructures can be characterized using rotationally-invariant probability functions
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PATTERN RECOGNITION (PR) STEPSPATTERN RECOGNITION (PR) STEPS
DATABASE CREATION
FEATURE EXTRACTION
TRAINING
PREDICTION
Datasets: microstructures from experiments or physical models
Extraction of statistical features from the database
Creation of a microstructure class hierarchy: Classification methods
Prediction of 3D reconstruction, process paths, etc,
PATTERN RECOGNITION : A DATA-DRIVEN OPTIMIZATION TOOL•Feature matching for reconstruction of 3D microstructures
Real-time
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POLYHEDRAL MICROSTRUCTURES: MC MODELPOLYHEDRAL MICROSTRUCTURES: MC MODEL
1
( )
1
(1 )
s
n
i j
N
ii
N i
i s sj
H J H
H
Potts Hamiltonian (H)Algorithm (1 Monte Carlo Step): • Calculation of the free energy of a randomly selected node (Hi)• Random choice of a new crystallographic orientation for the node• New calculation of the free energy of the element (Hf)• The orientation that minimizes the energy (min(Hf,Hi)) is chosen.
Ns: Total No. of nodes
Nn(i) : No. of neighbors of node ‘i’
Microstructure Database
Classes of microstructures based on grain size feature
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POLYHEDRAL MICROSTRUCTURES : GRAIN SIZE FEATUREPOLYHEDRAL MICROSTRUCTURES : GRAIN SIZE FEATURE
Intercept lengths of parallel network of lines with the grain boundaries are recorded at several anglesThe intercept length (x-axis) versus number of lines (y-axis) histogram is the measure of grain size (Heyn intercept histogram).
Slice
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FEATURE BASED CLASSIFICATIONFEATURE BASED CLASSIFICATION
Class - 1
3D Microstructures3D Microstructures
Class - 2
LEVEL - 1 LEVEL - 2
Class - 1
Class - 2
Class - 3
Class - 4
Rose of intersectionsHeyn int. Histogram
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180 0
20 40 60 80
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180 0
1 2 3 4 5 6 7 8 9 10111213140
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1 2 3 4 5 6 7 8 9 10111213140
5
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1 2 3 4 5 6 7 8 9 10111213140
5
10
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1 2 3 4 5 6 7 8 9 10111213140
5
10
15
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CCOORRNNEELLLL U N I V E R S I T Y
RECONSTRUCTION OF POLYHEDRAL MICROSTRUCTURERECONSTRUCTION OF POLYHEDRAL MICROSTRUCTURE
Polarized light micrographs of Aluminum alloy AA3002 representing the rolling plane
(Wittridge & Knutsen 1999)
A reconstructed 3D image
Comparison of the average feature of 3D class and the 2D image
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STEREOLOGICAL ESTIMATES OF 3D GRAIN SIZESSTEREOLOGICAL ESTIMATES OF 3D GRAIN SIZES
The stereological integral equation for estimating the 3D grain size distribution from a 2D image for polyhedral microstructures
Na,Fa(s) : density of grains and grain size distribution in 2D image
Nv,Fv(u) : density of grains and grain size distribution in 3D microstructure
: rotation average of the size of a particle with maximum size = 1
Gu(s): Size distribution function of the section profiles under the condition that a random size ‘U’ equals the 3D particle mean size (u).
Remark: Sizes are defined as the maximum calliper diameter of a grain
b
Numerical Scheme:
Let
Then
: Mean number/volume of grains with size ui
Let yk be the mean number per unit area of section profiles with size between [sk-1,sk]
And let ui = ai and sk = ak, then, y = Pwhere
P is a matrix formed from a set of coefficients ( based onthe shape assumption of grains
( , ) (1 ( ))up u s bu G s (1 ( )) ( , )a a i i iN F s p u s
i
1( ( ) ( ))k a a k a ky N F s F s
ii iba
0
[1 ( )] (1 ( )) ( )a a u v vN F s bu G s N dF u
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STEREOLOGICAL DISTRIBUTIONS (GEOMETRICAL)STEREOLOGICAL DISTRIBUTIONS (GEOMETRICAL)
3D reconstruction2D grain profile
3D grain
3D grain size distribution based on assumption that particles are randomly oriented cubes ( )3 / 2b
0
[1 ( )] (1 ( )) ( )a a u v vN F s bu G s N dF u
Na,Fa(s) : density of grains and grain size distribution in 2D image
Nv,Fv(u) : density of grains and grain size distribution in 3D microstructure
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STATISTICAL CORRELATION MEASURESSTATISTICAL CORRELATION MEASURES
MC Sampling: Computing the three point probability function of a 3D microstructure(40x40x40 mic)
S3(r,s,t), r = s = t = 2, 5000 initial points, 4 samples at each initial point.
Rotationally invariant probability functions (Si
N ) can be interpreted as the probability of finding the N vertices of a polyhedron separated by relative distances x1, x2,..,xN in phase i when tossed, without regard to orientation, in the microstructure.
When a voxel solidifies, liquid is expelled to its neighbors, creating solute concentration (ci,j,k) gradients. Movement of solute to minimize concentration gradients is modeled using fick’s law
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MC MODEL FOR TWO-PHASE MICROSTRUCTURESMC MODEL FOR TWO-PHASE MICROSTRUCTURES
Weights (w) of neighbors
Face neighbors = 1
Edge neighbors = 1
2Solid voxels
Microstructure is represented using voxels.Probability of solidification (P) depends on1) Net weight (w) of the No. of neighbors of a solid voxel:• If w >= 8.6568: voxel solidifies (P = 1)• If 3.8284 < w < 8.6568, P = 0.1• If weight < 3.8284, the voxel remains liquid (P = 0)
2) The solute concentration: A linear probability distribution with P = 0 at critical concentration and P = 1 when concentration is 0.
Final state
Where (i,j,k) is a voxel coordinate, n is the time step and D is the diffusion coefficient
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TWO PHASE MICROSTRUCTURE: CLASS HIERARCHYTWO PHASE MICROSTRUCTURE: CLASS HIERARCHY
Class - 1
3D Microstructures
Feature vector : Three point probability
function
3D Microstructures
Class - 2
Feature: Autocorrelation
function
LEVEL - 1 LEVEL - 2
r m
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EXAMPLE: 3D RECONSTRUCTION USING SVMSEXAMPLE: 3D RECONSTRUCTION USING SVMS
Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure
3 point probability function
Autocorrelation function
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MICROSTRUCTURE ELASTIC PROPERTIESMICROSTRUCTURE ELASTIC PROPERTIES
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0 200 400 600 800 1000Temperature (deg-C)
You
ngs
Mod
ulus
(G
Pa)
HS boundsBMMP boundsExperimentalFEM
3D image derived through pattern recognition
Experimental image
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WHAT IS MICROSTRUCTURE DESIGN
Initial microstructure
processing sequence? Final microstructure/
property
Microstructure?Known operating conditions
Known property limits
Initial Microstructure
Known operating conditions
Property?
Direct problem
Design problems
Use finite elements, experiments etc.
Design for best microstructure
Design for best processes
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CCOORRNNEELLLL U N I V E R S I T Y
SUPERVISED VS UNSUPERVISED LEARNING
Supervised classification for design:
1. Classify microstructures based on known process sequence classes
2. Given a desired microstructure, identify the processing stages required through classification
3. Drawback: Identifies a unique process sequence, but we that find many processing paths to lead to similar properties!
UNSUPERVISED CLASSIFICATION
1. Identify classes purely based on structural attributes
2. Associate processes and properties through databases
3. Explores the structural attribute space for similarities and unearths non-unique processing paths leading to similar microstructural properties
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K MEANSK MEANS
Suppose the coordinates of points drawn randomly from this dataset are transmitted.
You can install decoding software at the receiver.
You’re only allowed to send two bits per point.
It’ll have to be a “lossy transmission”.
Loss = Sum Squared Error between decoded coords and original coords.
What encoder/decoder will lose the least information?
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K MEANSK MEANS
Idea OneIdea One
00
1110
01
Break into a grid, decode each bit-pair as the middle of each grid-cell
QuestionsQuestions
• What are we trying to What are we trying to optimize?optimize?
• Are we sure it will find Are we sure it will find an optimal clustering?an optimal clustering?
Break into a grid, decode each bit-pair as the centroid of all data in that grid-cell
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K MEANSK MEANS
Find the cluster centers {C1,C2,…,Ck} such that the sum of the 2-norm distance squared between each feature xi , i = 1,..,n and its nearest cluster center ch is minimized.
1 2 2( )
1
( , ,.., ) ( )i
Rk
i encodei
J c c c
xx cCost Function =
OwnedBy( )
1
| OwnedBy( ) |j
j iij
c
c xc
Cost function minimized by
transmitting centroids
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THE EXPECTATION-MAXIMIZATION (EM) ALGORITHMTHE EXPECTATION-MAXIMIZATION (EM) ALGORITHM
What properties can be changed for centers c1 , c2 , … , ck have when distortion is not minimized?
Expectation step: Compute expected centers
(1) Change encoding so that xi is encoded by its nearest center
Maximization step: Compute maximum likelihood values of centers
(2) Set each Center to the centroid of points it owns.
There’s no point applying either operation twice in succession.
But it can be profitable to alternate.
…And that’s K-means!
2ENCODE( )
1
Distortion ( )i
R
ii
xx c
EM algorithm will be dealt with later
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K-MEANSK-MEANS
1. Ask user how many clusters they’d like. (e.g. k=5)
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K-MEANSK-MEANS
1. Ask user how many clusters they’d like. (e.g. k=5)
2. Randomly guess k cluster Center locations
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K-MEANSK-MEANS
1. Ask user how many clusters they’d like. (e.g. k=5)
2. Randomly guess k cluster Center locations
3. Each datapoint finds out which Center it’s closest to. (Thus each Center “owns” a set of datapoints)
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K-MEANSK-MEANS
1. Ask user how many clusters they’d like. (e.g. k=5)
2. Randomly guess k cluster Center locations
3. Each datapoint finds out which Center it’s closest to.
4. Each Center finds the centroid of the points it owns
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K-MEANSK-MEANS
1. Ask user how many clusters they’d like. (e.g. k=5)
2. Randomly guess k cluster Center locations
3. Each datapoint finds out which Center it’s closest to.
4. Each Center finds the centroid of the points it owns…
5. …and jumps there
6. …Repeat until terminated!
often unknown (is dependent on the features used for microstructure representation)
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SHORTCOMINGS OF K-MEANS AND REMEDIESSHORTCOMINGS OF K-MEANS AND REMEDIES
1) K-MEANS gives hyper-spherical clusters: Not always the case with data
2) Number of classes must be known apriori: Beats the reasoning for unsupervised clusters – we do not know anything about the classes in the data
3) May converge to local optima – not so bad
We will discuss about new strategies to get improved clusters of microstructural features
1) Gaussian mixture models and Bayesian clustering
2) Later, an improved k-means algorithm called X-means which uses a Bayesian information criterion
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PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES
• A is a Boolean-valued random variable if A denotes an event, and there is some degree of uncertainty as to whether A occurs.
Examples
• A = You win the toss
• A = Probability of failure of a structure
0 <= P(A) <= 1
P(True) = 1
P(False) = 0
P(A or B) = P(A) + P(B) - P(A and B)
P(~A) + P(A) = 1
P(B) = P(B ^ A) + P(B ^ ~A)
Discrete Random VariablesDiscrete Random Variables
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PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES
P(A ^ B) P(A|B) = ----------- P(B)
P(A ^ B) = P(A|B) P(B)
P(A ^ B) P(A|B) P(B)P(A ^ B) P(A|B) P(B)
P(B|A) = ----------- = ---------------P(B|A) = ----------- = ---------------
P(A) P(A)P(A) P(A)
Corollary: The Chain Rule
Definition of Conditional ProbabilityDefinition of Conditional Probability
Bayes Rule
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PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES
• MAP (Maximum A-Posteriori Estimator):predict
1 1argmax ( | )m mv
Y P Y v X u X u
predict1 1argmax ( | )m m
vY P X u X u Y v
What if Y = v itself is very unlikely?
• MLE (Maximum Likelihood Estimator):MLE (Maximum Likelihood Estimator):
Includes P(Y = v) information through Bayes rule (P(Y = v) is called as ‘prior’)
Class of data = argmaxi P(data | class = i)
Class of data = argmaxi P(class = i | data)
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PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES
1 1
1 1
1 1
1 1
( | )
( | ) ( )
( )
1( | ) ( )
m m
m m
m m
m m
P Y v X u X u
P X u X u Y v P Y v
P X u X u
P X u X u Y v P Y vc
• MAP (Maximum A-Posteriori Estimator):predict
1 1argmax ( | )m mv
Y P Y v X u X u
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PROBABILITY PRELIMINARIESPROBABILITY PRELIMINARIES
Bayes Classifiers in a nutshellBayes Classifiers in a nutshell
predict1 1
1 1
argmax ( | )
argmax ( | ) ( )
m mv
m mv
Y P Y v X u X u
P X u X u Y v P Y v
1. Learn the distribution over inputs for each value Y.
2. This gives P(X1, X2, … Xm | Y=vi ).
3. Estimate P(Y=vi ). as fraction of records with Y=vi .
4. For a new prediction:
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NAÏVE BAYES CLASSIFIERNAÏVE BAYES CLASSIFIER
predict1 1argmax ( | ) ( )m m
vY P X u X u Y v P Y v
In the case of the naive Bayes Classifier this can be simplified:
predict
1
argmax ( ) ( | )Yn
j jv j
Y P Y v P X u Y v
The independent features
assumption
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Notation change:
The naïve Bayes classifier
0 ,( ) ( , ), ( | ) ( , )c i c i j c j i cP v p x v P x v p x v
New Bayes classifier
NAÏVE BAYES CLASSIFIER IS AN SVM?
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Bayes classifier with feature weighting
NAÏVE BAYES CLASSIFIER IS AN SVM?
wj = 1 (for naïve Bayes)
But, features may be correlated!
A two class classifier
Decision function given by the sign of fWBC given by
Class t
Class f
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NAÏVE BAYES CLASSIFIER IS AN SVM?
Class t
Class f
SVM classifier!
Feature space of a naïve Bayes classifier
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INTRO TO BAYESIAN UNSUPERVISED CLASSIFICATIONINTRO TO BAYESIAN UNSUPERVISED CLASSIFICATION
( | ) ( )( | )
( )
p y i P y iP y i
p
xx
x
/ 2 1/ 2
1 1exp
(2 ) || || 2( | )
( )
T
k i i k i imi
p
P y ip
x μ Σ x μΣ
xx
Gaussian Mixture Models
Assume that each feature is generated as:
Pick a class at random. Choose class i with probability P(wi).
The feature is sampled from a Gaussian distribution : N(i, i )
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GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL
1
3
• There are k components. The i’th component is called yi
• Component yi has an associated mean vector i
• Each component generates data from a Gaussian with mean i and covariance matrix i
2
1
Assuming features in each class can be modeled by a Gaussian distribution, identify the parameters (means,variances etc.) of the distributions
Probabilistic extension of K-MEANS
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GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL
• We have x1 x2 … xn features of a microstructure
• We have P(y1) .. P(yk). We have σ.
• We can define, for any x , P(x|yi , μ1, μ2 .. μk)
• Can we define P(x | μ1, μ2 .. μk) ?
• Can we define P(x1, x2, .. xn | μ1, μ2 .. μk) ?
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GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL
Given a guess at Given a guess at μμ11, , μμ22 .... μμ k,k,
We can obtain the probability of the unlabeled data given We can obtain the probability of the unlabeled data given those those μμ‘s.‘s.
Inverse Problem: Find Inverse Problem: Find ’s given the points x’s given the points x1,1,xx22,…x,…xkk
The normal max likelihood trick:
Set d log Prob (….) = 0
d μi
and solve for μi‘s.
Using gradient descent, Slow but doable
Use a much faster and recently very popular method…
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EM ALGORITHM REVISITEDEM ALGORITHM REVISITED
We have unlabeled microstructural features We have unlabeled microstructural features xx11 xx22 … … xxRR
• We know there are k classesWe know there are k classes
• We know P(yWe know P(y11), P(y), P(y22), P(y), P(y33), …, P(y), …, P(ykk))
• We We don’tdon’t know know μμ11 μμ22 .. .. μμkk
• We can write P( data | We can write P( data | μμ11…. …. μμkk) )
1 1
11
111
2
211
p ... μ ...μ
p μ ...μ
p ,μ ...μ P
1K exp μ P
2σ
R k
R
i ki
R k
i j k jji
R k
i j jji
x x
x
x y y
x y
Maximize this likelihood
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GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL
1
11
11
For Max likelihood we know log Pr ob data μ ...μ 0μ
Some algebra turns this into: "For Max likelihood, for each j,
,μ ...μμ
,μ ...μ
ki
R
j i k ii
j R
j i ki
P y x x
P y x
This is n nonlinear equations in μj’s.”
If, for each xi we knew that for each yj the prob that μj was in class yj is P(yj|xi,μ1…μk) Then… we would easily compute μj.
If we knew each μj then we could easily compute P(yj|xi,μ1…μj) for each yj and xi.
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GAUSSIAN MIXTURE MODELGAUSSIAN MIXTURE MODEL
Iterate. On the Iterate. On the tt’th iteration let our estimates be’th iteration let our estimates be
{ { μμ11(t), (t), μμ22(t) … (t) … μμcc(t) }(t) }
• E-stepE-step
Compute “expected” classes of all datapoints for each classCompute “expected” classes of all datapoints for each class
2
2
1
p , ( ), ( )p , PP ,
p p , ( ), ( )
k i i ik i t i ti k t c
k tk j j j
j
x y t p tx y yy x
x x y t p t
I
IM-step.
Compute Max. like μ given our data’s class membership distributions
P , μ 1
P ,
i k t kk
ii k t
k
y x xt
y x
Just evaluate a Gaussian at xk
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GAUSSIAN MIXTURE MODEL: DENSITY ESTIMATIONGAUSSIAN MIXTURE MODEL: DENSITY ESTIMATION
Features in 2D
Complex PDF of the feature space
Classification + Probabilistic quantification of results
Ambiguity + Anomaly detection – Very popular in Genome mapping
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DATABASE FOR POLYCRYSTAL MICROSTRUCTURES
Statistical Learning
Feature Extraction
Multi-scale microstructure
evolution models
Process design for desired properties
RD
R-v
alue
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90
Angle from rolling direction
InitialIntermediateOptimalDesired
TDProcess Process parameters Values ..Tension Strain rate, time 0.56Forging Forging velocity ,Initial Temperature 2.13
Meso-scale database COMPONENTS
TD
You
ngs
Mod
ulus
RD0 20 40 60 80
144
144.1
144.2
144.3
144.4
144.5
144.6
144.7
Database
Divisive Clustering
Class hierarchies
Class Prediction
Driven by distance based (or) Probabilistic clustering
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DATABASE FOR POLYCRYSTAL MICROSTRUCTURES
Statistical Learning
Feature Extraction
Multi-scale microstructure
evolution models
Process design for desired properties
RD
R-v
alue
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90
Angle from rolling direction
InitialIntermediateOptimalDesired
TDProcess Process parameters Values ..Tension Strain rate, time 0.56Forging Forging velocity ,Initial Temperature 2.13
Meso-scale database COMPONENTS
TD
You
ngs
Mod
ulus
RD0 20 40 60 80
144
144.1
144.2
144.3
144.4
144.5
144.6
144.7
Database
Divisive Clustering
Class hierarchies
Class Prediction
Cluster based on similar microstructural features
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DATABASE FOR POLYCRYSTAL MICROSTRUCTURES
Statistical Learning
Feature Extraction
Multi-scale microstructure
evolution models
Process design for desired properties
RD
R-v
alue
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90
Angle from rolling direction
InitialIntermediateOptimalDesired
TDProcess Process parameters Values ..Tension Strain rate, time 0.56Forging Forging velocity ,Initial Temperature 2.13
Meso-scale database COMPONENTS
TD
You
ngs
Mod
ulus
RD0 20 40 60 80
144
144.1
144.2
144.3
144.4
144.5
144.6
144.7
Database
Divisive Clustering
Class hierarchies
Class Prediction
Associate process/ property info from database
Cluster based on similar microstructural features
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ORIENTATION DISTRIBUTION FUNCTION
Any macroscale property < χ > can be expressed as an expectation value if the corresponding single crystal property χ ( ,t) is known.
• Determines the volume fraction of crystals within a region R' of the fundamental region R• Probability of finding a crystal orientation within a region R' of the fundamental region• Characterizes texture evolution
ORIENTATION DISTRIBUTION FUNCTION – A(r,t)
– reorientation velocity
ODF EVOLUTION EQUATION – EULERIAN DESCRIPTION
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FEATURES OF AN ODF: ORIENTATION FIBERS
1(
1 .r h y+ (h+y))
h y
Points (r) of a (h,y) fiber in the fundamental region
angle
Crystal Axis = h
Sample Axis = y
Rotation (R) required to align h with y
(invariant to , )
Fibers: h{1,2,3}, y || [1,0,1]
{1,2,3} Pole FigurePoint y (1,0,1)
0 0
h||y
R.h=h, h||y
1P(h,y) = (P (h,y)+P (-h,y))
21
P(h,y) = 2
Ad
Integrated over all fibers corresponding to crystal direction h and sample direction y
For a particular (h), the pole figure takes values P(h,y) at locations y on a unit sphere.
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SIGNIFICANCE OF ORIENTATION FIBERS
Uniaxial (z-axis) Compression Texture
z-axis <110> fiber BB’
z-axis <100> fiber AA’
z-axis <111> fiber CC’
Predictable fiber
development
Important fiber families: <110> : uniaxial compression, plane strain compression and simple shear.
<111>: Torsion, <100>,<411> fibers: Tension
fiber (ND <110> ) & fiber: FCC metals under plane strain compression
close affiliation with processes
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LIBRARY FOR TEXTURES
[110] fiber family
DATABASE OF ODFsUni-axial (z-axis) Compression Texture
z-axis <110> fiber (BB’)
Feature:
fiber path corresponding to crystal direction h and sample direction y
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SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES
Given ODF/texture
Tension (T)
Stage 1
LEVEL – 2 CLASSIFICATIONPlane strain compression
T+P
LEVEL – I CLASSIFICATIONTension identified
Sta
ge 2
Stage 3
Multi-stage classification with each class affiliated with a unique process
Identifies a unique processing sequence:
Fails to capture the non-uniqueness in the
solution
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UNSUPERVISED CLASSIFICATION
Find the cluster centers {C1,C2,…,Ck} such that the sum of the 2-norm distance squared between each feature xi , i = 1,..,n and its nearest cluster center Ch is minimized.
21 2
21,..,1
1( , ,.., ) ( )
2minn
k hi
h ki
J c c c x C
Identify clusters
Clusters
DATABASE OF ODFs
Feature Space
Cost function Each class is affiliated with multiple processes
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ODF CLASSIFICATION
Desired ODF
Search path
Automatic class-discovery without class labels.
• Hierarchical Classification model
•Association of classes with processes, to facilitate data-mining
•Can be used to identify multiple process routes for obtaining a desired ODF
File index Process Description Number of parameters Process parameters Values ---------->1 Tension 2 (Strain rate, time, velocity gradient) 1 0.12 Plane Strain Compression 2 (Strain rate, time, velocity gradient) 1 0.43 Forging 7 (Forging velocity ,Time,Initial Temperature ) 1 -0.2
Data-mining for Process information with ODF Classification
ODF 2,12,32,97 One ODF, several process paths
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PROCESS PARAMETERS LEADING TO DESIRED PROPERTIESY
oung
’s M
odul
us (
GP
a)
Angle from rolling direction
CLASSIFICATION BASED ON PROPERTIES
Class - 1 Class - 2
Class - 3Class - 40.5 0.25 0
0.25 -1.25 00 0 0.75
0.5 0 00 0.75 00 0 -1.25
Velocity Gradient
Different processes, Similar properties
Database for ODFs
Property Extraction
ODF Classification
Identify multiple solutions
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K-MEANS ALGORITHM FOR UNSUPERVISED CLASSIFICATION
•User needs to provide ‘k’, the number of clusters.
( )
( )
1 2( ), 1,..,
( ) ( )( ) 1
2
( ) 0 (for a minimum)
Thus at a minimum, ( , ,.., )
x c
x c
x c
x c x cc
c c
x c
c mean x x x
i j
i j
i j
Ti j i j
clusterj
j j
Ti j
cluster
j ncluster i n
J
Lloyds Algorithm:
1. Start with ‘k’ randomly initialized centers
2. Change encoding so that xi is owned by its nearest center.
3. Reset each center to the centroid of the points it owns.
Alternate steps 1 and 2 until converged.
But, No. of clusters is unknown for the
texture classification problem
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SCHWARZ CRITERION FOR IDENTIFYING NUMBER OF CLUSTERS
The number of clusters chosen maximizes the Bayesian information criterion given by:
Where is the is the log-likelihood of the data taken at the maximum likelihood point, p is the number of free parameters in the model
Maximum likelihood of the variance assuming Gaussian data distribution
Probability of a point in cluster i
Log-likelihood of the data in a cluster
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CENTROID SPLIT TESTS
X-MEANS algorithm:
• Start with k clusters found through k-means algorithm
• Split each centroid into two centroids, and move the new centroids along a distance proportional to the cluster size in an arbitrarily chosen direction
• Run local k-means (k = 2) in each cluster
•Accept split cluster in each region if BIC(k = 1) < BIC(k = 2)
• Test for various initial values of ‘k’ and select the ‘k’ with maximum overall BIC
Split centers Run local k-means (k = 2) in each cluster New clusters based on BIC
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COMPARISON OF K-MEANS AND X-MEANS
Local Optimum produced by the kmeans algorithm with k = 4
Cluster configuration produced by k-means with k = 6: Over-estimates the natural number of clusters
Configuration produced by the x-means algorithm: Input range of k = 2 to 15. x-means found 4 clusters from the data-set based on the Bayesian Information Criterion
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MULTIPLE PROCESS ROUTESMULTIPLE PROCESS ROUTES
0 10 20 30 40 50 60 70 80 90144
144.5
145
145.5
Angle from the rolling direction
You
ngs
Mod
ulus
(G
Pa)
Desired Young’s Modulus distribution
Magnetic hysteresis loss distribution
0 10 20 30 40 50 60 70 80 901.205
1.21
1.215
1.22
1.225
1.23
1.235
1.24
Ma
gn
etic
hys
tere
sis
loss
(W
/kg
)
Stage: 1 Shear-1 = 0.9580
Stage: 2 Plane strain
compression ( = -0.1597 )
Stage: 1 Shear -1 = 0.9454
Stage: 2 Rotation-1 ( = -0.2748)
Stage 1: Tension = 0.9495
Stage 2: Shear-1 = 0.3384
Stage 1: Tension = 0.9699
Stage 2: Rotation-1
= -0.2408
Classification
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LIMITATIONS OF STATISTICAL LEARNING BASED DESIGN SOLUTIONSLIMITATIONS OF STATISTICAL LEARNING BASED DESIGN SOLUTIONS
Classification alone does not yield the final design solution
• Why? Since it is impossible to explore the infinite design space within a database of reasonable size.
• Use statistical learning for providing initial class of solutions
• Use local optimization schemes (details not given in this presentation) to identify the exact solutions
Response surface
Ob
ject
ive
to b
e m
inim
ized
Microstructure attributes
Stat Learning Design solutions
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DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEMDESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Iteration Index
No
rma
lize
d o
bje
ctiv
e fu
nct
ion
Initial guess, = 0.65, = -0.1
Desired ODF Optimal- Reduced order control
Full order ODF based on reduced order control parameters
Stage: 1 Plane strain compression ( = 0.9472)
Stage: 2 Compression ( = -0.2847)
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DESIGN FOR DESIRED MAGNETIC PROPERTYDESIGN FOR DESIRED MAGNETIC PROPERTY
Iteration Index
No
rma
lize
d o
bje
ctive
fu
nctio
n
5 10 150
0.2
0.4
0.6
0.8
1
h
Crystal <100> direction.
Easy direction of
magnetization – zero power
loss
External magnetization direction
0 20 40 60 80
1.21
1.215
1.22
1.225
1.23
1.235
Angle from the rolling direction
Ma
gn
etic
hys
tere
sis
loss
(W
/Kg
) Desired property distributionOptimal (reduced)Initial
Stage: 1 Shear – 1 ( = 0.9745)
Stage: 2 Tension ( = 0.4821)
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DESIGN FOR DESIRED YOUNGS MODULUSDESIGN FOR DESIRED YOUNGS MODULUS
Stage: 1 Shear ( = -0.03579)
Stage: 2 Tension
( = 0.17339)
Stiffness of F.C.C Cu in crystal frame
Elastic modulus is found using the polycrystal average <C> over the ODF as,
0 10 20 30 40 50 60 70 80 90143.6
143.8
144
144.2
144.4
144.6
144.8
145
145.2
145.4
Angle from the rolling direction
Yo
un
gs
Mo
du
lus
(GP
a)
Desired property distributionInitialOptimal (reduced)
1 2 3 4 5 6 7 80.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iteration Index
Nor
mal
ized
obj
ect
ive
func
tion
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WHAT WE SHOULD KNOW
How to “learn” microstructure/process/property relationships given computational and experimental data
•Be happy with probabilistic tools: Bayesian analytics and Gaussian mixture models
•Understand simple tools like K-MEANS that can be readily used.
•Understand SVMs as a versatile statistical learning tool: For both feature selection and classification
Apply statistical learning to perform real-time decisions under high degrees of uncertainty
Appreciate the uses and understand the limitations of statistical learning applied to materials
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USEFUL REFERENCESUSEFUL REFERENCES
• Andrew Moore’s Statistical learning course online:
http://www-2.cs.cmu.edu/~awm/tutorials/
• Books:
R.O. Duda, P.E. Hart and D.G. Stork, Pattern classification (2nd ed), John Wiley and Sons, New York (2001).
Example papers on microstructure/materials related applications for the tools presented in this talk:
• V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp. 4111-4119, 2004
• V. Sundararaghavan and N. Zabaras, "Classification of three-dimensional microstructures using support vector machines", Computational Materials Science, Vol. 32, pp. 223-239, 2005
• V. Sundararaghavan and N. Zabaras, "On the synergy between classification of textures and deformation process sequence selection", Acta Materialia, Vol. 53/4, pp. 1015-1027, 2005
• T J Sabin, C A L Bailer-Jones and P J Withers, Accelerated learning using Gaussian process models to predict static recrystallization in an Al–Mg alloy, Modelling Simul. Mater. Sci. Eng. 8 (2000) 687–706
•C. A. L. Bailer-Jones, H. K. D. H. Bhadeshia and D. J. C. MacKay, Gaussian Process Modelling of Austenite Formation in Steel, Materials Science and Technology, Vol. 15, 1999, 287-294.
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THANK YOUTHANK YOU