ordinal logistic regression analysis for statistical determination of forming limit diagrams b.m....

Ordinal Logistic Regression Analysis Ordinal Logistic Regression Analysis for Statistical Determination offor Statistical Determination of

Forming Limit DiagramsForming Limit Diagrams

B.M. ColosimoB.M. Colosimo

Università di CassinoUniversità di CassinoDip. Ingegneria IndustrialeDip. Ingegneria Industriale

http://webuser.unicas.it/tslhttp://webuser.unicas.it/tsl

ESAFORM 2006ESAFORM 2006

M. StranoM. Strano

Politecnico di MilanoPolitecnico di MilanoDip. Dip. MeccanicaMeccanica

http://tecnologie.mecc.polimi.ithttp://tecnologie.mecc.polimi.it

Ordinal logistic regression for FLD – Ordinal logistic regression for FLD – Strano, ColosimoStrano, ColosimoESAFORM 2006ESAFORM 2006

22/21/21

MotivationMotivation

• Scatter is usually quite large in FLD dataScatter is usually quite large in FLD data• Effective statistical tools are strongly Effective statistical tools are strongly

needed for a correct experimental needed for a correct experimental determination of formabilitydetermination of formability


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Some remarks on the FLDsSome remarks on the FLDs

An FLD taken from the literatureAn FLD taken from the literature[C.L. Chow, M.

Jie / I. J. Mech. Sc. 46 (2004)]

•Some points Some points will always will always fall outside fall outside the predicted the predicted FLDFLD

uncertaintyuncertainty


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Some remarks on the FLDsSome remarks on the FLDs

Another FLD taken from the literatureAnother FLD taken from the literature[D. Banabic et al. /

Modelling Simul. Mater. Sci. Eng. 13 (2005)]

•Experimental Experimental data are used data are used to compare to compare different modeldifferent model


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Uncertainty and multiple responseUncertainty and multiple response

• UncertaintyUncertainty– On the position and shape of the “true” FLDOn the position and shape of the “true” FLD

• Some use the concept of Some use the concept of safety regionsafety region or or forming limit bandforming limit band

• Statistical methods should be used to account Statistical methods should be used to account for uncertainty and a large number of experiments for uncertainty and a large number of experiments (replicates) should be conducted for each FLD(replicates) should be conducted for each FLD

• Multiple responseMultiple response– Experimental results are not simply safe and failed Experimental results are not simply safe and failed

but are generally classified in to 3 different setsbut are generally classified in to 3 different sets• SafeSafe• In the neck In the neck

fieldfield• NeckedNecked

• SafeSafe• NeckedNecked• FractureFracture

dd

eithereither oror


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• Uncertainty (40 papers in the literature)Uncertainty (40 papers in the literature)

• Multiple response (in the literature)Multiple response (in the literature)– Practically no paper deals (on an experimental and quantitative Practically no paper deals (on an experimental and quantitative

base) with the prediction of 2 different types of failurebase) with the prediction of 2 different types of failure


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• Proposed solution: Proposed solution: probability mapprobability map– A statistical tool for the determination or the A statistical tool for the determination or the

quantitative evaluation of FLDs can be useful, able quantitative evaluation of FLDs can be useful, able toto

• deal with deal with 33 different data categories different data categories

• provide the probability of failure associated with provide the probability of failure associated with each point on the each point on the ee11-e-e22 space space


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The probability mapThe probability map

[M. Strano B.M. Colosimo / Int. J. of Mach. Tools and Manuf., 46, 6 (2006) ]

Map obtained by Map obtained by binary logistic regressionbinary logistic regression Points are Points are

labeled only as labeled only as safesafe or or failedfailed

11 is the is the probability of a probability of a point being on point being on the safe sidethe safe side

The Forming The Forming Limit Band Limit Band ((FLBFLB) has been ) has been obtained by obtained by linear linear regression regression analysisanalysis


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The The binarybinary logistic regression logistic regression

• A new response variable is introduced, a (Bernoulli) A new response variable is introduced, a (Bernoulli) random variable random variable zz which assumes which assumes

– the value the value 11 with probability with probability 11 if the observed strains if the observed strains characterize a safe pointcharacterize a safe point

– the value the value 00 with probability with probability 00 if the observed strains induced a if the observed strains induced a failurefailure

• Binary logistic regression computes the probability of Binary logistic regression computes the probability of observing observing z=1z=1 as function of minor and major strains as function of minor and major strains ((yyee11,, x xee22))

1 1

1 10 1

ˆˆ ˆln ln1

q ri ji j

i ja c y d x

logitlogit link function link function polynomial modelpolynomial model


1010/21/21

The The binarybinary logistic regression logistic regression

are the maximum likelihood estimates of the true are the maximum likelihood estimates of the true coefficients and are obtained with an iterative coefficients and are obtained with an iterative weighted least squares algorithm implemented in weighted least squares algorithm implemented in most statistical software packagesmost statistical software packages

1 1

1 10 1

ˆˆ ˆln ln1

q ri ji j

i ja c y d x

logitlogit link function link function polynomial modelpolynomial model

ˆˆ ˆ, ( 1,..., ), ( 1,..., )i ja c i q d j r


1111/21/21

The The ordinalordinal logistic regression logistic regression

• A response random variable A response random variable z(x,y)z(x,y) which assumes which assumes– the value the value ss with probability with probability ss if the observed strains if the observed strains

characterize a safe pointcharacterize a safe point

– the value the value mm with probability with probability mm if the observed strains induced if the observed strains induced

an almost failed (or necked) pointan almost failed (or necked) point

– the value the value ff with probability with probability ff if the observed strains induced a if the observed strains induced a

failure (or fracture)failure (or fracture)

• The sum of the three probabilities is equal to oneThe sum of the three probabilities is equal to one

[[ss ((x,yx,y)+ )+ mm ((x,yx,y)+ )+ ff ((x,yx,y)]=1)]=1


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• Ordinal logistic regression computesOrdinal logistic regression computes

exp ,( , ) ;

1 exp ,

ss

s

a x yx y

a x y

2 21 2 1 2 1, ... ... ...x y b x b x c y c y d xy polynomipolynomi

al modelal model• Not all polynomial terms up to a given

degree must necessarily be included• Several alternatives should be tried until

the best model is found, while requiring the smallest number of terms (following a parsimony principle)


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diagnostidiagnostic c measuremeasuress

Material: Al 6022-T4; data in Fig. 1a Goodness-of-Fit Tests

Method 2 DF p-value Pearson 67.48 121 1

Deviance 54.02 121 1

Test that all slopes are zero:

Log-likelihood=-27 G=79.7; DF=3;

P-Value=0.0 Measures of Association

Pairs Number % Summary Measures Concordant 1206 94.7% Somers' D 0.89 Discordant 67 5.3% Goodman-Kruskal 0.89

Ties 1 0.1% Kendall's -a 0.58

Somers’ DSomers’ D is similar to is similar to rr22 in linear regression in linear regression


1414/21/21

Application of the method [1]Application of the method [1]

exp , exp , exp ,;

1 exp , 1 exp , 1 exp ,

s m ss m

s m s

a x y a x y a x y

a x y a x y a x y

2 2

1 2 1 2 1... ... ...b x b x c y c y d xy

Response Count s 28

m 14 f 21

Material: Al 6022-T4; data in [1]

Coefficients value Std. Error p-value

sa 21.223 4.656 0.000

ma 24.530 5.283 0.000

1b -73.78 19.47 0.000

2b 706.5 285.3 0.013

2c -449.31 98.74 0.000

modelmodel


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ss+ + mmx x ff

probability mapprobability map


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ss+ + mmx x ff

probability mapprobability map

5182-o5182-o


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Determination of a single FLD curveDetermination of a single FLD curve

A prescribed A prescribed minimum safety minimum safety probability probability ss

must be selected must be selected by the userby the user

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

0.05

0.06

0.07

0.08

0.09

0.1

Material: Al 5182-o

yy

xxss=0.9=0.9

ss=0.8=0.8ss=0.7=0.7


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Comparison with other FLDsComparison with other FLDs

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0.12Material: Al 6022-T4

yy

xx

ss

s+ mx f

•Any other FLD Any other FLD would most would most certainly cross certainly cross the iso-the iso-ss lines lines

• It is not It is not iso-iso-probabilisticprobabilistic

•Many interpret Many interpret the distance of the distance of a point from the a point from the FLD as a safety FLD as a safety factorfactor

•This is wrongThis is wrong


1919/21/21


•Probability maps are slightly differentProbability maps are slightly different•The most appropriate must be chosenThe most appropriate must be chosen

Binary vs. ordinal regressionBinary vs. ordinal regression

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0.12Material: Al 5182-o ss

Ordinal regression with 3 Ordinal regression with 3 data sets: data sets: ss, , mm, , ff

Binary regression with 2 Binary regression with 2 data sets: data sets: ss, , mm U U ff

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Ordinal regression with 3 Ordinal regression with 3 data sets: data sets: ss, , mm, , ff

Binary regression with 2 Binary regression with 2 data sets: data sets: ss, , mm U U ff

yyyy

xxxx


2020/21/21

ConclusionsConclusions

• The mathematical formulation of the logistic The mathematical formulation of the logistic regression model has been presented, as a regression model has been presented, as a method for experimental determination of method for experimental determination of FLDsFLDs

• The method canThe method can– provide a single, statistically determined, FLD curveprovide a single, statistically determined, FLD curve

• if a tolerable failure probability is fixedif a tolerable failure probability is fixed

– provide a probability map of failureprovide a probability map of failure– deal with binary or multiple response of deal with binary or multiple response of

experimentsexperiments– Give a quantitative indication of goodness of fit of Give a quantitative indication of goodness of fit of

any modelany model


2121/21/21

Contents of this presentationContents of this presentation

• Some remarks on the FLDsSome remarks on the FLDs– FLDs taken from the literatureFLDs taken from the literature– Uncertainty and multiple responseUncertainty and multiple response

• The probability mapThe probability map– The binary logistic regressionThe binary logistic regression– The ordinal logistic regressionThe ordinal logistic regression

• Application of the proposed methodApplication of the proposed method– ModelModel– Probability mapProbability map– Diagnostic measuresDiagnostic measures

• ConclusionsConclusions

ordinal logistic regression analysis for statistical determination of forming limit diagrams b.m....

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