Ordinal Logistic Regression Analysis for Statistical Determination of Forming Limit Diagrams

1. ESAFORM 2006 Ordinal Logistic Regression Analysis for Statistical Determination ofForming Limit Diagrams M. Strano B.M. Colosimo Politecnico di Milano Dip. Meccanica http://tecnologie.mecc.polimi.it Università di Cassino Dip. Ingegneria Industriale http://webuser.unicas.it/tsl

2. Motivation • Scatter is usually quite large in FLD data • Effective statistical tools are strongly needed for a correct experimental determination of formability

3. Some remarks on the FLDs uncertainty An FLD taken from the literature [C.L. Chow, M. Jie / I. J. Mech. Sc. 46 (2004)] • Some points will always fall outside the predicted FLD

4. Some remarks on the FLDs Another FLD taken from the literature [D. Banabic et al. / Modelling Simul. Mater. Sci. Eng. 13 (2005)] • Experimental data are used to compare different model

5. Uncertainty and multiple response • Safe • In the neck field • Necked • Safe • Necked • Fractured either or • Uncertainty • On the position and shape of the “true” FLD • Some use the concept of safety region or forming limit band • Statistical methods should be used to account for uncertainty and a large number of experiments (replicates) should be conducted for each FLD • Multiple response • Experimental results are not simply safe and failed but are generally classified in to 3 different sets

6. Uncertainty and multiple response • Uncertainty (40 papers in the literature) • Multiple response (in the literature) • Practically no paper deals (on an experimental and quantitative base) with the prediction of 2 different types of failure

7. Uncertainty and multiple response • Proposed solution: probability map • A statistical tool for the determination or the quantitative evaluation of FLDs can be useful, able to • deal with 3 different data categories • provide the probability of failure associated with each point on the e1-e2 space

8. The probability map Map obtained by binary logistic regression • Points are labeled only as safe or failed • p1 is the probability of a point being on the safe side • The Forming Limit Band (FLB) has been obtained by linear regression analysis [M. Strano B.M. Colosimo / Int. J. of Mach. Tools and Manuf., 46, 6 (2006) ]

9. The binary logistic regression polynomial model logit link function • A new response variable is introduced, a (Bernoulli) random variable z which assumes • the value 1 with probability p1if the observed strains characterize a safe point • the value 0 with probability p0 if the observed strains induced a failure • Binary logistic regression computes the probability of observing z=1 as function of minor and major strains (ye1, xe2)

10. The binary logistic regression polynomial model logit link function are the maximum likelihood estimates of the true coefficients and are obtained with an iterative weighted least squares algorithm implemented in most statistical software packages

11. The ordinal logistic regression • A response random variable z(x,y) which assumes • the value s with probability psif the observed strains characterize a safe point • the value m with probability pm if the observed strains induced an almost failed (or necked) point • the value f with probability pf if the observed strains induced a failure (or fracture) • The sum of the three probabilities is equal to one [ps(x,y)+ pm(x,y)+ pf(x,y)]=1

12. The ordinal logistic regression • Ordinal logistic regression computes polynomial 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)

13. The ordinal logistic regression diagnostic measures Somers’ D is similar to r2 in linear regression

14. Application of the method [1] model

15. Application of the method [1] x f + m s probability map

16. Application of the method [2] x f + m s probability map 5182-o

17. Application of the method [2] Determination of a single FLD curve A prescribed minimum safety probability ps must be selected by the user y ps=0.7 ps=0.8 ps=0.9 x

18. Application of the method [2] Comparison with other FLDs ps • Any other FLD would most certainly cross the iso-ps lines • It is not iso-probabilistic • Many interpret the distance of a point from the FLD as a safety factor • This is wrong y x f + m s x

19. Application of the method [2] Binary vs. ordinal regression y y x x • Probability maps are slightly different • The most appropriate must be chosen

20. Conclusions • The mathematical formulation of the logistic regression model has been presented, as a method for experimental determination of FLDs • The method can • provide a single, statistically determined, FLD curve • if a tolerable failure probability is fixed • provide a probability map of failure • deal with binary or multiple response of experiments • Give a quantitative indication of goodness of fit of any model

21. Contents of this presentation • Some remarks on the FLDs • FLDs taken from the literature • Uncertainty and multiple response • The probability map • The binary logistic regression • The ordinal logistic regression • Application of the proposed method • Model • Probability map • Diagnostic measures • Conclusions