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ESAFORM 2006
Ordinal Logistic Regression Analysis for
Statistical Determination of
Forming Limit Diagrams
M. Strano
B.M. Colosimo
Università di Cassino
Dip. Ingegneria Industriale
http://webuser.unicas.it/tsl
Politecnico di Milano
Dip. Meccanica
http://tecnologie.mecc.polimi.it
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Motivation
• Scatter is usually quite large in FLD data
• Effective statistical tools are strongly
needed for a correct experimental
determination of formability
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Some remarks on the FLDs
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
uncertainty
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
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
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Uncertainty and multiple response
• 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
either
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• Safe
• In the neck field
• Necked
or
• Safe
• Necked
• Fractured
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Uncertainty and multiple response
• Uncertainty (40 papers in the literature)
Number of tests
yes;
17.9
%
ea
r
no;
82.1
%
un
cl
>3
0
0
20.5%
17.9%
17.9%
20
-3
0
10
-2
<1
0
23.1%
20.5%
Use of statistics
• Multiple response (in the literature)
– Practically no paper deals (on an experimental and
quantitative base) with the prediction of 2 different types of
failure
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
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
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
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
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[M. Strano B.M. Colosimo / Int. J. of Mach. Tools and Manuf., 46, 6 (2006) ]
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
The binary logistic regression
• A new response variable is introduced, a (Bernoulli)
random variable z which assumes
– the value 1 with probability p1 if 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)
 p1
ln 
 p0
q
r

 p1 
i
j
ˆ
ˆ
ˆ

ln

a

c

y

d

x





i
j
1

p
i 1
j 1
1


logit link function
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polynomial model
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
The binary logistic regression
 p1
ln 
 p0
q
r

 p1 
i
j
ˆ
ˆ
ˆ

ln

a

c

y

d

x





i
j
1

p
i 1
j 1
1


logit link function
polynomial model
aˆ, cˆi (i  1,..., q), dˆ j ( j  1,..., r )
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
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
The ordinal logistic regression
• A response random variable z(x,y) which assumes
– the value s with probability
characterize a safe point
ps if the observed strains
– the value m with probability pm if the observed strains induced
an almost failed (or necked) point
– the value f with probability
failure (or fracture)
pf if the observed strains induced a
• The sum of the three probabilities is equal to one
[ps (x,y)+ pm (x,y)+ pf (x,y)]=1
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
The ordinal logistic regression
• Ordinal logistic regression computes
p s ( x, y ) 
polynomial
model
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exp  as    x, y  
1  exp  as    x, y  
;
  x, y   b1 x  b2 x 2 ...  c1 y  c2 y 2 ...  d1 xy  ...
• 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)
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
The ordinal logistic regression
diagnostic
measures
Material: Al 6022-T4; data in Fig. 1a
Goodness-of-Fit Tests
Test that all slopes
are zero:
Method 2
DF p-value
Log-likelihood=-27
Pearson 67.48 121 1
G=79.7; DF=3;
Deviance 54.02 121 1
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’ D is similar to r2 in linear regression
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [1]
ps 
exp  as    x, y  
1  exp  as    x, y  
; pm 
exp  am    x, y  
1  exp  am    x, y  

exp  as    x, y 
1  exp  as    x, y 
  b1x  b2 x2 ...  c1 y  c2 y 2 ...  d1xy  ...
model
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Response
s
m
f
Coefficients
as
am
b1
b2
c2
Count
28
14
21
value
21.223
24.530
-73.78
706.5
-449.31
Material: Al 6022-T4;
data in [1]
Std. Error
4.656
5.283
19.47
285.3
98.74
p-value
0.000
0.000
0.000
0.013
0.000
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [1]
probability map
x f
+m
s
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [2]
probability map
x f
+m
s
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5182-o
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [2]
Determination of a single FLD curve
A prescribed
minimum safety
probability ps must
be selected by the
user
Material: Al 5182-o
y 0.1
0.09
0.08
0.07
ps=0.7
ps=0.8
0.06
ps=0.9
0.05
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-0.06
-0.04
-0.02
0
0.02
x
0.04
0.06
0.08
0.1
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [2]
Comparison with other FLDs
• 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
ps
0.12
y
0.9
0.8
0.1
0.7
0.08
0.6
0.5
0.06
0.4
0.04
0.3
x f
+m
0.02
0
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Material: Al 6022-T4
0.2
s
-0.06 -0.04 -0.02
0
0.02
0.04
0.06
0.08
0.1
0.1
x
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
Application of the method [2]
Binary vs. ordinal regression
ps
Material: Al 5182-o
0.12
y
Material: Al 5182-o
0.9 0.12
0.8
0.1
y
0.1
0.7
0.08
0.6 0.08
0.5
0.06
0.06
0.4
0.04
0.3 0.04
Ordinal regression with 3
data sets: s, m, f
0.02
0
-0.06 -0.04 -0.02
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0
0.02
0.04
0.06
0.08
0.2
0.02
x
0.1
0.1
0
Binary regression with 2
data sets: s, m U f
-0.06 -0.04 -0.02
• Probability maps are slightly different
• The most appropriate must be chosen
0
0.02
0.04
0.06
0.08
x
0.1
ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
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
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ESAFORM 2006 Ordinal logistic regression for FLD – Strano, Colosimo
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
21/21