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Possibilistic Regression in False-Twist Texturing
S.M. TAHERI*, H. TAVANAI+, M. NASIRI+
*School of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156, IRAN (corresponding author), [email protected]
+Department of Textile Engineering, Isfahan University of Technology, Isfahan 84156, IRAN
Abstract: - A possibilistic linear regression, i.e. a linear regression with possibilistic coefficients, is explained.
The application of such possibilistic regression method for modeling of twist liveliness of false twist textured
nylon yarns as a function of percentage retraction has been studied, based on a few available data.
It turns out that possibilistic regression method is superior to conventional statistical regression, when a very
small number of observations are available. In such cases the basic assumptions, under which statistical
regression analysis is valid, can not be investigated. Based on some criterions, such as the total vagueness of
models and the mean of predictive capabilities, the optimum fuzzy model has been derived.
Key-Words: - Possibilistic regression, Texturing, Predictive capability
1 Introduction and Background Statistical regression analysis is a widely used
statistical tool to model the relationship among
variables to describe and/or predict the
phenomena. Statistical regression is useful in a
non-vague environment where the relationship
among variables is sharply defined.
On the other hand, fuzzy regression analysis
may be used wherever a relationship among
variables is imprecise and/or data are
inaccurate and/or the sample size is
insufficient. In such cases fuzzy regression
may be used as a complement or an alternative
to statistical regression analysis.
Fuzzy regression, for the first time, was
introduced and investigated by Tanaka et al. in
1982 [10]. They, especially, considered the
linear regression model with fuzzy
coefficients, and used linear programming
techniques to develop a model superficially
resembling linear regression. (A survey about
fuzzy regression can be found in [9]).
As mentioned above, one of the application
of fuzzy regression approaches is the cases in
which only a small amount of data is
available. It should be mentioned that classical
statistical regression makes rigid assumptions
about the statistical properties of the model;
e.g., the normality of error terms and the
independence of such errors [6]. These
assumptions, as well as, the aptness of the
model, are difficult to justify unless a
sufficiently large data set is available. The
violation of such basic assumptions could
adversely affect the validity and performance
of statistical regression analysis. Alternatively,
in such cases, fuzzy regression analysis can be
a useful tool [2].
After introducing and developing fuzzy set
theory, many attempts have been made to use
and apply this theory in textile researches. For
example, Raheel and Liu used fuzzy
comprehensive evaluation technique to predict
fabric hand [8]. Fuzzy cluster analysis was
used by Pan for fabric handle sorting [7].
Kokot and Jermini used fuzzy clustering for
estimating cotton damage when treated with
electro generated oxygen at different
temperatures [4]. Mujionemi and Mantysalo
tried to model the relationship between the dye
absorption and dye concentration in dyeing
leather with two dyestuffs by ANFIS [5], (see
also [14]). Tavanai et al. [13] investigated a
fuzzy regression approach for modeling of
colour yield in polyethylene terepthalate
dyeing.
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)
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In the present work, on the basis of a few
data on twist liveliness, we formulate a
possibilistic regression model and then
analyze the data based on such a model.
2 False-twist Texturing False-twist texturing imparts properties such
as bulk, better handle and stretch ability to the
thermoplastic filament yarns such as Polyester,
Polyamide or Polypropylene [12]. This is
fulfilled by heat setting a spirals form along
the filaments constituting the yarn.
Filament yarns with circular cross-section
feel cold, have a slippery surface and on the
whole do not enjoy properties that lead to
Comfort-in-Wear.
Bulked continuous filament and air jet
texturing are the other two main texturing
systems used widely depending on the final
application of the yarn. It must be pointed out
that air jet is not based on the thermoplasticity
of the yarn being textured and the air-jet
textured yarns do not show stretch ability.
In false-twist texturing machine, the
thermoplastic yarn enters the texturing zone
via the first feed rollers and then passes the
heating zone, cooling zone and false twist unit
respectively. The textured yarn finally leaves
texturing zone via second feed roller.
False-twist unit twists the yarn upstream, in
other words, the torque generated by the false-
twist unit is imparted to the yarn moving
downstream. As a result of twist build up, the
filaments twist around yarn axis and assume
helical forms. This leads to a build up of
torsional, compressional and shear stress in the
internal structure of the filaments. As the
twisted yarn enters the heater and moves along
it, the tensioned intermolecular bonds are
broken under the effect of heat and the new
state of the bonds in the internal structure
is set when the twisted yarn is cooled in the cooling zone. So, when the yarn reaches the
false-twist unit, the spiral form is heat-set.
When the yarn leaves the false-twist unit, a
detorque is imparted to the downstream and as
a result, the yarn is twisted by the same
amount but in opposite direction to the
upstream twist. Due to the uncurling of the
spiral form of the filaments, a spring form
yarn is obtained which enjoys the already
mentioned properties such as stretch ability.
Due to the untwisting action of the twisting
unit, the yarn shows a reaction to the imparted
detorque and as a result, it shows a tendency to
snarl when its two ends are brought near each
other. This tendency is called twist-liveliness
or residual torque. Residual torque can have
disadvantages for the fabrics knitted from a
twist-lively yarn. Of course, twist-liveliness
may also be an advantage for special purposes.
Retractive force as well as residual torque of a
false-twist textured yarn are the two main
characteristics of the stretch yarn affecting the
performance of the fabric produced from it.
Both of these factors are functions of the state
of spring form filaments. In other words they
are functions of the percentage retraction. A
fully stretched spring is considered to have a
zero percentage retraction.
It is the aim of this paper to represent the
dependence of the residual torque in a false-
twist textured 22f7 polyamide 66 yarn (22
decitex with 7 filaments) [12] on the
percentage retraction as a model with the help
of possibilistic regression.
3 Formulation of Possibilistic
Regression
3.1 Triangular fuzzy numbers and linear
operations
Definition 1 A fuzzy number A~ is called a
triangular fuzzy number if its membership
function can be expressed as
( )
( )
( )
+≤≤−+
<≤−−−
=R
R
R
L
L
L
saxas
xsa
axsas
sax
xA~
and write ( ) ,,,~
T
RL ssaA = where a is the mean
value of LsA,~
and Rs are called left and right
spreads, respectively. In special case, if
sss RL == then A~ is called symmetric
triangular fuzzy number and we write
( ) .,~
TsaA = Sometimes, we use ( ) ,,,~
T
L ksaA =
where k is a constant so that .LR kss =
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)
3
Linear operations on triangular fuzzy
numbers are easily constructed [1,3]:
Proposition 1 Let ( )T
R
a
L
a ssaA ,,~= and
( )T
R
b
L
b ssbB ,,~= be two triangular fuzzy
numbers. Then
1a) ( ) 0,,,~
>=⊗ λλλλλT
R
a
L
a ssaA
1b) ( ) 0,,,~
<−−=⊗ λλλλλT
L
a
R
a ssaA
2) ( )T
R
b
R
a
L
b
L
a ssssbaBA +++=⊕ ,,~~
3.2 Linear regression with fuzzy coefficients
The general model which is considered in this
study, can be stated in the following way
[10,11,15].
Given the set of observations
( ) ,,...,1,,...,, 1 mjxxy jnjj = find an optimal
fuzzy model such as
nn xAxAAY~
...~~~
110 +++= , (1)
where ( ) ,,...,1,0,,,~
nissaAT
R
i
L
i
c
ii == are
triangular fuzzy numbers.
Not that, based on Proposition 1, the
membership function of Y~ can be shown in
the following way:
( )( )( )
( ) ( ) ( )
( )( )
( ) ( ) ( )
+<≤−
−
<≤−−
−
=
xfxfyxfxf
xfy
xfyxfxfxf
yxf
yY
R
s
cc
R
s
c
cL
s
c
L
s
c
1
1
~
(2)
where
( ) n
c
n
ccc xaxaaxf +++= ...110 , (3)
( ) n
L
n
LL
s xsxssxf +++= ...110 , (4)
( ) n
R
n
RRR
s xsxssxf +++= ...110 . (5)
Here, the main problem is to determine
fuzzy parameters ,,...,1,0,~
niAi = such that
the model (1) has the best fitting with the
given data. In this manner, following [10] and
[15], two criteria were considered to determine
fuzzy coefficients in model (1):
I) For all observations ( ),,...,1 mj = the
membership value of jy (the j-th observed
value of the dependent variable) to its fuzzy
estimate jY~ be at least h, i.e.,
( ) ,,...,1,~
mjhyY jj =≥ (6)
where the value of h is selected by decision
maker for all .,...,1, mjj =
The value of h is between 0 and 1, is referred
to as the degree of fit of the estimated fuzzy
linear model to the given data set.
A physical interpretation of h is that jy is
contained in the support interval of jY~ which
has a degree of membership ,h≥ for all j .
Regarding membership function (2), the
condition (6) can be represented as a pair of
inequality constraints for each set of
observation j as follows:
( ) ( ) ( ) mjyxfxfh jj
c
j
L
s ,...,1,1 =−≥−− (7)
( ) ( ) ( ) mjyxfxfh jj
c
j
R
s ,...,1,1 =+≥+− (8)
II) The total fuzziness in the predicted values
of dependent variable ,,...,1,~
mjY j = must be
minimized. This can be achieved by
minimizing the sum of spreads of fuzzy output
for all the data sets, which is analogous to the
least squares criterion in statistical regression
analysis. Since the membership function of
each fuzzy output jY~ is a function of
( )nAAAA~
,...,~,
~~10= and ( ),,...,1 njjj xxx = the
sum of the spreads of fuzzy outputs is given
by
( ) ( )∑ ∑= =
+++=
n
i
m
j
ji
R
i
L
i
RL xssssmZ1 1
00 (9)
or
( ) ( )∑ ∑= =
+++=
n
i
m
j
ji
L
ii
L xskskmZ1 1
00 11
On the basis of two defined criteria I and II,
the problem of fitting a fuzzy model with
given data set ( ) ,,...,1,,...,, 1 mjxxy jnjj = can
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)
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be posed as an equivalent linear programming
problem as follows:
Find ( ),,...,0
c
n
c aaa = ( ),,...,0
L
n
lLsss = and
( )RnRRsss ,...,0= which
Minimize
( ) ( )∑ ∑= =
+++=
n
i
m
j
ji
R
i
L
i
RL xssssmZ1 1
00 (10)
subject to for all mj ,...,2,1=
jji
n
i
c
i
c
ji
n
i
L
i
L yxaaxshsh −≥−−−+− ∑∑== 1
0
1
0 )1()1(
(11)
jji
n
i
c
i
c
ji
n
i
R
i
R yxaaxshsh ≥++−+− ∑∑== 1
0
1
0 )1()1( (12)
where constraints (11) and (12) are obtained
by substituting (3),(4), and (5) in (7) and (8).
Remark 1 It should be mentioned that, Yen et
al. [15] consider the cost function Z as
( ) ( )∑ ∑= =
+++=
n
i
m
j
ji
R
i
L
i
RL xssssZ1 1
00 , which
does not present the total fuzziness of the
linear model (1). In other words their cost
function is not equal to the sum of spreads of
fuzzy outputs for all the data sets. It seems
that, the above mistake, made their models
unrealistic, see Examples 1 and 2 of [15], in which all the vagueness of the model
concentrated in the intercept, and so there is
no fuzziness in the coefficients of the
exploratory variables.
4 Evaluation of the Models To evaluate the goodness of fit of regression
model, we introduce one of the major uses of
regression analysis is the prediction of the
dependent (response) variable values given the
levels of independent variables. We propose
two indices to measure the predictive
capabilities in a fuzzy regression model.
The fist one is an extended version of the
usual MSE.
4.1 MSE (Mean Squared Error)
Definition 2 For the fuzzy regression model
such as (1), MSE is defined as
where jy
denoted the j-th observed value of the
dependent variable, and ( )jYdef~ is the
defuzzified value of ,~jY based on a
defuzzification method.
In the present work, we use the center of
maxima method [3] for defuzzification. In this
method, when ( ) ,,,~
T
RLc ssaA = then
( ) .~ caAdef =
4. 2 MPC (Mean of Predictive Capabilities)
Definition 3 In the fuzzy regression model (1),
MPC is defined as
( )jm
j
j yYm
MPC ∑=
=1
~1
The amount of MPC shows the average
degrees of membership of observed values iy
in fuzzy predictive values ,~iY which is
calculated on the basis of related amounts of
independent variables.
5 Case Study: Modeling Twist
Liveliness as a Function of Retraction One of the usual problem in texturing is
modeling of certain properties of yarns based
on other easily or cheaply measured
properties.
In this study, we try to model twist
liveliness of false twist textured nylon yarns as
a function of percentage retraction. Due to
some limits, to get the required data, only 11
experiments were carried out. Table 1 shows
the data related to retraction (as the
exploratory variable) and twist liveliness (as
the response variable).
But in this case (in which we have only 11
observations), we can not be certain about the
fulfillment of the basic assumptions of
statistical regression analysis (such as
normality of error, independence of errors, and
so on).
( )[ ]2
1
~1∑=
−=m
j
jj Ydefym
MSE
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)
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Table 1 The data related to twist liveliness.
Retraction (%) 5 10 15 21 30 35 40 45 50 55 65
Twist Liveliness (cN/cm) 1.6 1.75 2 2.25 2.5 2.65 2.85 3.1 3.25 3.35 3.4
In such a case, one may use the alternative
approaches. We employ the fuzzy regression
method, which does not need the above
mentioned conditions [2], to model and
analyze the observations on twist liveliness.
The fuzzy model with triangular fuzzy
coefficients for modeling of twist liveliness of
false twist textured nylon yarns as a function
of percentage retraction can be stated as
follows
( ) ( ) ,,,,, 111000 xksaksaYT
c
T
c +=
where Y is twist liveliness and x is percentage
retraction.
We are going to find the best model, with
credit level h=0.5.
Based on 11 data in Table 1, and adopting
relation (9), the objective function is
( ) ( )∑ ∑
+++=
i j
ji
R
i
L
i
RL xssssmZ 00
From a priori information we select, 4.10 =k
and .11 =k Then Z can be written as: LL ssZ 10 3714.15 += .
In addition, we must formulate 22 constraints
related to 11 observations, based on relations
(11), and (12). For example, two constraints
corresponding to the first observation, with
h=0.5, are:
6.1555.25.0 110010 −≥−+−++ ccccLL aaaass
6.1555.27.0 110010 ≥+++++ ccccRR aaaass
By minimizing the objective function Z
subject to 22 obtained constraints, with linear
programming methods, the coefficients of the
model are calculated as follows:
( ) ( ) ,005.0,033.0,7,0.0631.483,0.04 10 TT AA ==
Therefore the possibilistic regression model is:
( ) ( ) xY TT 005.0,033.0063,0,047,0,483.1 +=
We could select several different values for
0k and 1k , and derive the best model in each
case. The results are shown in Table 2. In this
Table, the variation of MSE, MPC, and Z
(total vagueness of the model) are given, too.
Table 2 Best models on the basis of different values for 0k and 1k .
No Condition Model MSE MPC Z
1
Symmetric
(k0=k1=1) Y=(1.478,0.056)+(0.033,0.005)x 0.0114 0.6499 2.630
2
Non Symmetric
(k0=1.2) Y=(1.481,0.051,0.061)+(0.033,0.005)x 0.0112 0.6422 2.685
3
Non Symmetric
(k0=1.4) Y=(1.483,0.047,0.066)+(0.033,0.005)x 0.0112 0.6347 2.731
4
Non Symmetric
(k0 =1.7) Y=(1.485,0.042,0.071)+(0.033,0.005)x 0.0111 0.6236 2.789
5
Non Symmetric
(k0=1.9) Y=(1.487,0.039,0.074)+(0.033,0.005)x 0.0111 0.6169 2.820
Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)
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Analyzing the above results, we find that:
1) As the MSE and MPC decrease, the value
of Z (total vagueness of the model) increases.
2) In non-symmetric cases, as the values of ik
increase, we are led to models with smaller
MSE and MPC, and larger amount for .Z
Finally, one would choose an optimal
model, regarding the three criterions: MSE,
MPC, and Z.
6 Conclusion This research employed possibilistic (fuzzy)
regression models for modeling twist
liveliness of false twist textured nylon yarns as
a function of percentage retraction. The
optimum model has been selected based on
some criterions, such as the total vagueness of
models, mean absolute errors and enabling the
mean of predictive capabilities. The sensitivity
analysis based on the credible level h, may be
one topic for the future researches.
Acknowledgement
This work was partially supported by the
CEAMA, Isfahan University of Technology,
Isfahan 84156, Iran. The authors also grateful
to the Fuzzy Systems and Its Applications
Center of Excellence, Shahid Bahonar
University of Kerman, Iran.
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Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp202-207)