possibilistic regression in false-twist texturing

1

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)

2

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 =


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


4

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


5

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


2

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.

References:

[1] Dubois, D., Prade, H., Fuzzy Sets and

Systems: Theory and Applications,

Academic, 1980.

[2] Kim, K.J., Maskowitz, H., Koksalan, M.,

Fuzzy versus statistical regression, Euro. J.

Oper. Res, Vol. 92, 1996, pp. 417-437.

[3] Klir, G., Yuan, B., Fuzzy Sets and Fuzzy

Logic, Theory and Applications, Prentice-

Hall, 1995.

[4] Kokot, S., Jermini, M., Characterizing

oxidatively damaged cotton fabric, Part I:

A method for estimating relative fabric

damage, Textile Research Journal, Vol. 64,

No. 2, 1994, pp. 100-105.

[5] Marjoniemi, M., Mantysalo, E., Neuro-

Fuzzy modeling of spectroscopic data, Part

A: Modelling of dye solutions, J.S.D.C,

Vol. 113, pp. 13-17, Part B: Dye

concentration prediction, pp. 64-67, 1997.

[6] Montgomery, D.C., Peek, C.A.,

Introduction to Linear Regression

Analysis, Sec. Ed., 1991.

[7] Pan, N., A new approach to the objective

evaluation of fabric handle from

mechanical properties, Part III: Fuzzy

cluster analysis for fabric handle sorting,

Textile Research Institute, 1988, pp. 565-

575.

[8] Raheel, M., Liu, J., An empirical model

for fabrics hand. Part I: Objective

assessment of light weight fabrics, Textile

Research Institute, 1991, pp. 31.

[9] Taheri, S.M., Trends in Fuzzy Statistics,

Austrian J. Stat., Vol. 32, No. 3, 2003, pp.

239-257.

[10] Tanaka, H., Uejima, S, Asai, K, Linear

regression analysis with fuzzy model,

IEEE Systems, Man & Cybernetics, Vol.

12, 1982, pp. 903-907.

[11] Tanaka, H., Lee, H., Exponential

possibility regression analysis by

identification method of possibilistic

coefficients, Fuzzy Sets. Syst., Vol. 106,

1999, pp.155-165.

[12] Tavanai, H., Denton, M.J., Direct

objective measurment of yarn-torque level,

J. Text. Inst., Vol. 1, 1996, pp. 87-95.

[13] Tavanai, H., Taheri, S.M., Nasiri, M.,

Modelling of colour yield in polyethylene

terephthalate dyeing with statistical and

fuzzy regression, Iranian Polymer J., Vol.

14, 2005, pp. 954-967.

[14] Um, J., A study of the emotional

evaluation models of color patterns based

on the adaptive fuzzy system and the

neural network, Color Research and

Application, Vol. 27, No. 3, 2002, pp. 208-

216.

[15] Yen, K.K., Ghoshray, S., Riog, G., A

linear regression model using triangular

fuzzy number coefficients, Fuzzy Sets

Syst, Vol. 106, 1999, pp. 166-177.


possibilistic regression in false-twist texturing

Documents