INTRODUCTION
Knitted fabrics are usually preferred for underwear, casual wear and sportswear
because their stretchability and elasticity which makes them comfortable and
provided more transpiration than other type of fabrics. Plated elastane yarns,
into knitted fabrics, have been used to enhance these properties. Thermal conductivity
is one of the main clothing comfort properties. It influences thermal comfort
to the wearer, also the ‘coolness’ and ‘warmness’ to touch.
This thermal characteristic becomes important depending on the season in which
the cloth is expected to be used. During the winter season, the fabric with
‘warm’ sensation will be appreciated by customers and vice versa (Oglakcioglu
and Marmarali, 2007; Ciukas et al., 2010).
Thermal comfort properties of fabrics are influenced by fiber type, yarn properties
and fabric structure (Le Pechoux et al., 2001).
The influence of yarn properties on the thermal comfort properties of several
fabrics has been reported by researchers such as Ozdil et
al. (2007), Das and Ishtiaque (2004), Du
et al. (2007), Ozcelik et al. (2007),
Oglakcioglu and Marmarali (2007). Majumdar
et al. (2010) compared the thermal conductivity of three weft knitted
fabric structures (plain, rib, interlock) prepared from regenerated bamboo,
cotton and blended (cottonbamboo) yarns. They found that the thermal conductivity
of knits increases when the proportion of bamboo fiber decreases. The thermal
conductivity was higher for fabrics made from thicker yarns. The interlock knitted
fabrics have the highest values of thermal conductivity followed by the rib
and plain knitted fabrics. Stankovic et al. (2008)
compared the thermal conductivity of plain knitted fabrics made from cellulosic
(cotton) and regenerated cellulosic (viscose) fibres. They found that viscose
knitted fabrics have the highest thermal conductivity. This is due to the further
hairy surface of the viscose yarn. The viscose knitted fabric interstices consist
of air and the fibers exceeding the yarn surface, thus it could be assumed that
both conduction and convection mechanisms occur. Though, it seems that the conduction
by fibers is a principal heat transfer mode, since the knitted fabrics made
from viscose fiber revealed the highest thermal conductivity.
There have been some researches studies the influence of elastane incorporation
on the thermal comfort properties of different fabrics (Gorjanc
et al., 2012; Cuden and Elesini, 2010; Tezel
and Kavusturan, 2008). The influence of some types of fibers of the socks
on the thermal conductivity of plain knits and plated with textured polyamide
or elastane thread was studied by Ciukas et al. (2010).
It was founded that thermal conductivity was lower for socks (knits) made from
pure yarns and those with Lycra thread: higher for socks with textured polyamide
thread. The thermal conductivity coefficient slightly decreases when the linear
density greatly decreases; Lycra thread changes the porosity, area density,
thickness and thus thermal conductivity of knitted fabrics. They also noted
that no linear correlation was found between the thermal conductivity and area
density or thickness when knitted fabrics made from pure yarns and plated with
polyamide or Lycra thread were used. Chidambaram et al.
(2011) investigated the influences of yarn linear density and loop length
on the thermal comfort properties of bamboo knitted fabrics. They found that
the thermal conductivity tended to decrease with an increase in loop length
but increase with the constituent yarn linear density.
These previous researches were studying the causeeffect relationships between
yarn and fabric parameters and thermal comfort properties using statistical
methods. But, these techniques have some limits. The most common problem faced
in statistical modeling is the nonlinear relationship between structural parameters
and functional properties. In addition, the greater parts of previous studies
haven’t considered the combinational effects of several factors. Without
considering the complex interactions of the various factors at the different
processing stages, the weight of each factor and their synergistic effect on
thermal conductivity cannot be fully understood.
Therefore, this complex industrial phenomenon depends on numerous factors that
handicap mathematical modeling. During the last decade, numerical simulations
based on mathematical models in the form of differential equations have been
commonly used in engineering fields and porous media area (Admon
et al., 2011; Du et al., 2007; Ganesh
and Krishnambal, 2006; Hasan et al., 2011;
Layeghi et al., 2010).
Numerical simulations help to reduce tooling’s costs and machine setup
times as well as optimize operating variables to achieve desired final functional
parameters. However, have several shortcomings: (1) A constitutive equation
must be used for adequately describing the nonlinearity between inputs and outputs,
(2) these equations suffer from limitations in terms of proper incorporation
of various factors affecting a complex industrial mechanisms, (3) modeling requires
numerous simplifying assumptions, consequently leading to a limited accuracy
of final results and (4) Numerical simulations have no ability to handle effects
of all inputs parameters at the same time, also they require too great a computational
effort for online use.
The artificial neural network has been applied widely to various fields. The
advantages of using neural networks technique over simulations based on numerical
analysis include: (1) capability to take into account the non linear relationship
of structural and functional parameters, (2) no need for constitutive equations,
(3) online prediction for industrial process, (4) universal function approximation,
(5) Accurateness and robustness, (6) no or a negligible number of simplifying
assumptions, (7) ability to learn from examples and (8) very reduced computation
time (Suzuki, 2011).
The ANN training is the process by which the ANN model is obtained. It is an
optimization process that involves reducing the slopes of the cost function
until a specified fit between desired and predicted output, is achieved. There
are several ways to formulate the training process and to minimize this cost
function (Bishop, 1995; Cichocki and Unbehauen,
1993) but in order to get the preferred result, two essential questions
must be answered: (1) to what level should stopping the training process? and
(2) how to select, among several models, the optimal one that will give the
best prediction with good accuracy?
Nowadays, several researchers have successfully used “Virtual Leave One
Out” approach to select the optimal ANN architecture to predict various
fabric properties (Alibi et al., 2012; Babay
et al., 2005; Bhattacharjee and Kothari, 2007;
Monari and Dreyfus, 2002). All these researchers have
obtained a high prediction accuracy of the ANN models.
The lack of an objective approach, in textile industry, for determining the
level of thermal clothing comfort which takes into account both operating parameters
and characteristics of yarn and fabric provide a strong motivation for the present
study. When studying the effect of each structural parameter on the functional
properties selected from the final stretch knitted fabric specifications, it
is difficult to produce a large number of samples. In practice, the quantity
of samples is constrained by the experiment or production costs. So it is necessary
to build a model to solve it.
In this study, an effort has been made to set up an optimal ANNbased model
to predict the thermal conductivity of stretch knitted fabrics made from pure
yarn cotton (cellulose) and viscose (regenerated cellulose) fibers and plated
knitted with elastane (Lycra) fibres according to their material, fabric construction
and clothing design. A smallscaled ANN models related to the product specificity
have been built and the optimal one has been selected via the “Leave One
Out” approach. According to the developed model, it would be feasible to
get the optimum combination of operating parameters and characteristics of yarn
and fabric, to attain a desired value of thermal conductivity before designing
a new stretch knitted fabrics.
MATERIALS AND METHODS
The focus of this research was conducted on pure cotton, pure viscose, viscose/Lycra
and cotton/Lycra^{®} plated knitted constructions. A series of 340
knitted fabrics commonly used in the clothing industry were produced by using
different industrial circular knitting machines (single jersey, double jersey,
interlock; tubular and largediameter; Diameter: 1634 inch, gauge: 1828).
Ground yarn was a 100% combed cotton (1) and 100% viscose yarn (2) (Nm: 2880)
and plating yarn was a Lycra^{®} monofilament (22, 33 and 44 dtex)
plated at half feeder. The fabric samples were comprised of nine different knitted
structures, (1) single jersey (2) single lacoste, (3) double lacoste, (4) polo
pique, (5) 1/1 rib, (6) 2/2 rib, (7) interlock, (8) visible molleton and (9)
invisible molleton. The fabric samples were conditioned in the testing laboratory
under standard atmospheric conditions of 20±2°C and 65±2%
relative humidity after a minimum period of 24 h conditioning in an NF ISO17025
certified laboratory. In this study, the tests carried out were concerning the
determination of these parameters according to the French national organization
for standardization (AFNOR). Table 1 shows the maximum, minimum,
average and standard deviation of knit fabric features used under study.
Table 1: 
The maximum, minimum, average and standard deviation of knit
fabric features 


Fig. 1: 
Apparatus for thermal conductivity measurement 
The function parameter, thermal conductivity λ of these samples, is obtained
using the apparatus illustrated in Fig. 1 according to Eq.
1 (Fayala et al., 2008):
Where:
r_{1} 
= 
Radius of heating resistance (m) 
r_{2} 
= 
r_{1}+E is the sample thickness (E) added to radius of heating
resistance (m) 
A 
= 
Area through which the heat is conducted (m^{2}) 
φ 
= 
Heat flow (W m^{2}) through the cylindrical sample (simulates
clothing covering arms, legs, or the human body in general) is used to simulates
the heat exchanges through fabric during wearing 
T_{sk} (K) 
= 
Temperature of the chamois leather (external surface of the heating resistance)
to simulate the thermal behaviour of human skin 
T_{Clo} (K) 
= 
Temperature of external surface of the cylindrical sample 
Here, the heat flow through the sample is:
where, U_{1} is the electric tension applied to resistance when it
was covered by the sample and R_{Ω} is the resistance of heating
element.

Fig. 2(ab): 
(a) General model including only public structural parameters
and (b) Special model including public and special structural parameters 
Modeling with artificial neural networks: In order to solve the problems
caused by the lack of learning samples, small scaled artificial neural networks
models are built according to previous researches (Huang
and Moraga, 2004; Raudys and Jain, 1991; Yuan
and Fine, 1998).
In our case, the structure of knitting fabrics varies with applied technology.
The corresponding stretch knitted fabrics are classified into families each
corresponding to one category of structure. Accordingly, all the operating parameters
and characteristics of yarn and fabric are divided into two groups. One group
contains public parameters available for all the families of knits and the other
group contains special parameters existing for each specific family. Therefore,
two ANN models are developed. The general model (Fig. 2a)
takes into account the public parameters as inputs. This model can be used by
all the families of knits. For each family, a special model was established
(Fig. 2b). It takes into account both the public and the special
parameters as inputs.
The LevenbergMarquardt learning procedure, based on a back propagation algorithm,
is used for calculating the unknown parameters of the general model from the
public learning data sets. In the special model, the weights connecting the
public inputs to the hidden neurons are kept invariable. Merely the weights
connecting the special input neurons to the hidden neurons are calculated during
the learning phase using the error back propagation algorithm.
Selecting the optimal model architecture: The fitted model is expected
not only to recall the observed data with the required accuracy but also to
produce acceptable predictions for unseen (test) data drawn from the same population
as the observed (training) data. Such a model is able to generalize (extrapolate)
well within the range of unseen data.
To estimate the generalization ability of the trained models, the “leaveoneout
score” E_{p} was used according to the following equation:
where, R_{k}^{(k)} is the prediction error on the example
k when the latter has been removed from the training dataset and the learning
phase has been performed with the rest of examples. The leaveoneout errors
R_{k}^{(k)} were computed by the “virtual leaveoneout”
method, described in (Oussar et al., 2004).
In this application, the model is based on p samples of knits fabrics. Training
of ANN was performed with the leave one out technique. After training, the optimal
model architecture was chosen by using a selection methodology (Alibi
et al., 2012; Golub and Van Loan, 1996; Monari
and Dreyfus, 2002; Vapnik, 1999).
RESULTS AND DISCUSSION
The network architecture used a three layered feedforward network with sigmoid
hiddenunit activity and a single linear output unit. There are six knitting
fabrics families different in the formation (simple or complex structure) and
the knitting technologies (simple and double needle machine or interlock machine).
In this study, the data were selected from the test results of yarn and knitted
fabrics properties of the last four years of knitting. A data of 340 samples
was used to train the neural networks. Among these properties we have selected
those having the highest influence on the thermal conductivity, such as the
Yarn Composition, Cotton Yarns Counts (Stankovic et al.,
2008; Ozdil et al., 2007; Das
and Ishtiaque, 2004; Du et al., 2007; Ozcelik
et al., 2007; Oglakcioglu and Marmarali, 2007),
Lycra Proportion and Lycra Yarn Count (Gorjanc et al.,
2012; Cuden and Elesini, 2010; Tezel
and Kavusturan, 2008).
The structure, loop length, weight per unit area, thickness of knitted fabrics
and the gauge of the circular knitting machine were also taken into account
in the input data, as process parameters (Le Pechoux et
al., 2001; Majumdar et al., 2010; Ucar
and Yilmaz, 2005; Chidambaram et al., 2011).
The 340 measurements were randomly divided into a training database of 244
values for training and model selection and a test database of 96 values for
the final evaluation of the generalization performance of the model.
Table 2: 
Statistical values of input and outputs parameters of training
and test set fabrics 

Table 2 presents the statistical values of inputs and output
parameters of training and test set fabrics.
When studying the effect of each structural parameter related to both operating
parameters and characteristics of yarn and fabric on the thermal conductivity,
the number of samples is quite limited because of their long production time
and high production and experiment cost.
Given these constraints, a small set of learning samples related to the classes
of stretch knitted fabrics with similar properties have been used to model the
relationship between structural parameters and thermal conductivity of materials.
On the other hand, in order to maintain the number of unknown parameters of
ANN no bigger than the number of available learning dataset, we should have
a sufficient amount of learning data. With increasing of the number of training
samples, the two error rates (training and testing) converge to the same value
and we have a good network performance.
Therefore, two types of ANN models are set up. The general model takes the
public variables as its inputs. This general model can be used by all the classes
of stretch knitted fabrics. For each specific class, a special model is set
up. It takes into account both the public and the special variables of this
class as inputs.
Thus we can’t use model from specific classes with similar properties
separately without considering the effect of all classes. The specific model
is built based on the architecture and parameters of general model. Just the
connections between the specific input neurons and hidden neurons are added
and weights are calculated. Consequently, we need to build these two types of
models to solve the problems related to the lack of learning samples.
In our case, a general model is built using an artificial neural network technique
for all the stretch knitted samples. A special model is built for the family
of knitting fabrics produced using a specific knitting technology (Exp: interlock
machine). Its architecture is built based on the corresponding general model
and then the Interlock Loop Length (for example) is added to the inputs of the
general model to build the special model corresponding to interlock knitting
family. Figure 3 shows the ANN architecture of the special
model to predict thermal conductivity (λ) with eight public parameters
(Knitted Structure’s, Yarn Composition, Cotton Yarns Counts, Gauge, Lycra
Proportion, Lycra Yarn Count, Weight per Unit Area and Thickness) as input variables.
The special structural parameter is then added to the set of these eight input
variables.
Once the architectures of models are built, we use the leave one out technique
to test the performance of the two types of models. This technique is described
as follows:
• 
We carry out 244 tests related to all training samples and
p tests corresponding to samples of each family 
• 
In each test, we remove one sample from the training dataset
for testing the models 
• 
The remaining 243 samples are used for training the general
model and the remaining p1 samples of the corresponding family are used
for training the special model. Then, for the removed sample, we compute
the difference between the experimental and predicted value of thermal conductivity
from both the general and special model 
• 
This technique is repeated for 244 times, thus all samples
can be used for testing the models 
Optimum neural network architecture: Models with an increasing of number
of Hidden Neurons (HN) (i.e., increasing complexity) was trained started from
zero HN (linear model) and the virtual leaveoneout score E_{p} of
each model was computed.

Fig. 3: 
Special model for the interlock knitting family 
Table 3: 
Optimization of the number of hidden neurons for the neural
networks 

The root mean square error (RMSE_{tr}) and coefficient of correlation
(R^{2}_{tr}) on the training datasets were also calculated;
results are shown in Table 3.
As expected, E_{p} decreases when the number of HN increases and starts
increasing when the number of parameters is big enough for overfitting to arise.
In the other hand, RMSE_{tr} on the training dataset decreases when
the number of HN increases (Table 3). Furthermore, it should
be noted that when the number of HN exceeds the eight, the learning task improved
but the generalization ability decreased. In fact, the generalization error
degraded and the overfitting phenomenon start to occur.
These results can be explained by the fact that ANN model with small number
of HN, has no sufficient complexity to extract the nonlinear relationship between
the input and outputs variables. On the other hand, an ANN model with too many
HN can perfectly adjust the learning samples and the model dedicated a large
part of its degrees of freedom to learn these samples. Therefore, it fit the
noise that exist in the data and bestow predictions deprived of significance
because its performance will depend for the largest part on the particular learning
set. Hence, the chosen model should exemplify the optimum tradeoff between training
aptitude and generalization ability. In our case, the generalization error does
not increase considerably when the number of HN exceeds the eight. Therefore,
with the purpose to reduce the number of model’s parameters, eight HN were
chosen. The optimized ANN architectures are shown in Fig. 3,
corresponding to 81 parameters.
According to these results, choose the ‘virtual leave one out’ approach
can be argued since it could takes over fitting into account based on the leverages
of the learning samples, i.e., on the influence that each sample has on the
parameters of the ANN model. Therefore it permits to resolve during the learning
of ANN the overfitting phenomenon.
With the purpose of comparing modeling with an ANN technique and a conventional
method used in the previous works (Ciukas et al.,
2010), a multiple linear regression model was developed to predict thermal
conductivity using the same training data set. The scatter plots for thermal
conductivity prediction, on the training set, from both ANN and linear model
together are shown on Fig. 4. Contrarily to the linear model,
the ANN model gave improved prediction results: In fact, the regression coefficient
between the experimental and the ANN predicted thermal conductivity is higher.

Fig. 4: 
The prediction of thermal conductivity from both neural network
and linear regression models for training dataset 

Fig. 5: 
The prediction of thermal conductivity from both neural network
and linear regression models for validation dataset 
In addition, as shown in Table 3, the learning and the generalization
errors are minimized with about 62% and 58% respectively.
Model validation: To test the generalization performance of the optimal
trained network, validating processes was applied using the test database (Table
2). The important quality indicator of an ANN modeling is its generalization
capability to predict the output from unseen data with good accuracy. The experimental
versus predicted values of validation dataset by both models together is shown
in Fig. 5. The coefficient of correlation (R_{T}^{2})
and mean absolute relative errors (MARE_{T}) on the validation dataset
were computed for both neural and linear model and results are summarized in
Table 4.

Fig. 6: 
Comparison general/special models, related to thermal conductivity 
Table 4: 
Summary of validation results of both linear and neural models 

Table 5: 
Results of comparing of general versus special model on thermal
conductivity for all product families 

It can be observed that the ANN model performs greater (average error = 4%).
It is also important to note that the maximum error is lowest in the neural
model and smaller than errors that generally occur due to experimental deviation
and instrumentation precision.
As shown in Fig. 5, the correlation coefficients (R^{2}_{tr})
between the experimental and the predicted thermal conductivity were 0.96 and
0.093, respectively for the ANN and the linear models. As it can be observed,
the predictability of ANN fits very well.
Furthermore, in all cases (training and testing) the neuronal predictions were
higher than the linear predictions. Therefore, the ANN model provides better
performers than the linear model for prediction the thermal conductivity of
stretch knitted fabrics. The ANN model was greater since the linear one was
unable to take into account the nonlinear relationship and the complex interaction
that exists between raw materials properties, operating parameters and thermal
conductivity, something the ANN technique does.
Prediction assessment of the product functional properties: Figure
6 compares the predicted values of thermal conductivity obtained from both
the general and special models and the corresponding experimental measures.
It demonstrated good agreement from special models (R^{2}_{tr}>0.95).
In Table 5 the experimental results on the thermal conductivity
(λ) and the corresponding predicted results obtained from the two models
were presented. According to these results, we can notice that the special models
give better prediction (averaged error: 4%) compared to the general models (averaged
error: 9%). This can be explained as follows: (1) the learning phase corresponding
to general model use samples from numerous families which differ from each other
in several aspects while the special model learning only uses samples from the
same family. The general model cannot benefit from the specificity of each stretch
knitted fabrics family; (2) the special model is developed based on the same
neural network as the general model. Merely the weights connecting the specific
input neurons to hidden neurons are added.

Fig. 7: 
Experimental and neural network predict values of thermal
conductivity for different knitted structure’s 

Fig. 8: 
Experimental and neural network predict values of thermal
conductivity for different yarn composition 
Hence, this model benefit from both the specificity of each stretch knitted
fabrics family and the generality of all these families.
Typical plots of the experimental and ANN predicted values of selected product
are presented and the results was discussed as following: whatever the knitted
structure’s the predictability of ANN fits very well, solving the lack
of samples of some knitted structure’s due to production constraints (Fig.
7). At the same time, the model accurately predicted the expected thermal
conductivity at high and lower values. While for some materials (i.e., cotton,
viscose) the predicted values of thermal conductivity closely matched that of
experimental values (Fig. 8), showing the prediction ability
of ANN model whatever the type of raw materials and the available amount of
data. Besides, it was able to predict thermal conductivity values with acceptable
accuracy whatever the value of gauge (Fig. 9). The developed
ANN model is expected to be used for different industrial circular knitting
machines.
The simulation results show robustness with good accuracy using the special
models for extreme value of Yarn Count (Fig. 10), Lycra Proportion
(Fig. 11) and Lycra Yarn Count (Fig. 12).
It should be pointed that this modeling procedure has proved its capability
to process the existing constraints in previous works (Admon
et al., 2011; Hasan et al., 2011;
Layeghi et al., 2010) such as initial and boundary
conditions.
This study exemplifies the feasibility of modeling the thermal conductivity
based to raw materials properties and operating parameters. This is mainly helpful
in selecting and blending structural parameters, then simulating and predicting
the thermal conductivity and finally deciding about the optimum combination
to design a new stretch knitted fabrics with the desired thermal property.

Fig. 9: 
Experimental and neural network predict values of thermal
conductivity for different gauge 

Fig. 10: 
Experimental and neural network predict values of thermal
conductivity for different yarn count 

Fig. 11: 
Experimental and neural network predict values of thermal
conductivity for different Lycra proportion 
Comparing the results of previous works (Fayala et al.,
2008; Alibi et al., 2012) to those of this
study, it should be pointed that the developed ANN model goes beyond the use
of thermophysical parameters to predict thermal comfort properties and helps
engineer to decide about designing a new product before the actual manufacturing
of stretch knitted fabrics by taken into account, mainly, the operating parameters,
a large types of knitted structure's and elastane characteristics.

Fig. 12: 
Experimental and neural network predict values of thermal
conductivity for different Lycra yarn count 
Besides, using elastane characteristics as inputs of ANN model to predict thermal
comfort properties of fabrics didn't investigated before.
CONCLUSION
In this study, a support system is proposed for modeling the thermal conductivity
of knitted fabrics made from pure yarn cotton (cellulose) and viscose (regenerated
cellulose) and plated knitted with elastane (Lycra) fibers using special models
of ANN to solve the constraints related to the lack of samples. The virtual
leave one out approach dealing with over fitting phenomenon and allowing the
selection of the optimal neural network architecture was used.
The selected ANN model was compared to linear analysis. The results prove the
superiority of ANN model, showing the lowest learning and generalization errors.
Besides, both errors were less than the experimental deviation and instrumentation
precision. The developed ANN model was tested on unseen data and has provided
greater results. In fact, the regression coefficient between the experimental
and the predicted thermal conductivity from neural model was equal to 0.96 while
that from the linear one was 0.093. Before designing a new product, the industry
can provide a possible combination of input variables and predict the expected
thermal conductivity value of the stretch knitted fabrics using the developed
ANN model. If the predicted value does not converge as close as possible to
a target value of thermal conductivity, then the industry can adjust the values
of the input variables, to attain the target value. Therefore, the desired thermal
conductivity of the stretch knitted fabric can be obtained more systematically,
substituting the classic hitandtrial approach.