2.1. Selection of relevant physical features using fuzzy sensitivity (FIS1 model)

The fuzzy sensitivity criterion FS for all the physical features related to a sensory descriptor is defined in Section 1 according to the same principles given by Blum and Langley¹⁰. These principles are transformed into a set of fuzzy rules for building a Fuzzy model in which the physical data variation Δx (distance between two normalized vectors in the physical feature space) and the sensory data variation Δy (distance between two normalized values of one specific sensory descriptor) are taken as two input variables, respectively, and the general sensitivity FS as output variable (model FIS1 in Fig. 1). Evidently, FS is a function of Δx and Δy, denoted as FS = FIS1 (Δx, Δy). This fuzzy model includes an interface of fuzzification, a base of fuzzy rules, an inference mechanism and an interface of defuzzification.

The fuzzification procedure aims to uniformly partition each of the two input variables into three fuzzy values: Small (S), Medium (M) and Big (B). The output variable is a fuzzy variable varying from 0 to 1 and composed either of three fuzzy values: Small (S), Medium (M) and Big (B) (illustrated in Fig. 2)³⁰.

Fig. 2.

Fuzzy values for each input and output data variable in FIS1 model

According to the experience of the quality experts on the relationship between physical and sensory parameters, a set of fuzzy rules are defined in Table 2. As the output variable includes fuzzy values, Mamdani’s fuzzy inference method is used for aggregating these fuzzy rules and obtaining defuzzified output values³¹.

Given a specific sensory descriptor y_β, for any pair of data samples (X_i, y_iβ) and (X_j, y_jβ) denoted as (i, j), the physical data variation d(X_i, X_j) is calculated as follows:

The corresponding sensitivity in the data pair (i, j) related to y_β, denoted as FS_β(i, j), can be obtained from the fuzzy model FIS1, i.e. FS_β (i, j) = FIS1(d(X_i, X_j), d(y_iβ, y_jβ)). FS_β(i, j) can be considered as a measure of information content of all the physical features in the pair (i, j) related to the sensory descriptor y_β.

When removing x_k from the whole set of physical features, the sensitivity of the remaining physical features in the data pair (i, j) related to the sensory descriptor y_β can be calculated in (2)

The general fuzzy sensitivity criterion FS_k,β for all the pairs of data samples when removing the physical feature x_k is defined in (3)

Clearly, when FS_k,β < FS_p,β for all p≠k, then the removed physical feature x_k is considered as the least relevant among all physical features.

The FS criterion FS_k,β is normalized in [0,1]. In this way, the closer the value of FS_k,β is to 1, the more the physical feature x_k is relevant to the sensory descriptor y_β. The closer the value of FS_k,β is to 0, the less the physical feature x_k is relevant to the sensory descriptor y_β.

Based on the fuzzy sensitivity criterion defined in (3), an algorithm for selecting the most relevant physical features and ranking all the physical features is proposed. The principle of this algorithm is illustrated in Fig. 3.

Fig. 3.

The algorithm for ranking physical features using fuzzy sensitivity criterion

2.2. Combination between FS and human knowledge (HK) rankings

The two previous ranking lists, i.e. the fuzzy sensitivity FS and the HK are combined in order to obtain a Global Relevancy (GR) value. At this stage, two aggregation techniques, i.e. Fuzzy Inference System (FIS2) and OWA (OWA3) (see Fig. 1) are applied in order to compare their results. In fact, the weights issued from the FIS2 aggregation is a result of a linear (triangular) membership functions managed by fuzzy rules whereas the weights obtained from the OWA3 are extracted drawing upon Zadeh’s concept of linguistic quantifiers³². These two aggregation techniques will be detailed right after. If both of them yield similar results, we consider that the obtained aggregation result is reliable. In both cases, the obtained GRs are robust and can effectively avoid the discrepancy in the ranking lists obtained from OWA1 and OWA2 due to the sensitivity of experimental data, the perceptual divergence of human evaluators and professional knowledge.

2.2.1. Using the FIS2 fuzzy model (FGR)

The combination is based on the fuzzy rules presented in Table 3, obtained from the common qualitative knowledge. All the scores have been normalized in the range of [0, 1] to eliminate the scale effects.

FGR scores (FIS2)	And	FS (OWA1)
		Very relevant	Relevant	Average relevant	Slightly relevant	Not relevant
HK (OWA2)	Very relevant	Very relevant	Very relevant	Very relevant	Very relevant	Not relevant
	Relevant	Very relevant	Relevant	Relevant	Slightly relevant	Not relevant
	Average relevant	Very relevant	Relevant	Average relevant	Slightly relevant	Not relevant
	Slightly relevant	Very relevant	Slightly relevant	Slightly relevant	Slightly relevant	Not relevant
	Not relevant	Not relevant	Not relevant	Not relevant	Not relevant	Not relevant

Table 3.

Fuzzy rules for aggregating FS and HK rankings

These fuzzy rules are used to build a fuzzy model in which FS and HK are taken as two input variables and the Fuzzy Global Relevancy score as output variable (FGR). The input and output variables are fuzzy variables each including five fuzzy values: Not Relevant (NR), Slightly Relevant (SR), Average Relevant (AR), Relevant (R) and Very Relevant (VR) (illustrated in Fig. 4).

Fig. 4.

Fuzzy values for each input and output data variable in the FIS2 model

Obviously, the output variable FGR varies in the interval of [0, 1]. The closer the value of FGR is to 1, the more the corresponding variable x_k is relevant.

2.3. Distinctness between relevant and irrelevant physical features

Until this stage, the role of the aggregation techniques used previously (Fuzzy or OWA) is restricted in building a single operation of aggregation and provide a ranking by relevance. However, for each ranking list, the scores of the physical features are usually closed each other. Therefore, we need to determine an appropriate threshold in order to separate the set of relevant physical features from irrelevant ones.

The key issue of the proposed percolation technique is to filter automatically and objectively the relevant features by creating a gap between scores of relevant and irrelevant physical features. It permits to automatically generate threshold that can effectively reduce human subjectivity and arbitrariness when manually choosing thresholds. For a specific sensory descriptor, the threshold is defined systematically by iteratively aggregating (n times) the ranking lists generated by OWA3 and FIS2.

The complete percolation technique algorithm is summarized below:

1. 1)

   Initialize the percolation technique algorithm by obtaining FGR and OWAGR rankings respectively from FIS2 and OWA3 models of the previews functional blocks as shown in Fig. 1.

2. 2)

   Repeat Until AE < ε

   1. a)

      Merge FGR and OWAGR rankings using FIS2 model to obtain a new FGR ranking

   2. b)

      Merge FGR and OWAGR rankings using OWA3 model to obtain a new OWAGR ranking

   3. c)

      Calculate the fuzzy and OWA Absolute Errors ‘AE’ for two consecutive iterations, i.e.

      (8)AE=|GRj+1−GRj|

3. 3)

   Generate the final FGR and OWAGR percolated ranking lists.

In our experiments, ε is chosen so that the error AE is less than or equal to 1% of the maximal values of FGR. Since the latter is normalized, then we may choose ε =0.01. In general, if the percolation algorithm converges very quickly, then ε can be smaller in order to obtain more precise results. However, if the convergence is slow, ε can be larger in order to reach a faster convergence.

The proposed percolation technique can effectively filter or select the relevant input variables because it’s capable of making a sharp difference between the scores of the relevant and irrelevant ones.

3.1. Aggregated ranking lists generated by FIS1-OWA1 and HK-OWA2

Using OWA1, we select the six most relevant features (N_s=6) from a total of m=20 physical features. Six evaluators (n=6) perform fabric hand evaluation on the selected samples. A physical features is considered relevant if the aggregated rank score is no smaller than m − N_s + 1 = 15. For a specific physical feature, if two-thirds of the evaluators (k=4) consider it as one of the p most relevant features (p=4), then we obtain from Eq.(5) and Eq.(6):

After interviewing the experts of Panel 2, we found that all of them agreed on the following principle: if a physical parameter has all but two excellent assigned values then his overall score should be about 3.75, which is a cut-off value for the relevant physical feature to a sensory one²⁷. In other words, by applying this principle to OWA2, a physical feature is considered as relevant if at least six (k=6) of the eight experts (n=8) in Panel 2 take it as relevant (the assigned value is 4). Then, the final score should be m − N_s + 1 = 3.75. In this case (application of OWA2), we have m=4 and p=1.

According to Eq. (5), we calculate the weights for all the sensory descriptors, shown in Table 5.

By aggregating the ranking lists given by different evaluators in Panel 1 and Panel 2, we obtain the scores and ranking lists for three sensory descriptors, presented in Table 6.

Table 6.

Aggregation results obtained from OWA1 (FS) and OWA2 (HK) for three sensory descriptors

For instance, when considering the sensory descriptor “Smooth”, the physical parameters (x₁₃, x₁₆, x₁₉, x₁₂, x₂) have been classified on top of relevant parameters by OWA1. When using OWA2 the same parameters have been considered as most relevant but in addition to the physical parameter x₁₅ which was classified on bottom of the list using OWA1.

In order to make more reliable decisions in selection of relevant physical features, it is necessary to further combine these two ranking lists. The combination is performed using two different approaches, i.e. FIS2 and OWA3.

3.2. Aggregated ranking lists generated by FIS2 and OWA3

Having obtained the aggregated ranking lists with OWA1 and OWA2 models, we further aggregate them to form the final ranking list of physical features for each specific sensory feature. This combination will permit to integrate both sources of information (perception of sensory evaluators and professional knowledge of experts) and overcome the sensitivity of experimental data and perceptual divergence of human evaluators.

The combination using fuzzy techniques (FIS3) takes FS and HK as fuzzy input variables and fuzzy global relevancy FGR as output variable. The five membership functions for these fuzzy variables are presented in Fig. 4.

The combination using OWA3 is based on the following rule: if a physical feature has, in the most divergent condition, a good score in one ranking list and bad score in another one, then its overall ranking score should be around 10. Therefore, by applying Eq. (6) and (7), we have:

According to the Eq. (5), we deduce the following weights (0.526 0.474) for aggregating the FS and HK ranking lists respectively, to obtain the OWA global relevancy OWAGR ranking list.

By using the two previous combination methods (FIS2 and OWA3), we obtain the values of FGR and OWAGR respectively and then can easily rank the 20 physical features from the most to the least relevant one. Indeed, in Table 7 and for “Smooth” sensory descriptor, FGR and OWAGR agreed that the eight most relevant physical features are (x₂, x₁₂, x₁₃, x₁₄, x₁₅, x₁₆, x₁₉, x₂₀) although with different classifications. However, we cannot sharply distinguish the relevant features from irrelevant ones due to very close aggregated scores between them (See Table 7). At the next step, we use the percolation technique to cause an abrupt variation in the scores of the relevant and irrelevant features.

3.3. Separation of the relevant physical features from irrelevant ones

Having applied the percolation method described in Section 2.3 with ε = 10⁻⁴, we can observe that the set of relevant physical features is separated as shown in Table 8. The number of iterations needed to achieve the convergence is around 26. When we change the percolation threshold to ε = 0.01, similar results are obtained within about 15 iterations.

Table 8.

Extraction of the set of the relevant physical features

Having applied the percolation technique, we separate the relevant physical features from irrelevant ones for each sensory descriptor. The results of three descriptors are shown in Table 8. From the final scores, we can clearly note that the relevant features are very different from the irrelevant ones. Besides, we can easily observe that the most relevant features have not changed when comparing Table 7 and Table 8. Nevertheless, apparently the order of the physical features changes after applying percolation, yet the scores several physical features are identical therefore, the order is indeed not different. For the presented descriptors, the numbers of relevant physical features vary: 8 selected relevant features for “Smooth” and 12 for “Supple” and “Elastic”. The originality and performance of the proposed relevant feature selection method can be shown from these various results. Instead of selecting identical numbers of features with a predefined threshold, the proposed method can be adapted to the specific natures of the complex relations between sensory descriptors and physical features, in order to propose lists of relevant features of different sizes for different descriptors.

In the real application of stone washed denim fabrics, the final physical feature selection results can be validated by the professional textile knowledge or results in textile research. Some analysis is given below. The relevant physical features for the descriptor “Smooth” include: surface thickness “St” (x₂), coefficient of friction “MIU” (x₁₃ and x₁₄), geometric roughness “SMD” (x₁₅ and x₁₆) and the elastic deformation of the surface texture (x₁₉ and x₂₀). Obviously, all the surface features are selected except the drape coefficient “F” (x₁₂). “F” is a parameter that describes the way in which fabric flows or falls with gravity. From the point of physical view, this feature is less related to surface smoothness. Nevertheless, the relevancy of “F” to smoothness can be validated by the study given by Hu and Chan³³, who found that the mean deviation of friction coefficient is highly correlated with the fabric drape coefficient.

The selected relevant features for the descriptor “Supple” are mainly physical parameters related to the suppleness, apart from (x₁₃) which describes fabric surface (see Table 8). However, the relevancy of x₁₃ to “Supple” can also be validated by the study of Hu and Chan³³.

According to the Australian Wool Textile Objective Measurement Executive Committee AWTOMEC³⁴, the descriptor “Elastic” is related to tensile, compression and shear. These three properties are represented by the set of physical features (x₅, x₆, x₉, x₁₀) for tensile, (x₂, x₁₉, x₂₀) for compression and (x₇) for shear. Indeed, all the stated physical features have been taken as the most relevant ones to the descriptor “Elastic” in Table 8. Also, conform to the conclusion obtained by Kim and Vaughn³⁵, the other selected relevant physical features of “Elastic” have a good correlation with the extensibility property.

The previous discussion on fabric mechanical properties show that the proposed relevant feature selection method is effectively conform to the conclusions obtained in the other textile research work and general textile standards.

3.4. Correlations between the selected physical parameters

In practice, there are often dependencies between input variables. It is more reasonable to choose the physical parameters that are more sensitive to the sensory descriptor, but less correlated with other input variables. For this reason, we propose to calculate the Bravais-Pearson coefficient R obtained from Eq. (9) for each pair of the relevant parameters (x_k, x_p) as:

After selecting the set of relevant parameters, the objective of the next step is to keep one of two highly correlated variables with other variables in the list of the relevant physical parameters according to the following algorithm:

1. 1)

   For each pair of the relevant parameters (x_k, x_p), calculate the coefficient (R);

2. 2)

   If x_k and x_p ∈ X_r; k ≠ p; |R(x_k, x_p)| ≥ t (correlation threshold) and FS_k,l> FS_p,l, x_p then must be removed from the list of the relevant parameters X_r.

3. 3)

   X_R = X_r \ {x_p}

Undergone this algorithm, the input variables obtained in the X_R final list are the most relevant variables to a specific sensory descriptor. The threshold is defined by the experts. The higher the threshold t, the less variable in the final list are correlated. The chosen threshold t is 0.75. Following this threshold value, we obtain the sets of most physical parameters as presented in Table 9.

After checking the correlations between physical parameters, the number of inputs was reduced according to the number of possible correlations between them (see Table 9).