The GLCM quantify the spatial relation of neighboring pixels in an image. It’s a comprehensive information of the image grayscale with regard to: the direction, the neighboring interval and the rangeability [16]. In simple terms, each element X d , θ ( i , j ) of the co-occurrence matrix represent the probability of occurrence to have the grayscale j and the grayscale i, at d-spatial distance and θ-orientation. Usually, orientation is chosen among four directions namely, horizontal, left diagonal, vertical, and right diagonal, respectively: 0˚, 45˚, 90˚ and 135˚. The spatial distance d belongs to the set of positive number and is usually one. Thus, for (d, θ) fixed, GLCM is a matrix whose elements X ( i , j ) are obtained for ( i , j ) ∈ ( N G , N G ) , where N G is the number of grayscale of image [17]. Table 1 present the 14 features of GLCM defined by Haralick in 1973, the 5 features of GLCM defined by Soh in 1999 and the feature of GLCM defined by Clausi in 2002.

The PCA is used for the purpose of dimensionality reduction of the high-di- mensional feature vector including the extracted texture features, due to the fact that the high-dimensional feature vector can degrade classification performance [20]. PCA algorithm select only the relevant principal components (linear transformation of the original features which are uncorrelated). Among these principal components, only the most significant are used to find GLCM features which are more correlated. Those features are named, relevant GLCM features and utilized as inputs to the classifier.

Sequential feature selection [21]. This method has two components. Firstly, an objective function, called the criterion, which the method seeks to minimize over all feasible feature subsets. The common criteria for classification models are misclassification rate. Secondly, a sequential search algorithm adds or removes features from a candidate subset while evaluating the criterion. Since an exhaustive comparison of the criterion value at all 2ⁿ subsets of an n-features data set is typically infeasible (depending on the size of n and the cost of objective calls), sequential searches move only in one direction, always growing or always shrinking the candidate set. During the process, SFE selects the best features among all features data which are able to discriminate each class from others.

Naive Bayesian is one of the classification methods using the similarity of the characteristics of an object. This method is classified as a fairly simple method, but is widely used in the fields of medicine, biometrics, text classification, and many more. Naive Bayesian uses the Gaussian distribution by considering (2) important parameters, namely the average µ and the variance σ [22]. In Naive Bayesian, Gaussian uses equation:

P ( X i = x i / Y = y i ) = 1 2 π σ i j exp [ − 1 2 ( x i − μ i j σ i j ) 2 ] (1)

where:

P = Probability of attribute x_i.

x_i = Attribute sought.

i= Index for the value of the attribute

j = class index.

Y = Represent the class sought

µ = The average value represented.

For variance (σ), find the Equation (2)

σ 2 = 1 n − 1 ∑ i − 1 n ( x i − μ ) 2 (2)

To classify using the Naive Bayesian method, it is necessary to calculate the average and standard deviation of each class for each characteristic. For the final stage, the test data is entered into each class to determine the opportunities that exist in each class so that it can be determined in which class the image has the greatest opportunity [22].

The diagnostic performance of the classifier can be evaluated by average classification accuracy (Acc1) which is calculated using (3). It’s the classifier success rate, where N_TP is the number of images in class c that are correctly classified as class c; N_images is the total number of images for all classes combined, and N_classes is the number of fault types or classes, N_{C_images} is the total number of images for class c [23]. For our study, we used two types of accuracy, namely: Acc1 and Acc2. Where, Acc2 is classifier identification rate between normal and faults vibration signal.

Acc 1 = ∑ N classes     N TP N images ⋅ 100 ;     Acc2 = N TP N C_images ⋅ 100 (3)

The aim of recognition is first of all to know how to find the positive examples “true positives”; it is also necessary to try to limit the number of false alarms “false positives”; these are objects that the system takes for normal but which are not.

Accurancy = TP + TN TP + TN + FP + FN (4)

Table 2 helps to illustrate the confusion matrix and Equation (4) is the recognition rate of all signals.

Since the literature proposes a wide range of features that can be computed on the co-occurrence matrix, and if they are all used at the same time for classification, it is more likely that the recognition rates are not consistent due to the presence of some redundant features. Therefore, a selection step of the most relevant features is necessary. The PCA and the SFE are merged to define the most relevant features that will be used in the classification to improve the recognition rates. Figure 3 represents the flowchart of the proposed methodology and the different steps of our work are organized as follows:

Step 1: Description of the vibration signal data of the bearings in normal and faulty conditions;

Step 2: The vibration signal can be split into random sub-samples, normalised and arranged in rows and columns to form a matrix; each matrix obtained is associated with a greyscale image of the vibration signal;

Step 3: The GLCM is calculated on each grey level image and its texture features are extracted on each GLCM;

Step 4: features selection is done first by the PCA method to define the variables corresponding to the most significant features; then, by the SFE method; and finally, by the proposed PCA/SFE fusion method to obtain the most relevant features.

Step 5: The multiclass-Naïves Bayesis used to validate the relevance of these features by an optimum recognition rate of the bearing defect classes.

Figure 4(a) shows the classification accuracy Acc2, which is the detection rate of the successful classifier between the normal signal and the faults. Figure 4(b) shows the classification accuracy Acc1, for several training sets, when we use the twenty GLCM attributes from Table 1 on the multiclass-Naïve Bayes. The training

set is randomly selected between fifty and ninety percent of the database (580 images from Table 1). For each training set, the model classification is obtained and tested on the remaining images in the dataset.

Figure 4(a) shows that there is an accuracy rate on the detection of normal or faulty faults above 95% from 60% of the training set. Thus, with the set of twenty GLCM features, we have a good distinction between the normal and defect signal as shown in the confusion table, where we have only one case of false positive and four cases of false negative in the 116 test images. Figure 4(b) shows that there is an accuracy rate of over 88% from 60% of the training set in the case of identification.

Table 4 shows the confusion matrix for the 20 attributes of the co-occurrence matrices in the case of defect detection for a training set greater than or equal to 70%.

The twenty GLCM attributes are realized on each of the five GLCMs obtained from five directions, namely 0˚, 45˚, 90˚, 135˚; and the average of these four directions (Mo). Thus, with twenty GLCM features calculated on five GLCMs, a set of one hundred attributes is obtained. The first principal component obtained from the set of features represents 99.77% of the data. Table 5 shows a

Accurancy = 95.68%.

higher correlation coefficient between the first principal component and the different features. Each feature is presented with the notation (see Table 1) followed by the direction of the GLCM used. For example, “ENERG_0” means the energy feature extracted from the GLCM obtained for the 0˚ direction. From these correlations, seven features of the GLCM most correlated with the first principal component are listed and represent the relevant features according to the PCA analysis.

Figure 5 shows the performance of the classifiers when we use the relevant PCA features. These relevant GLCM features are used on the classifiers and we observe a certain stability of the recognition rate from 89.65% for the training and test dataset greater than or equal to 60%.

Table 6 shows the top seven features of each class according to the SFE. Since we

are looking for the best features, we calculated the number of occurrences of each feature for all classes. Figure 6 shows the seven-high occurrence GLCM features that are the relevant features according to the SFE analysis.

Figure 6 shows the performance of the classifiers when we use the relevant attributes of the SFE. These GLCM relevant features are used on the classifiers

and we observe a certain stability of the recognition rate from 89.65% for the training and test dataset greater than or equal to 60%.

To take advantage of both feature selection methods (PCA and SFE), we can select the features that appear in the best selection features of the PCA and SFE presented in Figure 5 and Figure 6 shows the performance of the classifier when we use the relevant features of PCA and SFE respectively. Figure 7 show the performance of the classifiers when we use the relevant attributes of the fusion PCA/SFE. The relevant GLCM features of the fusion PCA/SFE show good classification performance with an accuracy above 99% for all training sets containing 60% or more data. Thus, the relevant GLCM features for bearing fault diagnosis among the twenty are the following four features: Energy, Entropy, Correlation and Maximum Probability.

The realisation of each classification system is based on the training and testing parameters. The classification system defined in this study is based on several training (50%; 60%; 70%; 80% and 90%) and testing (50%; 40%; 30%; 20% and 10%) samples. For each data item, an input vector is constructed by calculating the attributes of the GLCM. A study was first carried out on all 20 extracted features, then on the features by PCA and SFE and finally by merge PCA/SFE. The success rate of 89.65% was obtained on all training and test data sets of the 20 features (Figure 4(b)). This result already shows a feasibility in bearing diagnosis. Using the 04 relevant attributes obtained, the success rate is more than 98% on all data sets. We can observe the results of the classification of relevant features obtained by Naïve Bayes. We can observe the results based on the classification rate are listed here. It can be seen in Figure 7 that the recognition rate is equal to 98.27% for a set of 70% of the training data samples and 30% of the test samples. This result is more accurate than when we take into account all the features computed without selection and even without fusion.