Proposing a Fuzzy Soft‐max‐based classifier in a hybrid deep learning architecture for human activity recognition

IET BiometricsVolume 11, Issue 2 p. 171-186 ORIGINAL RESEARCHOpen Access Proposing a Fuzzy Soft-max-based classifier in a hybrid deep learning architecture for human activity recognition Reza Shakerian, Reza Shakerian Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, IranSearch for more papers by this authorMeisam Yadollahzadeh-Tabari, Corresponding Author Meisam Yadollahzadeh-Tabari m_tabari@baboliau.ac.ir orcid.org/0000-0002-5231-7611 Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, Iran Correspondence Meisam Yadollahzadeh-Tabari, Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, Iran. Email: m_tabari@baboliau.ac.irSearch for more papers by this authorSeyed Yaser Bozorgi Rad, Seyed Yaser Bozorgi Rad Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, IranSearch for more papers by this author Reza Shakerian, Reza Shakerian Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, IranSearch for more papers by this authorMeisam Yadollahzadeh-Tabari, Corresponding Author Meisam Yadollahzadeh-Tabari m_tabari@baboliau.ac.ir orcid.org/0000-0002-5231-7611 Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, Iran Correspondence Meisam Yadollahzadeh-Tabari, Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, Iran. Email: m_tabari@baboliau.ac.irSearch for more papers by this authorSeyed Yaser Bozorgi Rad, Seyed Yaser Bozorgi Rad Department of Computer Engineering, Islamic Azad University, Babol Branch, Babol, IranSearch for more papers by this author First published: 06 February 2022 https://doi.org/10.1049/bme2.12066AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Human Activity Recognition (HAR) is the process of identifying and analysing activities performed by a person (or persons). This paper proposes an efficient HAR system based on wearable sensors that uses deep learning techniques. The proposed HAR takes the advantage of staking Convolutional Neural Network and Long Short-Term (LSTM), for extracting the high-level features of the sensors data and for learning the time-series behaviour of the abstracted data, respectively. This paper proposed a Fuzzy Soft-max classifier for the dense layer which classifies the output of LSTM Blocks to the associated activity classes. The authors' decision for proposing this classifier was because sensor data related to the resembling human activities, such as walking and running or opening door and closing door, are often very similar to each other. For this reason, the authors expect that adding fuzzy inference power to the standard Soft-max classifier will increase its accuracy for distinguishing between similar activities. The authors were also interested in considering a post-processing module that considers activity classification over a longer period. Using the proposed Fuzzy Soft-max classifier and by the post-processing technique, the authors were able to reach the 97.03 and 85.1 rates of accuracy for the PAMAP2 and Opportunity dataset, respectively. 1 INTRODUCTION With the increase in the number of small and intangible sensors in living environments such as cars, smart rooms or public places, the importance of activity prediction methods to enhance reliable performance and provide self-adaptive services in these environments has greatly increased. Human Activity Recognition (HAR) [1] is the process of identifying and analysing activities performed by a person (or persons). Today, HAR methods are widely used in a variety of areas including the research community, healthcare [2], smart cities [3], and affordable mobile devices. Automatic HAR helps to measure physical activity and check if a person fulfils a certain physical activity recommendation or not. Based on the type of processed data, HAR methods can be divided into two categories: video-based HAR and sensor-based HAR [4, 5]. The video-based HAR includes the analyses of 2D or 3D images or videos received from the camera [6]. On the other hand, sensor-based HAR includes analyses of the raw data obtained from wearable sensors. In recent years, due to the advances in sensor technology and users' desire for protecting their privacy, sensor-based HAR methods have become increasingly popular. Most of the sensor-based HAR methods use body-worn inertial sensors and include the modules such as feature extraction from sensor data [7]. There is usually a large amount of sensor data in the first step of HAR that needs to be carefully scrutinised to extract knowledge (pattern discovery). Different activity prediction methods attempt to derive patterns that allow predicting future events using existing input data. In the literature of HAR, also some machine learning approaches have been applied [8, 9]. These approaches include Naïve Bayes [10], Markov Chains [11], Support Vector Machines (SVMs) [12], and k-Nearest Neighbours (k-NN) [13], but none of these traditional machine learning algorithms are not designed to work with the time-series data which is common in all HAR datasets. The development of deep learning methods (DL) and their uses in HAR led to significant success in sequential HAR. As an instance, feature extraction is performed automatically from the raw data using Convolutional Neural Networks (CNNs) [14, 15]. In CNNs, convolutional layers, used for feature learning, are combined with a Multi-Layer Perceptron for activity classification [16]. In this work, we will propose an efficient activity recognition model based on wearable sensors. The proposed HAR system is composed of three main modules. The first module normalises the acceleration signals of the raw data into the overlapped windows. The second module extracts high-level features at each window, using a deep learning structure, based on CNNs. Finally, a Long Short-Term Memory (LSTM) learning network layer for learning the time-series behaviour of the abstracted data structure is used. A fuzzy soft-max classifier was proposed for the dense layer which classifies the output of LSTM Blocks. Our decision for proposing this classifier was because sensor data related to the resembling human activities, such as walking and running or asc. stairs are often very similar to each other. For this reason, we expect that adding fuzzy inference power to the classifiers which are used in deep learning algorithms (Soft-max for instance) will increase its accuracy. We expect that this classifier performs classification better than the ordinal soft-max classifier. The third module integrates the window-level decision in a longer period by performing a post-processing step, obtaining a significant performance. Considering that each activity often lasts more than a unique window, it is interesting to consider a post-processing module to enhance activity classification in a longer period. The results show that the proposed method is capable of accurately recognising the user's activity. The overall architecture of the proposed HAR architecture has been illustrated in Figure 1. FIGURE 1Open in figure viewerPowerPoint The overall architecture of the proposed Human Activity Recognition architecture. CNN, Convolutional Neural Network; LSTM, Long Short-Term Memory In Section 2, an overview of the previous work is presented. In Section 3, a brief background related to the main method has been explained, including convoluted and recurrent LSTM networks and the Intuitionistic fuzzy sets (IFSs) theory. In Section 4, the architecture of the proposed method has been expressed and the innovated Fuzzy-Soft-max method. In Section 5, the lab environment, datasets description, experiments, and results are discussed and finally, in Section 6 the overall method is concluded briefly. 2 RELATED WORKS Experiments have shown the final performance of activity recognition is dependent on two steps: features extraction from raw data and their classification [2]. In the sensor-based HAR methods, features are either extracted manually (handcrafted features) or derived automatically. In traditional methods, features are handcrafted, which are made using methods such as symbolic representation [17], transform coding [18], and statistics of raw data such as average, standard deviation, or the Fourier transform of signals in frequency-domain [19]. Tian et al. [20] presented an ensemble-based filter feature selection approach using wavelet energy spectrum features. In recent years, the researchers found automatic feature extractor methods, such as CNN, as a more appropriate tool. In this case, we can mention to the research studies [7, 21-25] that experimented with advanced feature extractor algorithms on different HAR datasets. The results indicated more appropriate detection rate in comparison to previous research studies. In HAR methods, some machine learning algorithms usually have been proposed to classify extracted features. In this regard, the top well-known methods include the k-NN algorithm [26], artificial neural networks, SVM [27], the random forest (RF) [28], decision tree, and Naïve Bayes [4]. Reiss and Stricker in Ref. [13] extracted some hand-crafted features from raw data such as mean, standard deviation, or variance and used k-NN as a classification algorithm. In this research study, a Leave-One-Subject-Out (LOSO) nine-fold cross-validation was performed using eight subjects for training and one subject for testing. The system was optimised over the testing set directly. Bustoni et al. [5] compared the recognition performance of SVM, KNN, and Random Forest machine learning methods. The results showed that the SVM method with Support Vector Classifier and Radial Basis Function kernels can provide the highest precision, while the Random Forest offers the best performance. Gupta A and Semwal VB in Ref. [25] proposed a gait recognition method for different walking patterns using ELM classification, and the results were also compared with the other classification techniques such as SVM and KNN. Their proposed system reached the 84.06% rate of accuracy. Their proposed algorithm identified nine different types of walking pattern behaviour from the given dataset in less time. The paper has also implemented ensemble learning by providing the combined average results of the different classifiers. In contrast to machine learning methods with shallow statistical features, deep learning methods include high-level and meaningful features and have achieved excellent performance in HAR. Among the deep learning methods, the most important examples are Autoencoders (AEs), CNNs, Recurrent Neural Networks (RNNs), and LSTM. Ref. [29] proposed CNNs as a feature learning method for HAR. They examined the effect of the main hyperparameters of CNN on the discrimination of target activities in the feature space. Javier et al. [30] proposed the Deep ConvLSTM method consisting of convolutional, recurrent, and soft-max layers. This method obtained an F1-score of 69% performance using only signals obtained from accelerometers. This value was improved on average by 20% when fusing signals were obtained from accelerometers, gyroscopes, and magnetic sensors. In Ref. [2, 30], the authors proposed new approaches based on the combination of CNN and LSTM recurrent networks to perform activity recognition from wearable sensors. Their proposed methods define the boundaries of activities much more precisely than a standard CNN. Most of the previously proposed methods that use deep learning algorithms in HAR utilize the soft-max loss function in the last layer (fully connected layer) to classify the activities. However, the soft-max loss function is not sufficiently effective because it cannot desirably increase the intra-class compactness and inter-class discrepancy, and more discriminative features are needed when recognising different activities. In Ref. [31], Sun et al. proposed a hybrid deep framework, called CNN-LSTM-ELM, which used ELM in the last layer for classification purposes. An important limitation of this method is the instability of its performance in some situations; recently, some researchers have attempted to maximise. In Ref. [32], the authors propose a HAR system. The work includes three separate modules. One for segmenting the acceleration signals into overlapped windows and extracting information in the frequency domain and the second for detecting the performed activity at each window using CNNs. Finally, the third module integrates the window-level decision in longer periods using the HMM structure. The authors claim that they reach 96.62% of accuracy. As a more up-to-date research, Jain et al. in Ref. [23] presented a deep ensemble learning approach for distinguishing walking-based postures such as walking upstairs, walking downstairs, standing and so on. Their presented approach staked the ordinal CNN and LSTM structure in a simple but efficient way and executed on OI-ISIR, HAPT and m-health datasets. The authors in Ref. [21] also proposed a Deep Neural Network-based model, which uses CNN and Gated Recurrent Unit for performing automatic feature extraction and classification of the activities on wearable sensors data. They reported 96.20%, 97.21%, and 95.27% rate of accuracy on UCI-HAR, WISDM, and PAMAP2 datasets, respectively, which was considerable (It should be noted that they do not use any type of post-processing). 3 BACKGROUND 3.1 LSTM blocks LSTM is a type of network architecture that is designed to solve the RNN problem called 'unstable gradients', which will limit its usage for modelling long-term activity sequences and temporal dependencies in such data gained from sensors. In this way, the LSTM structure can learn long-term data dependency which is not possible by RNNs. The main element of LSTMs is the cell state. Using a combination of these memory cells, the LSTM can control the flow of input information [33, 34]. This is accomplished by the gate structures which can optionally allow information to be entered. The LSTM has three similar gates to control the cell state value. The first gate of the LSTM decides what information we want to clear from the cell state. The decision is made by a sigmoid layer called the 'forget gate'. The output of this gate is shown in Equation (1), where u t $\hspace*{.5em}{u}_{t}$ is an input vector at time t (current input), h t − 1 $\hspace*{.5em}{h}_{t-1}$ is the memory or history value from the previous time step (previous history), w ( u . ) , w ( h . ) ${w}_{(u.)\hspace*{.5em}},\hspace*{.5em}{w}_{(h.)}$ are the weight matrixes, respectively, related to the u , h $u\hspace*{.5em},\hspace*{.5em}h\hspace*{.5em}$ values and b ( . ) ${b}_{(.)}$ is the bias vector that defines the transformation of the particular gate. This gate outputs a value 0 or 1 for each number in the cell state c t − 1 $\hspace*{.5em}{c}_{t-1}$ regarding h t − 1 ${h}_{t-1}$ and u t $\hspace*{.5em}{u}_{t}$ . The value of one represents 'completely keeping this state' while the value of 0 represents 'completely forgetting of this state'. f t = σ f ( w u f u t + w h f h t − 1 + b f ) ${f}_{t}={\sigma }_{f}({w}_{uf}{u}_{t}+{w}_{hf}{h}_{t-1}+{b}_{f)}$ (1) The second gate is designed to decide which new information should be stored in each cell state. The process has two steps. In the first step, there is a sigmoid layer called the input gate that decides which values should be updated. This is shown in Equation (2). The next creates a vector called g t ${g}_{t}$ that can be added to the cell state using a hyperbolic tangent layer that is given as, i t = σ i ( w u i u t + w h i h t − 1 + b i ) ${i}_{t}={\sigma }_{i}({w}_{ui}{u}_{t}+{w}_{hi}{h}_{t-1}+{b}_{i})\hspace*{.5em}$ (2) g t = tanh ( w u g u t + w h g h t − 1 + b c ) ${g}_{t}=\mathrm{tanh}({w}_{ug}{u}_{t}+{w}_{hg}{h}_{t-1}+{b}_{c})$ (3) In the third gate, the old cell state C t − 1 ${C}_{t-1}$ will be updated with the new cell state g t ${g}_{t}$ . c t = f t ⋅ c t − 1 + i t ⋅ g t ${c}_{t}={f}_{t}\hspace*{.5em}\cdot {c}_{t-1}+{i}_{t}\cdot {g}_{t}$ (4) o t = σ ( w u o u t + w h o h t − 1 + b o ) ${o}_{t}=\sigma ({w}_{uo}{u}_{t}+\hspace*{.5em}{w}_{ho}{h}_{t-1}+{b}_{o})$ (5) Therefore, the value of the output parameter h t ${h}_{t}$ (new history) is updated in each step according to the output value o t ${o}_{t}\hspace*{.5em}$ and the value of cell state c t $\hspace*{.5em}{c}_{t}$ : [35] h t = o t ⋅ tanh ( c t ) ${h}_{t}={o}_{t}\cdot \mathrm{tanh}({c}_{t})$ (6) 3.2 Feature extraction based on the convolution network The CNN can be used for extracting meaningful features from raw data [36]. The main layers used in deep convolution networks are the Convolution layer, the Max-pooling layer, and the fully connected layer. In single-layer CNN networks, feature extraction is accomplished by applying a convolution operator with a filter (or kernel) on the input signal. In CNN, the activation of each unit represents the convolved filter or kernel through the input signal [37]. It is believed that the filters in the convolution layers in these networks act as feature extractors and gradually highlight specific features in the top layers of the network. Comparatively, when applying a 1D temporal sequence (such as a sensor signal), a one-dimensional kernel is utilised in temporal convolutions [36]. Formally, feature extraction is defined by using a two-dimensional displacement operation on the layers of the convolution layer as follows: a i l + 1 = σ ( ∑ j = 1 J l w j l . a i + j l + b i l ) ${a}_{i}^{l+1}=\sigma (\sum\limits _{j=1}^{{J}^{l}}{w}_{j\hspace*{.5em}}^{l}\hspace*{.5em}.\hspace*{.5em}{a}_{i+j}^{l}+\hspace*{.5em}{b}_{i}^{l})\hspace*{.5em}$ (7) In Equation (7), the parameter a i l + 1 $\hspace*{.5em}{a}_{i}^{l+1}$ is the feature map i $i$ to convolutional layer l + 1 $l+1$ and w j l ${w}_{j}^{l}$ is the weight matrix of kernel function in the convolutional layer l $l$ , which creates the next layer input by convolving with the output of the previous layer a i + j l $\hspace*{.5em}{a}_{i+j}^{l}$ . The parameter b i l ${b}_{i}^{l}$ is the bias vector. Another important layer in the convolution network is the pooling layer. This layer is a type of non-linear down-sampling, so it reduces the size of the data by combining the output associated with adjusted neurons in the convolution layer. After each layer of convolution, a pooling layer is placed in a period to summarise the output of the convolution layer on the network. 3.3 Intuitionistic fuzzy sets Atanassov [38] expanded Zadeh's Fuzzy sets idea by introducing the IFSs using the concept of dual membership degrees in each of the sets by giving both a degree of membership and a degree of non-membership, which is more or less independent of one another whereby the sum of these two grades should not be greater than one [39]. This idea, which is a natural generalisation of a standard fuzzy set, seems useful in modelling many real-life situations, which have uncertainty on its inherent [40]. An IFS A on a universe X (shown in Figure 2) is defined as an object of the following form: A = { ( x , μ A ( x ) , ϑ A ( x ) ) | x ∈ X } $A=\left\{(x,{\mu }_{A}(x),{\vartheta }_{A}(x))\vert \hspace*{.5em}x\in X\right\}$ (8)here μ A ${\mu }_{A}$ : X $X$ → [0, 1] defines the degree of membership and ϑ A ${\vartheta }_{A}$ : X $X$ → [0, 1] is the degree of non-membership of the element x $x$ ∈ X $X$ in A, and for every x $x$ ∈ X $X$ , 0 ≤ μ A ( x ) + ϑ A ( x ) ${\mu }_{A}(x)+{\vartheta }_{A}(x)$ ≤ 1. In the real world, humans may not express the non-membership value as 1-membership value. This is due to the presence of uncertainty or hesitation or the lack of knowledge in defining the membership function. Authors in Ref. [42] stated that the membership degree and non-membership degree of the IFS could be expressed better by the Intuitionistic Fuzzy Number (IFN) and the related ranking methods. Ref. [43] has given a method based on score function and accuracy function for any given IFNs α i = 〈 μ i , ϑ i 〉 ${\alpha }_{i}=\langle {\mu }_{i},{\vartheta }_{i}\rangle $ . We use the modified score function which was defined by Yu (2013a). This equation has the values between 0 and 1, which is shown by Equation (9). S ( α i ) = 1 + μ i − ϑ i 2 $S({\alpha }_{i})=\frac{1+{\mu }_{i}-{\vartheta }_{i}}{2}$ (9) FIGURE 2Open in figure viewerPowerPoint An intuitionistic fuzzy set description [41] Also, in Ref. [43], it is stated that if S ( α 1 ) > S ( α 2 ) $S({\alpha }_{1})\gt S({\alpha }_{2})$ then α 1 ${\alpha }_{1}$ is better than α 2 ${\alpha }_{2}$ . 4 THE PROPOSED METHOD The dataset used contains signals from peripheral Inertial Measurement Units (IMUs) clinging to the user body continuously collected in time periods, which are parameterised with symbol X $X$ in Figure 3. The output dataset probably contains a significant amount of unrelated data and unwanted user movements along. As shown in Figure 3, the raw signal first overlapped to the fixed tiny size windows, where we gain a correct understanding of the complex dependencies between the underlying data and the identification of user comprehensible activity. These overlapping windows are shown in Figure 3 with the parameters x 1 ${x}^{1}\hspace*{.5em}$ to x D $\hspace*{.5em}{x}^{D}$ . In the following, through the recursive steps of filtering and pooling phase, the high-level features of the primary data is discovered. In the convolution layer, a set of features with size H $H$ is extracted from the input signals by applying a filter with the size k $\hspace*{.5em}k$ , which is shown by the parameters K F 1 ${K}^{F1}$ to K F l ${K}^{Fl}$ . In the pooling layer, a feature vector with the size E are obtained by considering a window of size S $S$ applying maximum operation. Next, by applying it to the LSTM network, the associated activity class is determined for each time step window along the activity period. As an innovation in this article, the importance of each output in the LSTM network is determined fuzzily using a fuzzy soft-max classifier. The final human activity is determined by finding the maximum occurrence of activity class through the entire time window. To make the final result more resistant to noise, we used the post-processing stage during the test. The proposed method CNN-LSTM-FSC ignores the unwanted user activity prediction and performs the classification operation at high accuracy. In the following, the proposed Fuzzy Soft-max classifier is explained. The proposed deep architecture provides a vector with 12 scores, each one associated to a different activity. The scores then analysed with the proposed Fuzzy Soft-max classifier for assigning each of the window to a physical activity. FIGURE 3Open in figure viewerPowerPoint x 1 ${x}^{1}\hspace*{.5em}$ to x D ${x}^{D}$ are overlapping windows obtained from the input data. The feature extraction step includes the convolution layer and the max-pooling layer. k $k$ is filter size, K F 1 ${K}^{F1}$ to K F l ${K}^{Fl}$ are the filters considered in the convolution layer, l $l$ is the number of filters and d $d\hspace*{.5em}$ is the number of features extracted from the convolution layer, S $S$ is the window size in the max-pooling layer and E $E$ is the number of features extracted from the max-pooling layer. The vector V contains scores, each one associated to a different activity. The scores are then analysed with the proposed Fuzzy Soft-max classifier for assigning each of the window to a physical activity. LSTM, Long Short-Term Memory 4.1 Preprocessing step The input data of this paper include a set of RSS values received from subjects IMUs sampled at a specific time frequency. Hence, the input data has a time-based property. The input data consists of time-series data. To better understand this data and achieve high accuracy in the classification algorithm, each instance of input data is divided into the overlapping window with the appropriate size. The window size is chosen so that each item in it defines an understandable activity and consideration of the user. This will create optimal use of input data and increase accuracy in the classification algorithm. According to Figure 4, segmentation of the data is done by using window size N. Therefore, each record of the IMUs data entry is divided into D overlapping windows with size N. FIGURE 4Open in figure viewerPowerPoint Segmentation of the input data using the overlap window with length N. IMU, Inertial Measurement Unit 4.2 Fuzzy Soft-max Classifier The Soft-max function, as one of the generalisations of the logistic function, has been widely used in different probabilistic classification methods. It has been widely applied to many practical fields, such as machine learning. This function is great for classification problems, especially dealing with multi-class classification problems, as it will report back the 'confidence score' for each class. The predicted class is, therefore, the item in the list with the highest confidence score. Supposing having m different blocks in the LSTM network, we have the LSTM layer that has the output form as shown in Equation (10). The output of the LSTM layer is passed through a soft-max layer that transforms the raw output to class probabilities. The LSTM layer provides a vector with 12 scores; each score is associated with a different activity. The activity classification at every window should be done by selecting the activity with the highest score H V = ( h 1 . h 2 . … . h m ) $HV=({h}_{1}.{h}_{2}.{\mathellipsis}.{h}_{m})\hspace*{.5em}$ (10) By the Soft-max Classifier (SC), the final output of the user's activity is determined using Equation (11). y ˆ = S C ( H V ) = argmax c ∈ C ( exp ( T C j ( H V ) ) ∑ j = 1 m exp ( T c j ( H V ) ) ) $\widehat{y}=SC(HV)=\underset{c\in C}{\text{argmax}}(\frac{\text{exp}({T}_{{C}_{j}}(HV))}{{\sum }_{j=1}^{m}\text{exp}({T}_{{c}_{j}}(HV))})$ (11) T C j ( H V ) ${T}_{{C}_{j}}(HV)$ is the probability value of selecting C j ${C}_{j}$ class as the user's activity. Ref. [44] stated that equipping the soft-max function with fuzzy interference can significantly raise the differentiation ability of this classification function. Thus, we develop a novel fuzzy soft-max function that uses IFSs. The proposed classification method considers both the membership and non-membership value of the LSTM h state values to the appropriate classes. As stated in Equation (11), T C j ( H V ) ${T}_{{C}_{j}}(HV)$ is the probability value of selecting C j ${C}_{j}$ class. In the proposed Fuzzy Soft-max Classifier, this parameter is determined based on the fuzzy membership and fuzzy non-membership degree for each of the HV input vector to the associated classes. Equation (11) shows the T C j ( H V ) ${T}_{{C}_{j}}(HV)$ parameter. T c j ( H V ) = ∑ i = 1 n ( s c j ( h i ) f ( h i × v c j ( h i ) ) + b j ${T}_{{c}_{j}}(HV)=\sum\limits _{i=1}^{n}({s}_{{c}_{j}}({h}_{i})\hspace*{.5em}f({h}_{i}\hspace*{.5em}\times {v}_{{c}_{j}}({h}_{i}))+\hspace*{.5em}{b}_{j}$ (12) In Equation (12), v c j ( h i ) ${v}_{{c}_{j}}({h}_{i})\hspace*{.5em}$ points the weight of edge between the h i ${h}_{i}$ input node and the C j ${C}_{j}\hspace*{.5em}$ output class. f ( x ) $f(x)\hspace*{.5em}$ is an activation function. s C j ( h i ) ${s}_{{C}_{j}}({h}_{i})$ is the fuzzy significance of the h i ${h}_{i}$ input node, associated to the C j ${C}_{j}\hspace*{.5em}$ output class. This value is calculated by Equation (13) based on the fuzzy membership value μ C j ( h i ) ${\mu }_{{C}_{j}}({h}_{i})$ and the fuzzy non-membership value ϑ C j ( h i ) ${\vartheta }_{{C}_{j}}({h}_{i})\hspace*{.5em}$ of the h i ${h}_{i}\hspace*{.5em}$ input node associated to the C j ${C}_{j}\hspace*{.5em}$ output class. s C j ( h i ) = ( 1 + μ C j ( h i ) − ( ϑ C j ( h i ) ) λ 2 ) ${s}_{{C}_{j}}({h}_{i})=(\frac{1+{\mu }_{{C}_{j}}({h}_{i})-{({\vartheta }_{{C}_{j}}({h}_{i}))}^{\lambda }}{2})$ (13) The values of the parameters μ C j ( h i ) . ϑ C j ( h i ) ${\mu }_{{C}_{j}}({h}_{i})\hspace*{.5em}.\hspace*{.5em}{\vartheta }_{{C}_{j}}({h}_{i})\hspace*{.5em}\hspace*{.5em}$ are determined based on the weight vector V $\hspace*{.5em}V$ connectin