Artificial Neural Network Logic-Based Reverse Analysis with Application to COVID-19 Surveillance Dataset

Abstract

The Boolean Satisfiability Problem (BSAT) is one of the crucial decision problems in the fields of computing science, operation research, and mathematical logic that is resolved by deciding whether or not a solution to a Boolean formula exists. When there is a Boolean variable allocation that induces the Boolean formula to yield TRUE, then the SAT instance is satisfiable. The main purpose of this chapter is to utilize the optimization capacity of the Lyapunov energy function of Hopfield neural network (HNN) for optimal representation of the Random Satistibaility for COVID-19 Surveillance Data Set (CSDS) classification with the aim of extracting the relationship of dominant attributes that contribute to COVID-19 detections based on the COVID-19 Surveillance Data Set (CSDS). The logical mining task was carried based on the data mining technique of the energy minimization technique of HNN. The computational simulations have been carried using the different number of clauses in validating the efficiency of the proposed model in the training of COVID-19 Surveillance Data Set (CSDS) for classification. The findings reveals the effectiveness and robustness of k satisfiability reverse analysis with Hopfield neural network in extracting the dominant attributes toward COVID-19 Surveillance Data Set (CSDS) logic.

1. Introduction

The COVID-19 pandemic is still having a significant impact on people’s health and quality of life all around the world. Effectively identifying and isolating affected people is the most crucial step in stopping COVID-19. Clinical medicine can identify COVID-19 instances thanks to the discovery that patients with the infection exhibit anomalies in CT imaging. Additionally, CT scans can be used to determine the severity of an illness, which is useful for selecting the right course of action. Many deep learning-based COVID-19 case identification techniques have recently been put out, some of which have had good success. The biggest obstacle to increasing COVID-19’s classification accuracy is currently the small number of training instances and annotations. Additionally, due to the poor contrast of CT scans, deep learning-based classification systems struggle to comprehend ambiguous and imprecise information, such as pixels near boundaries and pictures related to COVID-19 instances. We suggest a belief function-based classification network to categorize COVID-19 cases using semi-supervised learning to address the aforementioned issues, and we acknowledge the research community’s open-source COVID-19 dataset [1, 2, 3].

Neural networks are used extensively in many fields of study originated in mathematical neurobiology. This can be because these networks are attempts to simulate human brain capabilities. Neural networks have been utilized since the last decade as a theoretically sound alternative to traditional statistical models. It is also effective in classifying data into identifiable groups or characteristics. When used in a hybrid framework with the many forms of predictive neural networks, the classification of neural networks becomes very efficient. Machine-learning methods including artificial neural networks (ANN) have been used in recent times as tools for decision, prediction, classification, and diagnosis [4, 5, 6, 7, 8, 9, 10, 11]. It has been used in medical approaches to digenesis, predict, and detect diseases using effects on development, such as fibrosis prediction, cirrhosis, and prediction of response therapy in patients with hepatitis C [12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

Artificial neural network (ANN) models have been widely used in data mining for a wide range of medical, science, engineering, and industry issues. Logic mining is one of the key data mining fields. It has been shown that knowledge in a logical or symbolic form can be represented [22]. Recent data mining studies have led to the emergence of various types of artificial neural network models such as Hopfield neural network (HNN), Radial Basis Neural Network, Convolution neural network, and other machine learning tools that can foster logic mining through knowledge extraction process [23, 24]. Consequently, data mining practitioners assimilate multidisciplinary knowledge such as Artificial neural networks, mathematics, artificial intelligence, machine learning, and statistics to create logic for data mining techniques for finding underlying information based on the behavior of databases or data sets. Therefore, data mining can be improved in the neural network to cater to various problems. As for this work, the incorporation of Random k Satisfiability (RANkSAT) propositional logic is utilized with the proposed to introduce a comprehensive model to solve real-life applications. Artificial neural network (ANN) possesses a comprehensive structure of training and testing stages thus, ANN has emerged as one of the most efficient tools extend in finding patterns and extracting information to solve real-life applications. Therefore, we presume that this research contributes to amplifying the efforts to enhance the capability of fundamental ANN with the inclusion of a recurrent type of network known as Hopfield neural network (HNN). Propositional logic based on Random-k Satisfiability (RANkSAT) is considered a suitable approach to represent logical rules in neural network for optimal classification of real-world problem. By considering only maximum of three literals per clause, the logical complexity in learning the relationship between the variables in real-life problem decreases.

In this work, Random k Satisfiability-based Reverse Analysis method (RANkSATRA) has been proposed to obtain the logical COVID-19 Surveillance Data Set (CSDS). The aforementioned studies, which offer a different perspective on describing the real data in the form of logical representation, have demonstrated the usefulness of diverse logic programming in HNN. In order to depict the behavior of the COVI-19 data sets, we require a very well, intelligent logical rule. However, there is no effort to bridge RANkSAT logical representation in Hopfield neural network for COVID-19 classification. This is crucial because an artificial neural network algorithm can cater variation and randomness in COVID-19 analysis and larger searching space. Therefore, the contributions of this research are presented as follows: The contributions of this work are as follows: (a) to convert the COVID-19 surveillance data set (CSDS) into intelligent systematic form based on RANkSAT logical clauses. (b) To propose random k satisfiability reverse analysis method as an alternative approach in extracting the relationships between the factors or attributes that contribute to the knowledge extraction based on COVID-19 surveillance data set (CSDS) obtained from a UCI machine learning repository. (c) To assess the capability and accuracy of three variants of the satisfiability proposed method based on Random k satisfiability, Random maximum k satisfiability, and Horn k satisfiability logical representation in completing the COVID-19 surveillance data set (CSDS) extraction with a different number of clauses. The performance evaluation metrics will be adopted to evaluate the effectiveness of both the proposed method and logical representations as an alternative data extraction method to the COVID-19 data set. The general implementation of random k satisfiability reverses the analysis method and HNN in extracting logic in COVID-19 data. The construction of our proposed model, would exhibit better performance in the training stage and successfully interprets real-life datasets to detect which factors are more prominent than others that contribute to the optimization problems. Our findings showed that the proposed model executing the best performance in terms of attaining small errors and efficient computational time compared to other existing models. This study has been organized as follows. Material and method including, Random k Satisfiability Logic, Hopfield Neural Network (HNN), and random k satisfiability-based reverse analysis method (RANkSATRA) have been described in Section 2. In Section 3 Implementation procedure for classifying the COVID-19 data set was presented. Section 4 reported model simulations and experimental setup, and section. 5 reported performance evaluation metrics Section 6 presented the result and discussion. The chapter concludes with future works presented in Final Section.

2. Materials and methods

2.1 Random k satisfiability logic

Propositional satisfiability logic can perceive as a logical rule that consists of clauses that contain literals or variables. Random k Satisfiability (RANkSAT) is a class of non-systematic Boolean logic representation, consists of a random number of literals (can be the negated literals) per clause. Non-systematic Boolean Satisfiability logic (RANkSAT) has been proven effective to represent simulated applications [25]. There is no study that utilizes the non-systematic behavior of Random k Satisfiability in discrete HNN for application real data set classification problems. The formulation of RANkSAT has the following properties:

1. A set of variables$\left\{x_1,x_2,x_3,\dots ,x_n\right\}$, in a clause $C_i$ where $i=1,2,3,\dots ,n$ that consists of $x\mathit{\text{or}}\neg x$.

2. Randomly select 2 variables from the set of $n$ variables with a 0.5 probability of negating each variable in the clause.

3. Each $x_i$ in $C_i$ is connected by a disjunction$\left(\vee \right).$

4. Each clause $C_i$ is connected by a conjunction$\left(\wedge \right)$.

The Boolean values for each $x_i$ are bipolar$x_i\in \left\{-1,1\right\}$ that exemplifies the notion of FALSE and TRUE respectively. The general formulation $F_{\mathit{RAN}3\mathit{SAT}}$ is represented in Eq. (1) as follows.

$$F_{\mathit{\text{RANkSAT}}}=\underset{i=0}{\overset{t}{\wedge }}{C}_i^{\left(3\right)}\underset{i=0}{\overset{n}{\wedge }}{C}_i^{\left(2\right)}\underset{i=0}{\overset{m}{\wedge }}{C}_i^{\left(1\right)}$$E1

where $t,n,m\in \left[1,2,\mathrm{..}k\right]$, $\forall t,n,m>0$. The clause $F_{\mathit{RAN}3\mathit{SAT}}$ is defined as a random 3-SAT which consists of a clause $C_i^{\left(k\right)}$ described in Eq. (2). as follows.

$$C_i^{\left(k\right)}=\left\{\begin{array}{l}\left({\tau }_i\vee {\alpha }_i\vee {\mathrm{\Psi }}_i\right),k=3\\ \left({\tau }_i\vee {\alpha }_i\right),k=2\\ {\lambda }_i,k=1\end{array}\right.$$E2

where ${\tau }_i\in \left[{\tau }_i,\neg {\tau }_i\right]$, ${\alpha }_i\in \left[{\alpha }_i,\neg {\alpha }_i\right]$, ${\mathrm{\Psi }}_i\in \left[{\mathrm{\Psi }}_i,\neg {\mathrm{\Psi }}_i\right]$_, and ${\lambda }_i\in \left[{\lambda }_i,\neg {\lambda }_i\right]$ represent literals and their negation respectively. In particular, the first and second-order clauses are denoted as $C_i^{\left(1\right)}$, $C_i^{\left(2\right)}$_, and $C_i^{\left(3\right)}$_, respectively. In this chapter, $F_r$ is a Conjunctive Normal Form (CNF) formula where the clauses are chosen uniformly, independently without replacement among all $2^r\left(\begin{array}{l}m+n\\ v\end{array}\right)$ nontrivial clauses of the length $r$. Note that, $A_i$ exists in the $C_i^{\left(k\right)}$, if the $C_i^{\left(k\right)}$ contains either $G_i$ or $\neg G_i$ and the mapping of $V\left(F_r\right)\to \left\{-1,1\right\}$ is called logical interpretation. The Boolean value for the mapping is expressed as 1 (TRUE) and − 1 (FALSE). In theory, the example of RANkSAT formula for $k\le 3$ is given as.

$$F_{\mathit{RAN}3\mathit{SAT}}=\left({\tau }_1\vee \neg {\tau }_2\vee {\tau }_3\right)\wedge \left(\neg {\alpha }_1\vee {\alpha }_2\right)\wedge \neg {\lambda }_1$$E3

According to Eq. (3), $F_{\mathit{\text{RANkSAT}}}$ comprises of Eq. (4)–(6) as follows.

$$C_i^{\left(3\right)}=\left({\tau }_1\vee \neg {\tau }_2\vee {\tau }_3\right)$$E4

$$C_2^{\left(2\right)}=\left(\neg {\alpha }_1\vee {\alpha }_2\right)$$E5

$$C_1^{\left(1\right)}=\neg {\lambda }_1$$E6

Therefore, the outcome of Eq. (3) is satisfied if.

Eqs. (4)–(6) are satisfied. i.e.,

$$C_i^{\left(3\right)}=C_1^{\left(2\right)}=C_2^{\left(1\right)}=1$$E7

In this study, $F_{\mathit{\text{RANkSAT}}}$ will be embedded in the proposed model based on reverse analysis technique for COVID-19 data classification. $F_{\mathit{\text{RANkSAT}}}$ will cater the modified networks to unveil the true pattern or behavior of the real data sets involved. Note that $F_{\mathit{\text{RANkSAT}}}$ is a symbolic form representation thus it is appropriate to be integrated in these networks as HNN is a nonsymbolic platform.

2.2 Hopfield neural network (HNN)

Hopfield type of neural network (HNN) is a recurrent neural network (RNN) that mimics the human biological brain learning structure. The architecture of HNN model consists of interconnected neurons and a powerful feature of content addressable memory that is crucial in solving various optimization and combinatorial tasks [26]. The system consists of structured $N$ neurons, each of which is represented by an Ising variable. The neurons in discrete HNN are utilized in bipolar representation whereby $S_i\in \left\{1,-1\right\}$, which strictly considers values of 1 and − 1 [27]. The fundamental overview for neuron state activation in HNN is shown in Eq. (4)

$$S_i=\left\{\begin{array}{l}1,\mathit{\text{if}}\sum _j{T}_{\mathit{ij}}{S}_j>\omega \\ -1,\mathit{\text{Otherwise}}\end{array}\right.$$E8

where $T_{\mathit{ij}}$ is the synaptic weight from unit $j$ to $i$. $S_j$ is the state of neuron $j$ and $\omega$ is the predefined threshold value. Barra et al. (2018) specified that $\omega =0$ to certify the network’s energy decreases. The connection in Hopfield net contains no connection with itself as follows.

$$T_{\mathit{ijk}}^{\left(3\right)}=T_{\mathit{kij}}^{\left(3\right)}=T_{\mathit{kji}}^{\left(3\right)}$$E9

$$T_{\mathit{ji}}^{\left(2\right)}=T_{\mathit{ij}}^{\left(2\right)}$$E10

$$T_i^{\left(1\right)}=T_j^{\left(1\right)}$$E11

$$T_{\mathit{jj}}=T_{\mathit{ii}}=0$$E12

In resulting, HNN holds symmetrical features in terms of architecture. HNN model has similar intricate details to the Ising model of magnetism [28]. As the neuron state is termed in bipolar $S_i\in \left\{1,-1\right\}$ representation, the spin points follow in the direction of a magnetic field. This causes each neuron to flip until the equilibrium is reached. Thus, it follows the dynamics $S_i\to sgn\left[h_i\left(t\right)\right]$ where $h_i$ is the local field of the connection of the neurons. The sum of the field induced by each neuron is given as follows.

$$h_i=\sum _k^N\sum _j^N{T}_{\mathit{ijk}}{S}_j{S}_k+\sum _j^N{T}_{\mathit{ij}}{S}_j+T_i$$E13

The task of the local field is to evaluate the final state of neurons and generate all the possible RAN-SAT-induced logic that was obtained from the final state of neurons. One of the most prominent features of the HNN network is the fact that it always converges to stable states (Hopfield, 1982). The Lyapunov energy function (LEF) utilized in HNN for RANkSAT logic programming is presented as follows

$$H_F=\dots -\frac{1}{3}\sum _{i=1}^N\sum _{j\ne k}^N\sum _{k=1,i\ne k}^N{T}_{\mathit{ijk}}^{\left(3\right)}{S}_i{S}_j{S}_k-\frac{1}{2}\sum _{i=1,i\ne j}^m\sum _{j=1}^m{T}_{\mathit{ij}}^{\left(2\right)}{S}_i{S}_j-\sum _{i=1,i\ne j}^m{T}_i^{\left(1\right)}{S}_j$$E14

The energy function of the HNN model is especially critical since it will decide the interoperability of the network. The value obtained from the equation will be verified as global or otherwise. The network would generate the right response when the induced neuron state reached global minimum energy. There are minimal works to integrate HNN with RANkSAT as a single computational network.

2.3 Random k satisfiability reverse analysis (RANkSATRA)

The central emphasis of logic mining is to extract useful logical rules from the data sets provided. One of the extraordinary fields of data mining is logic mining proposed by [22]. It was shown that the information can be expressed in logical form. As a logical rule, the reverse analysis approach (RA) was implemented to derive useful knowledge from real-life data based on the horn clause [22]. In this study, Random k Satisfiability enhanced reverse analysis method or abbreviated as (RANkSATRA) is proposed to extract the optimum RANkSATRA logical rule to explain the behavior of the COVID-19 data sets. In this context, RANkSATRA is a logic extraction method that uses the HNN-RANkSAT model structure to extract from the data set the valuable logical rule (COVID-19 data set). Because of its non-systematic behavior, the RANkSAT logical rule would be used to describe and chart the data sets due to flexibility and convenience. In particular, the RANkSATRA approach can derive the ideal logic representing the relationship between the actual data set attributes of the COVID-19. Pursuing that, to be used in classification or estimation, the secret information in the data set is retrieved.

In our hybrid HNN model, RANkSATRA is performed out to represent a data classification framework in data mining. Inside the RANkSAT clauses, each of the attributes can be translated into atoms. To construct the RANkSAT logical rule, seven attributes from the data sets are then chosen by considering $k\le 3$. Logic mining is a method that extracts information from a data set using logic programming. In this regard, this section will clarify how our HNN-RANkSATRA model implements the logic mining technique called the random 3-satisfaction-based reverse analysis process (RAN3SATRA) to obtain the relationship of COVID-19 data entries. By acquiring the synaptic weight between 3 neurons, RAN3SATRA might be able to reveal the level of their connectedness.

Consider n attributes of the COVID-19 data sets $\left(S_1,S_2,S_3,\dots ,S_n\right)$, where $S_i\in \left[1,-1\right]$. All entries are represented in bipolar states. Since this chapter considers $F_{\mathit{RAN}3\mathit{SAT}}$, the arrangement of each $S_m$ consists of $S_i,S_j,S_k$ where $i\ne j\ne k$. For $S_m$ those leads ${P_{\mathit{\text{RANkSAT}}}}_{\mathit{\text{learn}}}^{}=1$, we assign

$$S_m=\left(S_i^{max\left[n\left(s_i\right)\right]}\vee S_j^{max\left[n\left(s_j\right)\right]}\vee S_k^{max\left[n\left(s_k\right)\right]}\right)$$E15

$$S_i=\left\{\begin{array}{l}{S}_i,S_i=1\\ \neg S_i,S_i=-1\end{array}\right.$$E16

Based on the obtained $S_m$, we can formulate ${P_{\mathit{\text{RANkSAT}}}}_{\mathit{\text{best}}}^{}$:

$${P_{\mathit{\text{RANkSAT}}}}_{\mathit{\text{learn}}}^{}={\vee }_{m=1}^k{S}_m$$E17

For example, we will choose $G_1=\left(S_1\vee \neg S_2\vee \neg S_3\right)$ if $S_1^{max\left[n\left(S_1\right)\right]}=S_1$, $S_2^{max\left[n\left(S_2\right)\right]}=\neg S_2$_, and $S_3^{max\left[n\left(S_3\right)\right]}=\neg S_3$. ${P_{\mathit{\text{RANkSAT}}}}_{\mathit{\text{best}}}^{}$ will be embedded into HNN. Henceforth, we will obtain the states of $S_i$ that correspond to $E_{{F_{\mathit{RAN}3\mathit{SAT}}}_{\mathit{\text{best}}}^{}}\to 0$. By comparing Eq. (3) with Eq. (14), the corresponding $T_{\mathit{ijk}}$ will be obtained. During the testing phase, the induced states, $S_i^B$, will be obtained by obeying Eq. (13). Subsequently, the induced logic $P_i^B$ will be constructed according to logical rule given in Eq. (2). Finally, the chosen induced logic obeys $P_i^B=P_i^{\mathit{\text{test}}}$ (Training data). Figure 1 demonstrates how RANkSATRA has been implemented in the HNN model to classify COVID-19 surveillance data set (CSDS). In this study, RANkSATRA, is used to determine the relationship among the data set. In learning COVID-19 surveillance data set, {detected, not detected} would be converted into bipolar representation {1,-1}, respectively. Each objective taken would be represented in terms of neurons in RANkSATRA. Hence, there would be a total of seven neurons being considered in this data set. Each neuron will be represented with entries COVID-19 surveillance data set.

Figure 1.

Implementation of RANkSATRA for COVID-19 surveillance data set.

2.4 RANkSATRA experimental setup

The simulation has been developed to explore the capacity of the Random kSatisfiability reverse analysis (RANkSATRA) in Hopfield neural network for COVID-19 surveillance data set classification. Sixty percent (60%) of the data points in the databases were used for learning data collected, and four ty percent (40%) were used for testing. Microsoft Visual C++ applications running on Windows 8.1, 64-bit, 4.40 GHz CPU, 4GB RAM, and 400 GB hard drive specifications were adopted. For both learning and testing, the overall CPU time is 24 hr. If the model crosses the recommended processor time threshold, the recommended algorithm structure cannot train HNN-based RANkSATRA using real life. In terms of the logical rule that will be embedded inside HNN, the existing work of Sathasivam and Abdullah [29] that implemented HORNSAT in their proposed reverse analysis method. RANMAXkSAT has been proposed [30] and RANkSAT has been proposed in [31]. Both of these proposed models were considered the only existing logic mining in the literature.

2.5 Implementation of COVID-19 surveillance data set

In this chapter, COVID-19 surveillance data set was occupied in RANkSATRA, HORNkSATRA, and RANMAXkSATRA for classification COVID-19 data set collected from UCI machine learning repository. It contains information about the data set and contains different purposes. The original data contains 7 instances with nine attributes with two classes. The classes are detected and not detected. But, to find out more relevant features from a COVID-19 data set, feature selection methods are applied to the data set. In this experiment, our aim is to that utilize the same data set. The details of COVID-19 are shown in Table 1.

Details of each attribute		Output $F_{\mathit{\text{RANkSAT}}}$
T1	Fever or history of fever	To classify between; $F_{\mathit{\text{RANkSAT}}}=1$(Detected) and $F_{\mathit{\text{RANkSAT}}}=-1$ (not detected)
T2	signs and symptoms of respiratory; distress e.g., cough, cold, and sore throat
T3	Acute and/or severe pneumonia or; Respiratory Infections (ARI)
T4	There are no other causes based on convincing clinical descriptions
T5	In the last 14 days before the; symptoms have a history of; travel or living abroad who; reported local transmission
T6	In the last 14 days before the; symptoms have a history of; travel or stay in the local; transmission area in Indonesia
T7	Contact with Coronavirus Disease; 2019 (COVID-19) confirmation cases in the last 14 days before symptoms

Table 1.

List of attributes for COVID-19 surveillance data set (CSDS) [32].

3. Performance evaluation metrics

In this section, simulation experiments were performed to assess the performance of the proposed logical rule model on a different number of clauses. The performance evaluation indicators are deployed to analyze the effectiveness of our SATRA model in extracting important logical rule for CSDS. The metrics used in this study measure the performance of the training phase of HNN models. The metric solely indicates the performance of the retrieved neuron state that contributes to the best CSDS classification. During the learning phase, the performance of the RANkSAT representation that governs the network will be evaluated based on the following fitness equation:

$$f_k=\sum _{k=1}^{\mathit{NC}}{C}_k$$E18

NC is the number of clauses for any given $P_k^B$ . According to Eq. (18)$C_k$ is defined as follows.

$$C_k=\left\{\begin{array}{l}1\mathit{\text{True}}\\ 0\mathit{\text{False}}\end{array}\right.$$E19

The measurements are evaluated based on the accuracy, and errors accumulation that reflects the network complexity based on the number of neurons using the following formula.

$$\mathit{\text{TRANING}}\_\mathit{MAE}=\sum _{i=1}^n\frac{1}{n}\left|f_{\max }-f_k\right|$$E20

$$\mathit{\text{TRAINMING}}\_\mathit{\text{RMSE}}=\sum _{i=1}^n\sqrt{\frac{1}{n}{\left(f_{\max }-f_k\right)}^2}$$E21

where $f_{\max }$ and $f_k$ are the output value and target output value, respectively, and $n$ is a number of the iterations.

$$\text{TRAINING}\_\mathrm{BIC}=\mathit{nIn}\left(\mathit{MSE}\right)+\mathit{pa}.\mathit{\text{In}}\left(n\right)$$E22

where n, pa, and MSE indicate the number of solutions obtained, their parameters, and the mean square error used in the model, respectively. Since the HNN is free from any pa [33], the equation is re-written as follows.

$$\text{TRAINING}\_\mathrm{BIC}=\mathit{nIn}\left(\mathit{MSE}\right)$$E23

where the MSE is measured in calculating BIC and n depicts the number of iterations during the simulation. Hence, the formula of MSE is given as follows,

$$\mathit{\text{TRAINING}}\_\mathit{MSE}=\frac{1}{n}\sum _{i=1}^n{\left(f_{\max }-f_k\right)}^2$$E24

where $f_i$ and $f_{\max }$ describe the fitness value observed during each of the execution and the maximum fitness, respectively.

$$\mathit{\text{TRAINMING}}\_\mathit{\text{ACCURACY}}=\frac{P_{\mathit{\text{induced}}}^{\mathit{\text{Correct}}}}{N_{P_{\mathit{\text{test}}}}}\times 100\%$$E25

The performance of the HNN on the proposed logical rule is presented in Table 2.

Logical rule	MAE	RMSE	ACCURACY
RANkSATRA	1.6140	0.466	96.1
RANMAXkSTARA	3.921	1.170	83.3
HORNkSATRA	2.145	0.845	88.5

Table 2.

Training error and accuracy for all HNN models.

4. Result and discussion

The performance of the simulated program with different complexities for neurons in the HNN-RANkSATRA model will be evaluated with the existing models HNN-HORNSAT [29] and RANMAXkSAT [34] in terms of root mean square error (RMSE), mean absolute error (MAE), Bayesian information criterion (BIC), accuracy, and CPU time. Figures 2 and 3 illustrate the root mean square error (RMSE) and mean absolute error (MAE) of HNN models during the training process. This analysis only considers $1\le \mathit{NC}\le 10$. The COVID-19 surveillance data set (CSDS) data have successfully been embedded into the network and forming variants of learnable Boolean k Satsifiability logic, RANkSAT with the existing HORNSAT and RANMAXkSAT. A comparison has been made between the variants of satisfiability logic for COVID-19 surveillance data set (CSDS) data classification. As seen in Figure 2, HNN-RANkSATRA with NC = 1 until NC = 10 has the best performance in terms of RMSE compared to HNN-HORNkSATRA and RANMAXkSATRA. This is because the HNN-RANkSAT utilizes random logical inconsistencies to derive the optimum synaptic weight for HNN. Optimal synaptic weight is a building block for optimum CSDS classification. The RMSE result from Figure 2 has been supported by the value of MAE in Figure 3.

Figure 2.

MAE evaluation of HNN models for COVID-19 classification.

Figure 3.

RMSE evaluation of HNN models for COVID-19 classification.

The investigation of a model’s performance is separated into two parts. The first significant part is to examine the quality of the solution generated by different searching techniques by employing suitable training errors. Secondly is to analyze the robustness and efficiency of the proposed model by comparing CT and Q needed to execute the models’ mechanism. There are five performance evaluation metrics involved to analyze the training and testing stage of our modified models as presented in the performance evaluation section. Therefore, this research’s main contribution is to portray the competency of HNN in SAT in outperforming the existing models.

In Figure 2, HNN-RANkSATRA with NC = 2 has the best performance in terms of MAE. It can be observed that MAE for NC = 2 is equal to 0.027 compared with NC = 10 which recorded 1.3515. The searching process of HNN for HORNkSTRA and RANMAXkSTRA displayed a similar error trend with RANMAXkSTRA has the highest error at NC = 8. In Figure 3, HNN-RANkSATRA has the best performance in terms of RMSE. It can be observed at NC = 1, the RMSE is equal to 0.0323 and 1.3995 was recorded at NC = 10, which are lower than HNN-HORNSAT and HNN-RANMAXkSTRA. The HNN-RANMAXkSTRA recorded the highest RMSE with 0.832 at NC = 1 and 10.4602 at NC = 10. The searching process of HNN for RANkSTRA and HORNkSATRA displayed a similar RMSE trend with closed merging. As the number of neurons increased, the learning phase of HNN models was convoluted because the network required to find the consistent interpretation for the best logic for optimal COVID-19 surveillance data set (CSDS) classification. In this case, the learning phase of HNN for both RANMAXkSTRA and HornkSATRA was trapped in the trial and error search process, which resulted in high RMSE and MAE accumulations. However, the RMSE and MAE recorded by HNN-HORNSAT and RANMAXkSTRA were to some extent higher than the RANkSTRA. At NC = 5, the value of MAE and RMSE are approximately 57% times bigger than NC = 10 because for the network to converge into full fitness (learning completed), more iterations are needed. Therefore, a similar trend can be seen in the HNN model as the complexity increases. Thus, HNN model works optimally in learning the variants of satisfiability logic entrenched to the network before being stored into content addressable memory. The complete learning process will ensure the network generates the best logic to represent the characteristic of the HNN for optimal COVID-19 surveillance data set (CSDS) classification.

In contrast, the learning phase in HNN-HORNSAT and HNN- RANMAXkSTRA were computationally expensive as more iterations were needed leading to higher RMSE and MAE values compared to HNN-RANkSATRA. All in all, SATRA contributes to generating the best logic to represent the relationship between each instance and the verdict of HNN for COVID-19 surveillance data set (CSDS) classification. The MAE and RMSE results displayed in Figures 2 and 3 were supported by BIC in Figure 4. The “best” model will be the one that neither under-fits nor over-fits. Although the BIC will choose the best model from a set of models, it will not say anything about the absolute quality of the model. However, the HNN-RANkSATRA is the best choice for COVID-19 surveillance data set (CSDS) classification. In terms of BIC, HNN-RANkSATRA outclasses other models. The accumulation of MSE tends to penalize the values of BIC. The BIC for HNN-RANkSATRA is, therefore, the lowest compared to the other two models.

Figure 4.

BIC evaluation of HNN models for COVID-19 classification.

Table 3 displays the CPU Time results for the HNN models, respectively. To assess the robustness of the models in logic mining, CPU time is recorded for the learning and retrieval phase of HNN. HNN less CPU time is required to complete one execution of learning and testing for CSDS classification when the number of NC deployed is less. As it stands, HNN-RANkSATRA model, when the complexity is higher, models take a long time to finish the learning process. Overall, the HNN continues to be proficient at reducing the k Satisfiability inconsistencies and computing the global solution in a reasonable amount of time on the CPU. Because there are more instances to handle during the learning and testing phase of HNN, the CPU Time for HNN-RANMAXkSAT is consistently greater than HNN-RANkSATRA and HNN-HORNSATRA. However, the CPU time recorded for the existing methods was higher due to more iterations needed in generating the best logic for the HNN.

NC	RANk; SATRA	RANMAXk; SATRA	HORN; SATRA
1	0.643	1.8232	1.735
2	0.835	2.0279	2.8006
3	1.9707	5.3258	6.1323
4	3.2358	11.1566	13.004
6	9.7422	19.434	24.9292
7	18.46	22.5552	35.1675
8	25.2815	36.9081	42.005
9	37.958	65.6528	58.0088
10	72.312	91.41	76.234

Table 3.

Computation (CPU) time(s) for the HNN model.

Table 2 displays the testing error and accuracy data collected for each model when the HNN was tested. As a result, for each of NC = 1 through NC = 8, the testing RMSE, MAE, and accuracy recorded for HNN-RANkSATRA, HNN-RANMAXkSAT, and HNN-HORNSAT were consistently identical. The ability of our suggested network, SATRA and HNN-kSAT, to produce the optimal logic during the learning phase, which contributes to a very low error, is thus demonstrated by this. According to performance evaluation metrics captured during simulation, the learning mechanism in SATRA is capable of deriving the best logic to map the relationships between the characteristics in HNN. 96.30 percent accuracy was attained by the suggested model, as measured by the accuracy that each model recorded. The existing works on logic mining in HNN can still achieve up to 80% of the CSDS classification accuracy. To sum up, HNN-RANkSATRA is the best model for learning and testing HNN due to lower values of RMSE, MAE, and the highest accuracy for a logical rule. Hence, the logical rule obtained from HNN-RANkSATRA will benefit the healthcare sector in the CSDS classification problem.

5. Conclusion

In this research, the formulation of constructing HNN proved to be adequate to represent the mechanism in RANkSAT the implementation of EA with standalone HNN framework with reverse analysis proved to be effective in solving real-life data sets. In order to optimally express the relationship in a data set, we successfully turned the COVID-19 data set into an ideal logical representation in the form of RANkSAT representation in this chapter. Furthermore, we used COVID-19-RANkSATRA as a substitute method for identifying correlations between the variables that correspond to classification between $F_{\mathit{\text{RANkSAT}}}=1$ (Detected) and $F_{\mathit{\text{RANkSAT}}}=-1$ (not detected) of the COVID-19 data set. To work, the neural network needed to be trained. Because the structure of a neural network (NN) differs from the structure of microprocessors, it must be imitated. For large neural networks, it required a lot of processing time and space in weight training and adjustment, system adaptation for determining the number of layers, node transfer functions, retrieval phase, and learning rules.

One of the limitations of Hopfield type of artificial neural network is that it sometimes gets trapped at the local solution (premature convergence) instead of the global one. Our Future direction is to incorporate novel metaheuristics algorithms such as Election algorithm, genetics algorithm, dragonfly algorithm, etc. to enhance the performance of Hopfield type of artificial network for better training and retrieval process in preventing the Hopfield type of artificial network in settling in the local solution for better searching and classification problems. We will further utilize other variants of Boolean satisfiability such as k-SAT, Rondom Half-SAT, MAX-kSAT, Random NAE-SAT, and XOR-SAT for better optimization problems. Various type of data set, such as agriculture, financial, actuarial as well as environmental data set will also be used in our future studies.