ER rule classifier with an optimization operator recommendation

Abstract

It is difficult for many classic classification methods to consider expert experience and classify small-sample datasets well. The evidential reasoning rule (ER rule) classifier can solve these problems. The ER rule has strong processing and comprehensive analysis abilities for diversified mixed information and can solve problems with expert experience effectively. Moreover, the initial parameters of the classifier constructed based on the ER rule can be set according to empirical knowledge instead of being trained by a large number of samples, which can help the classifier classify small-sample datasets well. However, the initial parameters of the ER rule classifier need to be optimized, and choosing the best optimization algorithm is still a challenge. Considering these problems, the ER rule classifier with an optimization operator recommendation is proposed in this paper. First, the initial ER rule classifier is constructed based on training samples and expert experience. Second, the adjustable parameters are optimized, in which the optimization operator recommendation strategy is applied to select the best algorithm by partial samples, and then experiments with full samples are carried out. Finally, a case study on a turbofan engine degradation simulation dataset is carried out, and the results indicate that the ER rule classifier has a higher classification accuracy than other classic classifiers, which demonstrates the capability and effectiveness of the proposed ER rule classifier with an optimization operator recommendation.

Keywords

Evidential reasoning rule (ER rule)optimization operator recommendation classification turbofan engine degradation status

1 Introduction

Currently, with the amount of data growing explosively in many application fields, quickly and accurately mining valuable information has become a challenge, and data analysis has become a research hotspot [1]. Data analysis, which includes data mining, predictive analysis, visual analysis, semantic analysis, data quality management, etc., aims to acquire knowledge from data. As an important part of data analysis, classification is applied in a wide range of fields, especially in complex systems, such as pattern recognition, medical diagnosis, and fault diagnosis. It is vital to carry out classification research because of its wide range of applications.

To date, many classic classifiers have been proposed, such as the decision tree (DT), support vector machine (SVM), artificial neural network (ANN), classification based on association (CBA), and Bayesian classification. DT is a typical classifier that applies induction algorithms to generate readable rules and decision trees for classification [2, 3]. SVM follows the minimization criterion of structural risk and adopts mathematical methods and optimization techniques [4, 5]. ANN simulates the human neural network to process information, and it possesses strong fault tolerance and self-learning function [6, 7]. CBA realizes classification by generating classification association rules [8, 9]. The Bayesian classifier trains sample subsets with class labels, generalizes the classification functions, and generates classifiers to realize the classification of unclassified samples [10, 11]. These classic classification approaches have achieved some success in solving the classification of complex systems; however, these classifiers have several common disadvantages.

In the process of solving practical problems, the existing classification methods still encounter some challenges: (1) In the analysis and processing of data in some fields, expert experience and knowledge are required. For instance, the fault diagnosis of large equipment should combine the experience and professional knowledge of professional technicians. Expert experience and knowledge can also significantly improve interpretability. However, most of the existing classification methods are purely data-driven methods, which are difficult to combine with expert experience and knowledge. (2) Existing classification methods cannot classify small-sample datasets well since purely data-driven methods need large datasets to set the parameters. The sample size of some important cases is small because large numbers of training samples cannot be obtained due to the high cost or harsh conditions. Therefore, it is difficult to achieve the high-precision classification of small-sample datasets, and it is difficult to train the parameters with a small number of samples.

Considering the above problems, the evidential reasoning rule (ER rule) [12] developed from Dempster-Shafer (D-S) theory [13, 14] performs well in classification problems. As a more general Bayesian rule, the ER rule has strong processing and comprehensive analysis abilities for diversified mixed information, such as subjective/objective and qualitative/quantitative information, and can effectively solve problems with expert experience [15]. In addition, the initial parameters of the classifier constructed based on the ER rule can be set according to empirical knowledge, which can help the classifier classify small-sample datasets well. Therefore, a classifier based on the ER rule is proposed to perform classification in this study.

However, the initial ER rule classifier may not be able to accurately perform classification due to the assumptions of the initial values of the adjustable parameters. To achieve a higher classification accuracy, the initial parameters need to be optimized. That is, parameter learning by training samples is necessary when training samples are available. Thus, choosing the best optimization algorithm to achieve high efficiency is a challenge. The problem of selecting the best algorithm from among various feasible algorithms is common in theoretical research, especially regarding the selection of optimization algorithms. To make the right choice, the commonly adopted solution is to conduct complete experiments based on complete samples for all candidate algorithms individually, evaluate the algorithms by comparing the results, and finally choose the best algorithm. Although this approach can accurately select the best algorithm, it takes a long time to perform experiments with full samples, resulting in low efficiency. Considering this problem, an optimization operator recommendation strategy is developed in this study [16]. The core idea of the optimization operator recommendation is to select the best algorithm with partial samples and then perform experiments with full samples, which greatly improves the efficiency.

By comprehensively analyzing classification problems, an ER rule classifier with an optimization operator recommendation based on the original ER rule classifier [17] is proposed in this paper. First, the initial classifier is constructed by training data. Then, the optimization operator recommendation is applied to parameter optimization to select the best optimization algorithm with partial data. Finally, the optimized ER rule classifier can classify data with high accuracy.

The remainder of this paper is organized as follows. The basic concept of the ER rule and the construction of the original ER rule classifier are introduced in section 2. Section 3 describes the optimization steps of the ER rule classifier in detail, proposes the optimization operator recommendation strategy, and describes the process of the ER rule classifier and optimization operator recommendation. In section 4, the feasibility and effectiveness of the proposed ER rule classifier and the optimization operator recommendation strategy are verified by using a turbofan engine degradation simulation dataset. Finally, this study is concluded in section 5.

2 The basics of the ER rule

This section briefly reviews the development of the ER rule and the relevant research based on the ER rule and introduces the reasoning algorithm of the ER rule, which constitutes the theoretical basis of this study. Then, the modeling process of the ER rule classifier, which is consistent with that proposed by Xu et al. [17], is described. The process can be divided into three steps: evidence acquisition, reliability and weight calculation, and combination of activated evidence.

2.1 The basic concept of the ER rule

The concept of ER first appeared when Gordon and Shortliffe applied D–S theory to evidence-aggregation processes in a hypothesis space [18]. In 2002, to address both numerical data and qualitative information with uncertainty in multiple attribute decision analysis (MADA), Yang and Xu proposed an ER approach supporting decision analysis, the kernel of which is the ER algorithm developed based on D-S theory [19, 20].

The kernel of ER is D-S theory, while the kernel of D-S theory is Dempster’s rule, which originates from probability theory and constitutes the inference process of conjunctive probabilistic [21 –23]. Dempster’s rule generalizes Bayes’s rule and was initially used as the only evidence combination rule (ECR) to combine completely reliable independent evidence. Dempster’s rule can achieve the combination of completely reliable evidence, but it also has shortcomings: (1) Dempster’s rule lacks definition and cannot be applied to the special case where two pieces of evidence completely conflict, such as the case where two pieces of evidence support different propositions. (2) Dempster’s rule assumes that every piece of evidence is completely reliable and can reject any proposition. This means that if a piece of evidence does not support a proposition, the proposition is completely excluded. In other words, Dempster’s rule only accumulates consensus support, and if a proposition is opposed by any evidence, regardless of whatever support it may receive from any other evidence, Dempster’s rule will completely reject the proposition.

Considering the existing problems and deficiencies of Dempster’s rule, on the basis of Dempster’s rule, Yang et al. proposed the ER rule, a new rule combining the weight and reliability of multiple pieces of independent evidence [12]. Yang proposed a new concept of weighted belief distribution (WBD) and extended it to weighted belief distribution with reliability (WBDR) to represent the characteristics of the belief distribution (BD) introduced in D-S evidence theory. The ER rule is obtained by the orthogonal sum of WBDs and WBDRs. The most important property of the ER rule is that it constitutes a general conjunctive probability inference process or a generalized Bayesian inference process. In addition, when the reliability equals the weight of evidence and all the weight of evidence is normalized, the original ER algorithm is a special case of the ER rule. When every piece of evidence is completely reliable, Dempster’s rule is also a special case of the ER rule. The ER rule improves Dempster’s rule by identifying new reliability perturbations to combine highly conflicting or completely conflicting yet completely reliable pieces of evidence.

Since it was proposed, the ER rule has shown its excellent performance in various applications. Xu et al. solved the fusion decision problem of uncertain fault characteristic information and the data classification problem based on the ER rule [24]. Cheng et al. applied the ER rule to evaluate the network security situation [25]. Yang studied the fault diagnosis of high-speed train bogies based on the ER rule and hierarchical classification method [26]. Gao and Xu have applied ER rule to aggregating financial analysts’ opinions for decision making in stock investment [27].

2.2 Reasoning algorithm of the ER rule

Θ ={ θ₁, θ₂, …, θ_N } is a frame of discernment, which is a set of mutually exclusive and collectively exhaustive proposition. The power set of Θ is composed of 2^N subsets of Θ [12 , 28]: $\begin{matrix} P (Θ) = 2^{Θ} = {\emptyset, θ_{1}, \dots, θ_{N}, {θ_{1}, θ_{2}}, \\ \dots, {θ_{1}, θ_{N}}, \dots, {θ_{1}, \dots, θ_{N - 1}}, Θ} \end{matrix}$ (1)

A piece of evidence e_j is defined as follows: $e_{j} = {(θ, p_{θ, j}) ∣ \forall θ \subseteq Θ, \sum_{θ \subseteq Θ} p_{θ, j} = 1}$ (2)

In ER rule, a reliability r_j and weight w_j are considered for each piece of evidence. The belief distribution of evidence with reliability and weight is shown as follows: $m_{j} = {(θ, {\tilde{m}}_{θ, j}) ∣ \forall θ \subseteq Θ; (P (Θ), {\tilde{m}}_{P (Θ), j})}$ (3) ${\tilde{m}}_{θ, j} = {\begin{matrix} 0 & θ = \emptyset \\ c_{rw, j} m_{θ, j} & θ \subseteq Θ, θ \neq \emptyset \\ c_{rw, j} (1 - r_{j}) & θ = P (Θ) \end{matrix}$ (4) where c_rw,j = 1/(1 + w_j - r_j), m_θ,j = w_jp_θ,j, and 0 ≤ w_j ≤ 1, 0 ≤ r_j ≤ 1.

When multiple pieces of independent evidence are integrated, the combined belief degree to jointly support the proposition can also be expressed as follows: $\begin{matrix} m_{θ, e (i)} & = [m_{1} \oplus \dots \oplus m_{i}] (θ) \\ = {\begin{matrix} 0 & θ = \emptyset \\ \frac{{\hat{m}}_{θ, c (i)}}{\sum_{D \subseteq Θ} {\hat{m}}_{D, e (i)} + {\hat{m}}_{P (θ), e (i)}} & θ \subseteq Θ, θ \neq \emptyset \end{matrix} \end{matrix}$ (5) $\begin{matrix} {\hat{m}}_{θ, e (i)} = & [(1 - r_{i}) m_{θ, e (i - 1)} + m_{P (Θ), e (i - 1)} m_{θ, i}] \\ + \sum_{B \cap C = θ} m_{B, e (i - 1)} m_{C, i} \end{matrix}$ (6) ${\hat{m}}_{P (Θ), e (i)} = (1 - r_{i}) m_{P (Θ), e (i - 1)}$ (7)

2.3 ER rule classifier

2.3.1 Evidence acquisition

First, likelihood normalization is applied to obtain evidence from the training sample dataset. The relation between each attribute and each class is obtained from the input data to generate the set S: $\begin{matrix} S = {[x^{k}, y_{k}] ∣ x^{k} = (x_{1}^{k}, \dots, x_{i}^{k}, \dots, x_{M}^{k}), \\ x_{i}^{k} \in S_{i}, y_{k} \in Θ, k = 1, 2, \dots, K_{s}} \end{matrix}$ (8)

The relationship between attribute x_i and class y needs to be approximately transformed into the relationship between the specific referential values $A_{i} = {A_{j}^{i} ∣ j = 1, \dots, J_{i}}$ of attribute x_i and class y. For a specific attribute value $x_{i}^{k}$ , the similarity distribution $S_{I} (x_{i}^{k})$ of the referential value A_i can be calculated by the following information transformation:

$\begin{matrix} S_{I} (x_{i}^{k}) = {(A_{j}^{i}, α_{i, j}) ∣ j = 1, \dots, J_{i}; i = 1, \dots, M} \\ α_{i, j} = \frac{A_{j + 1}^{i} - x_{i}^{k}}{A_{j + 1}^{i} - A_{j}^{i}}, α_{i, j + 1} = 1 - α_{i, j} A_{j}^{i} \leq x_{i}^{k} \leq A_{j + 1}^{i} \\ α_{i, j^{'}} = 0, j^{'} = 1, \dots J_{i}, j^{'} \neq j, j + 1 \end{matrix}$ (9)

Thus, a sample pair $(x_{i}^{k}, y_{k})$ in set S can be transformed to a similarity distribution (α_i,j, α_i,j+1) of class y_k. a_n,j is the sum of the similarity degree of the sample pairs, whose attribute value $x_{i}^{k}$ matches the referential value $A_{j}^{i}$ and which also belong to the class y_n, $δ_{n} = \sum_{j = 1}^{J_{i}} a_{n, j}$ , $η_{j} = \sum_{n = 1}^{N} a_{n, j}$ , and $\sum_{n = 1}^{N} δ_{n} = \sum_{j - 1}^{J_{i}} η_{j} = K_{s}$ . When the class y_n is known, we can construct the value of the attribute x_i as the likelihood of the referential value $A_{j}^{i}$ , denoted as c_n,j: $c_{n, j} = p (A_{j}^{i} ∣ y_{n}) = \frac{a_{n, j}}{δ_{n}}$ (10)

Therefore, a piece of evidence $e_{j}^{i}$ can be defined, and the belief degree distribution of each class is given as follows: $β_{n, j}^{i} = \frac{c_{n, j}}{\sum_{k = 1}^{N} c_{k, j}}$ (11)

More precisely, $β_{n, j}^{i}$ is the probability that a sample is believed to belong to the class y_n given that the input attribute x_i takes the referential value $A_{j}^{i}$ .

2.3.2 Reliability and weight calculation

The reliability R ={ r_i ∣ i = 1, 2, …, M } reflects the ability of the information that generates the evidence to accurately solve specific problems, and it can be considered that the reliability of the evidence can represent the classification ability of its corresponding attributes. Therefore, the reliability r_i of attribute x_i can be defined as follows [17]: $r_{i} = \frac{Q_{i}}{{max}_{l, \in {1, 2, \dots, M}} (Q_{l})}$ (12) where Q_i is the number of samples that can be directly classified by attribute x_i. The initial weight of each evidence is set as the reliability, and then the weights are optimized in the following steps.

2.3.3 Combination of activated evidence

For a certain sample with M attribute values $x^{k} = (x_{1}^{k}, \dots, x_{i}^{k}, \dots, x_{M}^{k})$ , if the value of the attribute $x_{i}^{k}$ is in interval $[A_{j}^{i}, A_{j + 1}^{i}]$ , $x_{i}^{k}$ will activate the adjacent two pieces of evidence: evidence $e_{j}^{i}$ with the weight $w_{j}^{i}$ and evidence $e_{j + 1}^{i}$ with the weight $w_{j + 1}^{i}$ . The evidence e_i obtained from the attribute $x_{i}^{k}$ can be calculated as a weighted sum of $e_{j}^{i}$ and $e_{j + 1}^{i}$ :

$e_{i} = {(y_{n}, p_{n, i}), n = 1, \dots, N}$ (13) $p_{n, i} = α_{i, j} β_{n, j}^{i} + α_{i, j + 1} β_{n, j + 1}^{i}$ (14) where p_n,i refers to the belief of the sample belonging to class y_n when $x_{i}^{k}$ activates $e_{j}^{i}$ and $e_{j + 1}^{i}$ . The weight w_i of evidence e_i can be approximately calculated as the weighted sum of $w_{j}^{i}$ and $w_{j + 1}^{i}$ : $w_{i} = α_{i, j} w_{j}^{i} + α_{i, j + 1} w_{j + 1}^{i}$ (15)

After all the M pieces of evidence about M attributes have been obtained by Eqs. (13-15), the ER rule in Eqs. (5-7) can be used to combine the M pieces of evidence with weight and reliability, and the fusion results are shown in Eq. (16): $O (x^{k}) = {(y_{n}, p_{n, e (m)}), n = 1, \dots, N}$ (16)

According to the results O (x^k), sample x^k belongs to the class y_n that possesses the maximum belief p_n,e(m).

3 Parameter optimization based on an optimization operator recommendation strategy

In this section, the parametric optimization model of the ER rule classifier is introduced, and then an optimization operator recommendation strategy is proposed to select the optimization algorithm to optimize the adjustable parameters of the initial ER rule classifier. Finally, the overall process of the ER rule classifier with an optimization operator recommendation strategy is described.

3.1 Parametric optimization model of ER rule classifier

(1) Parametric optimization model

The construction of the initial ER rule classifier has been completed by 2.3.1-2.3.3 sections. The initial ER rule classifier may not be able to accurately model the complex causal relationship between the attributes x_i = (i = 1, …, M) and the actual classes due to the initial value assumptions of the adjustable parameters. Therefore, it is very important to train these adjustable parameters by the sample dataset to improve the classification accuracy of the classifier. The parametric optimization model to optimize the adjustable parameters is shown as Eqs.(17-18). $min f (A_{j^{'}}^{i}, w_{j}^{i})$ (17) $s . t . {\begin{matrix} A_{j^{'} - 1}^{i} < A_{j^{'}}^{i} < A_{j^{'} + 1}^{i}, & i = 1, \dots, M \\ j^{'} = 2, \dots, J_{i} - 1 \\ 0 \leq w_{j}^{i} \leq 1, & i = 1, \dots, M, \\ j = 1, . ., J_{i} \end{matrix}$ (18)

I. The adjustable parameters to be optimized

The parameters of the ER rule classifier include the referential values of attributes $A_{i} = {A_{j}^{i} ∣ j = 1, \dots, J_{i}}$ and weights of evidence $W = {w_{j}^{i} ∣ i = 1, \dots,$ M ; j = 1, … , J_i }. $P = {A_{j^{'}}^{i}, w_{j}^{i} ∣ i = 1, \dots, M;$ j′ = 2, …, J_i - 1 ; j = 1, …, J_i} are the parameters to be optimized, and the remaining parameters ( $A_{1}^{i}$ and $A_{J_{i}}^{i}$ ) are fixed boundary values ( $min_{k, k \in S_{x_{i}}} (x_{i}^{k})$ and $max_{k, k \in S_{x_{i}}} (x_{i}^{k})$ ). Eq.(18) shows the constraint conditions that the parameters must satisfy. The parameters P are shown clearly in Fig. 1, including M · (J_i - 2) referential values of attributes and M · J_i weights of evidence.

Fig. 1

The adjustable parameters of the ER rule classifier

II. The objective function

To optimize the parameters, Xu proposed the objective function $min_{P} ξ (P)$ based on the MSE (mean squared error) [17]. Considering that the MSE is more suitable for regression problems, this paper adopts the objective function $f (A_{j^{'}}^{i}, w_{j}^{i})$ based on the classification accuracy, as this function can more intuitively and accurately reflect the classification ability of the classifier: $min f (A_{j^{'}}^{i}, w_{j}^{i}) = Accuracy = - \frac{rightNum}{totalNum}$ (19) where rightNum is the number of samples correctly classified in the training dataset, and totalNum is the total number of samples in the training dataset. The objective function is obtained by taking a negative value of the classification accuracy since the value the target function pursues is the minimum value.

(2) The optimization algorithms

Due to the large number of optimization parameters, it is easy to fall into local optimum. Therefore, to avoid the local optimum solution, we use a global optimization algorithm. This study applies intelligent optimization algorithms to optimize the parameters globally, while Xu utilizes SLP (sequential linear programming) to obtain the local optimal solution [17]. The encoding of parameters to be optimized is shown in Fig. 2.

Fig. 2

The encoding of parameters

Since which optimization algorithm has the best optimization effect is still uncertain, three swarm intelligence optimization algorithms are chosen as candidates: particle swarm optimization (PSO) [29], the genetic algorithm (GA) [30], and the seeker optimization algorithm (SOA) [31]. The final optimization algorithm for the ER rule classifier is selected by the optimization operator recommendation strategy.

3.2 Optimization operator recommendation strategy

In practical applications, it is common to choose the algorithm that is most suitable and can improve the problem solving ability of the research method. To make the right choice, the commonly adopted method is to conduct complete experiments for all candidate algorithms in turn, evaluate the advantages and disadvantages of various algorithms by comparing the results obtained from specific experiments, and finally choose the best algorithm as the algorithm for the research method.

The general method for selecting the best algorithm uses all the sample data for complete experiments. Although the results obtained from experiments based on complete data are more accurate and can make the correct algorithm selection, the selection process involves a large workload and low efficiency, especially when the sample size is large. Therefore, it is necessary to propose a method to improve the efficiency of algorithm selection.

Considering the above problems, this study proposes an optimization operator recommendation strategy. The main idea is as follows: first, partial samples are randomly selected from the whole sample dataset, and the size of the partial sample dataset can be determined according to specific problems and scenarios. Then, all candidate algorithms are used to conduct experiments on partial samples, and the experimental results under different algorithms are obtained. Next, the experimental results are compared, and the optimal algorithm is selected as the algorithm used in the research method. Finally, the whole sample dataset is tested with the selected optimization algorithm. A schematic diagram of the optimization operator recommendation strategy is shown in Fig. 3.

Fig. 3

Schematic diagram of the optimization operator recommendation strategy

The randomly selected partial data can reflect the characteristics of the entire sample dataset and replace the complete sample set to select the most appropriate algorithm and can effectively improve the efficiency, which can also be verified by experiments.

As shown in Fig. 3, compared with the general algorithm selection method, the proposed optimization operator recommendation strategy does not require all algorithms to conduct experiments on the whole sample dataset, which can greatly reduce the amount of data to be processed and effectively improve the work efficiency.

3.3 ER rule classifier and optimization operator recommendation strategy

By applying the optimization operator recommendation strategy to the parameter optimization process of the ER rule classifier, the best optimization algorithm can be selected, and the best classification result can be obtained effectively. This process consists of three parts: acquisition and preprocessing of experimental data, training of the ER rule classifier, and testing of the ER rule classifier. The training of the classifier also comprises the construction of the initial classifier and the optimization of adjustable parameters. The test of the ER rule classifier is to classify the test dataset by the classifier, and the classification accuracy reflects the performance of the classifier. A schematic diagram of the whole process is shown in Fig. 4.

Fig. 4

The process of the ER rule classifier and optimization operator recommendation

Step 1: Preparing the experimental data

First, the experimental dataset is obtained from public datasets or from actual experimental operations, after which the data need to be preprocessed. Data preprocessing includes standardization of the data format, the elimination of abnormal data, the correction of errors, and determining the training and testing datasets.

Step 2: Training of the ER rule classifier

Step 2.1: Evidence acquisition

According to the experimental data and the evidence acquisition method in section 2.3.1, each piece of evidence $e_{j}^{i}$ and its belief degree distribution can be obtained based on Eqs.(9-11). That is, when attribute x_i has the referential value $A_{j}^{i}$ , the belief $β_{n, j}^{i}$ of the sample belonging to class y_n can be obtained.

Step 2.2: Reliability and weight calculation

The reliability of the attribute and the initial weight of the evidence can be determined by Eq.(12) described in section 3.1.2. According to belief value interval of different samples on each attribute and the number of samples in each interval, the individual classification ability of each attribute is clarified, the reliability of each attribute is generated, and the initial weight of evidence is determined by the reliability.

Step 2.3: Test the classification accuracy of the classifier

The above steps have generated evidence with reliability and weight, which can be combined to classify the test dataset. In other words, the initial ER rule classifier has been constructed. The evidence from the training dataset can generate the belief distribution of the estimated classes for each input by activating and integrating the evidence, and the class with the highest belief is determined to be the class of the input. By comparing the classification results obtained by the initial classifier with the actual results, the classification accuracy of the unoptimized classifier can be obtained.

Step 2.4: Optimization of the adjustable parameters

According to the constraint conditions and objective function described in section 3.1.4, the initial classifier can be globally optimized based on various intelligent optimization algorithms. For instance, PSO, GA, SOA, and other classic intelligent optimization algorithms can be applied to optimize parameters. In the process of optimization, new reliabilities and weights of evidence are generated, which will generate a new classification accuracy for the training dataset and improve the classification ability of the ER rule classifier.

Step 3: Testing of the ER rule classifier

Step 3.1: Classification of the test dataset

After step 2, the adjustable parameters of the ER rule classifier have been optimized by various optimization algorithms; that is, the optimized ER rule classifiers are obtained. In this step, the optimized classifiers are applied to classify the test dataset, and the results obtained will be used as the criterion to select the best optimization algorithm.

Step 3.2: Choosing the best optimization algorithm

By analyzing the classification results of the test dataset, the best optimization algorithm with the highest accuracy is selected as the ER rule classifier’s optimization algorithm, after which the optimized ER rule classifier is constructed. The result obtained by the optimized ER rule classifier is the final result of the dataset.

4 Case study: turbofan engine degradation status classification

In this section, the experimental steps and data information are first explained in detail, and second, the dataset is classified based on the optimization operator recommendation strategy. Then, experiments with complete datasets are conducted to verify the feasibility and effectiveness of the optimization operator recommendation strategy, and finally, the experimental results are analyzed.

4.1 Turbofan engine degradation simulation data

As the engine is the core of any large piece of equipment, the health of the engine is crucial to the operation of the equipment. Determining the engine’s health and then predicting potential failures is essential to improve the reliability, safety and availability of equipment, so it is necessary to classify the health status of the engine. The dataset adopted in the experiment is a turbofan engine degradation simulation dataset provided by the Prognostics Center of Excellence (PCoE) at National Aeronautics and Space Administration (NASA) Ames [32, 33]. The engine degradation simulation was carried out by recording several sensors to characterize the fault evolution using C-MAPSS. For each engine with a different state of initial wear, a set of 26-dimensional data is recorded for each turn number until failure, and a dataset with 20,631 samples is obtained. Each 26-dimensional sample in time series format contains the engine number, engine turn number, 3 operating condition setting parameters and 21 sensor parameters. The general information of each sensor parameter is shown in the appendix.

Through the observations of the sample dataset, it can be found that 7 parameters of the sensors, namely, 1, 5, 6, 10, 16, 18 and 19, are constants, the physical meanings of parameters 8 and 13 are the same, and those of parameters 9 and 14 are also the same. Therefore, this paper considers only 12 parameters, 2, 3, 4, 7, 11, 12, 13, 14, 15, 17, 20, and 21, as the attributes of the dataset.

The health of the engine can indicate the performance of the equipment and predict potential failure, which helps to maintain the equipment and improve its reliability and safety. The health status is usually measured by the health index (HI), which is a continuous value in the interval [0, 1]. HI is 1 when the device operates normally and 0 when the device is completely destroyed. In this paper, the Mahalanobis distance-based method proposed by Wang is selected to calculate the health index [34]: $H I_{i} = \frac{\sqrt{(X_{ti} - X_{ts}) {(X_{ti} - X_{ts})}^{T}}}{max \sqrt{(X_{ti} - X_{ts}) {(X_{ti} - X_{ts})}^{T}}}$ (20) where HI_i represents the health index of the device at moment i, X_ti represents the monitoring parameter vector identified at this moment, and X_ts represents the monitoring parameter vector when the device has completely failed. Once the health index of the engine has been calculated, the classification of the health status can directly determine whether the equipment needs maintenance, which can increase its lifespan in practical applications. Based on accumulated experience and knowledge in the field, this paper considers that the device is healthy when HI is in the interval [0, 0.5) and unhealthy when HI is in the interval 0.5, 1].

Therefore, the dataset used in this experiment contains 20,631 samples, including 12 attributes and 2 class labels.

4.2 Experimental results analysis

4.2.1 Experiment based on the optimization operator recommendation strategy

In this part, the optimization operator recommendation strategy is used to select the optimization algorithm for the ER rule classifier. To verify the effectiveness of the ER rule classifier on a small-sample dataset and the feasibility and effectiveness of the optimization operator recommendation strategy, 300 samples are randomly selected from the whole dataset for the experiment. The experimental results of each step are briefly listed as follows:

Step 1: Preparing the experimental data

The 300 samples are randomly selected from the 20,631 samples, 2/3 and 1/3 of which are used as the training dataset and test dataset, respectively.

Step 2: Training of the ER rule classifier

Step 2.1: Evidence acquisition

First, according to expert experience and historical data, the initial referential values of attributes $A_{i} = {A_{j}^{i} ∣ j = 1, \dots, J_{i}}$ are set as shown in Table 1.

Table 1
The initial referential values of the attributes

$A_{1}^{i}$ $A_{2}^{i}$ $A_{3}^{i}$ $A_{4}^{i}$ $A_{5}^{i}$ $A_{6}^{i}$

A ₁ 641.5000 642.1060 642.7120 643.3180 643.9240 644.5300

A ₂ 1575.7800 1581.6160 1587.4520 1593.2880 1599.1240 1604.9600

A ₃ 1390.3500 1398.3900 1406.4300 1414.4700 1422.5100 1430.5500

A ₄ 550.9800 551.8180 552.6560 553.4940 554.3320 555.1700

A ₅ 2387.9500 2388.0320 2388.1140 2388.1960 2388.2780 2388.3600

A ₆ 9028.2100 9056.7420 9085.2740 9113.8060 9142.3380 9170.8700

A ₇ 46.9800 47.2440 47.5080 47.7720 48.0360 48.3000

A ₈ 519.2400 519.9580 520.6760 521.3940 522.1120 522.8300

A ₉ 8.3485 8.3868 8.4251 8.4635 8.5018 8.5401

A ₁₀ 390.0000 391.6000 393.2000 394.8000 396.4000 398.0000

A ₁₁ 38.2600 38.4480 38.6360 38.8240 39.0120 39.2000

A ₁₂ 23.0369 23.1496 23.2623 23.3751 23.4878 23.6005

	$A_{1}^{i}$	$A_{2}^{i}$	$A_{3}^{i}$	$A_{4}^{i}$	$A_{5}^{i}$	$A_{6}^{i}$
A ₁	641.5000	642.1060	642.7120	643.3180	643.9240	644.5300
A ₂	1575.7800	1581.6160	1587.4520	1593.2880	1599.1240	1604.9600
A ₃	1390.3500	1398.3900	1406.4300	1414.4700	1422.5100	1430.5500
A ₄	550.9800	551.8180	552.6560	553.4940	554.3320	555.1700
A ₅	2387.9500	2388.0320	2388.1140	2388.1960	2388.2780	2388.3600
A ₆	9028.2100	9056.7420	9085.2740	9113.8060	9142.3380	9170.8700
A ₇	46.9800	47.2440	47.5080	47.7720	48.0360	48.3000
A ₈	519.2400	519.9580	520.6760	521.3940	522.1120	522.8300
A ₉	8.3485	8.3868	8.4251	8.4635	8.5018	8.5401
A ₁₀	390.0000	391.6000	393.2000	394.8000	396.4000	398.0000
A ₁₁	38.2600	38.4480	38.6360	38.8240	39.0120	39.2000
A ₁₂	23.0369	23.1496	23.2623	23.3751	23.4878	23.6005

Then, when attribute x_i is the referential value $A_{j}^{i}$ , the belief $β_{n, j}^{i}$ of the sample belonging to class y_n can be obtained by Eqs.(9-11). The results are shown in Table 2-13.

Table 2

The belief distribution of attribute x₁

	$β_{n, 1}^{1}$	$β_{n, 2}^{1}$	$β_{n, 3}^{1}$	$β_{n, 4}^{1}$	$β_{n, 5}^{1}$	$β_{n, 6}^{1}$
y ₁	1	0.9650	0.6188	0.2070	0.0325	0
y ₂	0	0.0350	0.3812	0.7930	0.9675	1

Table 3

The belief distribution of attribute x₂

	$β_{n, 1}^{2}$	$β_{n, 2}^{2}$	$β_{n, 3}^{2}$	$β_{n, 4}^{2}$	$β_{n, 5}^{2}$	$β_{n, 6}^{2}$
y ₁	1	1	0.7958	0.5007	0.1715	0.0113
y ₂	0	0	0.2042	0.4993	0.8285	0.9887

Table 4

The belief distribution of attribute x₃

	$β_{n, 1}^{3}$	$β_{n, 2}^{3}$	$β_{n, 3}^{3}$	$β_{n, 4}^{3}$	$β_{n, 5}^{3}$	$β_{n, 6}^{3}$
y ₁	1	0.9938	0.8371	0.5106	0.0892	0.0274
y ₂	0	0.0062	0.1629	0.4894	0.9108	0.9726

Table 5

The belief distribution of attribute x₄

	$β_{n, 1}^{4}$	$β_{n, 2}^{4}$	$β_{n, 3}^{4}$	$β_{n, 4}^{4}$	$β_{n, 5}^{4}$	$β_{n, 6}^{4}$
y ₁	0	0.0712	0.3621	0.8510	0.9592	1
y ₂	1	0.9288	0.6379	0.1490	0.0408	0

Table 6

The belief distribution of attribute x₅

	$β_{n, 1}^{5}$	$β_{n, 2}^{5}$	$β_{n, 3}^{5}$	$β_{n, 4}^{5}$	$β_{n, 5}^{5}$	$β_{n, 6}^{5}$
y ₁	1	0.9144	0.5537	0.2634	0.0142	0
y ₂	0	0.0856	0.4463	0.7366	0.9858	1

Table 7

The belief distribution of attribute x₆

	$β_{n, 1}^{6}$	$β_{n, 2}^{6}$	$β_{n, 3}^{6}$	$β_{n, 4}^{6}$	$β_{n, 5}^{6}$	$β_{n, 6}^{6}$
y ₁	0.4627	0.5916	0.4334	0.1136	0	0
y ₂	0.5373	0.4084	0.5666	0.8864	1	1

Table 8

The belief distribution of attribute x₇

	$β_{n, 1}^{7}$	$β_{n, 2}^{7}$	$β_{n, 3}^{7}$	$β_{n, 4}^{7}$	$β_{n, 5}^{7}$	$β_{n, 6}^{7}$
y ₁	1	1	0.8295	0.3307	0.0615	0
y ₂	0	0	0.1705	0.6693	0.9385	1

Table 9

The belief distribution of attribute x₈

	$β_{n, 1}^{8}$	$β_{n, 2}^{8}$	$β_{n, 3}^{8}$	$β_{n, 4}^{8}$	$β_{n, 5}^{8}$	$β_{n, 6}^{8}$
y ₁	0	0.0287	0.2656	0.8584	1	1
y ₂	1	0.9713	0.7344	0.1416	0	0

Table 10

The belief distribution of attribute x₉

	$β_{n, 1}^{9}$	$β_{n, 2}^{9}$	$β_{n, 3}^{9}$	$β_{n, 4}^{9}$	$β_{n, 5}^{9}$	$β_{n, 6}^{9}$
y ₁	1	1	0.9508	0.5130	0.0890	0
y ₂	0	0	0.0492	0.4870	0.9110	1

Table 11

The belief distribution of attribute x₁₀

	$β_{n, 1}^{10}$	$β_{n, 2}^{10}$	$β_{n, 3}^{10}$	$β_{n, 4}^{10}$	$β_{n, 5}^{10}$	$β_{n, 6}^{10}$
y ₁	1	0.9711	0.7437	0.2739	0.0485	0
y ₂	0	0.0289	0.2563	0.7261	0.9515	1

Table 12

The belief distribution of attribute x₁₁

	$β_{n, 1}^{11}$	$β_{n, 2}^{11}$	$β_{n, 3}^{11}$	$β_{n, 4}^{11}$	$β_{n, 5}^{11}$	$β_{n, 6}^{11}$
y ₁	0	0.1005	0.2974	0.6851	0.9832	1
y ₂	1	0.8995	0.7026	0.3149	0.0168	0

Table 13

The belief distribution of attribute x₁₂

	$β_{n, 1}^{12}$	$β_{n, 2}^{12}$	$β_{n, 3}^{12}$	$β_{n, 4}^{12}$	$β_{n, 5}^{12}$	$β_{n, 6}^{12}$
y ₁	0.1115	0.1570	0.602	0.9569	1	1
y ₂	0.8885	0.8430	0.398	0.0431	0	0

Step 2.2: Reliability and weight calculation

Based on the calculation method and Eq.(12), the reliability of each attribute is R ={ r_i ∣ i = 1, 2, linebreak … , M }= { 0.7150, 0.5492, 0.6114, 0.4093, 0.2902, 0.0725, 0.9689, 1, 0.8653, 0.5648, 0.6166, 0.5907 }, and the initial weight of each evidence is set as the corresponding reliability.

Step 2.3: Test the classification accuracy of the classifier

The initial ER rule classifier has been constructed after the above steps. The classification accuracies of the training dataset and test dataset based on the initial ER rule classifier are 92.5% and 93%, respectively.

Step 2.4: Optimization of the adjustable parameters

In this experiment, three intelligent optimization algorithms are adopted to optimize adjustable parameters: PSO, GA, and SOA. According to the constraint conditions and objective function in Eqs.(17-18), the initial classifier can be globally optimized based on PSO, GA and SOA. After optimization, the classification accuracies of the training dataset based on PSO, GA and SOA are 93%, 94.5%, and 93.5%, respectively.

Step 3: Testing the ER rule classifier

Step 3.1: Classification of the test dataset

The classification accuracy of the ER rule classifier obtained by the three optimization algorithms (PSO, GA, SOA) on the test dataset is shown as follows.

Table 14 shows that the classification accuracy of the initial ER rule classifier without optimization is 93%, while the classification accuracy of the classifier optimized by PSO, GA and SOA is 94%, 96% and 95%, respectively. Fig. 5 shows in detail the classification results of different classifiers for 100 test samples, among which the white rectangles indicate that the classification results of the corresponding samples are wrong. The results indicate that the initial classifier and the three optimized classifiers all show good classification performance. However, the classification accuracy of the classifier optimized by GA is the highest, 3%, 2% and 1% higher than the initial classifier, the classifier optimized by PSO and the classifier optimized by SOA, respectively.

Table 14

Comparison of results from three optimization algorithms

Optimization algorithm	Initial classifier	PSO	GA	SOA
Classification accuracy (%)	93.00	94.00	96.00	95.00

Fig. 5

Detailed classification results of different classifiers

Step 3.2: Choosing the best optimization algorithm

Analyzing the results in step 3.1 indicates that the optimization results of GA are superior to those of PSO and SOA, so GA is more suitable as the optimization algorithm of the ER rule classifier in this case. Therefore, GA (having the highest accuracy) is selected as the classifier’s optimization algorithm, after which the optimized ER rule classifier is constructed. This finding also confirms that the ER rule classifier can classify small-sample data well.

4.2.2 Experiment with a complete dataset

To verify the feasibility and effectiveness of the optimization operator recommendation strategy and ER rule classifier, experiments with complete datasets are conducted in this section. In this experiment, 2,000 samples are randomly selected as the training dataset, and the remaining samples comprise the test dataset. The classification accuracy of the ER rule classifier obtained by the three optimization algorithms (PSO, GA, SOA) on the turbofan engine degradation simulation dataset is shown as follows:

The classification accuracy of the initial ER rule classifier is 92.34%, while the classification accuracy of the classifier optimized by PSO, GA and SOA is 92.94%, 94.78% and 92.98%, respectively. The result of GA is obviously better than the results of SOA and PSO, which indicates that GA is the most effective optimization algorithm in this case.

The results obtained from the experiment with the complete dataset in this section are consistent with those obtained by the experiment with the partial dataset based on the optimization operator recommendation strategy in section 4.2.1. Both experiments indicate that GA is the most effective optimization algorithm among the three candidates. Therefore, the randomly selected partial data can reflect the characteristics of the entire sample data and select the most appropriate algorithm, which indicates that the optimization operator recommendation strategy proposed in this paper is feasible and effective.

4.2.3 Comparison between the ER rule classifier and classic classification algorithm

The classification accuracy of the ER rule classifier with GA is compared with the accuracies obtained by 4 classic classification methods (naive Bayes (NB), k-nearest neighbor (KNN), SVM, and DT), as shown in Table 16. Similarly, 2,000 samples are randomly selected as the training dataset, and the remaining samples comprise the test dataset when the dataset is classified by 4 classic classification methods.

Table 15
A comparison of the results from the three optimization algorithms

Optimization algorithm Initial classifier PSO GA SOA

Classification accuracy (%) 92.34 92.94 94.78 92.98

Optimization algorithm	Initial classifier	PSO	GA	SOA
Classification accuracy (%)	92.34	92.94	94.78	92.98

Table 16

A comparison among classification methods

Classification method	NB	KNN	SVM	DT	ER rule (GA)
Classification accuracy(%)	82.14	93.74	91.65	93.84	94.78

Table 16 shows that the ER rule classifier has the highest classification accuracy among the classification methods, where the accuracy is higher than that of the naive Bayes, KNN, SVM, and decision tree classifiers by 12.64%, 1.04%, 3.13% and 0.94%, respectively. The results show that the ER rule classifier achieves good classification performance and is an effective and feasible classification method in this case.

In this section, the turbofan engine degradation simulation dataset is applied for an experiment. First, the dataset is introduced in detail, and the parameters such as the attributes and classes are defined. Then, 300 samples are randomly selected, and the operator recommendation strategy is adopted to optimize the initial ER rule classifier. The results show that GA is the best, and it is verified that the ER rule classifier can classify small samples well. Then, to verify the feasibility and effectiveness of the operator recommendation strategy, the complete dataset is used for the experiment, and the results also show that the classifier optimized by GA possesses the highest classification accuracy. Finally, compared with other classic classification methods, the classification accuracy of the ER rule classifier optimized by GA is the highest. Therefore, the feasibility and effectiveness of the ER rule with the operator recommendation strategy are verified in the experiment.

5 Conclusion

The ER rule classifier with an optimization operator recommendation based on the original ER rule classifier has been proposed in this study that can consider expert experience, classify small-sample datasets well, and improve the efficiency of algorithm selection. An initial ER rule classifier is first constructed, and then the optimization algorithm is selected to optimize adjustable parameters globally based on the optimization operator recommendation strategy. Finally, a numerical study on a turbofan engine degradation simulation dataset demonstrates the capability and effectiveness of the proposed ER rule classifier with an optimization operator recommendation. Specifically, the experiment based on the optimization operator recommendation strategy shows that the ER rule classifier can classify a small dataset well, and the result of an experiment with a complete dataset proves the applicability and capability of the optimization operator recommendation strategy. A comparison between the ER rule classifier and classic classification algorithm demonstrates the feasibility and effectiveness of the ER rule classifier with an optimization operator recommendation.

In further research, other datasets can be taken into account, demonstrating that the ER rule classifier with an optimization operator recommendation is universal among different datasets. Moreover, the reliability and weight are the core parameters of the ER rule classifier, and research on their calculation method will be a focus of future research. Studying the operator recommendation strategy to improve the efficiency of algorithm selection will also be necessary in further research.

Footnotes

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under Grant 71901212 and Grant 71690233.

Appendix

Table 17

General information of 21 Parameters

Number	Parameters	Unit
1	Temperature at fan inlet	°R
2	Temperature at LPC outlet	°R
3	Temperature at HPC outlet	°R
4	Temperature at LPT outlet	°R
5	Pressure at fan inlet	psia
6	Pressure in bypass-duct	psia
7	Pressure at HPC outlet	psia
8	Physical fan speed	rpm
9	Engine core speed	rpm
10	Engine pressure ratio	–
11	Static pressure at HPC outlet	psia
12	Ratio of fuel flow to Ps30	pps/psi
13	Corrected fan speed	rpm
14	Corrected core speed	rpm
15	Bypass ratio	–
16	Burner fuel-air ratio	–
17	Bleed enthalpy	–
18	Demanded fan speed	rpm
19	Corrected demanded fan speed	rpm
20	HPT coolant bleed	lbm/s
21	LPT coolant bleed	lbm/s

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ER rule classifier with an optimization operator recommendation

Abstract

Keywords

1 Introduction

2 The basics of the ER rule

2.1 The basic concept of the ER rule

2.2 Reasoning algorithm of the ER rule

2.3.1 Evidence acquisition

3.1 Parametric optimization model of ER rule classifier

4.1 Turbofan engine degradation simulation data

4.2.1 Experiment based on the optimization operator recommendation strategy

4.2.3 Comparison between the ER rule classifier and classic classification algorithm

Table 15 A comparison of the results from the three optimization algorithms Optimization algorithm Initial classifier PSO GA SOA Classification accuracy (%) 92.34 92.94 94.78 92.98

Footnotes

Acknowledgment

Appendix

References

Table 15
A comparison of the results from the three optimization algorithms

Optimization algorithm Initial classifier PSO GA SOA

Classification accuracy (%) 92.34 92.94 94.78 92.98