A novel similarity measurement for triangular cloud models based on dual consideration of shape and distance

Definitions and notions

Here, we provide definitions, relationships, and necessary lemmas for the normal cloud, triangular cloud model, and triangular fuzzy numbers. We then describe the variance of triangular cloud model and EW-type closeness.

Definition 1. Let U be a non-empty infinite set expressed by an accurate numerical value, and C is a qualitative concept on U. If there is an accurate numerical value x ∈U, and the mapping $y={\mu }_C\left(x\right)\in \left[0,1\right]$ of x to C is a random number with a stable law, then the distribution of (x, y) on the universe U is called a cloud, and each (x, y) is called a cloud drop (Li, Han & Shi, 1998).

Definition 2. The three characteristic parameters (Ex, En, He) of the cloud model are the quantitative embodiment of its qualitative concept. The expectation (Ex) of the cloud is the representation of the expected value of the cloud in the non-empty infinite set expressed and it is also the center of gravity corresponding to the maximum value of the membership degree Y; entropy (En) is a measure of the uncertainty of cloud model, which reflects the expected dispersion of cloud droplets and the fuzziness of cloud model data. Hyper-entropy (He) is the entropy of En, which is a measure of the uncertainty of cloud model entropy. Its value can represent the thickness of cloud, reflecting the randomness of cloud model data.

Definition 3. If the random variable x satisfies x ∼ N(Ex, En^‘2), where En^‘ ∼ N(En, He²), and the certainty of x to the qualitative concept satisfies: (1)${\displaystyle {\mu }_C\left(x\right)=exp\left(-\frac{{\left(x-Ex\right)}^2}{2{\left(E{n}^{\prime }\right)}^2}\right)}$ then that distribution of x on the non-empty infinite set U is normal cloud (Li, Han & Shi, 1998).

Definition 4. If the random variable x satisfies x ∼ N(Ex, En^‘2), where En^‘ ∼ N(En, He²), and the certainty of x to the qualitative concept satisfies: (2)${\displaystyle {\mu }_C\left(x\right)=\left\{\begin{array}{c}\frac{x-\left(Ex-3E{n}^{\prime }\right)}{3E{n}^{\prime }},x<Ex\hfill \\ 1-\frac{x-Ex}{3E{n}^{\prime }},x\ge Ex\hfill \end{array}\right.}$ then that distribution of x on the non-empty infinite set U is triangle cloud.

Definition 5. If the random variable x satisfies x ∼ N(Ex, En^‘2), where En^‘ ∼ N(En, He²), and En ≠0. At this time, the Eq. (3) exists: (3)${\displaystyle y=\left\{\begin{array}{c}\frac{x-\left(Ex-3En\right)}{3En},x<Ex\hfill \\ 1-\frac{x-Ex}{3En},x\ge Ex\hfill \end{array}\right.}$ then y is called the expected curve of triangular cloud. The expected curve is obtained from the distribution law of cloud droplets in the horizontal direction. The expected curve can intuitively describe the shape characteristics of triangular cloud and all cloud droplets fluctuate randomly around the expected curve.

Definition 6. Fuzzy numbers are convex fuzzy sets defined on real numbers R (Wu & Zhao, 2008). For a certain fuzzy number, its membership degree satisfies Eq. (4): (4)${\displaystyle F\left(x\right)=\left\{\begin{array}{c}\frac{x-r^l}{r^m-r^l},r^l\le x\le r^m\hfill \\ \frac{x-r^u}{r^m-r^u},r^m\le x\le r^u\hfill \\ \begin{array}{c}0,\text{other}\hfill \end{array}\hfill \end{array}\right.}$ then $\tilde{r}=\left(r^l,r^m,r^n\right)$ is called triangular fuzzy number. The membership function of $\tilde{r}$ is $F\left(x\right):R\to \left[0,1\right]$, where x ∈ R and R is a real number field. r^l, r^m, r^u are the lower bound, median, and upper bound of triangular fuzzy numbers, respectively, and r^l ≤ r^m ≤ r^u. When they are equal, $\tilde{r}$ degenerate into real values.

Definition 7. For $\underset{\text{\_}}{a},\overline{a}\subseteq R$, and $\underset{\text{\_}}{a}\le \overline{a}$, then $a=\left[\underset{\text{\_}}{a},\overline{a}\right]$ is called interval number. The relationship between fuzzy numbers and interval numbers is shown in Fig. 1. The whole number of intervals is recorded as [R ]. for a ∈ [R], the Eq. (5) exists: (5)${\displaystyle E\left(a\right)=\frac{\overline{a}+\underset{\text{\_}}{a}}{2},W\left(a\right)=\frac{\overline{a}-\underset{\text{\_}}{a}}{2}}$

E(a) and W(a) are respectively the expected value and width of interval number a (Bao, Peng & Zhao, 2013).

Definition 8. For u ∈ F₀, F₀ is a fuzzy number space (Bao & Bai, 2018). The r-cut set with fuzzy number u is a closed interval, as shown in Eq. (6): (6)${\displaystyle {\left[u\right]}^r=\left[\underset{\text{\_}}{u}\left(r\right),\overline{u}\left(r\right)\right],r\in \left[0,1\right].}$

The r-cut interval number of triangular numbers is shown in Fig. 2.

The order relation on F₀ is defined as:

u ≤ v, and if and only if for any $r\in \left[0,1\right],\overline{u}\left(r\right)\le \overline{v}\left(r\right)$ and . $\underset{\text{\_}}{u}\left(r\right)\le \underset{\text{\_}}{v}\left(r\right)$.

u ≥ v, and if and only if for any $r\in \left[0,1\right],\overline{u}\left(r\right)\ge \overline{v}\left(r\right)$ and $\underset{\text{\_}}{u}\left(r\right)\ge \underset{\text{\_}}{v}\left(r\right)$.

Definition 8 can be regarded as a bridge between fuzzy numbers and interval numbers, and it is also the theoretical basis for the transformation of interval number closeness to fuzzy number closeness in EW-type closeness.

Lemma 1. If the random variable x satisfies x ∼ N(Ex, En′²), where En′ ∼ N(En, He²), and En ≠ 0. (7)${\displaystyle D\left(x\right)=E{n}^2+H{e}^2}$

The D (x) is called the variance of cloud droplets in the cloud model. He determines the thickness of clouds, and En determines the dispersion degree of cloud droplets. The larger the difference between He and En, the smaller the shape similarity between two clouds model.

Lemma 2. Set function: (Bao & Bai, 2018) (8)${\displaystyle g\left(x\right)=\frac{1}{1+x^n},n>0,x\in \left[\right.0,+\infty \left.\right)}$

The mapping f: $\left[R\right]\times \left[R\right]\to \left[\right.0,+\infty \left.\right)$ is defined as: (9)${\displaystyle f\left(u,v\right)={\left(|E\left(u\right)-E\left(v\right){|}^p+\frac{1}{3}|W\left(u\right)-W\left(v\right){|}^p\right)}^{\frac{1}{p}},p\ge 1,\forall u,v\in \left[R\right]}$

then $g\left(f\left(u,v\right)\right)=n_{EW}\left(u,v\right)$ is the closeness of the interval number u and v.

Lemma 3. Let the mapping $N_{EW}^p:F_0\times F_0\to \left[0,1\right]$ be defined as: (10)${\displaystyle N_{EW}^p\left(u,v\right)={\left({\int }_0^1{\left(n_{EW}\left({\left[u\right]}^r,{\left[v\right]}^r\right)\right)}^{p}dr\right)}^{\frac{1}{p}},p\ge 1,\forall u,v\in F_0}$ then $N_{EW}^p\left(u,v\right)$ is called EW-type closeness of triangular fuzzy numbers u and v (Bao & Bai, 2018).

A novel similarity measure for triangular cloud model

Many studies consider only one aspect to measure the similarity of cloud models, therefore, these methods have some shortcomings. In fact, the similarity of cloud models can be observed from two aspects: shape and distance. By combining these two perspectives with scientific methods, a more complete similarity measurement method for cloud models can be obtained. The EPTCM method proposed here is formed from this perspective. The EPTCM method consists of a combination of two methods (the ETCM and PCM methods). Since the ETCM method introduces only Ex and En, it is considered as the distance similarity between the cloud models. However, the PCM method is exactly related to the shape of the two cloud models (only En and He are considered). After establishing these two methods, a scientific empowerment method is designed in this article to combine these two methods scientifically. The ETCM method and PCM method are combined by empowerment to obtain the final EPTCM method.

Similarity measure of expected curve of cloud model based on EW-type closeness

Fuzzy closeness is an important concept for triangular fuzzy numbers (Jiang et al., 2019). Compared with the traditional Hausdorff-distance (Hossein-Abad, Shabanian & Kazerouni, 2020) and P-distance (You & Yan, 2017) formulas, the EW-type distance formula $\left(d_{EW}^P\right)$ (Bao, Peng & Zhao, 2013) used in EW-type closeness (Bao & Bai, 2018) considers the expected difference of two interval numbers and takes into account the difference of their widths. The simulation results from Bao, Peng & Zhao (2013) show that this method describes the distance of interval numbers more comprehensively and meticulously, and the utilization rate of information is greatly improved. Compared with the traditional exponential closeness (Eq. (11)), EW-type closeness introduces the degree of the interval number closeness to participate in the calculation. Bao & Bai (2018) show that EW-type closeness has better discrimination and practicability. (11)${\displaystyle d\left(A,B\right)=e^{-\sqrt[p]{|a^l-b^l{|}^p+|a^m-b^m{|}^p+|a^r-b^r{|}^p/q}}}$

According to the “3En” rule of the triangle cloud model (Li, Han & Shi, 1998), more than 90% of cloud droplets fall in the range of [Ex − 3En, Ex + 3En]. Therefore, when calculating the similarity of the triangular cloud model, we only need to consider the cloud droplets in this range and the expected curve. As shown in Fig. 3, the forward triangle cloud “3En” rule is introduced to the expected curve in Definition 5, and the transformed curve formula: (12)${\displaystyle y^{\prime }\left(x\right)=\left\{\begin{array}{cc}\hfill \frac{x-\left(Ex-3En\right)}{Ex-\left(Ex-3En\right)},\hfill & \hfill \left(Ex-3En\right)\le x\le Ex\hfill \\ \hfill 1,\hfill & \hfill x=Ex\hfill \\ \hfill \frac{\left(Ex+3En\right)-x}{Ex+3En-Ex},\hfill & \hfill Ex\le x\le \left(Ex+3En\right)\hfill \\ \hfill 0,\hfill & \hfill \text{other}\hfill \end{array}\right.}$

Equation (12) conforms to the definition of the triangular fuzzy number in Definition 6 and is denoted as y′ =  < Ex − 3En, Ex, Ex + 3En >. Therefore, EW-type closeness can be applied to y′. If Ex − (Ex − 3En) = (Ex + 3En) − Ex = 3En, y′ is called the symmetric triangular fuzzy number based on the expected curve (Hwang & Yang, 2007), it is noted as ${\tilde{y}}^{{}^{\prime }}={\left(Ex,3En\right)}_T$, where Ex and 3En are the expectation and ambiguity (also called width) of the triangular fuzzy number based on the expected curve, respectively.

According to Definitions 7 and 8, the upper bound and lower bound of the r-truncated closed interval ${\left[u\right]}^r$ of the expected curve can be obtained from triangular fuzzy number y′ =  < Ex − 3En, Ex, Ex + 3En >: (13)${\displaystyle \left\{\begin{array}{c}\hfill Upperbound:\overline{u}\left(r\right)=Ex+3En\left(1-r\right)\hfill \\ \hfill Lowerbound:\underset{\text{\_}}{u}\left(r\right)=Ex-3En\left(1-r\right)\hfill \end{array}\right.}$

The expectation and width of interval u are: (14)${\displaystyle \left\{\begin{array}{c}\hfill E\left(u\right)=\frac{\overline{u}+\underset{\text{\_}}{u}}{2}=Ex\hfill \\ \hfill W\left(u\right)=\frac{\overline{u}-\underset{\text{\_}}{u}}{2}=3En\left(1-r\right)\hfill \end{array}\right.}$

Therefore, the triangular fuzzy number v = 〈Ex′ − 3En′, Ex′, Ex′ + 3En′〉 of another group of cloud expectation curves is set. The similarity of the expectation curves of the two groups of cloud models based on EW-type closeness can be obtained from Lemma 3. (15)${\displaystyle Sim\left(ETCM\right)=N_{EW}^p\left(u,v\right)={\left({\int }_0^1{\left(n_{EW}\left({\left[u\right]}^r,{\left[v\right]}^r\right)\right)}^{p}dr\right)}^{\frac{1}{p}},p\ge 1.}$

The ETCM algorithm constructs two sets of cloud model expectation curves under the restriction of “3En” principle into two triangular fuzzy numbers. The EW-type closeness can be used to calculate the similarity of the two triangular fuzzy numbers under r ∈ [0, 1]. Because the ETCM method does not introduce He into the calculation, it was considered to be the distance similarity of the triangular cloud model. The larger the Sim (ETCM) value, the higher the distance similarity between the two triangular cloud models, and vice versa.

Shape similarity measurement of triangular cloud model based on cloud drop variance

All clouds can be translated to the position x = 0, therefore, the shape of the cloud has nothing to do with the cloud’s expectation (Ex). As mentioned earlier, the cloud’s En and He reflect the shape of the cloud and describe the conceptual expansion of the variables. It is clear that the basic skeleton of the cloud model shape is determined by En with the “3En” rule. In addition, the He controls the dispersion of the thickness or conceptual extension of the cloud. Based on the above theory, we sought to determine the relationship mathematically.

According to Lemma 1, the variance D(x) = En² + He² of the triangular cloud model consists of En and He. Although the variance does not consider the Ex (the location relationship of the cloud models) it fully reflects the shape similarity between cloud models. The greater the En difference between two clouds, the smaller the shape similarity between two clouds, so consider introducing the mean square error of cloud model to measure the shape similarity of cloud model. If there are two groups of triangular cloud models C_i(Ex_i, En_i, He_i) and C_j(Ex_j, En_j, He_j), their shape similarity is expressed as: (16)${\displaystyle Sim\left(PCM\right)=\frac{min\left(\sqrt{D{\left(x\right)}_i},\sqrt{D{\left(x\right)}_j}\right)}{max\left(\sqrt{D{\left(x\right)}_i},\sqrt{D{\left(x\right)}_j}\right)}}$

Regardless of the difference between the Ex of the two cloud models, if the En and He of the two clouds are equal, their shape similarity Sim (PCM) = 1. Although the Ex of triangular cloud models C₂ and C₃ are different, the shapes of the two clouds are consistent (Fig. 4). The PCM method has better authenticity and timeliness compared with the similarity method based on the maximum boundary curve of the cloud model in Li, Guo & Qiu (2011) whose model exaggerates the proportion of He in calculating shape similarity.

The integrated similarity measurement of the triangular cloud model

As previously mentioned, the ETCM method does not consider the influence of He, while the PCM method does not consider the influence of cloud model Ex. We defined a weighted calculation method, which combines the two methods to enrich the completeness and authenticity of cloud model similarity in order to create an integrated approach that incorporated the three main characteristic parameters of the cloud model (Ex, En, He). Referring to the analytic hierarchy process (AHP) and entropy weight method proposed in the reference (Ruan et al., 2017), we defined a method to determine the similarity weight of the integrated cloud model in Eq. (17). (17)${\displaystyle \left\{\begin{array}{c}d{\left(\alpha ,\beta \right)}^2=\sqrt{{\left(Sim\left(ETCM\right)-Sim\left(PCM\right)\right)}^2}\hfill \\ d{\left(\alpha ,\beta \right)}^2={\left(\alpha -\beta \right)}^2\hfill \\ \alpha +\beta =1\hfill \\ \alpha \ge \beta .\hfill \end{array}\right.}$

The calculated weight coefficients (α and β) were used to weight the triangular cloud ETCM method and PCM method, respectively. Thus, a similarity measurement algorithm for triangular cloud models based on the dual consideration of shape and distance is defined. As shown in Eq. (18): (18)${\displaystyle Sim\left(\text{EPTCM}\right)=\alpha Sim\left(\text{ETCM}\right)+\beta Sim\left(\text{PCM}\right)}$

Equation (18) is the final expression of the proposed EPTCM method. The complete computational procedure of the EPTCM algorithm is shown in Algorithm 1. In order to verify that the EPTCM method was feasible and effective, scientific simulation experiments were conducted.

Cloud model discrimination simulation experiment

Four classical cloud models are given in Zhang et al. (2007) using the collaborative filtering algorithm as follows: ${\displaystyle \left[\begin{array}{c}{C}_1\left(1.5,0.62666,0.33900\right)\hfill \\ C_2\left(4.6,0.60159,0.30862\right)\hfill \\ C_3\left(4.4,0.75199,0.27676\right)\hfill \\ C_4\left(1.6,0.60159,0.30862\right)\hfill \end{array}\right]}$

These four groups of classical cloud models are used to perform simulation experiments as shown in Li, Guo & Qiu (2011) and Yu et al. (2021). In order to verify the advantages of the proposed EPTCM method over the existing methods (LICM, FDCM, OECM, MSCM), the four groups of classical cloud models were also used for our simulation experiments.

We classified C₁, C₄ as group A and C₂, C₃ as group B. Figure 5 shows that there is a big gap between the cloud models of group A and group B in terms of distance and shape similarity. However, within the group, the distance and shape similarity are slightly different. Therefore, the better similarity measurement method may more accurately reflect the intragroup differences between group A and group B. Equation (19) was used as the basis to measure the discrimination ability of the cloud model similarity measurement method. The following uses the above EPTCM method to measure the similarity of cloud models among these four cloud models. The EPTCM method was to compare and analyze the results with traditional methods such as LICM, OECM, MSCM and FDCM. The calculation results are shown in Table 2. The comparison chart of Discrimination of the five methods is presented in Fig. 6. In order to reduce the time complexity in the calculation process: n = 1, p = 1. (19)${\displaystyle Discrimination=\left|Sim\left(C_1,C_4\right)-Sim\left(C_2,C_3\right)\right|}$

Experiment on time series classification accuracy based on synthetic control chart dataset

Details of the synthetic control chart dataset

The time series represents the state of an object across different time periods and arranges them according to their time of occurrence, thus generating a data series. Time series is often used in engineering data classification, prediction, and clustering (Alcock & Manolopoulos, 1999). Here, the synthetic control chart dataset (SYNDATA) (Pham & Chan, 1998), a commonly used time series dataset in the UCI knowledge discovery database, was used to test the classification accuracy. The SYNDATA dataset contains 600 control chart samples synthesized by the processes by Alcock and Manolopoulos in 1999 (Pham & Chan, 1998). This dataset was chosen because it contains the time series’ of multiple trends, which are volatile and complex and can be a good test of the accuracy of the similarity measurement. There are six main category patterns in this dataset set; these include normal, circular, upward trend, downward trend, upward transition, and downward transition. Each category contains 100 rows of data, with 60 data in each row. All trends were collectively referred to as “abnormal trends” with the exception of the normal trend. All the abnormal trends must be corrected, therefore, it is important to detect the abnormal trends quickly and accurately for strict control processes and good product quality. Because time series datasets have high requirements for the correctness and time complexity of the algorithms, they are a logical choice for classification experiments that analyze and study the time series classification algorithms starting from the correctness and time complexity of the results.

It was important to understand the data structure before applying the SYNDATA dataset. The synthetic control chart dataset had a labeled dataset. The given dataset D_m×n is a matrix with m = 600 rows and n = 60 columns; every 100 rows of data in 600 rows is one category with a total of six categories. In Table 3, we describe in detail the composition of the dataset. When using the SYNDATA dataset for time series classification experiments, each record is treated as a separate query sequence. Each record needs to be calculated with the remaining 599 records for cloud model similarity, then the top k largest ones are selected according to the similarity ranking, and categorized according to the group to which the k numbers belong.

Table 3:

SYNDATA-the composition and content details of the data set.

DOI: 10.7717/peerjcs.1506/table-3

Detailed introduction of time series classification accuracy test

To verify the accuracy and rapidity of the EPTCM method in a time series classification, we used the following algorithm and the last 10 rows of data of each main category pattern were used to form the test set. Previous studies interlaced the extraction of grayscale images in order to improve the efficiency of grayscale image processing. Here, the first 90 lines of data of each main category pattern were interlaced, leaving only odd lines. After the interlaced extraction, there were only 45 lines of data of each main category in the training set. The remaining 45 lines of data in each main category were divided into three groups, which were each subdivided into three groups labeled A, B, and C. The groups of data labeled A, B, and C each contained the every “abnormal trends”. Next, the data groups, A, B, and C, were classified using the k-NN (k-nearest neighbor) (Fuchs et al., 2010; Lin et al., 2007) algorithm in the machine learning algorithm. The k-NN algorithm is easy to use and has a simple process. Every row of data in the training set and the test set can be quantitatively represented as a triangular cloud model by the reverse cloud generator, therefore, the EPTCM method can be introduced into the k-NN algorithm. The traditional k-NN algorithm determines the number of nearest neighbors of a certain eigenvalue by comparing the distances between the eigenvalues and resulting in the classification of a different eigenvalue. The EPTCM method was used to replace the relative distances between the measured eigenvalues in the traditional KNN algorithm for data classification and statistics. The structural framework of the time series classification accuracy experiment is shown in Table 4. The classification accuracy is shown in Eq. (20) and the classification results are shown in Figs. 7 and 8, and Table 5. (20)${\displaystyle P_X=\frac{Numberofcorrectlyclassifiedsamples}{Thetotalnumberofsamples}}$ (21)${\displaystyle AverageAccuracy\left(GroupX\right)=\frac{\sum _{i=1}^{10}\left(P_X\left(k=i\right)\right)}{10},X=A,B,C}$

Table 4:

Structural framework for time series classification accuracy experiments.

Steps	Content
Step 1: Calculate the feature vectors	The data in the dataset Dm×n are followed by the inverse cloud generator algorithm to obtain three numerical features for each set of data and use them as the feature vector for the i th record (time series):
	Ci = (Exi, Eni, Hei)
	This enables the dimensionality reduction of the dataset, where 1 ≤ i ≤ m;
Step 2: Determine the training set and test set	The last 10 rows of data for each major category pattern in the Dm×n were used to form the test set. The rest of the data is used as the training set. After interlaced extraction, there are only 45 lines of data of each main category in the training set. And the remaining 45 lines of data in each main category are divided into three groups, which are grouped into three groups: A, B and C in turn. Make the three groups of data A, B and C each contain all kinds of “abnormal trends”;
Step 3: Calculate the similarity matrix	The (EPTCM, LICM, FDCM, OECM, MSCM) method is used to calculate the similarity between the feature vectors corresponding to the data in the test set and the feature vectors corresponding to the data in the training set. The resulting similarity matrix is generated:
	$Sim=\left[\begin{array}{c}\hfill Sim\left(C_{Test1},C_{Train1}\right)\cdots Sim\left(C_{Test1},C_{Train45}\right)\hfill \\ \hfill ⋮\ddots ⋮\hfill \\ \hfill Sim\left(C_{Test60},C_{Train1}\right)\cdots Sim\left(C_{Test60},C_{Train45}\right)\hfill \end{array}\right]$
	$Sim\left(C_{Testi},C_{Trainj}\right)$ denotes the similarity between CTesti and CTrainj calculated using the EPTCM algorithm;
Step 4: Classification experiments are performed according to the nearest neighbor k values	First, each row of the similarity matrix $Sim\left(C_{Testi},C_{Trainj}\right)$ is sorted from largest to smallest, then the top 1 ≤ k ≤ 10 values of each row of the test set similarity matrix $Sim\left(C_{Testi},C_{Trainj}\right)$are selected according to the number of nearest neighbors k. Finally, the time series classification experiments are completed by categorizing the number of groups to which the k numbers belong.

DOI: 10.7717/peerjcs.1506/table-4

Table 5:

Average Accuracy (Eq. (21)) of different methods on various training sets when k = 1 ∼ 10.

Average Accuracy	EPTCM	LICM	OECM	FDCM	MSCM
Group A	0.887	0.578	0.640	0.638	0.650
Group B	0.853	0.607	0.533	0.602	0.532
Group C	0.873	0.647	0.587	0.645	0.580

DOI: 10.7717/peerjcs.1506/table-5

Simulation experimental result analysis

The similarity order of the cloud models calculated by the proposed EPTCM method is as follows: $\left(C_1,C_4\right)>\left(C_2,C_3\right)>\left(C_3,C_4\right)>\left(C_1,C_3\right)>\left(C_2,C_4\right)>\left(C_1,C_2\right)$ (Table 2). This is consistent with the visual impression in Fig. 5. Table 2 shows that the discrimination of the EPTCM method for the cloud models in group A and group B was 0.11. The EPTCM method had the highest discrimination among the five methods, which can better identify the similarity of the cloud models and better reflect the differences between cloud models A and B. The MSCM method had a discrimination of 0.0932 between two groups of cloud models, which is second only to the EPTCM method. However, the measurement range of the MSCM method is very limited. Table 2 reveals that the results of the other four groups of similarity were all 0 (negative values will revert to 0), with the exception of Sim(C₁, C₄) and Sim(C₂, C₃). These results are limited and are inconsistent with human cognition. The results of the OECM method were 0.0234 for the discrimination between the cloud models of groups A and B. This indicates that the discriminatory ability of the OECM method is poor compared to the EPTCM method. The discrimination of the FDCM method was poor with results of 0.0104. It is not difficult to see from the data in Table 2 that the Sim(C₂, C₄) of the FDCM method is NAN. This is due to the limitation of its algorithm itself. The method will lose the influence of the judgment expectation on the similarity when the En and He of both clouds are equal. We can see that the cloud similarities of LICM are close to 1 and the discrimination is not clear. These results are obviously inconsistent with intuitive human feeling. Through the above analysis, it can be reflected that the EPTCM method proposed in this paper has certain superiority compared with the four existing methods.

Accuracy of the experimental result analysis of time series classification

Figure 7 shows that the proposed EPTCM method has a good classification accuracy in the time series classification experiment. When k was 1, 3, 4, 5, or 7, the classification accuracy of EPTCM was over 90% in the time series classification experiment with group A. According to the analysis of the data in Table 5, EPTCM achieved Average Accuracy of over 85% in the time series classification experiment with three training sets. Figure 8 shows the classification accuracy of each method in the range of k = 1 ∼ 10 when the group C training set was used in the time series classification experiments. Figure 8 clearly shows that EPTCM was the best among the five methods in terms of classification accuracy. The classification accuracy of the EPTCM method also shows good stability with the change of k value in the group C training set. The proposed EPTCM method has obvious advantages compared with the existing methods that exhibit low classification accuracy and poor stability. Figure 9 shows that the EPTCM method is similar to the MSCM method in CPU overhead time. However, the Average Accuracy of EPTCM was better than that of MSCM. Table 5 shows that the average accuracy of the EPTCM method was significantly better than that of the four existing algorithms when groups A, B, and C were used separately.

A comparative analysis of cloud model similarity metrics

Studies by Li, Wang & Yang (2019) and Li et al. (2020) provide the evaluation metrics for cloud model similarity metrics. Table 6 shows how the EPTCM method compares with the extant methods in terms of discriminability, efficiency, stability, and interpretability. Discriminability refers to the ability of the similarity matrix to distinguish between the differences of two concepts that are not identical; efficiency refers to the time complexity of computing the similarity between two concepts; stability refers to the fact that the value of the similarity measure is constant over multiple calculations; interpretability means that the process of calculating the similarity metric is interpretable. LICM has high efficiency and stability, but it does not distinguish the differences between two concepts with the same proportion of numerical features. Additionally, considering numerical features as a vector does not reflect the relationship between the numerical features and this lacks interpretability in the calculation process. Simulation experiments also confirm its low discriminatory ability. MCM and OECM are similarity metrics based on the overlap of feature curves. They have high efficiency and stability like LICM, but they do not find the difference between two different concepts in some cases because the variation of He is ignored. Moreover, they are only partially interpretable due to the lack of relationship between En and He. The experimental results show that MSCM has a moderate discriminative power; it also contains an integral operation process with high time complexity and its metric results are not affected by the number of cloud drops and the number of experiments. According to the results of the simulation experiments conducted by Li et al. (2020), UDCM has good discriminative ability and theoretical interpretability. However, it retains many integral operations and is less efficient. Although the authors verify that the results are less variable compared to other algorithms, the results are not constant. The simulation experiments prove that the EPTCM method has good resolution ability and the metric results of the EPTCM method are constant and have good stability. These results are easily obtained from Algorithm 1. The EPTCM method performs a similarity metric from the shape and distance of the cloud model in both directions and has good theoretical interpretation; it also retains a small number of integral operations and has an average performance in terms of efficiency.

In summary, the proposed EPTCM method has certain advantages over the existing algorithms.