Computer Systems Science & Engineering DOI:10.32604/csse.2021.015628 | |

Article |

Generalized Normalized Euclidean Distance Based Fuzzy Soft Set Similarity for Data Classification

1Faculty of Computer and Information Technology, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Malaysia

2Department of Information Technology, Politeknik Negeri Padang, Padang, Indonesia

3Department of Information System, Universitas Ahmad Dahlan, Yogyakarta, Indonesia

4Department of Statistics, Universitas Negeri Makassar, Makassar, Indonesia

*Corresponding Author: Rahmat Hidayat. Email: rahmat@pnp.ac.id

Received: 30 November 2020; Accepted: 17 February 2021

Abstract: Classification is one of the data mining processes used to predict predetermined target classes with data learning accurately. This study discusses data classification using a fuzzy soft set method to predict target classes accurately. This study aims to form a data classification algorithm using the fuzzy soft set method. In this study, the fuzzy soft set was calculated based on the normalized Hamming distance. Each parameter in this method is mapped to a power set from a subset of the fuzzy set using a fuzzy approximation function. In the classification step, a generalized normalized Euclidean distance is used to determine the similarity between two sets of fuzzy soft sets. The experiments used the University of California (UCI) Machine Learning dataset to assess the accuracy of the proposed data classification method. The dataset samples were divided into training (75% of samples) and test (25% of samples) sets. Experiments were performed in MATLAB R2010a software. The experiments showed that: (1) The fastest sequence is matching function, distance measure, similarity, normalized Euclidean distance, (2) the proposed approach can improve accuracy and recall by up to 10.3436% and 6.9723%, respectively, compared with baseline techniques. Hence, the fuzzy soft set method is appropriate for classifying data.

Keywords: Soft set; fuzzy soft set; classification; normalized euclidean distance; similarity

Nowadays, Big Data is used in Tuberculosis (TBC) patient data in healthcare, stock data in economics and business fields, and BMKG data (containing weather, temperature, and rainfall data), etc. Data mining is the process of extracting knowledge from large amounts of data [1], and is done by extracting information and analyzing data patterns or relationships [2,3].

Classification is one of the data mining processes used to predict predetermined target classes with data learning accurately. The classification has been used in health [4–6], economics, and agriculture fields [7,8]. Classifying data is challenging and requires further research [9].

In 1965, Zadeh [10] introduced a fuzzy set in which each element object had a grade of memberships ranging between zero and one. In comparison, Molodtsov [11] introduced soft set theory to collect parameters from the universal set subsets (set U). Soft set theory is widely used to overcome the presence of elements of uncertainty or doubt, such as those found in decision-making. Roy developed fuzzy soft set theory by combining soft set theory and fuzzy set theory. This theory was then used in decision-making problems [12,13]. Majumdar and Samanta [14] presented a fuzzy soft set for similarity measurement between two generalized fuzzy soft sets for decision-making.

The fuzzy soft set, an extension of the classical soft set, was introduced by Maji [15]. There have been many works about fuzzy soft set theory in decision-making. Ahmad et al. [16] defined arbitrary fuzzy soft union and fuzzy soft intersection and proved Demorgan laws using fuzzy soft set theory. Meanwhile, Aktas and Cagman [17] studied fuzzy parameterized soft set theory, related properties, and decision-making applications. Rehman et al. [18] studied some fuzzy soft sets’ operations and gave fuzzy soft sets the fundamental properties. Finally, Celik et al. [19] researched applications of fuzzy soft sets in ring theory.

The critical issue in fuzzy soft sets is the similarity measure. In recent years, similarity measurement between two fuzzy soft sets has been studied from different aspects and applied to various fields, such as decision-making, pattern recognition, region extraction, coding theory, and image processing. For example, similarity measurement [20] has been researched in fuzzy soft sets based on distance, set-theoretic approaches, and matching functions. Sut [21] and Rajarajeswari [22] used the notion of the similarity measure in Majumdar and Samanta [20] to make decisions. Several similarity measurement [23] based on four types of quasi-metrics were introduced to fuzzy soft sets. Sulaiman [24] researched a set-theoretic similarity measure for fuzzy soft sets, and applied it to group decision-making. However, some studies haphazardly investigated the similarity measurement of fuzzy soft sets based on distance, resulting in high computational costs [20,23]. Feng and Zheng [25] showed that the similarity measure based on the Hamming distance and normalized Euclidean distance in the fuzzy soft set is reasonable. Thus, the similarity of generalized normalized Euclidean distance is applied in the present paper to a fuzzy soft set for classification. The similarity is used to classify the label of data. The experimental results show that the proposed approach can improve classification accuracy.

2 The Proposed Method/Algorithm

This section presents the basic definitions of fuzzy set theory, soft set theory, and some useful definitions from Roy and Maji [12].

Definition 2.1 [10] Let U be a universe. A fuzzy set A over U is a set defined by a function

where

The notion that the set of all the fuzzy sets over U was denoted by F(U).

Definition 2.2 [10] Let A be a fuzzy set, where A

Definition 2.3 [10] Let A, B be the fuzzy set, where A, B

for all x

Definition 2.4 [10] Let A,B be the fuzzy set, where A,B

for all x

Fuzzification is a process that changes the crisp value to a fuzzy set, or a fuzzy quantity into a crisp quantity [26]. This process uses the membership function and fuzzy rules. The fuzzy rules can be formed as fuzzy implications, such as (x1 is A1) ° (x2 is A2) ° … ° (xn is An); then Y is B, with ° being the operator “AND” or “OR”. B can be determined by combining all antecedent values [14].

Definition 2.5 [12] Let U be an initial universe set and E be a set of parameters. Let P(U) denote the power set of all fuzzy subsets of U, and A ⊆ E. ΓA is called a fuzzy soft set over U, where the function of

Here, the function

Note that the set of all the fuzzy soft sets over U was denoted by FS(U).

Example 1 [14] Let a fuzzy soft set

γA (e1) = {0.5|u1, 0.9|u2},

γA (e2) = {1|u1, 0.8|u2, 0.7|u3},

γA (e3) = {1|u2, 1|u5}.

The family {γA (ei); i = 1,2,3} of P(U) is then a fuzzy soft set

Definition 2.6 [14] Let ΓA, ΓB

Definition 2.7 [14] Let ΓA,

Definition 2.8 [14] Let ΓA, ΓB

Definition 2.9 [14] Let ΓA, ΓB

Definition 2.10 [14] Let ΓA,

The set of all cardinal sets of fuzzy soft set over

Classification involves learning a target function that maps each collection of data attributes to several groups of predefined classes. The purpose of the classification is to see the class’s target predictions as accurate as possible for each case in the data. The classification algorithm consists of two stages. In the training stage, the classifier is trained on predefined classes or data categories. An

A measurement of similarity or dissimilarity defines the relationships between samples or objects. Similarity measurements were used to determine which patterns, signals, images, or sets are alike. For the similarity measure, the resemblance is more critical when its value increases, but, conversely, for a dissimilarity measurement, the resemblance is more robust when its value decreases [27]. An example of the dissimilarity measure is a distance measure. Measuring similarity or distance between two entities is crucial in various data mining and information discovery tasks, such as classification and clustering. Similarity indicators calculate the degree that various patterns, signals, images, or sets are alike. A few researchers have measured the similarity between fuzzy sets, fuzzy numbers, and vague sets. Recently [14,20,28] studied the similarity measure of the soft set and fuzzy soft set. They explained the similarity between the two generalized fuzzy soft sets as follows.

Let

The similarity between

Therefore, M (F, G) = max Mi (F,G), where:

Furthermore,

If we use the universal fuzzy soft set, then

Example 2. In this example,

Here,

and M1(F,G) ≅ 0.73; M2(F,G) ≅ 0.43; M3(F,G) ≅ 0.50. Thus, max [ Mi(F,G) ] ≅ 0.73.

Hence, the similarity between the two GFSS

In this study, the fuzzy soft set was calculated based on the normalized Hamming distance [25]. We assume fuzzy soft sets (F,A) and (G,B) have the same set of parameters, namely, A = B. The normalized Hamming distance and normalized distance in Fuzzy Soft Set (FSS) are obtained using Eqs. (13) and (14).

Example 3. As in Roy and Maji [12], let U = {u1, u2, u3} be a set with parameters

Using Eqs. (13) and (14), respectively, the normalized Hamming distance and normalized distance in FSS between

and

.

Feng and Zheng [13] extended Eq. (14) into a generalized normalized distance in FSS:

If

From Eq. (14), it can be known that

In this section, the proposed approach and experimental results of the Fuzzy Soft Set Classifier (FSSC) using the normalized Euclidean distance are discussed.

This study proposed a new classification algorithm based on the fuzzy soft set; we call it the Fuzzy Soft Set Classifier (FSSC). This algorithm used the normalized Euclidean distance of similarity between two fuzzy soft sets to classify unlabeled data. Before training and classification steps, we first conducted fuzzification and created a fuzzy soft set.

The goal of training the algorithm is to determine the center of each existing class.

Let

Thus,

The new data of the training step results were used to determine the classes in the new data; that is, by measuring the similarity of two sets of fuzzy soft sets acquired in the class center vector and new data.

Given

We use the generalized normalized Euclidean distance for normalized Euclidean distance of the fuzzy set. With relation to Eq. (15), rather than the normalized Euclidean distance fuzzy set,

The generalized normalized Euclidean distance fuzzy soft set is as follows:

Thus, the formula for the similarity measure becomes:

After the value the similarity for each class was obtained, the algorithm looked for which class label is appropriate for new data

We conducted experiments using the University of California (UCI) dataset to assess the accuracy of the proposed data classification method. The dataset samples were divided into training (75% of samples) and test (25% of samples) sets. Experiments were performed in MATLAB R2010a software. Figs. 1–4 show the classification results obtained by our fuzzy soft set method and other baseline techniques.

As seen in Fig. 1, calculations using the normalized Euclidean distance method yield the highest accuracy results. Fig. 2 shows that the normalized Euclidean distance method obtains the second-highest precision; the highest precision is obtained by the comparison table method in MatLab.

Fig. 3 shows that the normalized Euclidean distance method produces the highest recall results, whereas Fig. 4 illustrates that the method has the highest computation time.

The fastest sequence is matching function, distance measure, similarity, normalized Euclidean distance. Comparisons are shown in Tab. 4.

In this study, a new classification algorithm based on fuzzy soft set theory was proposed. Experimental results show that the normalized Euclidean distance method improves accuracy by 10.3436% and increases by 6.9723%, compared to baseline techniques. We also find that all similarity measurements proposed in this paper are reasonable.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no interest in reporting regarding the present study.

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