The following six gene expression datasets are used in this article.

The six gene expression datasets are summarized in Table 1.

Let U = {x₁, x₂, …, x_n} be a nonempty finite set and denote a universe, I = [0, 1], I^U denotes all fuzzy sets on U.

Fuzzy sets are regarded as the extensions of classical sets (Zadeh, 1965).

F is a fuzzy set on U, i.e., F: U → I, then F(x_i) is the membership degree of x_i to F.

The cardinality of F ∈ I^U is |F|=∑i=1nF(xi).

Fuzzy-binary relation are fuzzy sets on two universes. I^U×U denotes all fuzzy-binary relations on U × U.

Fuzzy-binary relation R can be represented by

where r_ij = R(x_i, x_j) ∈ I is the similarity of x_i and x_j.

Definition 2.1 (Li et al., 2017). Let Ube a set of objects and A a set of attributes. Suppose that U and A are finite sets. If each attribute a ∈ A determines an information function a:U→V_a, where V_a is the set of function values of attribute a, then the pair (U, A) is called an information system.

Moreover, if A = C⋃D, C is a condition attribute set and D is a decision attribute set, then the pair (U, A) is called a decision information system.

If (U, A) is an information system and P ⊆ A, then an equivalence relation (or indiscernibility relation) ind(P) can be defined by (x, y) ∈ ind(P)⇔∀a ∈ P, a(x) = a(y).

Obviously, ind(P)=⋂a∈Pind({a}).

For P ⊆ A and x ∈ U, denote [x]_ind(P) = {y|(x, y) ∈ ind(P)} and U/ind(P) = {[x]_ind(P)|x∈U}.

Usually, [x]_ind(P) and U/ind(P) are briefly denoted by [x]_P and U/P, respectively.

According to the rough set theory, for P ⊆ A, X ⊆ U is characterized by P¯(X) and P¯(X), where P¯(X)=⋃{Y|Y∈U/P,Y ⊆ X} and P¯(X)=⋃{Y|Y∈U/P,Y⋂X≠ϕ}.

P¯(X) and P¯(X) are referred to as the lower and upper approximations of X, respectively.

X is crisp if P¯(X)=P¯(X) and X is rough if P¯(X)≠P¯(X).

Given F,G ∈ I^U. For x ∈ U, F(x) and G(x) are the membership degrees of x belonging to fuzzy sets F and G, respectively. F(x) and G(x) ∈ [0,1]. Actually, it is very difficult to ensure that the equation F(x) = G(x) holds. For this reason, we propose the following approximately equal relation of fuzzy sets.

Definition 2.2 Given A,B ∈ I^U. If there exists k ∈ N(k≥2) such that for any x ∈ U, A(x),B(x) ∈ [0,1/k) or A(x),B(x) ∈ [1/k,2/k)…or A(x),B(x) ∈ [(k−1)/k,1], then we say that A is approximately equal to B, and denote it by A≈kB, where k is regarded as a threshold value.

Definition 2.3 For each a ∈ U, define x^R:U→[0,1],x^R(a) = R(x,a)(x ∈ U), x^R is referred to as a fuzzy set that means the membership degree of a to x.

Definition 2.4 [x]R={y|xR(a)≈kyR(a),y∈U}, [x]_Ris referred to as the fuzzy equal class of x induced by the fuzzy relation R on U.

Definition 2.5 [x_i]_R(i = 1,2,…,|U|) is named as the fuzzy information granule induced by the fuzzy relation R on U.

Definition 2.6G(R) = {[x₁]_R,[x₂]_R,…,[x_n]_R} is referred to as the fuzzy-binary granular structure of the universe U induced by R.

It is easy to prove: P¯(X)={x|[x]R ⊆ X,[x]R∈G(R)}, P¯(X)={x|[x]R⋂X≠ϕ,[x]R∈G(R)}.

Hu et al. (2010) found that there are some relationships between rough sets and Gaussian kernel method, so Gaussian kernel is used to obtain fuzzy relations. Compared with Gaussian kernel, Laplacian kernel has higher peak, faster reduction and smoother tail. Therefore, Laplacian kernel is better than Gaussian kernel in describing the similarity between objects. In this article, we use Laplacian kernel k(xi,xj)=exp⁡(-||xi-xj||σ) to extract the similarity between two objects from decision information system, where ||x_i−x_j|| is the Euclidean distance between two objects x_i and x_j. In general, σ is a given positive value.

Obviously, k(x_i, x_j) satisfies:

Let R = (k(x_i, x_j))_n×n, then R is called the fuzzy relation matrix induced by Laplacian kernel.

In this part, the classification accuracy of different Laplacian kernel parameters values of σ is tested. For gene expression data, feature selection aims to improve classification accuracy by eliminating redundant genes. The different values of σ influence the size of granulated gene data, which affects the classification accuracy of selected genes. Therefore, the different values of σ should be set in the process of feature selection of gene expression data sets. Moreover, the different values of σ also affect the composition of the selected gene subset. To obtain a suitable σ and a good gene subset, the classification accuracy of the selected gene subset for different values of σ should be discussed in detail.

The corresponding experiments are performed to graphically illustrate the classification accuracy of FSACE under different values of σ. The results are shown in Figure 1, where the horizontal axis denotes σ ∈ [0.05,1] at intervals of 0.05, and the vertical axis represents the classification accuracy.

Figure 1 shows that σ greatly influences the classification performance of FSACE. σ is usually set to make the classification accuracy highest. Thus, the appropriate parameter values of σ can be obtained for each data set from Figure 1. In Figure 1A, for Leukemia1 data set, when σ is 0.95, the classification accuracy is the highest. In Figure 1B, for Leukemia2 data set, when σ is 0.55, the classification accuracy is the highest. In Figure 1C, for Brain tumor data set, when σ is 0.80, the classification accuracy is the highest. In Figure 1D, for 9-tumors data set, when σ is 0.75, the classification accuracy is the highest. In Figure 1E, for Robert data set, when σ is 0.60, the classification accuracy is the highest. In Figure 1F, for Ting data set, when σ is 0.75, the classification accuracy is the highest. Therefore, the appropriate values of σ for different data sets are determined.

The classification results obtained from the three classifiers (KNN, CART, and SVM) with 10-fold cross-validation are shown in Table 2 on the test data by FSACE.

Table 2 shows that FSACE not only greatly reduces the dimensionality of all six gene expression data sets, but also improves the classification accuracy.

The results of feature genes selection from six gene expression data sets are shown in Table 3 using FSACE.

To evaluate the performance of FSACE in terms of classification accuracy, FSACE algorithm is compared with several state-of-the-art feature selection algorithms, including EGGS (Chen et al., 2017), EGGS-FS (Yang et al., 2016), MEAR (Xu et al., 2009), Fisher (Saqlain et al., 2019), and Lasso (Tibshirani, 1996). According to the change trend of Fisher scores of six gene datasets, we select the top-200 genes as the reduction set for Fisher algorithm.

Tables 4–9 show the experimental results of six gene expression data sets using six different feature selection methods.

As shown in Tables 4, 5, FSACE has the highest average classification accuracy for Leukemia1 and Leukemia2, and exhibits better classification performance than the other five algorithms.

As shown in Tables 6, 7, MEAR cannot work on Brain Tumor data set and 9-tumors data set, its results are denoted by the sign –. FSACE obtains the highest average classification accuracy among the five feature selection algorithms for Brain Tumor data set and 9-tumors data set.

Tables 8, 9 shows that MEAR still can not work on Robert data set and Ting data set, which indicates that the algorithm is not stable. Our algorithm still has the highest classification accuracy among all the algorithms. Although the classification accuracy of our algorithm is only a little higher than lasso algorithm, the number of attributes reduced by our algorithm is much less than lasso algorithm.

Tables 4–9 show that the average number of attributes reduced by our algorithm is slightly more than that of MEAR, ECGS, and EGGS-FS, but the average classification accuracy is much higher than that of these three algorithms.

Therefore, FSACE can not only effectively remove noise and redundant data from the original data, but also improve the classification accuracy of gene expression data sets.