Self-organizing map with granular competitive learning: Application to microarray clustering

1.1 Motivation: Discovering gene clusters with different algorithms

Data mining tasks (supervised and unsupervised learning methods) associated with microarray include the following: Discovering subgroups of samples that share common features, determining gene expression profiles which are different between two or more groups of samples, identifying differentially expressed genes, describing genes with similar biological functions, and discovering subclasses of samples affected by the same disease. Gene selection is an extremely important pre-processing step in microarray data mining due to the high-dimensional nature of the data. In [35, 36], clustering methods are used in selecting important genes in microarray data.

Clustering microarray array can be categorized as sample-based clustering or gene-based clustering. Sample-based clustering involves discovering phenotypic structures or substructures of samples where samples are treated as patterns and genes are considered features. Gene-based clustering partitions gene expression profiles into subgroups of common biological characteristics [23, 32]. Neural network-based hierarchical clustering methods were used in [11, 12]. In [22], clustering methods have proven to be helpful in the identification of diseased or normal genes and understanding gene function, gene regulation and cellular processes. An overview of clustering algorithms for gene expression data can be found in [14].

In microarray data, cluster boundaries are ill-defined and are characterized by regions that are overlapping, highly connected or embedded. These characteristics result in uncertainty, imprecise (noisy), and incomplete information. Noise problem exists in microarrays when massive experiments based on complex procedures are conducted [16]. As a result, the set-theoretic methods, for example, [15, 18, 21], designed under granular computing framework, for gene expression data, have shown to be more effective than conventional clustering methods. In [15], the initial c -means are computed using a fuzzy entropy. In [18], the rough sets are used to reduce effect of neighbors of a sample in assigning it into a cluster. A rough-fuzzy c -means method [21] works on the principle of constructing the lower approximation and boundary region of a cluster using rough sets. Further, the boundary region is defined in terms of possibilistic and probabilistic memberships in order to quantify imprecision in a cluster definition.

1.2 Granular computing: Granular neural networks for clustering

Granular computing is an umbrella term that refers to computing with the following set theoretic methodologies: Fuzzy sets [37], rough sets [27] or a combination of fuzzy and rough sets [6]. The basic operations used for generating granules include indiscernibility, similarity, proximity, and functionality. Granulation using fuzzy sets refers to generalized fuzzy constraint that assigns memberships to samples in a set. In rough sets, crisp equivalence classes represent granules where indiscernibility between objects is considered. Here, the equivalence classes are used to approximate a set in terms of crisp lower and upper approximations. In this study, we use fuzzy rough set theoretic approach for granular computing. Here fuzzy equivalence classes (granules) are generated based on fuzzy reflexive relations so as to represent fuzzy information granules. These fuzzy equivalence granules are combined using fuzzy implication and T-norm to determine the memberships of their belonging to the lower and upper approximations of a cluster. That means, the cluster is approximated in terms of fuzzy lower and fuzzy upper approximations.

In [1, 26], different granular neural networks were developed by integrating rough set or fuzzy rough set methodologies and self-organizing map (SOM) [17]. In [25], rough sets were used to determine the initial connection weights (network parameters) of SOM. Using fuzzy rough sets, the initial connection weights and fuzzy rough neighborhood function were determined in [9] and [30], respectively. While the underlying structure in [9, 25, 30] is granular in nature, they differ in the design of the competitive learning process and the application case studies. The methodological differences are: Generation of rough rules from attribute reducts [25], evolution of information granules by exploiting α -cuts on fuzzy similarity matrix [9] and obtaining the information granules (clusters) with SOM for the first iteration [30]. Further, fuzzy information granule represented by the lower approximation memberships of a set is used in [9] and [30]. Applications include clustering real valued data [25], analysis of microarray gene expressions [9] and selection of genes based on sample clustering in microarray [30].

In contrast to the previous work in [9, 21, 25, 30] where a Euclidean distance was used as a measure of similarity, in this work, a new fuzzy Euclidean distance is used. Here, a fuzzy relation is also newly defined. While a degree of similarity or dissimilarity between two patterns is measured using the fuzzy Euclidean distance, the fuzzy relation determines the proximity between a pair of patterns corresponding to a feature. The proposed competitive learning strategy (SOMGCL) is described in Section 1.3. The effectiveness of SOMGCL is demonstrated using four different datasets with dimensions (features) ranging from 98 to 54675.

1.3 Overview of SOMGCL

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In this paper, we propose a different competitive learning strategy for training the SOMGCL model. This strategy is based on three concepts: Fuzzy initial connection weights (FICW), a fuzzy distance measure (DM) and the DM based granular neighborhood function (FGNF). This is in contrast to the learning strategies and distance functions discussed in [9, 25, 30]. There are two main differences between this work and one reported in [9]: i) in [9], the information granules were determined by α -cuts on fuzzy similarity matrix to define the network parameters. Whereas, in this article, the network parameters are based on the clusters obtained using SOM for the first iteration, ii) the model discussed in [9] uses competitive learning method involving the Gaussian neighborhood function to update its networks parameters. In contrast, in this paper, network parameters are updated using new granular competitive learning method involving the fuzzy distance based granular neighborhood function. Figure 1 shows initial architecture of SOMGCL. The proposed methodology is explained in two parts. In the first part, the fuzzy initial connection weights (FICW) of SOM are determined. These are based on the output clusters of SOM and fuzzy rough sets. Here, the SOM is trained through competitive learning for the first iteration and its clustering results are presented to a decision table. Subsequently, a dependency value for every feature in the decision table is computed using fuzzy rough sets and a new fuzzy strict order relation (FRS) as well as fuzzy decision classes. The dependency values of all features are incorporated into SOM as its initial connection weights.

The second part deals with a fuzzy distance measure (DM) and the distance based granular neighborhood function (FGNF) of the self-organizing map. Here, we use the above decision table. The fuzzy rough set (FRS) defines membership values belonging to lower approximation and boundary region of a set (cluster). The average membership values belonging to the lower approximation and boundary region of a set are computed. These average memberships are then used to compute distance between two output nodes (winner and any output node) which forms fuzzy distance measure (DM). A fuzzy granular neighborhood function (FGNF) is formulated by replacing the distance value in Gaussian neighborhood with the fuzzy distance (DM). The process of defining FGNF is repeated for every second and remaining iterations. Collectively the methodology of FICW and FGNF thus obtained forms granular competitive learning and the SOM is trained through this learning method. The contribution of this work is a novel granular competitive learning model (SOMGCL) for clustering microarray gene expression data that takes advantage of the strengths of the fuzzy rough methodology and the self-organizing map model. The performance of SOMGCL for clustering samples and gene expressions in microarray is found to be superior to five benchmark algorithms in terms of Rand, Jaccard, Fowlkes-Mallows, β , DB and Dunn indices with four datasets.

This article is organized as follows: Preliminaries of fuzzy rough sets are discussed in Section 2. Section 3 describes the architecture of self-organizing map and its training method. Section 4 explains the proposed fuzzy strict order relation and the method of granular competitive learning of self-organizing map (SOMGCL). In Section 5, the performance of SOMGCL, as compared to related clustering methods, is discussed in terms of confusion matrices, external and inter cluster evaluation measures and quantization errors. Functional groups of genes obtained using SOMGCL and biologically significant gene ontology terms associated with the group of genes under GO-slim biological categories are also presented in this section. Conclusions of the present investigation are provided in Section 6.

Let x → i denote n -dimensional vector corresponding to i th pattern. The membership of i th pattern to k th class, denoted by μ k ⁢ ( x → i ) , is defined as

μ k ⁢ ( x → i ) = 1 1 + ( Z i ⁢ k f d ) f e ,(2)

where Z i ⁢ k is a weighted distance, and f d and f e are the denominational and exponential fuzzy generators controlling the amount of fuzziness in the class membership lying in [ 0 , 1 ] . The values of f d and f e are chosen as positive integers as mentioned in [9]. The weighted distance Z i ⁢ k is defined as

Z i ⁢ k = ∑ j = 1 n [ x i ⁢ j - O k ⁢ j V k ⁢ j ] 2 ,(3) ⁢ 𝑓𝑜𝑟 ⁢ k = 1 , 2 , … , c ,

where O k ⁢ j and V k ⁢ j denote the mean and standard deviation of k th class, respectively.

Let k and u be the two classes. The membership values of patterns belonging to its own class are expressed as:

1.

The membership values of patterns in the k th class to its own class are represented as

D ⁢ D k ⁢ k = μ k ⁢ ( x → i ) , if ⁢ k = u , and(4)

Equations (4) and (5) define memberships to patterns belonging to decision classes k and u . Let R d denote a fuzzy relation with respect to a decision feature. For a pattern x 1 belonging to k th class, the fuzzy decision classes, denoted by R d ⁢ ( x 1 ) , using Eqs (4) and (5), are defined as

(6)

Each pattern corresponding to k th fuzzy decision feature, X k , k = 1, 2, …, c contains membership values belonging to its own class and other classes. The fuzzy lower and upper approximations of the set A ⊆ U , based on the fuzzy reflexive relations and fuzzy decision classes, are described as follows.

The fuzzy lower and upper approximations of a set, A , based on the fuzzy reflexive relations and fuzzy decision classes, are defined as [4, 28]

( R ℬ ↓ R d ) ( x 1 ) = inf x 2 ∈ U I ( R ℬ ( x 1 , x 2 ) , R d ( x 1 ) ) ,(7) ( R ℬ ↑ R d ) ( x 1 ) = sup x 2 ∈ U T ( R ℬ ( x 1 , x 2 ) , R d ( x 1 ) ) ,(8)

where a pattern x 1 ∈ U , R is a fuzzy relation and ℬ ⊆ 𝒜 is a subset of features. Equations (7) and (8) refer to the lower and upper membership values of pattern in a set A . Here, a fuzzy implication I in Eq. (7) and a t -norm T in Eq. (8) are used.

The dependency value for the subset of features is now computed.

Let γ ℬ denote a dependency value for a conditional feature ℬ ⊆ 𝒜 . It is computed as

γ ℬ = ∑ x ∈ U ( R ℬ ↓ R d ) ( x ) | U | ,(9)

where | ⋅ | denotes cardinality of a set U , and γ is 0 ⩽ γ ⩽ 1.

In a conventional SOM, initial connection weights are chosen randomly between 0 and 1. With the random initial connection weights, the SOM method has the following shortcomings: i) slower convergence speed results in large number of iterations in order to converge to a solution, ii) inability to handle uncertain decisions, iii) updating weights at each training instance. To avoid the above listed shortcomings, fuzzy initial connection weights (FICW) are determined. To define FICW, the self-organizing map is initially trained through competitive learning for the first iteration. The output clusters of SOM, thus obtained, are then presented to a decision table. The number of clusters is denoted by c . We use the decision table thereafter.

The initial connection weights (FICW) using fuzzy rough sets based on the proposed fuzzy relation defined in Eq. (4) are defined as follows:

1.

Compute the fuzzy relational matrix for every feature using Eq. (4). Every row of the matrix represents an information granule.

2.

Define fuzzy decision classes, corresponding to fuzzy decision features, using Eq. (6).

3.

Compute the lower membership values of patterns in a set (concept) using Eq. (7), based on fuzzy decision classes and fuzzy relational matrix, for every feature. A concept refers to a set of decision classes. The lower membership values represent exactness in class belongingness of the patterns.

4.

Find the average of the lower membership values of patterns in a concept, representing a dependency value, using Eq. (9) for every feature. The dependency values of all the features, indicating domain knowledge about data, are encoded into the SOM as its initial connection weights.

We describe the initialization process of the connection of weights into network as follows: Let { γ 1 k , γ 2 k , … , γ n k } be the dependency values, with respect to k th concept (set), for features { a 1 , a 2 , … , a n } ∈ ℬ . For a feature j and a concept k , the dependency value γ j k is defined as

γ j k = ∑ x ∈ U k ( R ℬ ↓ R d ) ( x ) | U k | , j = 1 , 2 , … , n ; k = 1 , 2 , … , c .(17)

The weight w j ⁢ k between nodes in the input layer and the output layer of SOM is initialized with γ j k .

4.2 Computation of DM to define neighborhood neurons

In a conventional self-organizing map, the Euclidean distance between the positions of a winning node (winner) and each of the nodes is arranged in a one or two dimensional (1D or 2D) array (see Eq. (13)). It is transformed into a Gaussian function. Here, we use a fuzzy distance function based on a strict order relation defined in Eq. (4). It is computed as follows:

Let v denote a winning node (or winning neuron) and k be any neuron in the output layer of the network. A neuron in the output layer which has minimum Euclidean distance to the input neurons denotes the winning node (winner).

1.

Compute the distance from a winning neuron v to a neuron k , denoted by ∧ v ⁢ k , using

∧ v ⁢ k = ∑ j = 1 n ( | | γ j v - γ j k | | 2 + | | B j v - B j k | | 2 ) ,(18)

where γ j v and γ j k are the averages of membership values belonging to lower approximations (using Eq. (9)), and B j v and B j k are the averages of membership values belonging to boundary regions, of sets obtained at the nodes v and k respectively for j th feature.

(a)

B j v is defined as

B j v = ( ∑ x ∈ U v ( R ℬ ↑ R d ) ( x ) | U v | - ∑ x ∈ U v ( R ℬ ↓ R d ) ( x ) | U v | ) ,(19)

where ( R ℬ ↑ R d ) ( x ) and ( R ℬ ↓ R d ) ( x ) are computed using the lower and upper approximations of Eqs (7) and (8), respectively. The proposed fuzzy strict order relation, Eq. (4), is employed in computing the lower and upper approximations of a set.

(b)

B j k is defined using Eq. (19), where v is replaced with k .

2.

A neuron k lies within the neighborhood of a winning node v (neighborhood neuron of v ), when

( ∧ v ⁢ k ϱ ) ⩽ σ 2 .(20)

–

Here, ∧ v ⁢ k is the Euclidean distance from a winning neuron (winner) v to a neuron k and it is calculated using Eq. (18). Parameters ϱ and σ are defined as follows.

ϱ = max ⁢ { N ⁢ 1 , N ⁢ 2 } ,(21)

where N ⁢ 1 and N ⁢ 2 represent the numbers of nodes arranged in the rows and columns of output layer of SOM, respectively.

4.3 Computation of FGNF for updating connection weights

The proposed granular neighborhood function, based on the fuzzy distance in Eq. (18), is defined as

𝑁𝐵𝐷 v ⁢ ( e ) = exp ⁡ ( - ⁣ ∧ v ⁢ k 2 2 ⁢ σ ⁢ ( e ) 2 ) ,(22)

where σ ⁢ ( e ) represents width of the neighborhood at iteration e . The value of σ in Eq. (22) is obtained using Eq. (14). The weights of a winning neuron and its neighborhood neurons are updated using

w k ⁢ j ⁢ ( e + 1 )(23) = { w k ⁢ j ⁢ ( e ) + α ⁢ ( e ) ⁢ N ⁢ B ⁢ D v ⁢ ( e ) ⁢ ( x j ⁢ ( e ) - w k ⁢ j ⁢ ( e ) ) , if ⁢ ( ∧ v ⁢ k ϱ ) ⩽ σ 2 , w k ⁢ j ⁢ ( e ) , else .

A learning rate α ⁢ ( e ) in Eq. (23) is chosen as in [17]

α ⁢ ( e ) = α 0 ⁢ exp ⁡ ( - e τ 2 ) , e = 2 , 3 ⁢ … ,(24)

where τ 2 is another time constant. The value of α 0 is chosen between 0 and 1. The value of α ⁢ ( e ) monotonically decreases as the number of iterations e gradually increases.

The proposed granular competitive learning of self-organizing map (SOMGCL) satisfies properties of maximality and convergence. An algorithm is said to be convergent when its confusion matrix has dominant element in each row of the matrix. An algorithm attains maximum value 1 if the distance from winning neuron (winner) to its neighborhood neuron in neighborhood function is 0. An algorithm satisfies maximality condition if the value of neighborhood is 1. One can refer to [30] for the details of maximality and convergence.

Pseudo code of the proposed SOMGCL is shown in Algorithm 4 as follows:

Computational complexity of Algorithm 4:

1.

In step 4, the complexity is O(1).

2.

In step 5, the while loop runs e times. /*This is an outer loop for steps 6–17*/.

3.

In step 6, the if condition has O(1) time. /*This is an outer loop for steps 7–9.*/

4.

In step 7, for a sample, the complexity in computing the distance from the output nodes ( c ) to the input nodes ( n ) is O( c ⁢ n ), where n and c denote the number of features and classes (see Section 3). The complexity for determining a winning node among all the output nodes ( c ) is O( c ⁢ n + c ) (see Section 3). The complexity for updating the connection weights between the output nodes ( c ) to the input nodes ( n ) is O( c ⁢ n ) (see Section 3). The computational complexity of SOM is O( c ⁢ n + c ⁢ n + c + c ⁢ n + K ). Here, a constant K is the number of operations performed for determining neighborhood nodes of a winning node and updating the connection weights. For all the samples ( m ), the computational cost of SOM is O( m ⁢ c ⁢ n + m ⁢ c ⁢ n + m ⁢ c + m ⁢ c ⁢ n + K ). The asymptotic complexity of SOM is O( m ⁢ c ⁢ n ).

: Pseudo code of SOMGCL for clustering[1] Inputs: Mixture audio signal y ⁢ ( m ) . Output: Trained SOMGCL and output clusters. Method: e ← 0 while( the total number of iterations){ e ← e + 1, if e == 1, { Self-organizing map (SOM) is trained through competitive learning (see Section 3) and c number of clusters are attained at the nodes of SOM’s output layer. Present the output clusters to a decision table as its decision classes. Define initial connection weights of the SOM using the method of fuzzy initial connection weights (FICW) (see Section 4.1). }/*end if*/ else{ Repeat the following steps 13–17 to perform training of the SOM based on granular competitive learning (SOMGCL). Present an n -dimensional vector x j ⁢ ( e ) at the input nodes of SOMGCL. Compute the Euclidean distance using Eq. (10) between the input. vector x j ⁢ ( e ) and the weight vector w j ⁢ k ⁢ ( e ) for k th output node of SOMGCL. Find a winning neuron v in the output layer of SOMGCL using Eq. (11). Find neighborhood neurons of the winner v using Eq. (20). Modify the connection weights of the winner v and it’s neighborhood neurons of SOMGCL using Eq. (23). } /*end else*/ }. /*end while*/

5.

In step 8, for all the samples ( m ) with features ( n ) and classes ( c ), the computational complexity of fuzzy decision classes is O( c ⁢ ( ( m c ⁢ n + m c ⁢ n + m c ⁢ n + m c ) + ( m c ⁢ ( c - 1 ) ) ) ) (see Section 2.1). Here, O( m c ⁢ n ), O( m c ⁢ n ), O( m c ⁢ n ) and O( m c ) represent the complexity for calculating mean, variance of a class (cluster), the weighted distance from a class to the mean and defining memberships to the samples in a class, respectively, where m c denotes the number of samples in a class. The last term O( m c ⁢ ( c - 1 ) ) represents the complexity for fuzzy decision classes computed using Eq. (6). The asymptotic complexity is O( c ⁢ ( m c ⁢ ( n + c ) ) ). One can refer to [30] for further details of complexity for fuzzy decision classes.

6.

In step 9, for defining fuzzy initial connection weights of SOM, fuzzy relational matrix and fuzzy lower approximations of all the classes corresponding to every feature are computed.

(a)

For a feature, the complexity of fuzzy relational matrix of size m × m is O(cm c + 2m 2 ). Here, m c is the number of samples in c th class and m is total number of samples in all the classes. The asymptotic complexity is O( c ⁢ m c + m 2 ). The complexity for computing the membership values of all the samples ( m ) in the lower approximations of all the classes( c ), based on the above fuzzy decision classes, is O( c ⁢ m c ⁢ ( n + c ) ) + O( ( c ⁢ m c + m 2 ) + m 2 + m ⁢ m c + m ⁢ ( m - m c ) 2 + m ⁢ ( m - m c ) ).

(b)

For all the features ( n ), the computational complexity of lower approximations of all classes is O( c ⁢ m c ⁢ ( n + c ) ) + O( n ⁢ ( c ⁢ m c + m 2 ) + m 2 + m ⁢ m c + m ⁢ ( m - m c ) 2 + m ⁢ ( m - m c ) ). The asymptotic complexity is O( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ). Note that, the number of operations performed for computing the lower approximations (Eq. (7)) is equal to those in the upper approximations (Eq. (8)). So, the complexity of the upper approximation is same as the lower approximation.

(c)

For a feature, the complexity for computing the average of membership values in the lower approximation of a class is O( m c ). For all features ( n ) and classes ( c ), the complexity is O( n ⁢ c ⁢ m c ).

(d)

The complexity of fuzzy initial connection weights is O( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 + n ⁢ c ⁢ m c ). The asymptotic complexity is O( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ).

7.

The complexity of SOM with fuzzy initial connection weights is O( m ⁢ c ⁢ n + c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 )./* if condition is terminated in step 10*/

8.

In step 13, the computational complexity of a sample presented at the input nodes ( n ) of SOMGCL is O( n ).

9.

In step 14, for a sample, the complexity in calculating the distance from the output nodes ( c ) to the input nodes ( n ) of SOMGCL is O( n + c ⁢ n ). The asymptotic complexity is O( c ⁢ n ).

10.

In step 15, for a sample, the complexity in determining a winning node (winner) among the output nodes ( c ) of SOMGCL is O( c ⁢ n + c ). For all the samples ( m ), this is O( m ⁢ c ⁢ n + m ⁢ c ). The asymptotic complexity is O( m ⁢ c ⁢ n ).

11.

In step 16, we first calculate the computational cost of fuzzy distance from a winning node to an output node. The complexity of the fuzzy distance based neighborhood function (determining neighborhood nodes of the winning node) is then computed. This involves the above computational cost of the lower and upper approximations of all classes that is

O ( 2 ( c m c ( n + c ) + n m 2 ) ) ( s e e s t e p 6 . ( b ) ) .

(a)

For a feature, the complexity in computing the average memberships in the boundary regions of a winning node and ( c - 1 ) output nodes using Eq. (19) is O(1 + ( c - 1 )). Here the average membership of boundary region of an output node is the difference between the average memberships of the upper and lower approximations.

(b)

The complexity in computing fuzzy distance from a winning node to c - 1 output nodes, based on the average memberships of their lower and boundary regions, using Eq. (18) is O((1 + ( c - 1 ) ) + ( 1 ( c - 1 ) ) ). For all features ( n ), the complexity is O( ( n + n ⁢ ( c - 1 ) ) + ( n ⁢ ( c - 1 ) ) ). For all the samples ( m ), this is O( m ⁢ ( n + 2 ⁢ n ⁢ ( c - 1 ) ) ). The asymptotic complexity is O( m ⁢ n ⁢ ( c - 1 ) ).

12.

The computational complexity for determining neighborhood nodes of a winning node using Eq. (20) is O(( m c n ) + ( 2 ( c m c ( n + c ) + n m 2 )) + ( m ⁢ n ⁢ ( c - 1 ) ) + K ). Here, O( m ⁢ c ⁢ n ), O(2( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ) and O( m ⁢ n ⁢ ( c - 1 ) ) denote the complexities of a winning node, the lower and upper approximations of all classes and the fuzzy distance respectively. A constant K represents the complexity of the total number of operations in Eqs (20) and  (21) (maximum) (division and multiplication). The asymptotic complexity is O( m ⁢ c ⁢ n + ( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ) ).

13.

In step 17, for a sample, the complexity for updating connection weights between the input nodes ( n ) and output nodes ( c ) of SOM is O( c ⁢ n ). For all the samples ( m ), the complexity is O( m ⁢ c ⁢ n ).

14.

The complexity of Algorithm 4 involving SOM with fuzzy initial connection weights for 1 iteration and granular competitive learning of SOM (determining neighborhood nodes of a winning node and updating the connection weights) for ( e - 1 ) iterations is O( m c n + ( c m c ( n + c ) + n m 2 ) + ( e - 1 )( ( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ) + m ⁢ c ⁢ n + m ⁢ c ⁢ n )). Therefore, the asymptotic complexity is O( e ⁢ ( ( c ⁢ m c ⁢ ( n + c ) + n ⁢ m 2 ) + m ⁢ c ⁢ n ) ). /* while loop is terminated*/

.

Similarity and Difference betweenEqs. (13) and (22) The nodes in the output layer are arranged in 2D array. Equations (13) and (22) are based on the Gaussian neighbor function.

In Eq. (22), the Euclidean distance ∧ v ⁢ k from a winner ( v ) to a neuron in the output layer ( k ) is defined as fuzzy. Further, the membership values of neurons v and k , belonging to the lower approximation and the boundary region of a set (output cluster attained at either node v or k ) are determined using fuzzy rough sets, based on the proposed fuzzy strict order relation.

In contrast, the distance D v ⁢ k in Eq. (13) is calculated using the positions of a winning neuron ( v ) and an output neuron ( k ) in the output layer.

.

Difference between Eqs (23) and (13)Equations (23) and (13) use the same parameters except distance measures. In Eq. (23), ∧ v ⁢ k represents a fuzzy Euclidean distance which is computed between winning neuron (winner) v and a neuron k . Whereas, in Eq. (13), D v ⁢ k is a Euclidean distance from a winner v to a neuron k .

.

Difference between Eqs (17) and (9)Equations (17) denotes a dependency value of a feature with respect to a class. Whereas, Eq. (9) represents the average of the dependency values of all the classes for a feature. Here, the average of membership values of patterns belonging to the lower approximation of a set denotes the dependency value.

5. Experimental results

The proposed self-organizing map based on granular competitive learning (SOMGCL) is implemented in C-language using Intel Core i5-2430M CPU at 2.40 GHZ processor and 16 GB RAM. The following datasets are used in our experiments: breast cancer [34], multi-A [13], GDS5218 and GDS5499.1

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Breast cancer: The data has 98 samples of three breast cancer types, distant metastates, disease free and BRCA1 germline mutations. There are 1214 gene expressions for each sample. As an example, a three dimensional (3D) plot of 98 samples in 3 classes of breast cancer in F 1 -F 2 -F 3 space is shown in Fig. 2.

Multi-A: The multiple tissue type-A (Multi-A) data contains 103 samples belonging to four different tissue types (classes or groups), breast, prostate, lung and colon. Here, 5565 gene expressions are available for each sample.

GDS5218: The data consists of 110 muscle biospy samples of female young, male young, female old and male old adults. Each sample has 54675 gene expressions.

GDS5499: The data is based on 140 samples of patients from four disease groups, idiopathic pulmonary arterial hypertension (IPAH), systemic sclerosis (SSc), SSc associated PAH (SSc-PAH), and SSc complicated by interstitial lung disease and PH (SSc-PH-ILD). Every sample contains 48803 gene expressions.

In this study, a new method of granular competitive learning for self-organizing map (SOMGCL) is developed using fuzzy rough sets. A fuzzy strict order relation, which is an extension to the fuzzy reflexive relation, is proposed. The concepts of implication and t-norm are utilized in max and min composition of the proposed fuzzy relation. The proposed relation is incorporated into the lower and upper approximations of a set to define memberships belonging to their approximate regions. The lower and upper memberships of a set and the memberships of a class induced by the order relation represent information granules. The lower and upper memberships of a set are used to define the proposed SOMGCL involving a new fuzzy distance and the distance based granular neighborhood function, and the initial connection weights. The information granules embedded in SOMGCL help in efficient handling of uncertainty. The performance of SOMGCL for clustering of samples and genes is demonstrated on four well-known microarray datasets with dimensions ranging from 98 to 54675.

The performance of SOMGCL for clustering samples for all the datasets is found to be superior to GSOM, SOM, RRFCM, CEM and c -medoids with respect to confusion matrices, Rand, Jaccard, Fowlkes-Mallows, β and DB indices. The superiority of SOMGCL, when compared with the 5-benchmarked methods, in terms of Dunn-index, except GSOM for GDS5218 and GDS5499, is achieved. The SOMGCL obtains quantization error lower than GSOM and SOM for breast and multi-A datasets. Moreover, it is found that the SOMGCL identifies biologically significant gene clusters from the breast cancer and mutli-A datasets. The significant gene ontology terms associated with a group of genes are presented for four clusters. Although the significant gene terms are presented for only four clusters, similar results are observed for four other values of c (clusters) discussed in this paper.

This work demonstrates the effectiveness of SOMGCL for clustering microarray with different characteristics that include large dimensions and more overlapping classes. The advantages of the proposed SOMGCL include less convergence speed and better clustering results for data having more overlapping classes. The limitation of the proposed SOMGCL is that it takes higher computational time. Future work will involve investigation of SOMGCL for clustering real life data with different sizes and multiple classes, where reducing computation time and finding biologically meaningful gene groups are the major tasks. Designing a granular model for clustering by considering feature selection techniques may constitute another research problem, where maintaining low computational cost and high performance is important. Future work may also include testing the effectiveness of the proposed algorithm with different distance functions and similarity relations. Selection of appropriate norm and fuzzy implication for the similarity relations is another research direction. It is also important to recommend the results of the proposed method to the experts in the field from which the data originates in terms of their usefulness in practice.

Availability of data and material (data transparency)

Data and material would be available with the manuscript.

Code availability

Code would be available with the manuscript.

Compliance with ethical Standards

(In case of Funding) Funding: No funding was granted to this study.(In case animals were involved) Ethical approval: No animals were involved in this study.(And/or in case humans were involved) Ethical approval: No humans were involved in the study.(If articles do not contain studies with human participants or animals by any of the authors) Ethical approval: The article does not contain any studies with human participants or animals performed by any of the authors.(In case humans are involved) Informed consent: No human participants were involved in the study. There was no need to obtain informed consent from the participants.