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Computational Biology Journal
Volume 2013 (2013), Article ID 898090, 12 pages
Two-Stage Approach for Protein Superfamily Classification
Department of Computer Science & Engineering, N.I.T., Rourkela, Odisha 769008, India
Received 26 February 2013; Revised 3 June 2013; Accepted 3 June 2013
Academic Editor: Qianzhong Li
Copyright © 2013 Swati Vipsita and Santanu Ku. Rath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We deal with the problem of protein superfamily classification in which the family membership of newly discovered amino acid sequence is predicted. Correct prediction is a matter of great concern for the researchers and drug analyst which helps them in discovery of new drugs. As this problem falls broadly under the category of pattern classification problem, we have made all efforts to optimize feature extraction in the first stage and classifier design in the second stage with an overall objective to maximize the performance accuracy of the classifier. In the feature extraction phase, Genetic Algorithm- (GA-) based wrapper approach is used to select few eigenvectors from the principal component analysis (PCA) space which are encoded as binary strings in the chromosome. On the basis of position of 1’s in the chromosome, the eigenvectors are selected to build the transformation matrix which then maps the original high-dimension feature space to lower dimension feature space. Using PCA-NSGA-II (non-dominated sorting GA), the nondominated solutions obtained from the Pareto front solve the trade-off problem by compromising between the number of eigenvectors selected and the accuracy obtained by the classifier. In the second stage, recursive orthogonal least square algorithm (ROLSA) is used for training radial basis function network (RBFN) to select optimal number of hidden centres as well as update the output layer weighting matrix. This approach can be applied to large data set with much lower requirements of computer memory. Thus, very small architectures having few number of hidden centres are obtained showing higher level of performance accuracy.
Protein superfamily classification deals with the prediction of family membership of newly discovered proteins. This helps the drug analyst for discovery of new drugs required in the treatment of new diseases. This classification helps in predicting the protein function and/or structure of the unknown sequence, thus avoiding the expensive biological (wet) experiments in the laboratory. Once a particular sequence S causing disease D is classified to a superfamily , the researchers can design some new drugs by trying some combination of existing drugs for family . Thus, this classification problem helps the researchers in treatment of diseases by discovering new drugs . Among the major techniques developed in the past, the popular BLAST tool  represents the simplest nearest neighbour approach and exploits pairwise local alignments to measure sequence similarity. Another type of direct modelling method is based on hidden markov models (HMMs) , (Karplus et al., 1998). After constructing an HMM for each family, protein queries can be easily scored against all established HMMs. This can be done by calculating the log-likelihood of each model for the unknown sequence and then selecting the class label of the most likely model. The Motif Alignment and Search Tool (MAST) (Bailey and Gribskov, 1998) is based on the combination of multiple motif-based statistical score values. According to this scheme, groups of probabilistic motifs discovered by the MEME (Multiple Em for Motif Elicitation) algorithm (Bailey and Elkan, 1994) are used to construct protein profiles for the families of interest.
As there is an exponential growth in size of biological database, computational intelligent techniques possess the real challenge to handle this huge database and retrieve valuable knowledge from it. These intelligent techniques are mostly applied for feature selection, dimensionality reduction as well as in the design and optimization of the classifier. Feature extraction and feature selection have a great impact on the performance accuracy of the classifier. Singular value decomposition (SVD) for n-gram feature reduction and feedforward neural network (FFNN) trained using backpropagation (BP) algorithm are implemented in . PCA is used to reduce the size of n-gram feature vector in . Besides the global features, a linear correlation coefficient (LCC) is evaluated as well as local features are extracted from motif content, and Bayesian neural network (BNN) as a classifier is implemented in . A generalized radial basis function (GRBF) neural network architecture that generates fuzzy classification rules is used for protein sequence classification in , and comprehensibility of fuzzy rule-based classifier with comparable classification accuracy is shown in . Principal component null space analysis (PCNSA) linear classifier in  reported excellent results compared to those of neural networks and support vector machines (SVM) which efficiently handled high-dimensional data thereby maximizing accuracy to much extent. Extreme learning machine (ELM) needed up to four orders of magnitude less training time compared to BP network was used to classify protein sequences with ten classes of superfamilies downloaded from a public domain database in . Feature selection based on low relative entropy values using SVM as classifier is shown in . Feature selection based on feature ranking method is implemented in  where a novel steady-state genetic algorithm for extracting fuzzy classification rules from data is used to generate a compact set of interpretable fuzzy rules that is used for protein superfamily classification. An implementation of subtractive clustering in standard RBF and modular RBF is shown in . Based on probabilities of occurrences of the amino acids in the different positions of the sequences, a genetic fuzzy clustering approach evolves a set of prototypes representing each class. The nearest neighbour rule is then used to classify an unknown sequence into a particular class, based on its proximity to the prototypes . In [14, 15], the classification problem is mapped as a multiobjective problem (MOP) having two objectives such as minimization of number of features selected and minimization of misclassification error rate which are solved using the weighted sum approach.
This paper provides insight to solve the protein superfamily classification problem in two major stages. In the first stage, we have implemented PCA-NSGA-II, a hybridized approach which tries to solve the trade-off problem in selecting optimal number of significant eigenvectors by maintaining a good level of performance accuracy of the inductive algorithm. The encoding of chromosome becomes a very difficult task as feature vector extracted from amino acid sequence is too high. So to overcome this problem, eigenvectors having nonzero eigenvalues are encoded in the chromosome, and GA helps in searching the eigenspace to select the distinguishing eigenvectors. Probabilistic neural network (PNN) is used as an inductive algorithm, and the evaluation function used in this wrapper approach is the minimization of the misclassification error rate of the PNN over the test samples. (A detailed description of wrapper approach and inductive algorithm is discussed in .) The major limitations of not using PNN as a classifier are that all the training patterns need to be stored for classifying a new pattern. As the size of the training set increases, the network becomes too complex as every node in the pattern layer represents a training sample. So, radial basis function network (RBFN) is preferred to be a classifier, but the major issue lies in finding out the exact number of hidden centres as well as the update matrix connecting the hidden and the output layer. Recursive orthogonal least square algorithm (ROLSA) finds the real solution to the above-mentioned problems.
The rest of the paper is organized as follows: Section 2 describes data acquisition and preprocessing steps from amino acid sequence. Section 3 describes in detail the feature extraction from eigenspace using PCA-NSGA-II with relevant algorithms. Section 4 describes the implementation of ROLSA used for training of RBFN. Section 5 describes the experiment details, and Section 6 discusses the results of numerical simulations. Section 7 concludes the paper.
2. Data Acquisition and Preprocessing from Amino Acid Sequence
In sequence-based classification of proteins the data acquisition and data preprocessing stage involves the retrieval of amino acid sequences from the publicly available database such as Uniprot, Pubmed, and Protein Information Resource (PIR). The data are downloaded and saved in FASTA format. The sequence-based classification methods can be divided into three broad categories such as feature-based classification [17, 18] where feature vectors are built from long amino acid sequences and these features help in classification of protein to their superfamily. In sequence distance-based classification , the distance function determines the similarity between sequences, and the quality of classifier can then be measured. The third category is model-based classification where hidden markov model (HMM)  or some other statistical models are used to classify the sequences . Out of these three categories, we emphasize on feature-based classification from amino acid sequences for protein superfamily classification. The schematic representation of data acquisition and preprocessing is shown in Figure 1.
In general, the genetic code specifies 20 standard amino acids such as
For protein feature selection, the two gram features such as are selected. The total number of possible bigrams from a set of 20 amino acids is , that is, 400. The two gram features represent the majority of the protein features. Two grams have the advantages of being length invariant, insertion/deletion invariant, not requiring motif finding and allowing classification based on local similarity . Apart from this, bigrams reflecting the pattern of substitution of amino acids are also extracted. For this purpose, equivalence classes of amino acids that substitute for one another are derived from the percent accepted mutation matrix (PAM) . Exchange grams are similar but are based on a many-to-one translation of the amino acid alphabet into a six letter alphabet that represents six groups of amino acids, which represent high evolutionary similarity. Generally the exchange groups used are
The exchange groups statistically describe the probability of one amino acid replacing another over time. The total number of possible bigrams on these six substitution groups is , that is, 36. Thus the overall bigram features extracted compute to 436 values, 400 corresponding to the consecutive pairs of amino acids and 36 corresponding to the consecutive pairs of substitution groups. Besides that, the amino acid distribution and exchange group distribution , some other features are also taken into account.
Therefore, for every amino acid sequence, the 470 features were processed to build the fixed dimension feature vector as follows: = atomic composition (C, H, N, O, S), = molecular weight, = isoelectric point, = average mass of protein sequence, amino acid distribution, two gram distribution, exchange group distribution, two gram exchange group distribution.
If we assume total “” number of instances, then the matrix size becomes which is a matrix having large number of sparse entries.
3. Feature Extraction Using PCA-NSGA II
3.1. Dimension Reduction of Long Feature Vector
Jain et al. in  had clearly distinguished the concept of feature selection from feature extraction. Feature selection refers to some algorithm or technique which selects best subset of features from the original feature set. But feature extraction refers to some techniques which performs some transformation of the original feature matrix to create new feature matrix by discarding the features having low discrimination ability. In other words, feature extraction can be described as, given an pattern matrix ( points in a -dimensional space), an pattern matrix is derived, such that where and is a transformation matrix.
3.2. Feature Extraction Using Principal Component Analysis (PCA)
The concept of PCA was developed by Karl Pearson in 1901. Principal component analysis (PCA) is a statistical technique used to transform a data space of high dimension into a feature space of lower dimension having the most significant features. PCA rigidly rotates the axes of the -dimensional space to new positions (principal axes) such that principal axis 1 has the highest variance, axis 2 has the next highest variance, and so on. The covariance among each pair of the principal axes is zero so the principal axes are uncorrelated.
3.2.1. Basic Steps of PCA
(i) Calculate mean as the average value across each dimension and then subtract the mean value from each of the data dimension thereby producing data set with zero mean.(ii) Compute the covariance matrix as and find out the eigenvalues.(iii) Sort the eigenvalues in a decreasing order and let them be denoted as . Let the corresponding eigenvectors be denoted as .(iv) Select the first eigenvectors from vectors such as . The cumulative percentage of variance guides in selecting the top eigenvectors from the covariance matrix whose threshold value is fixed by the user: (v) Finally, based on selected eigenvectors, project the data set into lower dimension as given by If is a test point
3.3. Multiobjective Genetic Algorithm (MOGA)
Most real world problems are complex; in the sense they may be nonlinear, multimodal, and stochastic, and there may be more than one parameter which are needed either to be minimized or to be maximized. These types of problems are referred to as multiobjective problems (MOP) which can be solved by various approaches. A survey of various multiobjective evolutionary techniques is described in . The most simple method among all approaches is to form a composite objective function as the weighted sum of various objectives, and the weight value is assigned as per the priority of individual objective. This technique is otherwise referred to as preference-based multiobjective optimization in which a multiobjective problem is mapped as a single objective which is a composite of more than one objective. This can be denoted as where denotes the various objective functions. In the above weighted sum approach, the major practical problem lies in the proper selection of weights. So to handle MOP, pareto optimal solutions are mostly preferred as they can handle the trade-off problem arising due to various objective functions. A MOP should achieve the following three goals .(i)The best known pareto front should be as close as possible to the true pareto front.(ii)The best known solutions in the pareto front should be uniformly distributed and diverse enough to show the exact concept of trade-off due to conflicting objectives.(iii)The best known pareto front should capture the whole spectrum of the pareto front. The main objective is to extend the pareto fronts at both ends to explore new extreme solutions.
In 2001, Rudolph introduced the concept of elitism to MOP. Nondominated sorting using NSGA-II was suggested by Deb et al. in , which requires time complexity of where is the number of objectives and is the population size. It is shown that NSGA-II was able to maintain a better spread of solutions and converge very near to the true pareto front in comparison to elitist multiobjective evolutionary algorithm such as Pareto archived evolution strategy (PAES) and strength pareto evolutionary algorithm (SPEA). Farina et al., in 2004 , have addressed the growing need of solving dynamic MOP in which the pareto optimal set changes with respect to time or with the iteration of the optimization process.
Generally, traditional PCA selects the top few eigenvectors having higher eigenvalues. But Balci and Atalay in  have illustrated that eigenvectors having large eigenvalues may not always have a great impact regarding the target concept of interest.
In , it is shown that, in face detection and vehicle detection experiments, the eigenvectors having low eigenvalues encode more lighting information and also encode some specific local features. So, GA is implemented which searches the eigenspace to select subset of eigenvectors. The two objectives such as selection of minimum number of significant eigenvectors and minimization of classification error rate are solved using weighted sum approach by giving proper weight values to the objective functions. Zhao et al. in  also applied the weighted sum approach for extracting features from motif content and protein content where support vector machine (SVM) is used as an inductive algorithm. Zhao et al. in  developed a hybrid GA/RBFNN technique which selects features from protein sequences and trains the RBF neural network simultaneously. The weight factors are assumed to be 40000 and 0.1 for the recognition rate and number of features removed from the original feature set, respectively.
In our approach, PCA is hybridized with the elitist nondominated sorting GA or NSGA-II for feature subset selection from the eigenspace. The encoding of chromosome is done as string of 0’s and 1’s. The trade-off between two objectives such as minimization of number of eigenvectors selected and minimization of misclassification error rate is solved by generating pareto fronts. The goodness of a chromosome is evaluated based on the number of 1’s in the chromosome string. In our problem, GA searches the eigenspace comprising of top 67 eigenvectors having nonzero eigenvalues. The eigenvectors above 67 have very small eigenvalues nearly equal to 0. So the length of chromosome is fixed at 67. After mapping the original feature space to lower dimension using the transformation matrix, the reduced feature matrix is given as input to the probabilistic neural network (PNN). PNN evaluates the second fitness value of chromosome, that is, the misclassification error rate obtained by the selected eigenvectors. The best solutions are derived from the lowest pareto front. The algorithm for implementing PCA-NSGA-II is described below, and the brief overview of the entire process is shown in Figure 2. The NSGA-II procedure is outlined in Algorithm 2. The crowded sorting of the solutions in a front is evaluated using crowding distance metric described in Algorithm 3. Here, the boundary solutions are assigned large distance values () whereas the intermediate solutions are assigned with crowding distance using the metric. Thus, the population is arranged in descending order of magnitude of the crowding distance values. The crowded tournament selection operator (described in Algorithm 4) decides which solution to enter into the mating pool and become the parents for next generation. The operator randomly chooses two solutions and returns the winner of the tournament based on two conditions. The first condition ensures the solution to be winner if it lies on a better nondominated front. The second condition resolves the tie if both solutions lie on the same front by selecting the solution with a larger value of crowding distance (Algorithm 1).
4. Radial Basis Function Network as a Classifier
4.1. Brief Concept and Architecture of RBFN
A RBF network consists of three layers, namely, the input layer, the hidden layer, and the output layer. The input layer broadcasts the coordinates of the input vector to each of the units in the hidden layer (Figure 3).
The inputs of hidden layer are the linear combinations of scalar weights and the input vector , where the scalar weights are usually assigned unity values. Each unit in the hidden layer then produces an activation based on the associated radial basis function. The output layer yields a vector for outputs by linear combination of the outputs of the hidden nodes to produce the final output: where is the final output, denotes the radial basis function of the th hidden node, denotes the hidden to output weight corresponding to the th hidden node, and is the total number of hidden nodes. A normalized Gaussian function is usually used as the radial basis function, that is, where denotes the input vector, denotes the th center vector, and represents the width parameter.
RBFN is an effective tool for pattern classification problem as it has good generalization and approximation ability with a simple network structure. This network has been successfully used in many applications such as pattern classification, system identification, nonlinear function approximation, adaptive control, and speech recognition. RBFN has a faster training procedure in comparison to multilayer perceptron (MLP) trained using backpropagation (BP) algorithm. In generalized RBF, the supervised learning of the center location as well as output layer weights and the Gaussian spread () are performed based on error correction learning rule using a gradient descent procedure. In , Neruda and Kudová presented three learning methods of RBFN such as gradient-based learning, three-step unsupervised learning, and evolutionary algorithms, and it was examined on few benchmark problems. It was observed that gradient-based learning performed better in terms of error measured on both the training and test data set. The three-step learning was the fastest but the convergence rate of GA was too high.
In the design and implementation of RBFN, a very critical issue lies in selecting the optimal number of hidden centres as the number of basis function controls the complexity and generalization ability of the network. If too many nodes are selected in the hidden layer, this may include redundant data which results in large network structure. Thus, the computational overhead is too high when we classify an unknown pattern. The network will also have poor generalization capability as it becomes over sensitive to the training data and thereby fails to recognize the noisy samples. On the contrary, very few number of hidden centres in the hidden layer may lower the generalization ability of the trained network. Many researchers used evolutionary optimization techniques to optimize the structure of RBFN.
Implementation of Genetic Algorithm (GA) for structure optimization of RBFN is shown in  where each network is coded as a variable length string with distinct integers, and both the single objective and multiobjective functions have been proposed to evaluate network fitness. In 2003, Guo et al.  used GA to optimize the parameters of RBFN, and a hybrid learning algorithm further adjusts the parameter values. González et al., in 2003 , implemented multiobjective evolutionary algorithm (MOEA) in which global mutation operators based on matrix transformation such as singular value decomposition (SVD) and orthogonal least square (OLS) have been used. Multiobjective structure selection method using MOGA is shown in  where the structure of RBFN is encoded as chromosome of GA, and the pareto optimal solutions are obtained from the pareto optimal fronts which solves the trade-off problem between model accuracy and complexity. In , GA with hybrid learning algorithm (HLA) has outperformed GA, ROLSA, and -means clustering with HLA. HLA mostly adjusts the centres and the widths. In , multiobjective PSO (MOPSO) is implemented to simultaneously optimize the architecture and the connection weights of RBFN. The RBF networks are encoded as particles in PSO, and the particle evolves towards pareto optimal front to solve the trade-off problem between model accuracy and complexity. In 2010, a two level learning method for designing an optimal RBFN using adaptive velocity update relaxation PSO (AVURPSO) and OLS is implemented in .
Besides evolutionary techniques, orthogonal decomposition method is used to solve the least square problem to determine the update weight matrix connecting the hidden and output layer. In 1991, Chen et al., in [37, 38], used orthogonal least square algorithm for the construction of RBFN, and the forward regression procedure provided a systematic approach for supervised selection of centres. In , besides forward center selection, backward centre selection is proposed on batch OLS where the centres are sequentially removed while minimizing the network output error. In [40, 41], Yu et al. have demonstrated recursive version in contrast to batch OLS. The recursive version requires less computer memory and also maintains robust property. ROLSA solves to find the output layer weights, and the approach is extended to select the minimum number of centres of RBFN. The approach of ROLSA was validated on two applications such as nonlinear time series and a real multiinput and multi-output (MIMO) chemical process.
4.2. Training of RBFN Using ROLSA (Pseudocode)
The backward center selection algorithm as proposed in  by Altschul et al. is implemented here to optimize the structure of RBFN. The pseudocode of the implementation is described below. The final updated weight matrix connecting the hidden and the output layer gave high performance accuracy when the network is used as a classifier for protein superfamily classification problem. As the orthogonal decomposition is numerically robust for solving least square problem; the final network so obtained is highly reliable with small architecture.
5. Experiment Details
5.1. Input Details
The four superfamilies considered in our experiment are globin (840), kinase (542), ribitol dehydrogenase (828), and ligase (678) from Uniprot database http://www.uniprot.org/. From each family, 60% of total data set formed the training set and the remaining 40% formed the test set. The redundant samples were eliminated during data preprocessing stage and the data were then normalized using min-max normalization equation as given below (Algorithm 5, Table 1):
5.2. Architecture of PNN as Inductive Algorithm
The number of nodes of the input layer of PNN was fixed based on the reduced feature matrix obtained from every chromosome. The pattern layer in the PNN architecture was built using fifteen samples from every class to estimate the Gaussian PDF (probability density function) value for every test sample. The summation layer has four nodes each representing an individual class. The single output node of the PNN evaluates the maximum Gaussian PDF estimate to classify a test pattern. 50 randomly chosen samples from every class formed the test matrix.
5.3. Parameter Details of PCA-NSGA-II
5.3.1. Initialization of Chromosome
The population was initialized by encoding the chromosomes as strings of 0’s and 1’s. The chromosome length was fixed on the basis of number of eigenvectors having nonzero eigenvalues. In our experiment, the chromosome string length was fixed at 63. Every one bit of chromosome represents selection of the eigenvector, and zero represents discard of the eigenvector.
The population size was fixed at ,
probability of Crossover ,
probability of mutation .
5.3.2. Evaluation of Fitness Function
The first objective function is to minimize the number of eigenvectors.
is computed as number of 1’s in the chromosome string. The second objective function is to minimize the misclassification error by implementing PNN as classifier, which is the measure of number of misclassified samples with respect to total number of samples.
This is calculated as
5.3.3. Stopping Criteria
The PCA-NSGA-II process terminates when 90% or more number of solutions are in the best nondomination level.
5.4. Parameter Details of RBFN-ROLSA
From the number of solutions obtained from the first pareto front , twenty-nine (29) eigenvectors showing a misclassification error of 0.57 are selected. Twenty five (25) randomly chosen samples from the training set of every class were selected as hidden centres in the hidden layer. The output layer consists of four nodes, each representing a class.
5.5. The Parameters Used for Measuring the Efficiency of Classifier
For any classification problem, the outcomes of the data are always labelled, that is, either positive (p) or negative (n). Based on the two outcomes there may be various combinations of outputs. If the outcome from a classifier is p and the actual outcome value is also p, then it is called true positive, but if the classifier output is p and the actual outcome is n, then it is false positive. Conversely, if the actual output and the classifier are both n then it is called true negative, and if the classifier output is n and the actual value is p we refer to it as false negative. The measures used by most of the researchers are as follows:(1),(2),(3),
where TP = number of true positive, TN = number of true negative, FP = number of false positive, and FN = number of false negative.
6. Results and Discussion
6.1. Results from 1st Stage
PCA-NSGA-II procedure took 2041 number of generations to meet the stopping criteria thereby generating three pareto fronts shown in Figure 4. Few solutions in the upper, lower and mid region of are shown in Table 2.
6.2. Results from 2nd Stage
The 29 selected eigenvectors (from stage 1) showing a misclassification error rate of 0.57 are selected from the mid region of which formed the reduced feature matrix. The implementation of backward center selection algorithm (ROLSA) on network finally yielded in reduced architecture. The final optimal weight matrix of size 45 × 4 on the reduced network shows phenomenal performance in accuracy when tested on the test data set. The loss function , final prediction error (FPE) versus number of centres is shown in Figure 5. The number of correctly classified samples with respect to individual superfamily is shown in Table 3.
-Fold Cross-Validation Technique. To measure the overall performance of the classifier, 10-fold cross-validation technique is implemented. The training set is randomly partitioned into ten equal disjoint sets: . The ten RBFN classifiers are trained on the complement , and each classifier is then tested on the corresponding unseen test set . The final cross-validation recognition rate is given by where is the recognition rate on using the RBFN classifier trained on [15, 18].
To show the efficiency of RBFN-ROLSA over standard neural networks, the comparison process is further extended by implementing some standard neural networks. Feedforward neural network (trained using backpropagation (BP) algorithm), PNN, and standard RBFN (trained using supervised gradient descent learning algorithm ) are implemented. The learning rate () and momentum () are the two main control parameters of BP algorithm. Keeping fixed and varying in the range of 0.1 to 1, variation in performance accuracies was observed. Similarly, the smoothing parameter () is the controlling parameter for PNN and RBFN which was varied to derive various performance accuracies. In RBFN-MOGA, after deriving the optimal ensemble of RBFN from the pareto optimal set, various accuracies were observed by varying values in the range of 0.1 to 1. The highest possible accuracies obtained by the networks are shown in Table 4 and Figure 6.
7. Conclusion and Scope of Future Work
From the above implementation of two-stage approach, it can be concluded that PCA-NSGA-II gave the most optimal number of significant features which was verified on PNN used as induction learning algorithm. The distinguishing features were derived from the pareto front which was the input to the RBFN. The backward center selection algorithm gave the most parsimonious structure having a very high performance accuracy value of 98.61% which is the mean recognition rate obtained from the 10-fold cross-validation technique. Generally, the standard neural networks (FFNN and RBFN) initialize the connecting weight matrix randomly due to which their performance varies when the algorithm is implemented number of times. This approach of structure optimization using ROLSA is very reliable in comparison to randomized evolutionary techniques. The RBFN so obtained is highly robust which works efficiently on large training and test data set. Many pattern classification problems can be efficiently solved by implementing RBFN with ROLSA as a classifier. Thus, the two-stage approach has efficiently minimized number of features in the first stage and also minimized number of hidden centres (of RBFN) in the second stage thereby updating the connecting weight matrix between the hidden and output layer.
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