Singular value decomposition (SVD) is a popular decomposition method for solving least squares estimation (LSE) problems. However, for large data sets, applying SVD directly on the coefficient matrix is very time consuming and memory demanding in obtaining least squares solutions. In this paper, we propose an iterative divide-and-merge-based estimator for solving large-scale LSE problems. Iteratively, the LSE problem to be solved is processed and transformed to equivalent but smaller LSE problems. In each iteration, the input matrices are subdivided into a set of small submatrices. The submatrices are decomposed by SVD, respectively, and the results are merged, and the resulting matrices become the input of the next iteration. The process is iterated until the resulting matrices are small enough which can then be solved directly and efficiently by SVD. The number of iterations required is determined dynamically according to the size of the input data set. As a result, the requirements in time and space for finding least squares solutions are greatly improved. Furthermore, the decomposition and merging of the submatrices in each iteration can be independently done in parallel. The idea can be easily implemented in MapReduce and experimental results show that the proposed approach can solve large-scale LSE problems effectively.

Karnik and Mendel proposed an algorithm to compute the centroid of an interval type-2 fuzzy set efficiently. Based on this algorithm, Liu developed a centroid type-reduction strategy to carry out type reduction for type-2 fuzzy sets. A type-2 fuzzy set is decomposed into a collection of interval type-2 fuzzy sets by -cuts. Then, the Karnik-Mendel algorithm is called for each interval type-2 fuzzy set iteratively. However, the initialization of the switch point in each application of the Karnik-Mendel algorithm is not a good one. In this paper, we present an improvement to Liu's algorithm. We employ the previously obtained result to construct the starting values in the current application of the Karnik-Mendel algorithm. Convergence in each iteration, except the first one, can then speed up, and type reduction for type-2 fuzzy sets can be carried out faster. The efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results.

We propose a novel approach for building a type-2 neural-fuzzy system from a given set of input-output training data. A self-constructing fuzzy clustering method is used to partition the training dataset into clusters through input-similarity and output-similarity tests. The membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster. Then a type-2 fuzzy Takagi-Sugeno-Kang IF-THEN rule is derived from each cluster to form a fuzzy rule base. A fuzzy neural network is constructed accordingly and the associated parameters are refined by a hybrid learning algorithm which incorporates particle swarm optimization and a least squares estimation. For a new input, a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type-2 fuzzy set, which is then defuzzified by applying a refined type reduction algorithm. Experimental results are presented to demonstrate the effectiveness of our proposed approach.

A novel Takagi-Sugeno-Kang (TSK) type fuzzy neural network which uses general type-2 fuzzy sets in a type-2 fuzzy logic system, called general type-2 fuzzy neural network (GT2FNN), is proposed for function approximation. The problems of constructing a GT2FNN include type reduction, structure identification, and parameter identification. An efficient strategy is proposed by using alpha-cuts to decompose a general type-2 fuzzy set into several interval type-2 fuzzy sets to solve the type reduction problem. Incremental similarity-based fuzzy clustering and linear least squares regression are combined to solve the structure identification problem. Regarding the parameter identification, a hybrid learning algorithm (HLA) which combines particle swarm optimization (PSO) and recursive least squares (RLS) estimator is proposed for refining the antecedent and consequent parameters, respectively, of fuzzy rules. Simulation results show that the resulting networks obtained are robust against outliers.

Using machine learning techniques such as naive Bayes, decision trees and support vector machines to automatically filter out spam e-mails has drawn many researchers' attention. Previous methods use keywords contained in e-mails to extract binary features from the corpus. However, since keywords of e-mails change from time to time, the performance of keyword-based solution is not stable. In this study, we use behaviors of spammers as the features for classifying e-mails. Such behaviors are first described by meta-heuristics and used as features of e-mails for classification. A total of 113 new features are extracted from the given meta-heuristics. Using existing machine learning techniques, the filtering performance is much better than that using keyword-based filtering. In addition, the training time is substantially reduced because of the low dimensional feature space and sparse feature vectors.