Purpose The purpose of this paper is to introduce some basic knowledge of statistical learning theory (SLT) based on random set samples in set‐valued probability space for the first time and generalize the key theorem and bounds on the rate of uniform convergence of learning theory in Vapnik, to the key theorem and bounds on the rate of uniform convergence for random sets in set‐valued probability space. SLT based on random samples formed in probability space is considered, at present, as one of the fundamental theories about small samples statistical learning. It has become a novel and important field of machine learning, along with other concepts and architectures such as neural networks. However, the theory hardly handles statistical learning problems for samples that involve random set samples. Design/methodology/approach Being motivated by some applications, in this paper a SLT is developed based on random set samples. First, a certain law of large numbers for random sets is proved. Second, the definitions of the distribution function and the expectation of random sets are introduced, and the concepts of the expected risk functional and the empirical risk functional are discussed. A notion of the strict consistency of the principle of empirical risk minimization is presented. Findings The paper formulates and proves the key theorem and presents the bounds on the rate of uniform convergence of learning theory based on random sets in set‐valued probability space, which become cornerstones of the theoretical fundamentals of the SLT for random set samples. Originality/value The paper provides a studied analysis of some theoretical results of learning theory.
In this paper, we have proposed a multilevel soft thresholding technique for noise removal in Daubechies complex wavelet transform domain. Two useful properties of Daubechies complex wavelet transform, approximate shift invariance and strong edge representation, have been explored. Most of the uncorrelated noise gets removed by shrinking complex wavelet coefficients at the lowest level, while correlated noise gets removed by only a fraction at lower levels, so we used multilevel thresholding and shrinkage on complex wavelet coefficients. The proposed method firstly detects strong edges using imaginary components of complex coefficients and then applies multilevel thresholding and shrinkage on complex wavelet coefficients in the wavelet domain at non-edge points. The proposed threshold depends on the variance of wavelet coefficients, the mean and the median of absolute wavelet coefficients at a particular level. Dependence of these parameters makes this method adaptive in nature. Results obtained for one-dimensional signals and two-dimensional images demonstrate an improved denoising performance over other related methods available in the literature.
The paper deals with a model of decision‐making given in a form of a fuzzy relational equation with relationships between imprecisely defined goals and constraints modelled by means of a fuzzy relation. The usefulness of the model and its novelty is clearly pointed out. Some basic problems concerned, e.g. with a construction of the relation of the model and ordering of goals and constraints with respect to their importance (viz. their influence on a fuzzy decision), are also discussed in a detailed manner.
A new time series prediction architecture is introduced using a fuzzy inference system (FIS) and a new framework for fuzzy relational clustering of time series. The FIS is used to predict future samples in a time series where recurrent neural networks comprise the consequents of the rules. The antecedents come in the form of fuzzy relations; however, previous approaches such as FCM build these antecedents in a Euclidean feature space which is very limiting and not well suited to the problem of clustering time series. Our approach to learning the antecedents of the rules involves clustering time series using proximity values, indicative of closeness. A variant of the classical correlation is used to measure proximity. Our objective is to investigate and evaluate the application of proximity fuzzy clustering in the domain of time series prediction by comparing its performance against several commonly used time series prediction models.
