Masaru Shimbo

Multidimensional curve classification using passing—through regions

Abstract A new method is proposed for classifying sets of a variable number of points and curves in a multidimensional space as time series. Almost all classifiers proposed so far assume that there is a constant number of features and they cannot treat a variable number of features. To cope with this difficulty, we examine a fixed number of questions like “how many points are in a certain range of a certain dimension”, and we convert the corresponding answers into a binary vector with a fixed length. These converted binary vectors are used as the basis for our classification. With respect to curve classification, many conventional methods are based on a frequency analysis such as Fourier analysis, a predictive analysis such as auto-regression, or a hidden Markov model. However, their resulting classification rules are difficult to interpret. In addition, they also rely on the global shape of curves and cannot treat cases in which only one part of a curve is important for classification. We propose some methods that are especially effective for such cases and the obtained rule is visualized.

A strong law of large numbers for fuzzy random variables

Abstract A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.

Lower solutions of systems of fuzzy equations

Lower solutions of a system of fuzzy equations are determined by lower solutions of each equation. The result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations.

Construction of class regions by a randomized algorithm: a randomized subclass method

A randomized algorithm is proposed for solving the problem of finding hyper-rectangles, sufficiently approximating the true region in each class. This method yields a suboptimal solution, but is more efficient than previous methods. The performance is analysed based on a criterion of PAC (Probably Approximately Correct) learning. Experimental results show that the proposed method can solve large problems which were not able to be solved previously.

Feature selection based on the structural indices of categories

Abstract A new technique is proposed to select features out of all available ones on the basis of structural indices of categories. In terms of hyper-rectangles including as many training samples of a category as possible, two characteristic indices are calculated which summarize its underlying distribution of samples. The hyper-rectangles and the indices are available in evaluating the degree of importance of features, and are used to increase the discrimination rates of discrimination rules by removing redundant features. The running time of the algorithm is linear order in the number of features. Experiments on artificial and real data attests its effectiveness.

FIELD THEORY AND MODAL LOGIC BY SEMANTIC FIELDS TO MAKE UNCERTAINTY EMERGE FROM INFORMATION

The new notion of semantic fields for modal logic is introduced, by which a degree of significance can be assigned to any possible world. Available information is the main source of the significance that shows the logical structure of the information itself. Thus, a guide is obtained to use information and also to discover and measure the uncertainty located in it. The worlds are useful for separating such information into its important parts. Any sentence is evaluated in worlds that are located at different levels of significance. An accessibility relation exists between the more significant and the less significant world. In this way, possible-worlds models of information are made possible using the semantic field. Uncertainty is given, normally, without any logic but with set theory. Thus, the semantic field is used to connect modal logic with set theory in order to obtain a better description of uncertainty.

MDL-Based Selection of the Number of Components in Mixture Models for Pattern Classification

A new method is proposed for selection of the optimal number of components of a mixture model for pattern classification. We approximate a class-conditional density by a mixture of Gaussian components. We estimate the parameters of the mixture components by the EM (Expectation Maximization) algorithm and select the optimal number of components on the basis of the MDL (Minimum Description Length) principle. We evaluate the goodness of an estimated model in a trade-off between the number of the misclassified training samples and the complexity of the model.

Sets of solution-set-invariant coefficient matrices of simple fuzzy relation equations

Abstract A set of coefficient matrices which are invariant to a solution set of a simple fuzzy relation equation is considered. The equipollency of the cardinal numbers of solution sets of simple fuzzy relation equations for two constant vectors is also shown.

Measure-Based Semantics for Modal Logic

A fuzzy-measure-based approach to semantics for modal logic is presented and its several properties are discussed. Measure-based models for modal logic are defined and the soundness and completeness theorems of several systems of modal logic are proved with respect to classes of finite measure-based models, particularly, formulated by fuzzy, possibility, necessity, probability, and Dirac measures.

Optimal subclasses with dichotomous variables for feature selection and discrimination

The authors present an efficient algorithm for finding optimal subclasses of a class whose members are represented by several dichotomous features with 0 or 1. Each subclass is expressed by a logical formula with common features among its members. It is shown that some typical subclasses, which contain a large number of samples from a class, consist of a few features. Thus one can select these features as a small subset of all features in problems of feature selection. The selection of best subclasses, when subclasses found by the algorithm is a moderate size, is discussed. >