The paper formulates new types of fuzzification in principal component analysis, which deal with subjective data obtained by evaluating objects intuitively. The first technique identifies fuzzy sets in the data space and the second one identifies fuzzy sets in the model parameter space. Both focus on preservation of differences between the feeling of evaluators, and give principal components with fuzzy numbers that reflect vagueness in evaluation. The first technique is mainly used for the analysis of objects by taking into account vagueness in evaluation, while the second one is mainly used for estimating fuzzy principal components, or comprehensive evaluation, for a new crisp data. The paper briefly shows a numerical example using the data obtained by evaluating local environment with linguistic expressions.

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In this paper, we propose a fuzzy ensemble learning method for pattern classification problems. In our fuzzy ensemble method, two different types of fuzzy rule-based systems are used. One is a fuzzy rule-based classification system that suggests a class for input patterns. Multiple fuzzy rule-based classification systems are included in the proposed ensemble learning method. The other type of fuzzy rule-based systems assigns a weight value to each of the suggested classes from the fuzzy rule-based classification systems. This type of fuzzy rule-based systems is referred to as a fuzzy rule-based ensemble system. Our fuzzy ensemble method consists of multiple fuzzy rule-based classification systems, a fuzzy rule-based ensemble system, and a gating node. A gating node is used to determine the final classification of an input pattern. By using the proposed method, the performance of fuzzy ensemble system improves comparing to any single fuzzy rule-based classification system. We show in our computer simulations that the proposed ensemble method works well for real-world pattern classification problems. We also discuss the disadvantage of the proposed ensemble method. That is, the number of fuzzy if-then rules becomes large since multiple fuzzy rule-based systems are used. In order to tackle with this issue, we use a genetic-algorithm-based rule selection method to design compact fuzzy rule-based classification systems.

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FCM-type fuzzy clustering approaches are closely related to Gaussian Mixture Models (GMMs) and the objective function of Fuzzy c-Means with regularization by K-L information (KFCM) is optimized by an EM-like algorithm. In this paper, we propose to apply probabilistic PCA mixture models to linear clustering following the discussion on the relationship between Local PCA and linear fuzzy clustering. Although the proposed method is a kind of the constrained model of KFCM, the algorithm includes a similar formulation with the Fuzzy c-Varieties (FCV) algorithm as a special case. Then the algorithm can be regarded as a modified FCV algorithm with regularization by K-L information, which makes it possible to tune the cluster shapes adaptively.

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Generalized Principal Component Analysis (Generalized PCA) is a useful extension of the PCA algorithm for estimating a suitable non-linear coordinate system when sample data points have non-linear distribution. The non -linear models derived by Generalized PCA is closely related to shell clustering that partitions data sets into several shell-shape fuzzy clusters by extracting local circles or ellipses as the prototypes of clusters. This paper proposes a robust shell clustering technique by generalizing a linear fuzzy clustering algorithm based on least absolute deviations. The proposed method is a hybrid technique of local minor component analysis and FCM-type fuzzy clustering in the enlarged data space and can be regarded as an application of Fuzzy c-Varieties (FCV) algorithm for capturing local non-linear singularities. The tuning of the trade-off parameter makes it possible to derive stable clustering results that are robust to the initial partitioning. Numerical example composed of a comparison with the possibilistic shell clustering method shows the characteristic properties of our method.

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In this paper, we propose a reinforcement learning method called a fuzzy Q-learning where an agent determines its action based on inference results by a fuzzy rule-based system. We apply the proposed method to a soccer agent that learns to intercept a passed ball, i.e., it tries to catch up with a passed ball by another agent. In the proposed method, a state space is represented by internal information that the learning agent maintains such as a relative velocity and a relative position of the ball to the learning agent. We divide the state space into a number of fuzzy subspaces. We define each fuzzy subspace by specifying the fuzzy partition of each axis of the state space. A reward is given to the learning agent if the distance between the ball and the agent becomes smaller or if the agent catches up with the ball. It is expected that the learning agent finally obtains the ball intercept skill after a process of trial and error.

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