The estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, is a central topic in causal inference. This study develops semiparametric efficient estimators for ATE in a setting where only a treatment group and an unlabeled group—consisting of units whose treatment status is unknown—are observed. This scenario constitutes a variant of learning from positive and unlabeled data (PU learning) and can be viewed as a special case of ATE estimation with missing data. For this setting, we derive the semiparametric efficiency bounds, which characterize the lowest achievable asymptotic variance for regular estimators. We then construct semiparametric efficient ATE estimators that attain these bounds. Our results contribute to the literature on causal inference with missing data and weakly supervised learning.

This study designs an adaptive experiment for efficiently estimating *average treatment effects* (ATEs). In each round of our adaptive experiment, an experimenter sequentially samples an experimental unit, assigns a treatment, and observes the corresponding outcome immediately. At the end of the experiment, the experimenter estimates an ATE using the gathered samples. The objective is to estimate the ATE with a smaller asymptotic variance. Existing studies have designed experiments that adaptively optimize the propensity score (treatment-assignment probability). As a generalization of such an approach, we propose optimizing the covariate density as well as the propensity score. First, we derive the efficient covariate density and propensity score that minimize the semiparametric efficiency bound and find that optimizing both covariate density and propensity score minimizes the semiparametric efficiency bound more effectively than optimizing only the propensity score. Next, we design an adaptive experiment using the efficient covariate density and propensity score sequentially estimated during the experiment. Lastly, we propose an ATE estimator whose asymptotic variance aligns with the minimized semiparametric efficiency bound.

Fuzzy multiset is applicable as a model of information retrieval because it has the mathematical structure that expresses the number and the degree of attribution of an element simultaneously. Therefore, fuzzy multisets can be used also as a suitable model for document clustering. This paper aims at developing clustering algorithms based on a fuzzy multiset model for document clustering. The standard proximity measure of the cosine correlation is generalized in the multiset model, and two nonlinear clustering techniques are applied to the existing clustering methods. One introduces a variable for controlling cluster volume sizes; the other one is a kernel trick used in support vector machines. Moreover, clustering by competitive learning is also studied. When the kernel trick has been used the classification configuration of data in a high-dimensional feature space is visualized by self-organizing maps. Two numerical examples, which use an artificial data and real document data, are shown and effects of the proposed methods are discussed. © 2008 Wiley Periodicals, Inc.

Many cluster validity measures have been proposed up to now, and it is realized that no universally best measure exists. In this paper we propose kernelized validity measures where a kernel means the kernel function used in support vector machines. Two measures are considered: one is the sum of the traces of the fuzzy covariances within clusters. Why we consider the trace instead of the determinant is that the calculation of the determinant will be ill-posed when kernelized, while the trace is sound and easily computed. The second is a kernelized Xie-Beni's measure. These two measures are applied to the determination of the number of clusters having nonlinear boundaries generated by kernelized clustering algorithms. Another application of the measures is the evaluation of robustness of different algorithms with respect to variations of initial values and changes of a parameter.

This paper describes a family of methods of fuzzy clustering handling objects with weights. Weighted objects easily appear when an individual is a representative of several data units. Fuzzy c-means and possibilistic clustering algorithms for weighted objects are proposed. Relationships as well as differences between solutions of possibilistic and fuzzy c-means methods are described. It is also shown that the methods for weighted objects and techniques handling cluster volumes are closely related. A feature in the present approach is a systematic development of a family of algorithms for weighted objects.

Recently kernel methods in support vector machines have widely been used in machine learning algorithms to obtain nonlinear models. Clustering is an unsupervised learning method which divides whole data set into subgroups, and popular clustering algorithms such as c-means are employing kernel methods. Other kernel-based clustering algorithms have been inspired from kernel c-means. However, the formulation of kernel c-means has a high computational complexity. This paper gives an alternative formulation of kernel-based clustering algorithms derived from competitive learning clustering. This new formulation obviously uses sequential updating or on-line learning to avoid high computational complexity. We apply kernel methods to related algorithms: learning vector quantization and self-organizing map. We moreover consider kernel methods for sequential c-means and its fuzzy version by the proposed formulation.

A class of hard c-means algorithms is derived from 'defuzzifying' objective functions of the entropy-based fuzzy c-means and the KL-information based fuzzy c-means. Namely an entropy term is deleted from the objective functions while other parameters of cluster sizes and covariances are preserved. As a result, the objective function becomes linear with respect to the membership whereby a hard c-means algorithm is derived. Variations of the basic hard c-means algorithms are moreover proposed and reduction of computation using iterative matrix inversion is considered. Numerical examples are shown to compare results of proposed algorithms. Finally, a cluster validity measure is used whereby stability of clusters by different algorithms is compared.

This paper aims at discussing clustering algorithm based on learning vector quantization (LVQ) using a kernel function in support vector machines. Furthermore, self-organizing map (SOM) using a kernel function is considered. Examples of clustering using different techniques are shown and effects of the kernel function are discussed.