Recently, a number of methods, in which sparse loadings are estimated, have been proposed in the factor analysis literature. Since most of them are based on a penalized estimation, such as LASSO (Tibshirani, 1996), their regularization parameters need endless tunings. To overcome that inconvenience, Adachi & Trendafilov (2015) proposed a cardinality-constrained procedure, referred to as CC-MDFA, which drastically reduces the number of candidates for the tuning parameters. Although this procedure is useful for exploring sparse loadings, it is used only for orthogonal factor models and is not able to estimate correlations between common factors. In this paper, we propose a new formulation of CC-MDFA which can estimate not only orthogonal models but also oblique models, and we show our formulation includes CC-MDFA as a special case.

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　In the statistical analysis of spatial data, spatial events such as the “mortality rate of the disease observed in each municipality” and “measured value of harmful substances at each measurement point” may be concentrated in a specific area. This point, indicates that “a cluster exists.” Herein, we propose a new method for detecting a spatial cluster. To date, many methods have been proposed; however, problems exist wherein only the clusters of limited shape or clusters having low-risk regions can be detected. Thus, we focus on the hierarchical spatial data structure and try to solve the problems of existing methods by extracting the upper hierarchy. Moreover, the proposed method can be applied to large-scale spatial data because it significantly reduces calculation costs. To verify the cluster-detection accuracy and effectiveness of the proposed method, we compare it with existing methods using numerical experiments.

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