We propose a new principal component analysis (PCA) for interval-valued data by using a covariance involving a fuzzy classification structure based on dissimilarity in higher dimensional space in which objects exist. The covariance for interval-valued data is obtained adaptively by evaluating the validity of the fuzzy classification structure based on the selection of an appropriate number of clusters. In order to select an appropriate number of clusters, we propose an alignment criterion to evaluate the obtained classification structure and prove the concentration of the criterion around the expected value with respect to variation of similarity among clusters. The merit of this PCA is to consider not only the projection of objects to a lower dimensional space, but also the dissimilarity of objects in a higher dimensional space by using a weighted covariance matrix. The weight is estimated as the degree of contribution for the fuzzy classification structure based on dissimilarity of objects in the higher dimensional space. A numerical example of interval-valued data consisting of human based subjective decisions shows a better performance when compared with a result of an ordinary PCA.
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