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In comparing clustering partitions, the Rand index (RI) and the adjusted Rand index (ARI) are commonly used for measuring the agreement between partitions. Such external validation indexes can be used to quantify how close the clusters are to a reference partition (or to prior knowledge about the data) by counting classified pairs of elements. To evaluate the solution of a fuzzy clustering algorithm, several extensions of the Rand index and other similarity measures to fuzzy partitions have been proposed. An extension of the ARI for fuzzy partitions based on the normalized degree of concordance is proposed. The performance of the proposed index is evaluated through Monte Carlo simulation studies.

This paper constitutes Part I of the contribution to the analysis of web visit histories through a new methodological framework. Firstly, web usage and web structure mining are considered as an unique mining process to detect the latent structure of the web navigation across the web sections of a single portal. We extend association rules theory to web data defining new concepts of web (patterns) association and preference matrices, as well as of (indirect and direct) sequence rules. We identify the most significant rules, according to a multiple testing procedure. In the literature, web usage patterns can be visualized in no-distance-based graphs describing the navigation behavior across web pages with sequential arrows. In the following, we introduce a geometrical visualization of sequence rules at any click of the web navigation. In particular, we provide two distance-based visualization methods for the static analysis of all data tout court and the dynamic analysis to discover the most significant web paths click by click. A real world case study is considered throughout the methodological description.