The random utility model is one of the most fundamental models in discrete choice analysis in economics. Although Falmagne (1978) obtained an axiomatization of the random utility model, his characterization requires strong observability of choices, i.e., the frequencies of choices must be observed from all subsets of the set of alternatives. Little is known, however, about the axiomatization when the frequencies on some choice sets are not observable. In fact, the problem of obtaining a tight characterization appears to be out of reach in most cases in view of a related NP-hard problem. We consider the following incomplete dataset. Let X be a finite set of alternatives. Let X* ⊆ X bea set of unobservable alternatives. Let D ⊆ 2X be the set of choice sets. We assume that the choice frequency ρ(D, x) is unobservable (i.e., not defined) if and only if x ∈ X* or D ∉ D. Let M* ≡ {(D,x)|x ∈ D ∈ 2X and [x ∈ X* or D ∉ D]} be the set of all pairs (D,x) such that ρ(D, x) is not observable. To state our theorem, for any ρ and (D, x) ∈ M ≡ {(D, x) ∈ D × X | x ∈ D}, define a Block-Marschak polynomial by K(ρ, D, x) = ΣE:E⊇D(−1)|E\D|ρ(E,x).

The data visualization, which reduces data dimension to make us easy to see data directly, is important in data mining. It is important that one category becomes one cluster (we define this as data cohesion) and the clusters are standoff to each other (we define this as cluster separation) in data visualization. In this paper, we propose a self-organizing method using data movement on a spherical surface for data visualization. The proposed method puts all data points on the spherical surface and each data point moves on the spherical surface under the force from all the other data points. The key features of the proposed method are using a spherical surface as output space and employing weighted inter-point distance which emphasizes similarity between these data points. The experimental results show that the proposed method visualizes data with high data cohesion and high cluster separation by dint of above features.

Analyzing users' Web log data and extracting their interests of Web-watching behaviors are important and challenging research topics of Web usage mining. Users visit their favorite sites and sometimes search new sites by performing keyword search on search engines. Users' Web-watching behaviors can be regarded as a graph since visited Web sites and entered search keywords are connected with each other in a time sequence. We call this graph a site-keyword graph. This paper describes a method for clarifying users' interests based on an analysis of the site-keyword graph. The method is for extracting subgraphs representing users' main interests from a site-keyword graph which is generated from augmented Web audience measurement data (Web log data). Experimental results show that our new method succeeds in finding subgraphs which contain most of the sites that users are interested in