We propose a method for clustering functional data obtained at several spatial locations, including unobserved ones, using multivariate longitudinal data. The observed data are transformed into functional data by .tting smooth functions and then the functions at un-observed locations are predicted by the ordinary kriging for functional data. This enables us to predict the data taking the spatial correlation into consideration. Furthermore, we apply the clustering based on the x-means method to the functional data at observed locations and unobserved locations which are predicted via the kriging. The proposed method enables clustering at arbitrary locations regardless of whether the data are observed or unobserved. Numerical experiments and real data analysis show the e.ectiveness of our method.

Although customer purchasing behavior is of great interest to both practitioners and researchers, the two primary components, purchase interval and purchase amount, have typically been handled separately. However, considering the tendency of actual purchase behavior, it is desirable to model them simultaneously. Thus, this paper proposes the method for stochastically estimating the joint mechanism of purchase time and amount while considering all the past purchasing behavior by applying the marked Hawkes process, a type of stochastic process, to the context of customer purchasing behavior in a super-market. After estimating the parameters of our model incorporating customers’ random e.ects and covariates such as gender and age group through the EM algorithm, we obtained several managerially important implications. In addition, our simulation revealed that the marked Hawkes process proposed in this paper had an extremely higher predictive power than traditional models.