Keiji Takai

Relation between stay-time and purchase probability based on RFID data in a Japanese supermarket

Radio Frequency Identification (RFID) technology uses radio waves to track an object to which a small tag is attached. In a Japanese supermarket, we attach the RFID device to the cart and collect data on purchase behavior. In this article, we clarify the relation between purchase probability and the time customers spend in the store section by analyzing the RFID data with main use of descriptive methods. We clarify the way how the stay-time explains the purchase probability and characteristics of each store section. Based on the result, some implications for business are made as well.

A framework for analysis of the effect of time on shopping behavior

Due to technological developments, data about how many items a customer buys and how long the customer spends in a supermarket are available. A major problem with the data, however, is that there is no framework that considers the heterogeneity hidden in the data. In this article, we propose a framework that considers heterogeneity in the number of items a customer buys. The first step of our framework is based on the Poisson mixture regression model using a stationary time in the department where the items are sold as its independent variable. This model finds latent homogeneous groups of customers and gives the regression models within each group. It simultaneously classifies the customers into the homogeneous groups. In the second step of our framework, a method to investigate whether another factor (variable) influences the classification into homogeneous groups is presented. This proposed framework is applied to real data collected from the customers, and the effectiveness of the framework is shown. The managerial implications are drawn from the result of the analysis.

Asymptotic Inference with Incomplete Data

This article discusses asymptotic theory for the maximum likelihood estimator based on incomplete data. Although much literature has implicitly assumed the basic properties of the estimator, such as consistency and asymptotic normality, it is hard to find their precise and comprehensive proofs. In this article, we first show that under MAR an estimator based on the likelihood function ignoring the missing-data mechanism is strongly consistent. The estimator is then shown to be asymptotically normal. When the data are NMAR and when the data are MAR without parameter distinctness, the consistency and the asymptotic normality are shown. Several examples are provided.

Finite-sample analysis of impacts of unlabeled data and their labeling mechanisms in linear discriminant analysis

ABSTRACT It is widely believed that unlabeled data are promising for improving prediction accuracy in classification problems. Although theoretical studies about when/how unlabeled data are beneficial exist, an actual prediction improvement has not been sufficiently investigated for a finite sample in a systematic manner. We investigate the impact of unlabeled data in linear discriminant analysis and compare the error rates of the classifiers estimated with/without unlabeled data. Our focus is a labeling mechanism that characterizes the probabilistic structure of occurrence of labeled cases. Results imply that an extremely small proportion of unlabeled data has a large effect on the analysis results.

On the use of the selection matrix in the maximum likelihood estimation of normal distribution models with missing data

ABSTRACT In this article, by using the constant and random selection matrices, several properties of the maximum likelihood (ML) estimates and the ML estimator of a normal distribution with missing data are derived. The constant selection matrix allows us to obtain an explicit form of the ML estimates and the exact relationship between the EM algorithm and the score function. The random selection matrix allows us to clarify how the missing-data mechanism works in the proof of the consistency of the ML estimator, to derive the asymptotic properties of the sequence by the EM algorithm, and to derive the information matrix.

Exploration of Dependencies among Sections in a Supermarket Using a Tree-Structured Undirected Graphical Model

In research of purchase behavior in a supermarket, it is important to understand dependencies among sections of the supermarket, with each section corresponding to a category of items. An undirected graphical model is a powerful tool for this purpose. A problem with the application of an undirected graphical model is that there are many variables and, thus, a lot of computation is needed. In this article, we first apply a tree-structured undirected graphical model to reduce the computational amount, and second, propose a method to impose a restriction, based on our needs, on the tree structured undirected graphical model. The variables we use are the length of time spent in each section and the number of items bought from each section. We found that some of the sections have influence on the adjacent sections and that some of the other sections have no influence on the adjacent sections, but do have influence on the nonadjacent sections. We also found that the number of items and the length of stationary time in the sections that influence a large number of sections are negatively related to those same variables in the other sections. Based on this result, managerial implications are described. Finally, we summarize this article and discuss some problems in the application of graphical models.