# Predicting Online Item-Choice Behavior: A Shape-Restricted Regression Approach
## 1 Introduction  
A growing number of companies are now operating e-commerce websites that allow users to browse and purchase a variety of items [1]. Within this context, there is great potential value in analyzing users' item-choice behavior from clickstream data, which is a record of user pageview (PV) histories on an e-commerce website. By grasping users' purchase intention as revealed by PV histories, we can lead users to target pages or design special sales promotions, providing companies with opportunities to build profitable relationships with users [2, 3]. Companies can also use clickstream data to improve the quality of operational forecasting and inventory management [4]. Meanwhile, users often find it difficult to select an appropriate item from among the plethora of choices presented by e-commerce websites [5]. Analyzing item-choice behavior can improve the performance of recommendation systems that help users discover items of interest [6]. For these reasons, a number of prior studies have investigated clickstream data from various perspectives [7]. In this study, we focused on closely examining the relationship between PV histories and item-choice behavior on an e-commerce website. It has been demonstrated that the recency and frequency of a user's past purchases are critical indicators for purchase prediction [8, 9] and sequential pattern mining [10]. Accordingly, [PERSON] et al. [11] developed a shape-restricted optimization model for estimating item-choice probabilities from the recency and frequency of a user's previous PVs. Their method creates a two-dimensional probability table consisting of item-choice probabilities for all recency-frequency combinations in a user's previous PVs. [PERSON] et al. [12] employed latent-class modeling to integrate item heterogeneity into a two-dimensional probability table. These prior studies demonstrated experimentally that higher prediction performance was achieved with the two-dimensional probability table than with commonmachine learning methods, namely, logistic regression, kernel-based support vector machines, artificial neural networks, and random forests. Notably, however, reducing PV histories to two dimensions (recency and frequency) can markedly decrease the amount of information contained in PV histories reflecting item-choice behavior. This study focused on PV sequences, which represent time series of the number of PVs for each user-item pair. In contrast to the two-dimensional probability table, PV sequences allow us to retain detailed information contained in the PV history. However, the huge number of possible PV sequences makes it extremely difficult to accurately estimate item-choice probabilities for all of them. To overcome this difficulty, we propose a shape-restricted optimization model that imposes monotonicity constraints on item-choice probabilities based on a partially ordered set (poset) for PV sequences. While this optimization model contains a huge number of constraints, redundant constraints can be eliminated according to the transitivity of the partial order. To accomplish this, we compute a transitive reduction [13] of a directed graph representing the poset. We demonstrate the effectiveness of our method through experiments using real-world clickstream data. The main contributions of this paper are as follows:  
* We propose a shape-restricted optimization model for estimating item-choice probabilities from a user's previous PV sequence. This PV sequence model exploits the monotonicity constraints to precisely estimate item-choice probabilities. * We derive two types of PV sequence posets according to the recency and frequency of a user's previous PVs. Experimental results show that the monotonicity constraints based on these posets greatly enhances the prediction performance of our PV sequence model. * We devise constructive algorithms for transitive reduction specific to these posets. The time complexity of our algorithms is much smaller than that of general-purpose algorithms.