# Predicting Online Item-Choice Behavior: A Shape-Restricted Regression Approach
## 7 Conclusions  
We presented a shape-restricted optimization model for estimating item-choice probabilities on an e-commerce website. Our monotonicity constraints based on tailored order relations could better estimate item-choice probabilities for all possible PV sequences. To improve the computational efficiency of our optimization model, we devised constructive algorithms for transitive reduction that remove all redundant constraints from the optimization model.  
We assessed the effectiveness of our method through experiments using real-world clickstream data. Experimental results demonstrated that transitive reduction enhanced the efficiency of our optimization model in terms of both computation time and memory usage. In addition, our method delivered a better prediction performance than did the two-dimensional monotonicity model [11] and common machine learning methods. Our method was also helpful in correcting prediction values computed by other machine learning methods.  
This study made three main contributions. First, we derived two types of posets by exploiting the properties of recency and frequency of a user's previous PVs. These posets allow us to place appropriate monotonicity constraints on item-choice probabilities. Next, we developed algorithms for the transitive reduction of our posets. These algorithms are more efficient than general-purpose algorithms in terms of time complexity for transitive reduction. Finally, our method further expanded the potential of shape-restricted regression for predicting user behavior on e-commerce websites.  
Once the optimization model for estimating item-choice probabilities has been solved, the obtained results can easily be put into practical use on e-commerce websites. Accurate estimates of item-choice probabilities will be useful for customizing sales promotions according to the needs of a particular user. In addition, our method can estimate user preferences from clickstream data, therefore aiding in the creation of high-quality user-item rating matrices for recommendation algorithms [6].  
Figure 6: Item-choice probabilities estimated from the 10%-sampled training set with \((n,m)=(5,6)\).  
In future studies, we will develop new posets that further improve the prediction performance of our PV sequence model. Another direction of future research will be to incorporate user-item heterogeneity into our optimization model, as in the case of latent-class modeling with a two-dimensional probability table [12]. It is also important to compare the prediction performance of our method with that of topline graph neural networks [37; 38].  
Conceptualization, Y.T. and J.I.; methodology, N.N. and N.S.; software, N.N. and J.I.; validation, N.N.; formal analysis, N.S.; investigation, Y.T.; resources, N.N.; data curation, N.N.; writing--original draft preparation, Y.T.; writing--review and editing, N.S. and N.N.; visualization, N.N.; supervision, J.I. and Y.T.; project administration, N.N.; funding acquisition, Y.T. All authors have read and agreed to the published version of the manuscript.  
This work was partially supported by a joint research of the University of Tsukuba and Toyota Motor Corporation.  
The data set we used is available online at [[https://www.dropbox.com/sh/dbzmqt2](https://www.dropbox.com/sh/dbzmqt2) ahzbj509/AACIdzQWbw-igKjcPTBI6 ZPAa?dl=0]([https://www.dropbox.com/sh/dbzmqt2](https://www.dropbox.com/sh/dbzmqt2) ahzbj509/AACIdzQWbw-igKjcPTBI6 ZPAa?dl=0) (accessed on 21 August 2023). This data set was created by [PERSON] and [PERSON] as a preprocessed version of the clickstream data collected from a Chinese e-commerce website, Tmall, which is also available online at [[https://tianchi.aliyun.com/dataset/](https://tianchi.aliyun.com/dataset/)]([https://tianchi.aliyun.com/dataset/](https://tianchi.aliyun.com/dataset/)) (accessed on 21 August 2023).  
The authors declare no conflict of interest.