# Predicting Online Item-Choice Behavior: A Shape-Restricted Regression Approach
## 6 Experiments
### Effects of the Transitive Reduction  
We generated constraints in Equation (8) based on the following three directed graphs:  
**Case 1 (Enumeration):**: All edges \((\mathbf{u},\mathbf{v})\) satisfying \(\mathbf{u}\prec\mathbf{v}\) were enumerated. **Case 2 (Operation):**: Edges corresponding to operations Up, Move, and Swap were generated as in Figures 1a and 2a. **Case 3 (Reduction):**: Transitive reduction was computed using our algorithms as in Figures 1b and 2b. Table 6 shows the problem size of our PV sequence model (7)-(9) for some \((n,m)\) settings of the PV sequence. Here, the \"#Vars\" column shows the number of decision variables (i.e., \((m+1)^{n}\)), and the subsequent columns show the number of constraints in Equation (8) for the three cases mentioned above. The number of constraints grew rapidly as \(n\) and \(m\) increased in the enumeration case. In contrast, the number of constraints was always kept smallest by the transitive reduction among the three cases. When \((n,m)=(5,6)\), for instance, transitive reductions reduced the number of constraints in the operation case to 63,798/195,510 \(\approx\) 32.6% for SeqUM and 85,272/144,060 \(\approx\) 59.2% for SeqUS. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Training**} \\ \cline{2-4}
**Pair ID** & **Start** & **End** & **Validation** \\ \hline
1 & 21 May 2015 & 18 August 2015 & 19 August 2015 \\
2 & 31 May 2015 & 28 August 2015 & 29 August 2015 \\
3 & 10 June 2015 & 7 September 2015 & 8 September 2015 \\
4 & 20 June 2015 & 17 September 2015 & 18 September 2015 \\
5 & 30 June 2015 & 27 September 2015 & 28 September 2015 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Training and validation periods. The number of constraints was larger for SeqUM than for SeqUS in the enumeration and operation cases. In contrast, the number of constraints was often smaller for SeqUM than for SeqUS in the reduction case. Thus, the transitive reduction had a greater impact on SeqUM than on SeqUS in terms of the number of constraints. Table 7 lists the computation times required for solving the optimization problem (7)-(9) for some \((n,m)\) settings of the PV sequence. Here, \"OM\" indicates that computation was aborted due to a lack of memory. The enumeration case often caused out-of-memory errors because of the huge number of constraints (see Table 6), but the operation and reduction cases completed the computations for all \((n,m)\) settings for the PV sequence. Moreover, the transitive reduction made computations faster. A notable example is SeqUM with \((n,m)=(5,6)\), for which the computation time in the reduction case (86.02 s) was only one-tenth of that in the operation case (906.76 s). These results demonstrate that transitive reduction improves efficiency in terms of both computation time and memory usage. Table 8 shows the computational performance of our optimization model (7)-(9) for some \((n,m)\) settings of PV sequences. Here, for each \(n\in\{3,4\ldots,9\}\), the largest \(m\) was chosen such that the computation finished within 30 min. Both SeqUM and SeqUS always delivered higher F1 scores than SeqEmp did. This indicates that our monotonicity constraint (8) works well for improving prediction performance. The F1 scores provided by