# Addressing the Cold-Start Problem in Recommender Systems Based on Frequent Patterns
## 3 Proposed Methodology
Thus, the following equations represent the adopted measures:

\[Sensitivity\ (Recall)=\frac{TP}{P} \tag{5}\]

\[Positive\ Predictive\ Value-PPV\ (Precision)=\frac{TP}{TP+FP} \tag{6}\]

\[F_{1}\ measure=2*\frac{Precision*Recall}{Precision+Recall} \tag{7}\]where \(TP\), \(FP\) is the true and false positives, respectively (predicted cell value = 1), and \(P\) is the number of all positives in the initial user-product matrix (existed cell value = 1). Obviously, the better achieved results are indicated by a value approaching 1 for both measures. Note that there is a difference in calculating the measure of sensitivity within this paper compared to other similar problems (e.g., in disease diagnosis), because of the existence of empty cells (null cell value). Since the proposed methodology, apart from adding positives (predicted cell value = 1), is allowed to leave empty cells within the test subset, the sensitivity measure could be low. This is because some positives in the initial input matrix (existing cell value = 1) can be left empty, which, although they are not false positives, cannot be considered true positives, as well. To evaluate the proposed methodology for the sparsity problem, we introduce three new measures. The plentifulness measure indicates how many initially empty cells were filled with respect to existing positives and it is defined as the proportion of added positives (predicted cell value is 1) into initially empty cells to the number of initial positives. The plentifulness rate measure indicates how many initially empty cells were filled with respect to all existing empty ones and it is defined as the proportion of added positives into initially empty cells to the number of initially empty cells. The plentifulness predictive value measure indicates how many initially empty cells were filled with respect to both the latter cells and the initially positives that were left empty and it is defined as the proportion of added positives into initially empty cells to the sum of the latter and the cells left empty while they were initially positive. Thus:

\[Plentifulness=\frac{NP}{P} \tag{8}\]

\[Plentifulness\ rate=\frac{NP}{E} \tag{9}\]

\[Plentifulness\ Predictive\ Value=PLPV=\frac{NP}{NP+EP} \tag{10}\]

where \(E\) is the number of empty cells (null cell value) in the initial user-product matrix; \(NP\) is the number of added positives (predicted cell value is 1) into previously empty cells and \(EP\) is the number of cells left empty while they were positives in the initial user-product matrix (existed cell value = 1). Obviously, the better achieved results are indicated by a value approaching 1 for all measures. We implemented the proposed methodology using Python 3.11 and all the experiments were performed on a common Personal Computer Intel Core TM i7-10750 CPU (2.59 GHz) with 16 GB RAM. The time complexity of the proposed methodology is dominated by the runtime of the Apriori algorithm (e.g., 37.65 s for minimum support set to 15% and 1.97 for minimum support set to 20%). Figures 2-6 present the experimental results obtained by the 10-fold cross-validation procedure. More specifically, Figures 2 and 3 show the _out-of-matrix prediction_ results obtained by the proposed methodology, since all the rows (users) included in the test set were cleared and then they were updated by the proposed methodology. In Figure 2, the cross-validation results (average over the \(k=10\) runs) for the evaluation measures are shown with respect to different values of \(\theta\) parameter while the minimum support is set to 15% and \(\varphi\) parameter to 0.001 (see Equations (2) and (3)). In Figure 3, the cross-validation results (average over the \(k=10\) runs) for the evaluation measures are shown with respect to different values of minimum support while the \(\theta\) parameter is set to 1 and \(\varphi\) parameter to 0.001. Note that a higher value of minimum support reduces the number of extracted frequent itemsets and hence that of discriminant itemsets.