## 5 Results  
We commence with the outcomes that derived from the mobile application data, as summarized in Table 3. In each model, we made predictions for the weekly login counts of the users. For Pareto/NBD (HB) and Pareto/GGG models, some login predictions exceeded 7. To align with the practical constraint, that is, 7 represents the upper limit for weekly logins per user, these predictions were adjusted to this maximum value. Additionally, Pareto/NBD (HB) and Pareto/GGG models yielded negative results, which we converted to 0. In Table 3, we did not include BG/CNBD-k and MBG/CNBD-k models because the optimum \(k\) paremeters for the BG/CNBD-k and MBG/CNBD-k models were 1, which means that they do not differ from BG/CNBD and MBG/CNBD. For GRU and LSTM models, we have three hyperparameters which are memory units, dense units and simulation numbers. The GRU model was configured with 256 memory units and 32 dense units, while the LSTM model employed 128 memory units and 128 dense units during training. Additionally, simulation numbers were taken as 23 for both models. The impact of these hyperparameters presented in Figures 3-5. It is apparent that the bias is influenced by the choice of the number of memory and dense units. When simulation numbers increase, SMAPE values increase. This pattern seems plausible as more simulations lead to slightly elevated predictions for zero values, contributing to a greater deviation in SMAPE calculations. Conversely, the stability of MAE and RMSE values remains consistent. The results emphasize the impact of simulation numbers and underscore the persistent advantages of GRU and LSTM, despite variations in simulation numbers compared to BTYD models. Upon reviewing Figure 6 and Table 3, GRU and the LSTM have the least bias and the lowest SMAPE values; and they capture the trends better than the BTYD models. GRU has the lowest difference between the actuals and estimations, lowest bias and second lowest SMAPE values. However, it is important to note that MAE and RMSE values are relatively higher than most BTYD models. Further, Table 4 presents the evaluation metrics for each model applied on Online Retail II dataset. It is noteworthy that the Pareto/NBD (HB) and Pareto/GGG models produced negative results, which we adjusted to zero. Figure 4: Evaluation metrics vary with the dense units for mobile application data. Figure 5: Evaluation metrics vary with the number of simulations for mobile application data. Figure 6: Aggregate weekly actual logins and predictions for mobile application data. The dotted vertical blue line in the upper line chart separates the calibration and holdout periods. In both the GRU and LSTM models, we employed 256 memory units and 32 dense units, conducting 23 training simulations. Upon examining Figure 7, and Table 4, it becomes apparent that both the GRU and LSTM models perform better than the BTYD models in capturing the underlying trend. GRU exhibits the smallest disparity between actual and estimated values, the least bias and the lowest SMAPE values although the MAE and RMSE values are higher compared to all the BTYD models. Comparisons between the weekly aggregated actual data and the corresponding predictions for the LSTM and GRU models are illustrated in Figure 7. This visualization clearly demonstrates that LSTM and GRU effectively capture the overall trend in the data. The MAE and RMSE metrics for the BTYD models were observed to be lower than those of the GRU and LSTM models. As in the user login dataset, users exhibit infrequent