# Shapley Values as a Strategy for Ensemble Weights Estimation
## 4 Results and Analysis
### Results of Homogeneous Model Ensembles
When the ensemble member count reached 8 members, the distinction between Mono and other strategies became apparent, but the difference remained minimal at only less than 0.1%. Of all tested cases, the Perf weighting approach showed the best performance of 0.236 \(BCE\). The weighting strategy comparison via the critical difference diagram in Figure 10 reveals that the posShape strategy had the best performance, while the  
Figure 8: Ranking of homogenous ensembles: decision tree in Python (**a**) and R (**b**) using BNG_credit-a dataset. Figure 7: Detection error by \(BCE\) for various homogenous ensemble sizes: decision tree in Python (**a**) and R (**b**) using BNG_credit-a dataset. Perf strategy gave the second-best results. Both posShape and Perf were ranked better than the Mono approach indicating a performance boost of weighting. Although the results indicate a clear performance boost of weighting strategies, the 0.4% gain compared to the monolith approach could be considered trivial this also explains no statistical significance between the most tested weighting strategies. Decision tree model ensembles for the Bank Marketing dataset (Figure 11) demonstrate that all weighting strategies surpassed the monolith approach, even in the case of very small ensembles. When the ensemble contained 3 or more members, the \(BCE\) of maxShap classifiers performance decreased and kept on diminishing with higher member counts. The results of Python implementation revealed that ensembles with member counts of 2 to 8 outperformed the monolith approach, and for larger ensembles of 13 members only, the posShape strategy exhibited better results of 0.264 \(BCE\) for Python and 0.274 \(BCE\) for R implementations. As for experiment implementation using R language Perf, Equal and posShape weighting strategies presented a lower \(BCE\) than the monolith approach for all  
Figure 10: Ranking of homogenous ensembles: logistic regression in Python (**a**) and R (**b**) using Bank Marketing dataset. Figure 9: Detection error by \(BCE\) for various homogenous ensemble sizes: logistic regression in Python (**a**) and R (**b**) using Bank Marketing dataset. evaluated model counts. The results also indicate that the Perf ensemble weighting strategy with member count 8 produced the best result of 0.251 _BCE_. The posShap approach was ranked first for the R implementation and second for the Python implementation, as can be seen in the critical difference diagram (Figure 12). In both implementations, the results were not statistically significant between Random and Roz strategies as well as the PosShap and Equal approaches in Python implementation. The results of the homogeneous ensembles experiment results presented in Table 4 indicate that ensembles composed of the decision tree classifier benefited more from weighting than the logistic regression classifier ensembles. Decision tree-based ensembles exhibited a performance increase when compared to the monolith approach by 1.9 % for Bank Marketing and 4.8% for BNG_credit-a datasets, whereas the performance of logistic regression ensembles resulted in only a trivial gain of 0.2 % and 0.002%. Although the benefit of weighting was not as effective in logistic regression ensembles, posShap weighting still performed very similarly to the Perf strategy, which outperformed other weighting strategies. The lowest _BCE_ of 0.236 for the Bank Marketing dataset was exhibited by the Perf weighting strategy with the largest ensemble size of 13, where the posShap  
Figure 11: Detection error by _BCE_ for various homogenous ensemble sizes: decision tree in Python (**a**) and R (**b**) using Bank Marketing dataset. Figure 12: Ranking of homogenous ensembles: decision tree in Python (**a**) and R (**b**) using Bank Marketing dataset. strategy presented the second-best result of 0.238 _BCE_. Similar experiment results were exhibited for the BNG_credit-a dataset where posShap and Perf performance was 0.317 and 0.318 _BCE_.