# Shapley Values as a Strategy for Ensemble Weights Estimation
## 4 Results and Analysis
### Results of Homogeneous Model Ensembles  
The homogeneous model ensembles were created using a single type of model. The two model types chosen were decision tree and logistic regression. The results of logis \begin{table}
\begin{tabular}{c c c} \hline
**Model Type** & **Common Parameters** & **Implementation-Specific Parameters** \\ \hline MLR3 LR 1 & \multirow{2}{*}{\begin{tabular}{c} E = e \(\times\) 10\({}^{8}\) \\ iterations = 25 \\ \end{tabular} } & \begin{tabular}{c} singular.ok = True; trace = False \\ \end{tabular} \\ PySpark LR 1 & \multirow{2}{*}{\begin{tabular}{c} regParam = 0.0; aggregationDepth = 2; \\ threshold = 0.5; elasticNetParam = 0.0; fitIntercept = True \\ \end{tabular} } \\ \hline MLR3 DT 2 & \multirow{2}{*}{\begin{tabular}{c} maxDepth = 30 \\ minInfoGain = 0.01 \\ \end{tabular} } & \
\begin{tabular}{c} minSplit = 20; maxcompete = 4; maxsurrogate = 5; \\ surrogatestyle = 0; usesurrogate = 2; xval = 10 \\ \end{tabular} \\ PySpark DT 2 & \multirow{2}{*}{
\begin{tabular}{c} minInstancesPerNode = 20; Standardization = False; \\ minWeightFractionPerNode = 0.0; minInstancesPerNode = 1; \\ maxBins = 32; minInfoGain = 0.0; impurity = ‘gini’ \\ \end{tabular} } \\ \hline \end{tabular}  
* LR—logistic regression. 2 DT—decision tree.  
\end{table} 
Table 3: Hyperparameter values for different model implementations. tic regression ensembles for the BNG-credit_a dataset are presented in Figure 5. These results indicate that the monolith approach had the lowest _BCE_ of 0.326. All ensemble weighting strategies exhibited similar results across all evaluated ensemble lengths. The critical difference diagram in Figure 6 shows an overview of the aggregated rankings for all tested ensemble configurations and notes that the equivalent model rankings were produced by both Python and R implementations. These model ranks specify that there was no statistical significance difference between the other weighting strategies and the Roz strategy indicating that the gains provided by ensembles using logistic regression model type we minimal. A Mono approach had the highest rank with posShap ranked as the second most successful. Homogeneous decision tree ensembles in Figure 7 indicate that all ensemble weighting strategies surpassed the performance of the Mono approach. The difference between weighted ensembles and the Mono approach increased on higher ensemble member counts. This difference increased from 0% to 4.8% for the best-performing posShap weighting strategy. If we compare the performance of only weighting strategies, the results reveal that if the number of models in an ensemble is small (2 or 3), all tested strategies perform similarly, when the number of models is greater than 5 posShap outperforms other approaches with the best result of 0.317 _BCE_. The performance difference between other  
Figure 5: Detection error by _BCE_ for various homogenous ensemble sizes: logistic regression in Python (a) and R (b) using BNG_credit-a dataset. Figure 6: Ranking of homogenous ensembles: logistic regression in Python (a) and R (b) using BNG_credit-a dataset. strategies and posShape also appears in the critical difference diagram in Figure 8 where the posShape strategy obtained the highest rank. The results also reveal that there is no statistical significance between Equal and Roz strategy results, as well as Perf and Roz strategies for the Python implementation. The results for the R language implementation produced the highest performance gains of all experimentally tested solutions. When compared to a single model approach Mono on the largest number of ensemble member (13) setting, ensemble weighting increased the performance with 4.1% and 4.8 % for Perf and posShape weighting strategies, respectively. Logistic regression ensembles for the Bank Marketing dataset (Figure 9) indicate similar results for ensembles with 2, 3 and 5 member counts over all tested strategies. For ensembles sized 8 and 13, there was a major increase in \(BCE\) for the maxShape weighting approach.