# Ensemble Deep Learning Models for Forecasting Cryptocurrency Time-Series
## 5 Numerical Experiments
Tables 10-12 present the confusion matrices of averaging\({}_{11}\), Bagging\({}_{11}\), Stacking\({}_{7}^{(\text{LR})}\) and Stacking\({}_{11}^{(\text{RNN})}\) using CNN-BiLSTM as base learner, regarding BTC, ETH and XRP datasets, respectively. The confusion matrices for BTC dataset demonstrated that both average\({}_{11}\) and bagging\({}_{11}\) presented the best performance while stacking\({}_{7}^{(\text{LR})}\) was biased, since most of the instances were misclassified as \"Down\". Regarding ETH dataset, both average\({}_{11}\) and bagging\({}_{11}\) were considered biased since most \"Up\" instances were misclassified as \"Down\". In contrast, both stacking ensembles presented the best performance, with stacking\({}_{11}^{(\text{RNN})}\) reporting slightly considerably better trade-off between sensitivity and specificity. Regarding XRP dataset, bagging\({}_{11}\) presented the highest prediction accuracy and the best trade-off between true  
In the rest of this section, we evaluate the reliability of the best reported ensemble models by examining if they have properly fitted the time series. In other words, we examine if the models' residuals defined by  
\[\hat{e}_{t}=y_{t}-\hat{y}_{t}\]  
are identically distributed and asymptotically independent. It is worth noticing the residuals are dedicated to evaluate whether the model has properly fitted the time series. For this purpose, we utilize the AutoCorrelation Function (ACF) plot [47] which is obtained from the linear correlation of each residual \(\hat{e}_{t}\) to the others in different lags, \(\hat{e}_{t-1},\hat{e}_{t-2},\dots\) and illustrates the intensity of the temporal autocorrelation. Notice that in case the forecasting model violates the assumption of no autocorrelation in the errors implies that its predictions may be inefficient since there is some additional information left over which should be accounted by the model. Figures 4-6 present the ACF plots for BTC, ETH and XRP datasets, respectively. Notice that the confident limits (blue dashed line) are constructed assuming that the residuals follow a Gaussian probability distribution. It is worth noticing that averaging\({}_{11}\) and bagging\({}_{11}\) ensemble models violate the assumption of no autocorrelation in the errors which suggests that their forecasts may be inefficient, regarding BTC and ETH datasets. More specifically, the significant spikes at lags 1 and 2 imply that there exists some additional information left over which should be accounted by the models. Regarding XRP dataset, the ACF plot of average\({}_{11}\) presents that the residuals have no autocorrelation; while the ACF plot of bagging\({}_{11}\) presents that there is a spike at lag 1, which violates the assumption of no autocorrelation in the residuals. Both ACF plots of stacking ensemble are within 95% percent confidence interval for all lags, regarding BTC and XRP datasets, which verifies that the residuals have no autocorrelation. Regarding the ETH dataset, the ACF plot of stacking\({}_{77}^{(\text{LR})}\) reported a small spike at lag 1, which reveals that there is some autocorrelation of the residuals but not particularly large; while the ACF plot of stacking\({}_{11}^{(\text{kNN})}\) reveals that there exist small spikes at lags 1 and 2, implying that there is some  
Figure 7-9 present the ACF plots of averaging\({}_{11}\), bagging\({}_{11}\), stacking\({}_{97}^{(\text{LR})}\) and stacking\({}_{111}^{(\text{RNN})}\) ensembles utilizing CNN-BiLSTM as base learner for BTC, ETH and XRP datasets, respectively. Both averaging\({}_{11}\) and bagging\({}_{11}\) ensemble models violate the assumption of no autocorrelation in the errors, relative to all cryptocurrencies, implying that these models are not properly fitted the time-series. In more detail, the significant spikes at lags 1 and 2 suggest that the residuals are not identically distributed and asymptotically independent, for all datasets.