## 6 Experiments, Results, and Analysis
### Dealing with Anomalous GTSMLFM Forecasts (Experiment 2 and Experiment 4)  
Once the GTSMLFM was trained, we proceeded to identify outliers produced by the GTSMLFM using an ensemble of time series anomaly detectors (COPOD [131], ABOD [132], and KNN [133]), and considering a given forecast anomalous only when all the detectors agreed it should be considered as such. Next, we used the LIME [75] to compute features relevance for each anomalous forecast.  
The most similar instances between the feature vector generating an anomalous prediction and data available in the training set were obtained, computing the cosine distance over a subset of the five most relevant features to each prediction. In Experiment 2 and Experiment 4, we computed such similarity against the data instances in the train set for all GTSMLFM anomalous forecasts. For Experiment 2, we considered a vector built with the most relevant features to a specific forecast, while in Experiment 4, we enlarged that vector with the target value for the train set instances and the predicted value for the reference vector. The intuition behind including such value in the vector is that instances most likely influence the model with similar input values and teach a target value close to the observed prediction.  
While we experimented with multiple similarity thresholds, we finally adopted a threshold of 0.9999999, for which, in most cases, just a handful of train instances were identified as similar to the feature vectors producing anomalous forecasts. However, due to some exceptional cases resulting in a relatively high number of similar instances despite the tight threshold, we decided to provide an additional bounding criterion, collecting, at most, ten top train instances for each anomalous forecast. This limit was established considering the relationship between the size of the test set, the number of anomalies, and an upper bound to the number of instances to be removed. While such a procedure helped us identify and remove instances that distorted the GTSMLFM's training, future work will develop a better procedure to determine the parameters and identify such instances.  
When analyzing the results, we observed, in both cases, that the model's performance decayed (MASE measurements for GM(GBMR) and GM(GBMR)+naive), while the discrepancy between GM(GBMR) predictions and target values that were considered anomalous reduced, indicating a better adjustment to the real data behavior for unexpected values. While the MASE improvement was slight, it must be noted that such improvement was consistent and thus is improbable that it could be attributed to models' variance. Furthermore, we expect that the forecast improvements are minor in terms of MASE: reducing the number of anomalies across a test set of 24,000 records for dataset M4 CD, and 28,000 for dataset KWWTFCD, should not have a substantial impact on the overall MASE. While the MASE is indicative of the overall models' forecast adjustment, the improvement regarding anomalous forecasts should be considered in terms of a reduction of the target discrepancy (the model better adjusts to the observed target values in the test set). We observe that the number of anomalous forecasts is reduced in both cases, showing that the new model better adjusts to the time series behavior.