## 5 Case Study  
In this research, we considered two open datasets that are widely used in time series forecasting research: M4 competition dataset (M4 CD) [5], and the Kaggle Wikipedia Web Traffic forecasting competition dataset (KWWTFCD) [123]. To ensure that the performance of the global models does not depend on the knowledge of a particular domain or characteristics of the time series data, we defined the same set of generic features for both cases. In this section, we describe the characteristics of each dataset, provide details on how we sampled instances from them, and the preprocessing steps we performed before training the GTSMLFM. The M4 CD comprises 100,000 time series selected from the ForeDeCk database, corresponding to multiple business domains, such as industries, government, transport, household, and natural resources. While the dataset includes time series at different frequencies (from hourly to yearly frequency), we focused on those provided monthly, which account for 48,000 time series. When analyzing their metadata and the actual time series length, we found some discrepancies. We clarified them with the authors from [124], who researched how representative it is of the reality. In private correspondence, the authors confirmed that such discrepancies existed, attributing them to the original public sources from which they were obtained. Since such time series represented only a tiny proportion of the total dataset, we ignored them. The KWWTFCD was provided by Google and introduced in a Kaggle competition in 2017. The dataset comprises time series accounting for the daily views of approximately 145,000 Wikipedia articles, starting from 1 July 2015, until 11 September 2017. The dataset distinguishes between zero and missing values. We randomly sampled 2000 time series for each dataset and selected only a subsequence to minimize the number of missing values (first 45 values for M4 CD and first 56 values for the KWWTFCD). We considered 2000 time series as a big enough sample to evaluate diverse time series and provide meaningful results, while avoiding processing the whole dataset. The need for sampling is grounded in the fact that a feature vector is created per time step for each time series, requiring extensive computational resources and time to train the models. When designing the experiment, we considered that a global model is more likely to produce anomalous forecasts if trained over a dataset that contains two types of time series: (i) the ones whose values remain in the same order of magnitude, and (ii) the ones whose values comprehend different orders of magnitude (TSDM). We thus analyzed the proportion of time series with such properties in each dataset and ensured that the sampling respected those proportions. The new datasets comprised 903 of type (i) and 1097 of type (ii) time series for the M4 CD, and 512 type (i) and 1488 type (ii) time series for the KWWTFCD. We provide further details on the original and reduced dataset in Table 1, and describe the target values distribution in Figure 3.  
Figure 3: Target values distribution. (**A**) describes values for M4 CD, while (**B**) describes values for KWWTFCD. To describe the time series, we created a set of features, presented in Table 2. We considered three types of features: (i) the features that describe the values observed for a given time series (e.g., minimum, mean, median values, along with the standard deviation), (ii) the features that describe the time series shape (e.g., skew, kurtosis, the number of peaks we observed, and the number of values above the mean), and (iii) the features that describe the context close to the forecasted value (e.g., last observed value--which can be used as a naive forecast). We compute thirty-three features for each dataset. For the global models, we do not perform feature selection, given that in all cases, we observe the number of features satisfies the Equation (2), as suggested in [125].