# Conditional Temporal Aggregation for Time Series Forecasting Using Feature-Based Meta-Learning
## 4 Empirical Evaluation  
### Data Sets  
To evaluate the performance of the proposed approach both for the case of fast- and slow-moving data, we consider two different data sets that are widely used in the forecasting literature for benchmarking. The first data set includes the monthly series of the M4 competition [4], while the second the daily product-store series of the M5 competition [29].  
The M4 competition involves 48,000 monthly time series that originate from several domains, namely microeconomic, industry, macroeconomic, finance, demographic, and others. Moreover, compared to other public data sets, the competition consists of diverse series that sufficiently represent several real-world forecasting applications [51]. The lengths of the series differ, ranging from 5 to about 82 years, with the majority of the series covering a period of about 18 years. Note that M4 originally involved the prediction of 100,000 time series that covered in addition to the monthly series, yearly, quarterly, weekly, daily, and hourly data. Our choice to focus on the monthly data of the competition is twofold. First, temporal aggregation becomes more meaningful for series of relatively high frequencies. The lower the frequency, the fewer the temporal aggregation levels that can be constructed and used by the meta-learner to improve forecasting accuracy. For instance, yearly data cannot be meaningfully aggregated further, while quarterly data can construct just three temporal levels (quarterly, semi-yearly, and yearly data), in contrast to monthly that can construct six temporal levels (monthly, bi-monthly, quarterly, four-monthly, semi-yearly, and yearly data). Second, the weekly, daily, and hourly series of the M4 competition were considerably fewer in number compared to the monthly data, thus limiting the learning potential of the proposed meta-learner and the significance of our results.  
The M5 competition involves 30,490 time series that represent the hierarchical unit sales of the largest retail company in the world, Walmart. Specifically, the series cover 3490 products, which can be classified into three product categories (hobbies, foods, and household), sold across 10 stores, located in the states of California, Texas, and Wisconsin. The series are daily and cover a period of about 5.5 years, but their lengths differ, ranging from 124 to 1969 days (about 1810 days on average). According to [PERSON] et al. [51], 73% of the series are intermittent, 17% are lumpy, 3% are erratic, and 7% are smooth [14]. This makes M5 a representative data set for evaluating the performance of the proposed approach on slow-moving series. Moreover, [PERSON] et al. [24] suggest that the M5 data sufficiently represent retailers that operate in different regions, sell different product types, and consider different marketing strategies than Walmart. Thus, we expect our results to be relevant for various retail firms around the globe.  
Both data sets are publicly available and can be downloaded from the GitHub repository ([[https://github.com/Mcompetitions](https://github.com/Mcompetitions)]([https://github.com/Mcompetitions](https://github.com/Mcompetitions)), accessed on 10 April 2023) devoted to the M competitions.