# Conditional Temporal Aggregation for Time Series Forecasting Using Feature-Based Meta-Learning
## 1 Introduction
Multiple temporal aggregation, which is effectively a forecast combination subject to linear constraints [17], has several advantages over focusing on a single aggregation level [18]. Similar to standard ensemble strategies, widely used in the literature to blend forecasts produced by different models or variants of them, it avoids selecting a single \"best\" aggregation level, which is challenging to do in practice [19], and mitigates model and parameter uncertainty by exploiting the merits of combining for a non-systematic review on temporal aggregation, please refer to [6]. In practice, the main difference between standard forecast combination and multiple temporal aggregation is that in the former case, the forecasts to be ensembled are reported just at the original frequency of the series, while in the latter, they are reported at various data frequencies, thus exploiting the potential benefits of temporal aggregation in addition to those of forecast combination. Nevertheless, multiple temporal aggregation also comes with some challenges. First, when the series is characterized by seasonality, combining seasonal base forecasts (typically produced at lower aggregation levels) with non-seasonal base forecasts (typically produced at higher aggregation levels) may lead to an unnecessary seasonal shrinkage that deterio  
Figure 1: Visual examples where multiple new temporally aggregated time series are created based on the original data. The quarterly data (black line) are temporally aggregated to semesterly (red line) and yearly (blue line) data. For fast-moving data, the aggregation uncovers the trend of the series while for slow-moving data it filters out randomness. rates accuracy. [PERSON] et al. (2016) and [PERSON] et al. (2018) elaborate on this issue and propose some pre-processing techniques and heuristic rules to mitigate its effect. Second, some multiple temporal aggregation methods suggest that all levels should contribute equally to the final forecasts ([PERSON] et al., 2018), an approach that may be proved sub-optimal in practice. Simply put, the information available at some of the levels may be more critical for improving forecasting accuracy, meaning that more weight should be assigned to the respective base forecasts. [PERSON] et al. (2018) and [PERSON] et al. (2018) suggest accounting for error variances contributing to the forecast error at some or multiple aggregation levels to weight base forecasts more appropriately. However, these estimates still rely on in-sample one-step-ahead forecast errors (residual errors), which may neither represent post-sample performance precisely nor be correlated with the objective function ultimately used to assess forecasting performance. More importantly, these approaches will still combine the base forecasts linearly, thus possibly failing to account for complex data relationships. To tackle these issues, [PERSON] et al. (2018) suggested using cross-validation to estimate the forecast errors and weight the base forecasts accordingly, while [PERSON] et al. (2018) employed machine learning (ML) models to nonlinearly combine cross-sectional hierarchical forecasts using weights that explicitly focus on post-sample accuracy, thus allowing for more flexible and selective ensembling. In this paper, we offer a dynamic perspective to the problem of (multiple) temporal aggregation to address the aforementioned challenges, i.e., to avoid seasonal shrinkage, dynamically compute combination weights, and adjust said weights using post-sample accuracy records. Inspired by the recent advances in the field of ML, we propose the use of a classification model to either select the most appropriate temporal aggregation level or to derive the weights that properly combine the base forecasts computed at various levels. To do so, we extract several time series features at multiple temporal aggregation levels ([PERSON] et al., 2018) and use the features as leading indicators to estimate the importance of each level in improving forecasting accuracy.