# Conditional Temporal Aggregation for Time Series Forecasting Using Feature-Based Meta-Learning
## 1 Introduction  
Forecasting is essential for supporting decisions at strategic, tactical, and operational levels. Accurate forecasts can assist companies and organizations in reducing costs, avoid risks, and exploit opportunities, thus finding application in a variety of settings. Nevertheless, univariate forecasting is a challenging task that usually requires identifying patterns in time series data and selecting the most appropriate model for extrapolating them in time. As a result, numerous approaches have been proposed to perform model selection [1]. Still, this selection process involves significant uncertainty, especially when the patterns of the series are complex or the data are noisy [2]. To deal with model uncertainty, forecasters have been combining (ensembling) forecasts of multiple models, each making different assumptions about the distribution and correlation of the data [3]. The practice of combining has been proved particularly effective in various forecasting studies and competitions [4] and many ensemble strategies have been suggested to exploit its full potential. From the combination methods found in the literature, those that build on data manipulation and transformation are probably the most promising. Instead of training different models on the examined series and combining their forecasts, these methods transform the series to create new ones, each highlighting particular characteristics (typically called time series features) of the original data. Then, the same or different models can be used to forecast the transformed series. More often than not, combining these forecasts (typically called base forecasts) results in better forecasting accuracy. [PERSON] and [PERSON] [5] discuss how the \"wisdom of the data\" can be used to extract useful information from series and review various approaches that exploit its benefits in univariate forecasting settings. Temporal aggregation is one of the transformations that utilize the \"wisdom of the data\" concept to identify patterns that may be difficult to capture when analyzing the original series directly [6]. Specifically, temporal aggregation is the transformation of a time series from one frequency to another, lower frequency. For instance, a quarterly time series of length \(n\) can be transformed to a semi-yearly series of length \(n/2\) by using equally sized time buckets of two periods each or a yearly series of length \(n/4\) by aggregating (summing)the observations by four, as shown in Figure 1. By changing the original frequency of the data, the apparent series characteristics also change and, as a result, different patterns can be observed and modeled to produce more accurate forecasts [7]. Thus, temporal aggregation becomes relevant for predicting both slow- [8; 9; 10; 11] and fast-moving [12; 13] series. In the first case, where the data typically display intermittency or erraticness [14], the aggregation filters out randomness and reveals the underlying signal of the series. In the second case, the transformation uncovers the trend patterns of the series, while allowing seasonality and level to be modeled at different levels. Temporal aggregation can be applied both in an non-overlapping and an overlapping manner. However, the former approach has become more popular since its results are easier to interpret and avoids introducing autocorrelations [15]. Moreover, non-overlapping aggregation allows the series created to be organized in temporal hierarchies [13] and, as a result, forecasts can be produced using methods that have been proved successful when predicting cross-sectional hierarchies. Apart for improving accuracy, forecasting with temporal hierarchies also has the advantage that it generates reconciled predictions that support aligned decisions at different planning horizons [16]. In addition, it provides a structured framework for combining forecasts from multiple temporal levels, an approach called multiple temporal aggregation.