A study on forecasting high -frequency time series with multiple seasonal patterns
The present study aimed to examine the forecasting performance of various univariate approaches to forecast high-frequency time series which contains more than one seasonal pattern in hourly data. To achieve the purpose of the study, we performed the following procedures with major findings.
First, we produced point forecasts of hourly data series using benchmark forecasting approaches—the traditional (single seasonal) Holt-Winters exponential smoothing methods—and also made additional sets of forecasts by applying non-benchmark approaches with double seasonality, such as Taylor's double seasonal Holt-Winters method, innovation state-space models, and unobserved components models. We then compared forecasting performance or accuracy across those suggested forecasting approaches. In this empirical study, we found that the benchmark forecasting approaches were outperformed by the non-benchmark approaches when we applied those forecasting approaches to our two data sets of hourly time series on emergency room arrivals at a hospital and on electricity loads in New England area of the U.S. This indicates that forecast accuracy was able to be improved by incorporating two different types of seasonal patterns simultaneously. We also found that the innovations state-space models with a single source of error appeared not to outperform the generalized version of the state-space models with multiple sources of error (i.e., unobserved components models). This observation may imply that an assumption on the structure of disturbances in a time-series model affects an improvement of forecasting performance. We favor the unobserved components models as a structural time-series modeling approach with a general structure of error for both of the two data series, in the present empirical study.
Second, we repeated the investigation of forecasting performance across the forecasting approaches at the aggregated (partially summed up to four hours) level, while the previous empirical results were made based on the disaggregate (hourly) level. We took advantage of two approaches to produce aggregate point forecasts: direct (top-down) and indirect (bottom-up) approaches. We examined the effectiveness of the two approaches and observed that the data series on emergency room arrivals showed somewhat mixed empirical results on the forecasting performance. On the other hand, the series of electricity loads showed a strong and consistent support for the indirect approach to aggregate forecasts. In this study we favor unobserved components modeling approach within the indirect approach to forecasting at the aggregate level.
Finally, we constructed prediction intervals for the two time series data sets using two model-based approaches such as the innovations state-space models and the unobserved components models. In the framework of the innovation state-space model, we employed an analytical approach to obtain a (standard) statistical formula of prediction intervals, which depend on the estimates of model parameters and seasonal factors (also, level and trend) in more complicated specifications. Using both innovations state-space models and unobserved components models, we obtained considerably stable ranges of the prediction intervals with data on emergency room arrivals, but we observed divergent widths of the prediction intervals for the hourly electricity load series from the both models. In particular, the prediction intervals of the electricity loads series got fairly wider over long lead time based on the innovations state-space model. Furthermore, the prediction intervals based on the unobserved components model appeared to seriously diverge even with short lead time (less than 10 hours), which may imply that the extremely wide ranges of prediction intervals are not appropriate in practice for actual electricity loads data, although the general state-space models produced better point forecasts than the innovations state-space models did.