HMM-Based Mixture of Gaussian Process Functional Regressions for Seasonal Time Series Forecasting Under Multimodal Distributions: A Case Study on the Hourly Prediction of Land Surface Temperature

Seasonal patterns are commonly found in realworld time series. However, despite having a seasonal pattern, the cycle in time series may not recur on a strict-consistent basis, as there might be changes during certain periods. People often refer to this as multiple seasonalities, with the assumption that a series has multiple seasonal cycles of different lengths. The concept of multiple seasonalities itself is still limited to situations where the changes happen at regular intervals; while in real life, these changes can appear at any time. Thus, we employ a more general term in this paper, i.e., 'multimodal distributions', in which we assume that the seasonal time series has a probability distribution containing multiple modes. We use the HMMBased Mixture of Gaussian Process Functional Regressions (HM-GPFR) to forecast the time series in this case. HM-GPFR utilizes K-GPFR models simultaneously to capture the diversity of cycle patterns, while also implementing an HMM to simulate their transitions. In our numerical implementation, HM-GPFR was able to generate highly precise predictions, even for the long-term, with a MAPE of around 5 percent.