  The Hidden Markov Model (HMM) can predict the future value of a time series
based on its current and previous values, making it a powerful algorithm for
handling various types of time series. Numerous studies have explored the
improvement of HMM using advanced techniques, leading to the development of
several variations of HMM. Despite these studies indicating the increased
competitiveness of HMM compared to other advanced algorithms, few have
recognized the significance and impact of incorporating multistep stochastic
states into its performance. In this work, we propose a Pyramidal Hidden Markov
Model (PHMM) that can capture multiple multistep stochastic states. Initially,
a multistep HMM is designed for extracting short multistep stochastic states.
Next, a novel time series forecasting structure is proposed based on PHMM,
which utilizes pyramid-like stacking to adaptively identify long multistep
stochastic states. By employing these two schemes, our model can effectively
handle non-stationary and noisy data, while also establishing long-term
dependencies for more accurate and comprehensive forecasting. The experimental
results on diverse multivariate time series datasets convincingly demonstrate
the superior performance of our proposed PHMM compared to its competitive peers
in time series forecasting.
