Some complex intelligent systems such as for tackling the COVID-19 pandemic involve coupled multivariate time series (MTSs), where both target variables (such as COVID-19 infected, confirmed, and recovered cases) and external factors (such as virus mutation and infectivity, vaccination, and government intervention influence) are coupled. Forecasting such MTSs with multiple external factors needs to model interactions within and between MTSs and handle their uncertainty, heterogeneity, and dynamics. Moreover, COVID-19 case time series across multiple countries whose covariates may have high volatilities caused by missing samples. The primary challenges in modeling coupled and volatile multivariate time series (CVMTS) include examining both inter- and intra-MTS interactions, handling volatile covariates, dealing with multiple external factors, and ensuring robust probabilistic predictions. However, existing shallow to deep MTSs modelers, including regressors, deep recurrent neural networks such as DeepAR, deep state space models, and deep factor models, do not jointly characterize these issues in a probabilistic manner across MTSs and they cannot explicitly model intra- and inter-MTS couplings and effectively handle volatile covariates in multiple multivariate time series.

To tackle the challenge of modeling MTSs with multiple external factors and conducting robust probabilistic forecasting, chapter 3 proposes an end-to-end deep probabilistic cross-MTS learning network (MTSNet). It incorporates a tensor input consisting of scaled targeting and external MTSs. It then vertically and horizontally stacks long-short memory networks for encoding and decoding target MTSs and enhances uncertainty modeling, generalization, and forecasting robustness by residual connection, variational zoneout, and probabilistic forecasting. The tensor input is projected
to a probability distribution for target MTS forecasting. MTSNet outperforms the SOTA deep probabilistic MTS networks in forecasting COVID-19 confirmed cases and ICU patient numbers for six countries by involving virus mutation, vaccination, government interventions, and infectivity.

To explicitly model intra- and inter-MTS couplings and effectively handle volatile covariates, chapter 4 proposes deep spectral copula mechanisms (DSCM) to adapt CVMTS. Specifically, DSCM incorporates a singular spectral analysis (SSA) module to reduce the volatility of multiple covariates. It applies an intra-MTS coupling module to explicitly model the temporal couplings within a single set of multivariate time series and transforms target variables into joint probability distributions via Gaussian copula transformation to establish inter-MTS couplings across multiple multivariate time series. Substantial experiments on COVID-19 time-series data from multiple countries indicate the superiority of DSCM over state-of-the-art approaches.