In recent years, tensors have garnered significant attention from researchers across the domains of statistics, applied mathematics, and machine learning. The inherent multi-linear structure of tensors renders them an efficient means of representing high-dimensional data. The technological revolution in data collection and processing has led to the emergence of tensorial datasets across numerous scientific applications, such as neuroimaging, collaborative filtering, and longitudinal data analysis. In this thesis, we focus specifically on the analysis of tensors with spatial and temporal dimensionality, commonly referred to as spatio-temporal tensors. We leverage the efficient tensor representation to analyze large-scale spatio-temporal data and integrate intricate spatio-temporal dependencies into the tensor model. Inspired by scientific applications in space weather monitoring, we introduce novel statistical methodologies addressing four distinct challenges.

I) The first part investigates the missing value imputation of spatio-temporal tensors with locally dependent missingness. Traditional low-rank matrix/tensor completion methods cannot provide reasonable imputations at locations where almost all data are missing in the neighborhood. We adopt the classic low-rank matrix completion framework and improve it by giving a tensor completion estimator exhibiting spatial and temporal continuity. We establish the convergence guarantee of the new method and apply it extensively to the global Total Electron Content (TEC) reconstruction problem.

II) The second part dives into the uncertainty quantification (UQ) of tensor completion. Literature on the UQ of tensor completion relies heavily on the assumption that data is missing uniformly at random or at least independently and only applies to a restricted class of the completion method. We circumvent these restrictions by introducing a conformal prediction framework for the UQ. The resulting confidence intervals are constructed by properly accounting for the missing propensity of each tensor entry, which is estimated by a low-rank tensor Ising model that can account for the dependent data missingness. We establish the theoretical coverage guarantee and validate the method through extensive simulations and an application to the global TEC reconstruction problem.

III) The third part focuses on the forecasting problem of matrix-valued spatial time series with auxiliary vector-valued, non-spatial time series covariates. Existing works on matrix autoregression cannot handle such settings with predictors of non-uniform modes and spatio-temporal dimensions. We propose a novel semi-parametric matrix autoregression model incorporating the vector covariates with spatially smooth tensor coefficients. We establish the joint asymptotics of the autoregressive and tensor parameters under fixed and high-dimensional regimes and apply our method to the global TEC forecast problem.

IV) The last part is dedicated to a scalar-on-tensor regression problem with multi-modal imaging tensor covariate. We encapsulate a tensor dimension reduction step and a Gaussian Process regression model in a single framework and introduce a total-variation regularization to capture spatially contiguous predictive signals. The new model complements the current literature by accounting for the interplay among different data modalities in an interpretable fashion. We apply our model to forecast the intensity of solar flares with multi-channel solar imaging data.