Time series analysis is widely discussed in fields such as finance, economy, brain imaging etc. Among all types of data, categorical and multivariate time series maintain both of challenges and promising applications. In this dissertation, we propose some statistical approaches to model binary and multivariate time series and thus provide alternative solutions of statistical inference and prediction.

We first focus on binary time series. Classical methods do not differentiate between exogenous and endogenous exploratory variable, which leads to invalid statistical inference. We develop a close form of the Fisher information matrix of logistic autoregressive model and demonstrate that it yields narrower confidence intervals while maintaining nominal type I error rate. We also propose a framework of predicting binary time series using Gaussian process. The approach comprises of a linear part that captures the effects from covariates and a stochastic process that characterizes the information not covered by the linear part. Both the simulation and the real data examples demonstrate the high predictive power and appropriate interpretability.

Next, we discuss on the problems of multivariate time series. In an illustrative example of analyzing Local Field Potentials (LFPs) signals, existing methods such as Independent Component Analysis (ICA), Principal Component Analysis (PCA) have limitations in modeling spatial-temporal dependencies across trials (epochs). To address these issues, we introduce Evolutionary State Space Model (E-SSM) allowing the latent signals evolve during the experiment. By fixing the phase of the AR polynomial roots, the framework is able to model the evolution for pre-specified frequency bands. As the last part of this dissertation, we characterize multivariate time series as 2 - dimensional tensors. By introducing a penalized mixture matrix normal model, we are able to uncover the ``latent" mean spatial-temporal structures across trials (epochs) and capture the sparsity simultaneously. Some theoretical results are established to show the consistency of the constrained maximum likelihood estimator.

This dissertation studies the notion of simplicial distance on Bruhat-Tits buildings. That is a measure of proximity between vertices in the simplicial structure. The purpose of this research is three-fold: (i). to provide a concrete characterization of the simplicial distance; (ii). to better understand simplicial balls, and (iii). to derive a formula for the simplicial volume and explore its asymptotic growth.To accomplish these goals, a comprehensive examination of vertices becomes necessary. They are analyzed using three frameworks: root systems, norms, and lattices. By leveraging concave functions, we interpret simplicial balls as fixed-point sets of Moy-Prasad subgroups and deduce a formula for the simplicial volume. Additionally, the theory of ?-exponential polynomials is developed to facilitate the asymptotic study. Through this research, we focus on split classical types (namely, types of ??, ??, ??, ??, and any combination thereof) over a local field. The presented findings contribute to the advancement of our understanding of Bruhat-Tits buildings.

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