## 4 Proposed Methods: Autoregressive TP Models
### Review and Notation for Student-\(t\) Processes  
We say that \(\mathbf{f}\in\mathbb{R}^{n}\) follows a _multivariate Student's-\(t\)_ distribution with degrees of freedom \(v\in\mathbb{R}^{+}\), location \(\mathbf{\mu}\in\mathbb{R}^{n}\), and positive definite scale matrix \(\mathbf{K}\in\mathbb{R}^{n\times n}\) if and only if it has the following density:  
\[p(\mathbf{f})=\frac{\Gamma((v+n)/2)}{(v\pi)^{n/2}\Gamma(v/2)|\mathbf{K}|^{1/2} }\bigg{(}1+\frac{1}{v}(\mathbf{f}-\mathbf{\mu})^{{}^{\prime}}\mathbf{K}^{-1}( \mathbf{f}-\mathbf{\mu})\bigg{)}^{-\frac{v+n}{2}}, \tag{23}\] \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Model** & **Characteristics** & **Pros** & **Cons** \\ \hline \multirow{8}{*}{GP-NARX} & 1. Target is a non-linear, & \multirow{8}{*}{1. Fast to train} & \multirow{8}{*}{1. Incapable of handling} \\  & autoregressive function of & & \\  & observed past values & & 2. Non-parametric \\  & and exogenous predictors. & GP prior & 2. Assumes a \\  & 2. Trained via Type II MLE & 3. Forecasts attained via & 3. Predictive uncertainty & Gaussian likelihood \\  & simple Monte & & \\  & Carlo sampling & & \\ \hline \multirow{8}{*}{GP-RLARX} & 1. Target variable is & \multirow{8}{*}{1. Robust to} \\  & assumed to equal a latent & \multirow{8}{*}{1. Somewhat GP-NARX} \\  & state plus noise & & \\  & 2. Autoregressive behavior & & \\  & captured through & heavy-tailed noise & \\  & latent state dynamics & & \\  & 3. Exogenous predictors & & \\  & can be placed at observed & & \\  & and latent level & & \\  & 4. Trained using variational & & \\  & Bayesian and sparse & & \\  & GP methods & & \\  & 5. Forecasts by sampling & & \\  & from approximate posterior & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of GP-NARX and GP-RLARX methods. which can be written succinctly as \(\mathbf{f}\sim\mathcal{T}_{n}(v,\mathbf{\mu},\mathbf{K})\). Now, suppose that we have \(\mathbf{f}\in\mathbb{R}^{n}\) and \(\mathbf{f}_{*}\in\mathbb{R}^{m}\) with joint density  
\[ \begin{bmatrix}\mathbf{f}\\ \mathbf{f}_{*}\end{bmatrix} \sim\mathcal{T}_{m+n}\bigg{(}v_{\rho} \begin{bmatrix} \mathbf{\mu}_{\mathbf{t}}\\ \mathbf{\mu}_{*}\end{bmatrix}\begin{bmatrix}\mathbf{K}_{\mathbf{ff}}&\mathbf{K}_{ \mathbf{fs}}\\ \mathbf{K}_{*\mathbf{f}}&\mathbf{K}_{**}\end{bmatrix} \bigg{)}. \tag{24}\]  
By properties of the multivariate Student-\(t\) distribution, we have  
\[\mathbf{f}\sim\mathcal{T}_{n}(v,\mathbf{\mu}_{\mathbf{f}},\mathbf{K}_{\mathbf{ff}}) \;\;\text{and}\;\;\mathbf{f}_{*}|\mathbf{f}\sim\mathcal{T}_{m}\bigg{(}v+n,\bar {\mathbf{\mu}}_{*},\frac{v+\beta-2}{v+n-2}\tilde{\mathbf{K}}_{**}\bigg{)} \tag{25}\]  
where  
\[\bar{\mathbf{\mu}}_{*}=\mathbf{\mu}_{*}+\mathbf{K}_{*\mathbf{f}}\mathbf{K}_{\mathbf{ff} }^{-1}(\mathbf{f}-\mathbf{\mu}_{\mathbf{f}}),\;\beta=(\mathbf{f}-\mathbf{\mu}_{\mathbf{ f}})^{{}^{\prime}}\mathbf{K}_{\mathbf{ff}}^{-1}(\mathbf{f}-\mathbf{\mu}_{\mathbf{f}}), \tilde{\mathbf{K}}_{**}=\mathbf{K}_{**}-\mathbf{K}_{*\mathbf{f}}\mathbf{K}_{ \mathbf{ff}}^{-1}\mathbf{K}_{\mathbf{fs}}. \tag{26}\]  
Finally, we say that \(f(\cdot)\) follows a Student-\(t\) process on \(\mathcal{X}\), denoted \(\mathcal{TP}(v,\mu(\cdot),k(\cdot,\cdot))\), where \(v>2\) denotes the degrees of freedom, \(\mu(\cdot)\in\mathbb{R}\) denotes the mean function, and \(k(\cdot,\cdot)\in\mathbb{R}\) is the covariance function, if for any finite collection of function values, we have \(\mathbf{f}=(f(\mathbf{x}_{1}),\ldots,f(\mathbf{x}_{n}))^{\prime}\sim\mathcal{T }_{n}(v,\mathbf{\mu},\mathbf{K})\). While less popular than GP models, Student-\(t\) processes have still been employed in a number of contexts. For instance, [PERSON]. [27] proposed an online time series anomaly detection algorithm that employs TP regression to simultaneously learn time series dynamics in the presence of heavy-tailed noise and identify anomalous events.