A Hierarchical Spatiotemporal Statistical Model Motivated by Glaciology

Abstract

In this paper, we extend and analyze a Bayesian hierarchical spatiotemporal model for physical systems. A novelty is to model the discrepancy between the output of a computer simulator for a physical process and the actual process values with a multivariate random walk. For computational efficiency, linear algebra for bandwidth limited matrices is utilized, and first-order emulator inference allows for the fast emulation of a numerical partial differential equation (PDE) solver. A test scenario from a physical system motivated by glaciology is used to examine the speed and accuracy of the computational methods used, in addition to the viability of modeling assumptions. We conclude by discussing how the model and associated methodology can be applied in other physical contexts besides glaciology.

Appendices

A Derivation of the Exact Likelihood and Computational Simplifications

As is shown in Appendix B of Gopalan et al. (2018), the covariance matrix for the observed data can be written as $\text {U} \otimes \text {V} + \sigma ^2\text {I}$, where $\text {U}_{ab} = k \min (a,b)$ with $\text {U} \in \mathbb {R}^{N \times N}$, and $\text {V} = \text {A}(\Sigma )\text {A}^{\intercal }$. It can be verified that $\text {U}^{-1}$ is tridiagonal, so it has bandwidth one—more specifically:

$$\begin{aligned} \text {U}^{-1}= & {} k^{-1} \begin{bmatrix} 2&\quad -1&\quad 0&\quad \dots&\quad&\quad \\ -1&\quad 2&\quad -1&\quad 0&\quad \dots&\quad \\ 0&\quad -1&\quad 2&\quad -1&\quad 0&\quad \dots&\quad \\ 0&\quad 0&\quad \ddots&\quad \ddots&\quad \ddots \\ 0&\quad \ldots&\quad&\quad -1&\quad 2&\quad -1 \\ 0&\quad \ldots&\quad&\quad&\quad -1&\quad 1 \end{bmatrix}. \end{aligned}$$

(18)

One useful property of the Kronecker product is that $(\text {U} \otimes \text {V})^{-1} = \text {U}^{-1} \otimes \text {V}^{-1}$. Therefore:

$$\begin{aligned} (\text {U} \otimes \text {V})^{-1}= & {} \text {U}^{-1} \otimes \text {V}^{-1} \end{aligned}$$

(19)

$$\begin{aligned}= & {} k^{-1} \begin{bmatrix} 2\text {V}^{-1}&\quad -\text {V}^{-1}&\quad 0&\quad \dots&\quad&\quad \\ -\text {V}^{-1}&\quad 2\text {V}^{-1}&\quad -\text {V}^{-1}&\quad 0&\quad \dots&\quad \\ 0&\quad -\text {V}^{-1}&\quad 2V^{-1}&\quad -\text {V}^{-1}&\quad 0&\quad \dots&\quad \\ 0&\quad 0&\quad \ddots&\quad \ddots&\quad \ddots \\ 0&\quad \ldots&\quad&\quad -\text {V}^{-1}&\quad 2\text {V}^{-1}&\quad -\text {V}^{-1} \\ 0&\quad \ldots&\quad&\quad&\quad \text {V}^{-1}&\quad \text {V}^{-1} \end{bmatrix}, \end{aligned}$$

(20)

whose bandwidth is O(m).

Let us denote $\text {U} \otimes \text {V}$ as $\text {W}$. By the matrix inversion lemma, it follows that $(\sigma ^2\text {I}+\text {W})^{-1} = \sigma ^{-2}\text {I}-\sigma ^{-2}(\text {W}^{-1}+\sigma ^{-2}\text {I})^{-1}\text {I}\sigma ^{-2}$. The matrix $\text {W}^{-1}+\sigma ^{-2}\text {I}$ has bandwidth O(m) since $\text {W}^{-1}$ has bandwidth O(m) as shown previously, so this expression can be computed in $O(Nm^3)$ (Rue 2001; Golub and Van Loan 2012).

Similarly, by the matrix determinant lemma, $\text {log}[\text {det}(\sigma ^2\text {I}+\text {W})]$ is $\text {log}[\text {det}(\text {I}+\sigma ^2\text {W}^{-1})\text {det}(\text {W}^{-1})^{-1}]$ = $\text {log}[\text {det}(\text {I}+\sigma ^2\text {W}^{-1})]$-$\text {log}[\text {det}(\text {W}^{-1})]$. Since both terms are log determinants of square matrices of dimension Nm and bandwidth O(m), this can be calculated in $O(Nm^3)$ due to the efficient Cholesky factorization of band-limited matrices (Rue 2001; Golub and Van Loan 2012).

B First-Order Spatiotemporal Emulators

In the examples of this paper, the function $\varvec{f}(.,.,.)$ (i.e., the computer simulator) can take one of the two forms: a numerical PDE solver for the SIA, or an emulator constructed from the numerical PDE solver for the SIA. The numerical method for solving the SIA PDE is as given in Gopalan et al. (2018), and the emulator is constructed based on the finite difference solver in a manner as suggested in Hooten et al. (2011), termed first-order emulation.

That is, we start with a set of plausible values for ice viscosity: $\{\theta _1,\theta _2,\ldots ,\theta _p\}$ and, for each time point there is collected data ck, we store a matrix $\text {M}_{ck}$, where the $q\text {th}$ column of matrix $\text {M}_{ck}$ is the output of the numerical solver using parameter value $\theta _q$ after running for ck time steps forward. Thus, each matrix $\text {M}_{ck}$ is of dimension n by p, and without essential loss of generality, we can assume that the number n is much larger than p, and each matrix $\text {M}_{ck}$ is of rank p.

For each matrix, $\text {M}_{ck}$, we compute a singular value decomposition (SVD), $\text {U}_{ck}\text {D}_{ck}\text {V}^{\intercal }_{ck}$. The goal is to find a (vector-valued) function $v_{ck}(\theta *)$ such that the emulated output at time ck for parameter value $\theta *$ is $\text {U}_{ck}\text {D}_{ck}v_{ck}(\theta *)$. To find the $q\text {th}$ element of $v_{ck}$, we train a random forest (Breiman 2001; Liaw and Wiener 2002) with $(\theta _1, (V^{\intercal }_{ck})_{q1}), (\theta _2, (V^{\intercal }_{ck})_{q2}),\ldots ,(\theta _p, (V^{\intercal }_{ck})_{qp})$ as training data, where $(V^{\intercal }_{ck})_{q1}$ is the first element of the $q\text {th}$ right singular vector, $(V^{\intercal }_{ck})_{q2}$ is the second element of the $q\text {th}$ right singular vector, and so on. Not all of the right singular vectors need be used in emulation, and a heuristic such as an elbow–scree plot or the randomization procedure of Friedman et al. (2001) can be used to determine the number of right singular vectors to keep. However, if the number of simulator runs (p) is much smaller than the dimensionality of the output (n), all of the right singular vectors can be utilized with computational savings, as is done in the experiments of this paper.

We have assumed the initial conditions and boundary conditions are known, since this is the case in the glaciology problems we have studied, where the boundary condition is that glacial thickness is nonnegative, and the initial glacier profile (i.e., a dome) is known. In general, however, $\varvec{\phi }$ may be incorporated into the analysis above by considering $\theta$ and $\varvec{\phi }$ jointly. Additionally, a variant is to directly emulate the likelihood function. However, since there is flexibility in the choice of $\Sigma$ (which enters into the likelihood), unless one is set on using a particular value of $\Sigma$, it is sensible to emulate the numerical solver as opposed to retraining a likelihood emulator for each potential choice of $\Sigma$.