Fast Update of Conditional Simulation Ensembles

Abstract

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at minimal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on $n\ge 0$ observations to an ensemble conditioned on $n+q$ observations, for arbitrary $q\ge 1$. Compared to direct approaches, the proposed formulae prove to substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the $q$ new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

Appendices

Appendix A: Universal and Simple Kriging

Let $Z$ be a $L^2$ random field defined on a bounded set $\mathbb {X}\subset \mathbb {R}^d$ with known (and not necessarily stationary) covariance function $k(\cdot ,\cdot )$ and unknown mean function $m(\cdot )$ such that $Z|m \sim \text {GRF}(m,k)$, where $\text {GRF}(m,k)$ denotes a GRF with mean function $m$ and covariance function $k$. A well-known Bayesian approach consists in writing $m$ as follows

$$\begin{aligned} m(\cdot ) = \sum _{i=1}^\ell \beta _i f_i(\cdot ), \end{aligned}$$

where $\ell \ge 1$, $f_1,\ldots ,f_\ell$ are $\ell$ known basis functions and $\varvec{\beta }= (\beta _1,\ldots ,\beta _\ell )$ has an improper uniform prior in $\mathbb {R}^\ell$. In these settings, known as the universal kriging settings, when $n$ observations $Z(\mathbf {x}_{1:n})$ are assimilated at points $\mathbf {x}_{1:n} := (\mathbf {x}_1,\ldots ,\mathbf {x}_n)\in \mathbb {X}^n$, it is known (O’Hagan 1978) that the posterior distribution of $Z$ is a GRF with posterior (or conditional) mean function $m^{UK}_n$ and covariance function $k^{UK}_n$ given by the so-called universal kriging equations

$$\begin{aligned} \varvec{\lambda }^{UK}(\mathbf {x})&= K^{-1} \left( \mathbf {k}(\mathbf {x})+ \mathbb {F}(\mathbb {F}^\top K^{-1} \mathbb {F})^{-1} (\mathbf {f}(\mathbf {x})- \mathbb {F}^\top K^{-1}\mathbf {k}(\mathbf {x})) \right) , \end{aligned}$$

(10)

$$\begin{aligned} m^{UK}_n(\mathbf {x})&= \varvec{\lambda }^{UK}(\mathbf {x})^\top Z(\mathbf {x}_{1:n}) = \mathbf {f}(\mathbf {x})^\top \widehat{\varvec{\beta }} + \mathbf {k}(\mathbf {x})^\top K^{-1} \left( Z(\mathbf {x}_{1:n}) - \mathbb {F}\widehat{\varvec{\beta }} \right) , \end{aligned}$$

(11)

$$\begin{aligned} k^{UK}_n(\mathbf {x},\mathbf {x}^\prime )&= k(\mathbf {x},\mathbf {x}^\prime ) - \mathbf {k}(\mathbf {x})^\top K^{-1}\mathbf {k}(\mathbf {x}^\prime )\nonumber \\&\quad +(\mathbf {f}(\mathbf {x})^\top - \mathbf {k}(\mathbf {x})^\top K^{-1}\mathbb {F})(\mathbb {F}^\top K^{-1}\mathbb {F})^{-1} (\mathbf {f}(\mathbf {x}^\prime )^\top - \mathbf {k}(\mathbf {x}^\prime )^\top K^{-1}\mathbb {F})^\top ,\nonumber \\ \end{aligned}$$

(12)

where $\widehat{\varvec{\beta }} := (\mathbb {F}^\top K^{-1} \mathbb {F})^{-1}\mathbb {F}^\top K^{-1}Z(\mathbf {x}_{1:n})$, $\mathbf {f}(\mathbf {x}):= (f_1(\mathbf {x}),\ldots ,f_\ell (\mathbf {x}))^\top$, $\mathbb {F}\in \mathbb {R}^{n\times \ell }$ is the matrix with row $i$ equal to $\mathbf {f}(\mathbf {x}_i)^\top$, $\mathbf {k}(\mathbf {x}):= (k(\mathbf {x},\mathbf {x}_1),\ldots ,k(\mathbf {x},\mathbf {x}_n)) ^\top$, $K$ is the covariance matrix at the observation points, $K := (k(\mathbf {x}_i,\mathbf {x}_j))_{1\le i,j\le n}$. The vector $\varvec{\lambda }^{UK}(\mathbf {x})$ is the vector of $n$ kriging weights of $\mathbf {x}_1,\ldots , \mathbf {x}_n$ for the prediction at point $\mathbf {x}$.

A well-known simpler setting is the case where the non-conditional mean function $m$ is already known. In that case, the Bayesian approach is no longer necessary and the conditional mean and covariance function of $Z$ are given by the so-called simple kriging equations, written here in the case where $m(\cdot )=0$

$$\begin{aligned} \varvec{\lambda }^\mathrm{SK}(\mathbf {x})&= \!\!\ K^{-1} \mathbf {k}(\mathbf {x}), \end{aligned}$$

(13)

$$\begin{aligned} m^\mathrm{SK}_n(\mathbf {x})&= \!\!\ \varvec{\lambda }^\mathrm{SK}(\mathbf {x})^\top Z(\mathbf {x}_{1:n}) = \mathbf {k}(\mathbf {x})^\top K^{-1} Z(\mathbf {x}_{1:n}), \end{aligned}$$

(14)

$$\begin{aligned} k_n^\mathrm{SK}(\mathbf {x},\mathbf {x}^\prime )&= \!\!\ k(\mathbf {x},\mathbf {x}^\prime ) - \mathbf {k}(\mathbf {x})^\top K^{-1}\mathbf {k}(\mathbf {x}^\prime ). \end{aligned}$$

(15)

If $m$ is not equal to zero, the simple kriging covariance function $k_n^\mathrm{SK}$ is unchanged and an application of Eq. (14) to the centred GRF $Z-m$ yields $m^\mathrm{SK}_n(\mathbf {x}) = m(\mathbf {x}) + \mathbf {k}(\mathbf {x})^\top K^{-1} ( Z(\mathbf {x}_{1:n}) - m(\mathbf {x}_{1:n}) )$.

Appendix B: Algorithms