On the Construction of Uncertain Time Series Surrogates Using Polynomial Chaos and Gaussian Processes

Abstract

The analysis of time series is a fundamental task in many flow simulations such as oceanic and atmospheric flows. A major challenge is the design of a faithful and accurate time-dependent surrogate built with a tractable sample set and a manageable number of degrees of freedom. Several techniques are implemented to handle the time-dependent aspect of the quantity of interest including uncoupled approaches, low-rank approximations, auto-regressive models and global Bayesian emulators. These approaches rely on two popular methods for uncertainty quantification: polynomial chaos and Gaussian process regression. The different techniques are tested and compared on the uncertain evolution of the sea surface height forecast at two locations exhibiting contrasting levels of variance. Two ensemble sizes are considered as well as two versions of polynomial chaos (ordinary least squares or ridge regression) and Gaussian processes (squared exponential or Matérn covariance function) in order to assess their impact on the results. The conclusions focus on the advantages and the drawbacks, in terms of accuracy, flexibility and computational costs of the different techniques.

Appendix A: Cross-Validation Technique

Cross-validation (CV) is a popular technique used in statistics and machine learning to assess the quality of the predictive capacity of a model. The principle of CV is to partition the data into two complementary subsets, then to build the model on one subset (the training one), and finally to test the model on the other subset (the validation one). This procedure is repeated several times with different partitioning of the data. In the leave-one-out version of CV, the predicted residual $e_{[i]}$ at $\varvec{\xi }^{(i)}$ is defined as

$$\begin{aligned} e_{[i]} = u(\varvec{\xi }^{(i)}) - \tilde{u}_{[i]}(\varvec{\xi }^{(i)}), \end{aligned}$$

where $u(\varvec{\xi }^{(i)})$ is the true value, and $\tilde{u}_{[i]}(\varvec{\xi }^{(i)})$ denotes the predicted value of the model $\tilde{u}_{[i]}$ built by removing the training point $\varvec{\xi }^{(i)}$ in the training set. The leave-one-out error $E_{\mathrm{loo}}$, a.k.a predicted residual error sum of squares, is estimated by an empirical mean square of the predicted residual,

$$\begin{aligned} E_{\mathrm{loo}} = \frac{1}{N}\sum _{i=1}^N e_{[i]}^2. \end{aligned}$$

In the general case, CV can be an expensive technique due to the construction of the N models $\tilde{u}_{[i]}$. However, a fast computation of $E_{\mathrm{loo}}$ is possible in linear regression models (Seber and Lee 2003) by using the relation

$$\begin{aligned} e_{[i]} = \frac{u(\varvec{\xi }^{(i)}) - \tilde{u}(\varvec{\xi }^{(i)})}{1-h_i}, \end{aligned}$$

where $\tilde{u}(\varvec{\xi }^{(i)})$ is the single model built with all the training points, and $h_i$ is the i-th diagonal term of the hat matrix $H=P(P^{\top }P)^{-1}P^{\top }$ with P the design matrix of the linear regression. In practice, the vector $\varvec{e}_{[]}$ of the predicted residual can be directly computed from the model outputs as follows

where $I=[\delta _{ij}]\in \mathbb {R}^{N,N}$ is the identity matrix, $\oslash$ denotes the component-wise division, and $\varvec{1}=[1]^{\top }\in {\mathbb {R}^N}$ and $\mathrm{diag}(H)$ represent the diagonal part of H.