Kernel Analog Forecasting: Multiscale Test Problems

Abstract

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale-separation is not present and the predicted variables are non-Markovian, are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.

Kernel-based prediction of non-Markovian time series

“…Finally, the RKHS structure allows one to evaluate the estimator on new data points using a classical interpolator, the Nyström projection method. It should be noted that this construction does not require that the covariate time series is Markovian, and is therefore well suited to forecasting under partial observations; e.g., see [4] for applications of KAF to prediction of slow components of multiscale systems exhibiting averaging or homogenization.…”

Learning to Forecast Dynamical Systems from Streaming Data

“…Yet, at the same time, the conditional expectation may not be a good approximation for actual dynamical trajectories, which makes direct regression approaches unsuitable for simulating the statistical behavior of the system (despite yielding RMSE-optimal forecasts). For further details, see the paper [13], which studies applications of KAF to multiscale systems with averaging and homogenization limits. A recent paper [38] has explored applications of kernel learning [63] to forecasting with kernel regression.…”

“…Case Study: L96. In our second set of experiments, we explore the performance of scalable KAF for the L96 system in the periodic, quasi-periodic, and chaotic regimes docu-mented in [13]. An increase in the forcing constant F generates more chaotic behavior and, unsurprisingly, reduces the time horizon for which KAF can make informative forecasts.…”

“…, x(9) of the coupled system using only the slow variables as input data. This setup is motivated by the experiments of [13], which studied KAF for multi-scale systems but did not investigate the scalability as a function of the amount of training data. 4.2.…”

Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization

“…To achieve this robust learning, we improve upon several leading kernel and system identification algorithms. Recent works have successfully applied kernel methods [26,27] to study data-driven dynamical systems [7,[28][29][30][31]. A key inspiration for the present work is kernel DMD (kDMD, [7]), which seeks to approximate the infinite-dimensional Koopman operator as a large square matrix evolving nonlinear functions of the original state.…”