Continuous-Discrete Differentiable Particle Filters for Irregular Time Series

Continuous-discrete state space models (CDSSMs) enable modelling of irregular time series by learning the underlying continuous-time dynamics with noisy measurements obtained at discrete timestamps. Recent studies have shown remarkable performance involving CDSSMs with neural networks. However, challenges still remain in the application of general non-linear non-Gaussian CDSSMs to irregular time series. To address these challenges, we propose a new method, named continuous-discrete differentiable particle filters (CD-DPFs), to model probabilistic irregular time series. Representing the latent state probability density function by a Gaussian mixture model (GMM), an adaptive Gaussian sum particle filter is applied into CDSSMs, and the weights of the GMM are optimised adaptively through solving a convex optimisation problem by direct matching the Fokker-Planck-Kolmogorov equation. Performance is evaluated on a stochastic Lorenz 63 model, a highly non-linear chaotic system. Compared with the state-of-the-art, the proposed method demonstrates significant improvement for forecasting whilst maintaining competitive performance on imputation.