Taylor Moment Expansion for Continuous-Discrete Gaussian Filtering

Abstract

The note is concerned with Gaussian filtering in non-linear continuous-discrete state-space models. We propose a novel Taylor moment expansion (TME) Gaussian filter which approximates the moments of the stochastic differential equation with a temporal Taylor expansion. Differently from classical linearisation or Itô-Taylor approaches, the Taylor expansion is formed for the moment functions directly and in time variable, not by using a Taylor expansion on the non-linear functions in the model. We analyse the theoretical properties, including the positive definiteness of the covariance estimate and stability of the TME filter. By numerical experiments, we demonstrate that the proposed TME Gaussian filter significantly outperforms the state-of-the-art methods in terms of estimation accuracy and numerical stability.

Deep state-space Gaussian processes

“…For the wrapping function g, we choose g(u) = exp(u). For the discretization of SS-DGP, we use the 3rd-order TME method (Zhao et al 2021). We control the smoothness of f and hyperparameter processes by using α = 1 and 0, respectively (see Eq.…”

“…The Taylor moment expansion (TME) is one way to proceed instead of Euler-Maruyama (Zhao et al 2021;Kessler 1997;Florens-Zmirou 1989). This method requires that the SDE coefficients Λ and β are differentiable and there exists an infinitesimal generator for the SDE (Zhao et al 2021). The deep Matérn process satisfies these conditions provided that the wrapping function g is chosen suitably.…”

“…In short, we formulate the (temporal) DGP regression as a state-estimation problem on a non-linear continuous-discrete state-space model. For this purpose, various well-established filters and smoothers are available, for example, the Gaussian (assumed density) filters and smoothers (Särkkä 2013;Särkkä and Sarmavuori 2013;Zhao et al 2021). For temporal data, the computational complexity of using filtering and smoothing approaches is O(N ), where N is the number of measurements.…”

State-Space Gaussian Process for Drift Estimation in Stochastic Differential Equations

“…In the case of adopting higher order Itô-Taylor expansion as constructed in Section 2, the measurement function becomes non-linear, and exact Kalman filtering and smoothing cannot be applied anymore. Instead, we can resort to general Gaussian filtering and smoothing approaches for nonlinear systems such as statistically linearized and sigma-point Kalman filters (e.g., the cubature or unscented Kalman filters) [21,22]. A particularly useful state-of-the-art tool for this purpose is the iterated posterior linearization filter (IPLF) and smoother (IPLS) [23].…”

“…This gives more accurate approximations with larger time intervals. Secondly, we transform the GP into an equivalent state-space representation such that the regression of the drift function reduces to a filtering and smoothing problem in state-space [6,[19][20][21]. When the Kalman filter and Rauch-Tung-Striebel (RTS) smoother are used, the computational complexity is linear in the number of measurements.…”