Continuous-time DC kernel — A stable generalized first order spline kernel

Abstract

The stable spline (SS) kernel and the diagonal correlated (DC) kernel are two kernels that have been applied and studied extensively for kernel-based regularized LTI system identification. In this note, we show that similar to the derivation of the SS kernel, the continuous-time DC kernel can be derived by applying the same "stable" coordinate change to a "generalized" first-order spline kernel, and thus can be interpreted as a stable generalized first-order spline kernel. This interpretation provides new facets to understand the properties of the DC kernel. In particular, we derive a new orthonormal basis expansion of the DC kernel, and the explicit expression of the norm of the RKHS associated with the DC kernel.Moreover, for the non-uniformly sampled DC kernel, we derive its maximum entropy property and show that its kernel matrix has tridiagonal inverse.

On the Coordinate Change to the First-Order Spline Kernel for Regularized Impulse Response Estimation

“…Moreover, the coordinate change functions X(t) are all e −αt : R 0+ → [0, 1] for these three kernels, while the kernel W is the second order spline kernel for the SS kernel, the first order spline kernel for the TC kernel, and a generalized first order spline kernel for the DC kernel, cf. [16]. In particular, the TC kernel,…”

“…We show such an orthonormal basis for H G 0 , which can yield a reasonable finite dimensional approximation of H G 0 and can make some computations easy and fast. In this section, we focus on (12) where G 0 (s) = 1 (s+α) n+1 with n = 1, 2, . .…”

“…Interestingly, these three kernels share some common properties [7,12]. For example, they can be derived by applying an exponential decay function as a coordinate change to different kinds of spline kernels [13] (cf.…”

FRF estimation using multiple kernel-based regularisation

“…Here lays the heart of kernel-based modelling. In both existing time-and frequency domain approaches (Chen et al, 2012) (Lataire & Chen, 2016), it is imposed that the impulse response is smooth, stable and causal and, hence, can be captured by the discrete-or continuous-time DC (diagonal correlated) or TC (tuned correlated) kernel, which are first order stable-spline kernels (Chen, 2018a(Chen, , 2018bPillonetto, Chiuso, & De Nicolao, 2011). In Scandella, Mazzoleni, Formentin, and Previdi (2020), the use of higher order stable-spline kernels is elaborated, where the order of the kernel is a hyper parameter that is derived from the data.…”

Estimation of sparse linear dynamic networks using the stable spline horseshoe prior

“…A maximum entropy derivation of this prior can be found in [8], while other important classes of covariances to model dynamic systems are e.g. described in [9, 7,36,35,10,13]. The use of the stable spline prior for linear system identification has shown some important advantages in comparison with other classical techniques like parametric prediction error methods [28].…”