Generalized system identification with stable spline kernels
This work provides a flexible framework for system identification with robust penalties and constraints, but it is incremental as it builds on existing stable spline kernel methods.
The paper extends linear system identification to a wide class of nonsmooth stable spline estimators using piecewise linear-quadratic penalties, such as 1-norm and Huber, and implements an interior-point solver (IPsolve) that is competitive with alternatives like TFOCS and FISTA, with improvements shown in robust formulations and constraints.
Regularized least-squares approaches have been successfully applied to linear system identification. Recent approaches use quadratic penalty terms on the unknown impulse response defined by stable spline kernels, which control model space complexity by leveraging regularity and bounded-input bounded-output stability. This paper extends linear system identification to a wide class of nonsmooth stable spline estimators, where regularization functionals and data misfits can be selected from a rich set of piecewise linear-quadratic (PLQ) penalties. This class includes the 1-norm, Huber, and Vapnik, in addition to the least-squares penalty. By representing penalties through their conjugates, the modeler can specify any piecewise linear-quadratic penalty for misfit and regularizer, as well as inequality constraints on the response. The interior-point solver we implement (IPsolve) is locally quadratically convergent, with $O(\min(m,n)^2(m+n))$ arithmetic operations per iteration, where $n$ the number of unknown impulse response coefficients and $m$ the number of observed output measurements. IPsolve is competitive with available alternatives for system identification. This is shown by a comparison with TFOCS, libSVM, and the FISTA algorithm. The code is open source (https://github.com/saravkin/IPsolve). The impact of the approach for system identification is illustrated with numerical experiments featuring robust formulations for contaminated data, relaxation systems, nonnegativity and unimodality constraints on the impulse response, and sparsity promoting regularization. Incorporating constraints yields particularly significant improvements.