Estimation of Smooth Time-Varying Parameters in State Space Models

Abstract

In this article, we propose a penalized likelihood method to estimate time-varying parameters in standard linear state space models. The time-varying parameter is modeled as a smoothing spline and then expressed as a state space model. The maximum likelihood method is used to estimate the smoothing parameter. The proposed method is assessed by a simulation study and applied to virological response data from an HIV-infected patient receiving antiretroviral treatment.

Semiparametric modeling of autonomous nonlinear dynamical systems with application to plant growth

“…Ramsay, Hooker, Campbell and Cao (2007) consider modeling a continuously stirred tank reactor. Zhu and Wu (2007) adopt a state space approach for estimating the dynamics of cell-virus interactions in an AIDS clinical trial. Poyton et al (2006) use the principal differential analysis approach to fit dynamical systems.…”

“…In a recent work, Cao, Fussmann and Ramsay (2008) model a nonlinear dynamical system using splines with predetermined knots for describing the gradient function. Most of the existing approaches assume known functional forms of the dynamical system; and many of them require data measured on a dense grid (e.g., Varah, 1982;Zhu and Wu, 2007).…”

“…For example, Ramsay et al (2007) consider modeling a continuously stirred tank reactor and propose a method called parameter cascading for model fitting. Zhu and Wu (2007) adopt a state space approach for estimating the dynamics of cell-virus interactions in an AIDS clinical trial. Silverman (2002, 2005) consider fitting dynamical systems given by systems of linear differential equations where the coefficients of the differential operator may be time varying.…”

Forecasting with time-varying vector autoregressive models

“…A recent advance in time series analysis is the development of autoregressive (AR) models, and more generally autoregressive moving average models, with time-varying coefficients. Such models are developed as in Gersch (1985, 1996), Dahlhaus (1997), West et al (1999), Prado and Huerta (2002), Andrieu et al (2003), Lundbergh et al (2003), Francq and Gautier (2004), Moulines et al (2005), Huerta and Prado (2006), Abramovich et al (2007), Triantafyllopoulos and Nason (2007), and Zhu and Wu (2007). All these studies refer to univariate time series; attempts to model vector time series with time-varying autoregressive (TV-VAR) models include Jiang and Kitagawa (1993), Sarantis (2006), Sato et al (2007), and Triantafyllopoulos (2007).…”