A0281
Title: Dynamic modelling of functional data using warped solutions of SDEs
Authors: Anders Tolver - University of Copenhagen (Denmark)
Niels Lundtorp Olsen - University of Copenhagen (Denmark) [presenting]
Abstract: Many models for functional data assume that temporal and amplitude variation can be separated. However, complex interactions may exist e.g. when data are obtained from biomechanical systems where changing the frequency will likely influence the amplitude of a signal. We suggest to jointly model the phase and amplitude variation of a function by noisy differential equations, i.e. stochastic differential equations, but crucially the SDEs are inhomogeneous with a warped time argument.We restrict ourselves to second-order SDEs on the form: $\dot{X}(t)= F({X}(t), \dot{X}(t), v(t)) dt+ \sigma dW_t$. Based on discrete and noisy observations of the signals solving the SDE we demonstrate how maximum likelihood estimation may be carried out. Under certain linearity conditions, the Kalman filter can be applied, and used together with MCMC methods for estimation. We demonstrate the method on simulated and real data. One important example is functional data with locally periodic structures arising from applying periodic SDEs with warped time arguments. The approach allows us to model the influence of the local internal speed of the signal on the dynamical relation between derivatives. In particular, it has the potential of predicting the functional response when altering the timing of the system.
