Functional State Space Models and the Kalman Filter

In this talk, we propose a state space model for functional time series data, which extends many time series models to the realm of functional data. Most notably, we introduce the Functional ARMAX process (FARMAX), which is developed in the fully functional setting, i.e. without relying on projection onto a finite number of basis functions. These models are fit via our fully functional variant of the Kalman filter and smoother methods. The theoretical soundness of this approach is proven using tools from the theory of Gaussian measures in locally convex spaces. As an application, we consider signals data collected from small wearable medical tri-axial accelerometers affixed to a patient's wrists or ankles. Each device collects three time series (x, y, z directions) at 100Hz and can continuously collect data for 14 days.