Clustering and Forecasting in Fully Functional Time Series Models -
Modern datasets often have a temporal feature that must be accounted for in modeling. This becomes more troublesome when the data in question is functional instead of univariate or multivariate.  For example, we can consider 8 hours of EEG signals collected from pediatric hospital patients broken into 30-sec chunks.  The result is a time series of 960 curves to be clustered according to the sleep state of the child (REM, non-REM, etc).  Another example is tracking the movement of a person's feet with accelerometers as they go for a walk.  We may need to smooth this noisy signal and consider forecasting these movements into the future.  While such problems can be addressed with univariate time series methods or by projecting functional data onto a finite basis, we argue that working with functional data in the proper infinite dimensional mathematical setting for such data can result in superior performance in statistical models.  We will discuss our recent and ongoing work on functional Hidden Markov Models as well as other time series models currently being investigated.