Combining Local and Global Models to Capture Fast and Slow Dynamics in Time Series Data

Many time series exhibit dynamics over vastly different time scales. The standard way to capture this behavior is to assume that the slow dynamics are a “trend”, to de-trend the data, and then to model the fast dynamics. However, for nonlinear dynamical systems this is generally insufficient. In this paper we describe a new method, utilizing two distinct nonlinear modeling architectures to capture both fast and slow dynamics. Slow dynamics are modeled with the method of analogues, and fast dynamics with a deterministic radial basis function network. When combined the resulting model out-performs either individual system. 1 Fast and slow dynamics Scalar time series often exhibit deterministic dynamics on very different time scales (see Fig. 1). For example, sound waves exhibit fast intra-cycle variation and slow inter-cycle fluctuations. It is difficult for a single model to describe both behaviors simultaneously. A standard method for treating such data is to first apply some statistical de-trending and to then model the residuals. In financial data analysis, one sees long term (cyclic) fluctuations and interor even intraday variability. Analysts will usually focus on either the long term trend or the rapid fluctuations, but not both. In this paper we describe an alternative approach that is capable of capturing both fast and slow dynamics simultaneously. To model the slow dynamics we embed the scalar time series [1] and use the method of analogues [2]. The method of analogues essentially predicts the future by using the temporal successors of the preceding observations that are most like the current state. Such techniques have found wide application and have been seen to capture long term dynamics well [3]. The short term dynamics are now nothing more than the model prediction errors of the method of analogues prediction. We model the short time dynamics with a deterministic and parametric model structure. The choice of model structure is arbitrary, but we choose minimum description length radial basis function networks [4] because this is what we are familiar with [5, 6]. We find that the result of this technique outperforms either of the standard methods for both experimental and artificial time series data. Moreover, this combined methodology allows us to produce realistic simulations of experimental time series data. In the next section, we describe the model structure. Following this, we present results for experimental and simulated time series data. 100 200 300 400 500 600 700 800 0 10 20 30 40 50 true (blue) data; linear (green) and final (black) model; and, error (red) One step predictions (RMS(final)=2.4267 & RMS(linear)=2.5437) 0 100 200 300 400 500 600 700 800 90