Estimation of drift and diffusion functions from unevenly sampled time-series data

Complex systems can often be modeled as stochastic processes. However, physical observations of such systems are often irregularly spaced in time, leading to difficulties in estimating appropriate models from data. Here we present extensions of two methods for estimating drift and diffusion functions from irregularly sampled time-series data. Our methods are flexible and applicable to a variety of stochastic systems, including non-Markov processes or systems contaminated with measurement noise. To demonstrate applicability, we use this approach to analyze an irregularly sampled paleoclimatological isotope record, giving insights into underlying physical processes.

Figure 1

Results for an Ornstein-Uhlenbeck process. Estimated functions $D^{\left(1\right)}\left(x\right)$ and $D^{\left(2\right)}\left(x\right)$ are shown in the top and bottom plots, respectively. Estimates from HBR are from interpolated data.

Figure 2

Results for a bistable system with correlated noise. As Fig. 1.

Figure 3

Climate variations in the Early Eocene, recorded in benthic foraminiferal ${\delta }^{13}\text{C}$ data [24]. A running mean of 1-Ma has been subtracted to remove longer-scale climate effects. Time-series data and a simulated trajectory are shown in the top and bottom plots, respectively. Histograms are shown in the right plot. By convention, axes for ${\delta }^{13}\text{C}$ are reflected.

Figure 4

Results for early Eocene ${\delta }^{13}\text{C}$ record. Estimated drift and diffusion functions $D^{\left(1\right)}\left(x\right)$ and $D^{\left(2\right)}\left(x\right)$ are shown in the top and bottom plots, respectively. Best estimates are plotted as black lines, and bootstrapped $95\%$ confidence intervals are shown as grey regions [57].