Modeling continuous processes from data

Experimental and simulated time series are necessarily discretized in time. However, many real and artificial
systems are more naturally modeled as continuous-time systems. This paper reviews the major techniques
employed to estimate a continuous vector field from a finite discrete time series. We compare the performance
of various methods on experimental and artificial time series and explore the connection between continuous
~differential! and discrete ~difference equation! systems. As part of this process we propose improvements to
existing techniques. Our results demonstrate that the continuous-time dynamics of many noisy data sets can be
simulated more accurately by modeling the one-step prediction map than by modeling the vector field. We also
show that radial basis models provide superior results to global polynomial models.