Minimal-entropy velocity estimation from GPS position time series

Abstract

Abstract We propose a nonparametric method based on minimal-entropy for estimating an optimal velocity and its realistic variance from a position time series of unknown noise. We show based on simulations that minimal-entropy derived velocity is accurate and its uncertainty is realistic in the presence of colored noise. We then show that entropy and hence the proposed method is unaffected by periodic loading effects and use 130 CORS GPS time series to numerically verify that the proposed method is even more conclusive, reaching less scattered results, than existing methods for short time series. Although the presence of discontinuities complicates everything, we demonstrate by simulation that minimal-entropy velocity estimation offers a theory for handling discontinuities if they are viewed as independent binary random variables. This offers a new tool for investigating the effect of increasing number of step discontinuities on velocity estimation. Finally, viewing the timing variable as a random variable, uniformly distributed in the case of no data gaps or not if gaps do exist, offers a tool for dealing with the effect of data gaps (or the so-called “censored data”) on velocity estimation. The entropy of a stochastic process is a single unique quantity of the process which expresses its irreducible complexity/uncertainty, beyond which there is no simplification (or “compression”). It is based on the probability density function (pdf) of the process rather than only one or two of its moments, thus it senses all stochastic properties related to variations within the series. Because it is nonparametric, the proposed method neither requires appriori knowledge the type of stochastic noise contaminating the position time series nor its pdf.