Time and ensemble averaging in time series analysis

Details Time and ensemble averaging in time series analysis Time and ensemble averaging in time series analysis

2010-04-13

arxiv.org/abs/1004.2153v1

Physics

Data Analysis, Statistics and Probability

4 pages, 3 figures

Scientific paper

In many applications expectation values are calculated by partitioning a single experimental time series into an ensemble of data segments of equal length. Such single trajectory ensemble (STE) is a counterpart to a multiple trajectory ensemble (MTE) used whenever independent measurements or realizations of a stochastic process are available. The equivalence of STE and MTE for stationary systems was postulated by Wang and Uhlenbeck in their classic paper on Brownian motion (Rev. Mod. Phys. 17, 323 (1945)) but surprisingly has not yet been proved. Using the stationary and ergodic paradigm of statistical physics -- the Ornstein-Uhlenbeck (OU) Langevin equation, we revisit Wang and Uhlenbeck's postulate. In particular, we find that the variance of the solution of this equation is different for these two ensembles. While the variance calculated using the MTE quantifies the spreading of independent trajectories originating from the same initial point, the variance for STE measures the spreading of two correlated random walkers. Thus, STE and MTE refer to two completely different dynamical processes. Guided by this interpretation, we introduce a novel algorithm of partitioning a single trajectory into a phenomenological ensemble, which we name a threshold trajectory ensemble (TTE), that for an ergodic system is equivalent to MTE. We find that in the cohort of healthy volunteers, the ratio of STE and TTE asymptotic variances of stage 4 sleep electroencephalogram is equal to 1.96 \pm 0.04 which is in agreement with the theoretically predicted value of 2.

Affiliated with

Also associated with

No associations

Rating

Time and ensemble averaging in time series analysis does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Time and ensemble averaging in time series analysis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Time and ensemble averaging in time series analysis will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-282257