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.
The ergodic hypothesis asserting the equivalence of time and ensemble averages began with Boltzmann's [1] conjecture that a single trajectory can densely cover a surface of constant energy in phase space. His proof of the hypothesis as well as many subsequent proofs were shown to be fatally flawed. It was not until metric decomposability was introduced by Birkoff [2] that a rigorous mathematical theory of ergodicity began to take shape. Kinchin, who wrote a seminal work on the mathematical foundations of statistical mechanics [3], offered a surprisingly pragmatic approach to the ergodic hypothesis. He proposed to sidestep potentially formidable proofs of ergodicity and judge the theory constructed on such assumption by its practical success or failure. This latter perspective is adopted by the vast majority of physicists when averaging over independent measurements is not possible.
Let us consider a stochastic process X(t) that can be the time series of the position of a particle undergoing the Brownian motion, inter-beat interval of human heart or a plethora of other time series generated by complex physical or physiological systems. In the absence of direct evidence to the contrary, it is assumed that such processes are ergodic so that their underlying dynamics can in principle be deduced from a single, very often historical, record X(t) measured over a sufficiently long time. In the classic paper Wang and Uhlenbeck wrote [4]:
…One can then cut the record in pieces of length T (where T is long compared to all periods occurring in the process), and one may consider the different pieces as the different records of an ensemble of observations. In computing average values one has in general to distinguish between an ensemble average and a time average. However, for a stationary process these two ways of averaging will always give the same result… This quote describes a ubiquitous process of generating a phenomenological ensemble by partitioning a single dataset. In the context of time series analysis, we also use the synonym single trajectory ensemble (STE) as a counterpart to a multiple trajectory ensemble (MTE) which is employed whenever independent measurements or realizations of a stochastic process are available. Wang and Uhlenbeck have implicitly linked the of validity of phenomenological (STE) ensembles to the ergodicity of the underlying dynamical system.
It is remarkable that Wang and Uhlenbeck’s postulate of equivalency of STE and MTE, which in fact provided justification of a half a century of empirical analyses, has not, to our knowledge, been directly tested. Motivated by our recent study [5], we revisit this postulate using the stationary and ergodic paradigm of statistical physicsthe Ornstein-Uhlenbeck (OU) Langevin equation:
where λ is the dissipation rate [6]. In the above equation, a zero-centered Gaussian random force η (t) is delta correlated in time
and the angular brackets denote an average over an ensemble of realizations of the random force. X(t) is also a Gaussian random process with a spectrum [4]:
Consequently, from the convolution theorem one obtains the autocorrelation function of X(t)
X(t) may be expressed as the formal solution to the firstorder, linear, stochastic differential equation ( 1)
In Fig. 1 we present a solution X(t) generated by numerical integration of Eq. ( 1) with a constant time step ∆t = 1 (λ=0.025 and σ η =7.8).
According to Wang and Uhlenbeck’s prescription, a long time series generated by the OU Langevin equation can be partitioned to yield a STE. We know that the solution to the OU Langevin equation is ergodic so that averages obtained using STE and MTE should coincide. Variance is the most commonly used measure of time series variability. Therefore, let us perform computer simulations to calculate this metric for both ensembles. In Fig. 2, σ 2 of the solution X(t) of the Langevin equation ( 1) is plotted as a function of the length τ of the data window. The MTE variance σ 2 M is denoted by opened squares and was calculated using n M = 3000 trajectories of length N M = 1000 originating at zero. The STE variance σ 2 S , represented by open circles, was computed by partitioning a single trajectory of length N S = n M N M into segments of length τ (we used a sliding window algorithm). It is obvious that the two ways of calculating the variance are not equivalent. 1) is plotted as a function of the length of the data window τ . The model’s parameters are the same as in Fig. 1. The numerical estimate of variance was calculated using the single (circles), multiple (squares) and threshold (filled squares) trajectory ensembles. The theoretical value of the variance is drawn for the MTE (solid line, cf. Eq. ( 6)) and the STE (dashed line, cf. Eq. ( 12)). The sliding window algorithm was used to generate the STE. The non-overlapping window partitioning yielded the same results. The ratio of the asymptotic variances for STE and MTE is is given by Eq. ( 13
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