Transient dynamics of Pearson diffusions facilitates estimation of rate parameters

Warning of a forthcoming collapse of the Atlantic meridional overturning circulation

“…where µ(λ) = m + |λ| and α(λ) = 2 |λ|. For fixed λ the process is stationary and the estimators of µ, ρ = e −α∆t and γ 2 = σ 2 /2α using MLE are given in the online material, see also [25]. The asymptotic variances of the estimators obtained by inverting the Fisher information are Var(μ)…”

“…Within a short enough window, [t−T w /2, t+T w /2], we may assume λ(t) to be constant and the noise small enough, that the process is well approximated by the linear process dX t = −α(X t − µ)dt + σdB t , where µ is the mean and α is the inverse correlation time. The EWSs are obtained from the observed time series by maximum likelihood estimation (MLE), which for mean µ, one-lag autocorrelation ρ = exp(−α∆t) and variance γ 2 = σ 2 /2α are given in closed forms (see Methods) [25].…”

“…( 1) are determined: α = − log ρ/∆t, σ 2 = 2αγ 2 , A = α/2(µ − m) and λ = (σ 2 /4γ 2 ) 2 /A. Closed form estimators for µ, γ 2 and ρ are obtained from the observed time series within a running window by maximum likelihood estimation (MLE) (supplementary text S1, see also [29]). The uncertainty is expressed through the variances of the estimators γ2 and ρ obtained from the observations within a time window T w .…”

“…This is a Gaussian process with well-known properties [29,35]. The variance is γ 2 = σ 2 /2α and the ∆t-lag autocorrelation is ρ = e −α∆t .…”

Excitable dynamics in neural and cardiac systems

“…However, achieving the regime where asymptotic properties are valid can take so long that well-founded theoretical results have no practical value. In [20] , three processes of the Pearson diffusion class are studied, which are examples of leaky integrate-and-fire models, that describe the electrical properties of a neuronal membrane.…”