Estimating Kramers–Moyal coefficients in short and non-stationary data sets

Multiscale reconstruction of time series

“…A second possibility to estimate these probabilities is to use the ansatz presented in [7,12,15,16]. The probability densities are now obtained as a solution of the Fokker-Planck equation:…”

Low-sampling-rate Kramers-Moyal coefficients

“…[6]). Aiming to simplify the notation, we denote the statistical averages conditioned to the initial value x 0 : · · · | x=x0 ≡ · · · , while we will reserve · · · u for usual stationary unconditioned averages.…”

In vitro chemotaxis and tissue remodeling assays quantitatively characterize foreign body reaction

“…Due to these findings, a biased random walk model for the migration angle was developed. Such continuous models enable unraveling complex empirical data (van Mourik et al, 2006). The investigated model parameters allowed assessing the biomaterials' potential to induce fibroblast migration.…”

“…When supposing a small constant time shift Δt between all changes in orientation, migration direction is either changed by a small angle Δα(t) or remains. From the conditional probability density P(α+∆α,t+∆t│α,t) of distinct time steps, the Kramers-Moyal coefficients can be calculated by to parameterize the respective random walk model (van Mourik et al, 2006). The parameterized model allows analyzing system dynamics and identification of migration stability and robustness.…”