Geometric and projection effects in Kramers-Moyal analysis

Steven J. Lade

Phys. Rev. E 80, 031137 – Published 24 September 2009

Abstract

Kramers-Moyal coefficients provide a simple and easily visualized method with which to analyze nonlinear stochastic time series. One mechanism that can affect the estimation of the coefficients is geometric projection effects. For some biologically inspired examples, these effects are predicted and explored with a nonstochastic projection operator method and compared with direct numerical simulation of the systems’ Langevin equations. General features and characteristics are identified, and the utility of the Kramers-Moyal method is discussed. Projections of a system are in general non-Markovian, but here the Kramers-Moyal method remains useful, and in any case the primary examples considered are found to be close to Markovian.

Received 27 May 2009

DOI:https://doi.org/10.1103/PhysRevE.80.031137

Authors & Affiliations

Steven J. Lade^*

Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Australian Capital Territory 0200, Australia

^*steven.lade@anu.edu.au

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Vol. 80, Iss. 3 — September 2009

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Images

Figure 1
Projected (top) drift and (bottom) diffusion coefficients for biased diffusion on a sphere. Semianalytical predictions from Eq. (10) are shown with solid lines, together with results from direct numerical simulation, using Eq. (2), for restoring force strength $k = 0$ (circles), 1 (triangles), and 5 (squares). Preferred binding angles (and reference for coordinate transformation) were $θ_{0} = π / 2, π / 4, π / 4$ , respectively. The diffusion coefficient on the surface was $D = 1$ and the radius of the sphere (length of the tether) $r = 1$ in all cases.Reuse & Permissions
Figure 2
Semianalytical and numerical estimates for projected (top) drift and (bottom) diffusion coefficients for compound diffusion on a sphere, as described in the text. Parameters for the first rod are the same as for Fig. 1, and with the same marker shapes. Surface diffusion coefficient and radius for the second rod (with respect to the first) are also $D = 1$ and $r = 1$ , respectively.Reuse & Permissions
Figure 3
Sample trajectory (top) and projected (left) drift and (right) diffusion coefficients for the projection $X = x_{1}$ of the system of Eqs. (11) with $a = 0$ , $b = 1$ , $c = - 1$ , $d = - 0.1$ , $D_{1} = 1$ , and $D_{2} = 10$ by direct numerical simulation (circles) and analytically (lines). Analytical predictions were calculated using Erickson’s results [19] for multidimensional Ornstein-Uhlenbeck processes, giving $D^{(1)} (X) = \frac{(a + d) (a d - b c) D_{1}}{(a d - b c + d^{2}) D_{1} + b^{2} D_{2}} X$ and $D^{(2)} (X) = D_{1}$ , under suitable stability conditions on the SDEs (11).Reuse & Permissions
Figure 4
Chapman-Kolmogorov equation for projected biased diffusion on a sphere (9) with $τ = 0.1$ . Contours, in logarithmic scale, of the left-hand side of the equation are in gray and of the right-hand side in black.Reuse & Permissions
Figure 5
(Left) Normalized autocorrelation and (right) probability density (circles) with Gaussian fit (solid line) of the reconstructed noise, as per the a posteriori test of Micheletti et al. [21], for projected biased diffusion on a sphere (9). The test reconstructs the effective noise $d W$ from the reconstructed drift and diffusion coefficients, assuming a Langevin equation of the form $d X = g (X) d t + h (X) d W$ holds. If $d W$ has delta-autocorrelation and Gaussian probability density, claims the test, the process is Markov.Reuse & Permissions
Figure 6
(Left) Chapman-Kolmogorov equation with $τ = 1$ for the projection onto $X = x_{1}$ of the Ornstein-Uhlenbeck process [Eqs. (11)] with parameters as in Fig. 3. Contours, in logarithmic scale, of the left-hand side are in gray and of the right-hand side in black. They clearly do not coincide. (Right) Autocorrelation of the reconstructed noise, as per the Micheletti test described in Fig. 5. The noise is clearly not delta-autocorrelated.Reuse & Permissions

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