Analysis of stochastic time series in the presence of strong measurement noise

An alternative approach for the analysis of Langevin-type stochastic processes in the presence of strong measurement noise is presented. For the case of Gaussian distributed, exponentially correlated, measurement noise it is possible to extract the strength and the correlation time of the noise as well as polynomial approximations of the drift and diffusion functions of the underlying Langevin equation.

Figure 1

Values $\mathrm{z̃}$ derived from experimental data (Ornstein-Uhlenbeck process) with and without measurement noise ($\sigma =1$ and $\sigma =0$, respectively). The correlation time $T$ is given in units of $\Delta t$. The process time scale is $100\Delta t$.

Figure 2

Probability density function for data set A. $\sigma$ is the standard deviation of the superimposed noise. Drift and diffusion functions are given by $D^{(1)}=-x$ and $D^{(2)}=2$.

Figure 3

Absolute error $\Delta \sigma =\mathrm{\sigma ̃}-\sigma$ in the estimation of the measurement noise of data set A. The dashed lines indicate the expected lower bound of the standard deviation of $\Delta \sigma$.

Figure 4

Estimated polynomial coefficients of $D^{(1)}$ for data set A with $a_0$ and $a_1$ according to Eq. (41).

Figure 5

Estimated polynomial coefficient $b_0$ of $D^{(2)}$ for data set A with $b_0$ according to Eq. (41).

Figure 6

Probability density function of data set B. $\sigma$ is the standard deviation of the superimposed noise. Drift and diffusion functions are given by $D^{(1)}=1-x$ and $D^{(2)}=2-2x+2x^2$.

Figure 7

Results for data set B. (a–e) Estimations of the polynomial coefficients $a_j$ and $b_j$ and (f) absolute error $\Delta \sigma$ in the estimation of the measurement noise. The superimposed noise is delta correlated.

Figure 8

Results for data set B. (a–e) Estimations of the polynomial coefficients $a_j$ and $b_j$, (f) correlation time $T$, and (g) absolute error $\Delta \sigma$ in the estimation of the measurement noise. The superimposed noise has a correlation time of $T=2.0$.

Figure 9

Results for data set B1. (a–e) Estimations of the polynomial coefficients $a_j$ and $b_j$ and (f) absolute error $\Delta \sigma$ in the estimation of the measurement noise. The superimposed noise is delta correlated.

Figure 10

Results for data set B1. (a–e) Estimations of the polynomial coefficients $a_j$ and $b_j$, (f) correlation time $T$, and (g) absolute error $\Delta \sigma$ in the estimation of the measurement noise. The superimposed noise has a correlation time of $T=2.0$.