Extracting strong measurement noise from stochastic time series: Applications to empirical data

It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time series even in the case when the amplitudes of measurement noise and uncontaminated signal are of the same order of magnitude. Our approach is based on a recently developed method for a nonparametric reconstruction of Langevin processes. Minimizing a proper non-negative function, the procedure is able to correctly extract strong measurement noise and to estimate drift and diffusion coefficients in the Langevin equation describing the evolution of the original uncorrupted signal. As input, the algorithm uses only the two first conditional moments extracted directly from the stochastic series and is therefore suitable for a broad panoply of different signals. To demonstrate the power of the method, we apply the algorithm to synthetic as well as climatological measurement data, namely, the daily North Atlantic Oscillation index, shedding light on the discussion of the nature of its underlying physical processes.

Figure 1

Langevin time series with different measurement noise strengths. Here we show (a) the probability density function $P\left(y\right)$ of the series with noise [see Eq. (3)], with the corresponding mean value $\mu$ and standard deviation $\theta$ in the inset, and the corresponding functions (b) ${\widehat{D}}_1\left(y\right)$ and (c) ${\widehat{D}}_2\left(y\right)$, see Eq. (6). In all cases, the assumed time series $x\left(t\right)$ without measurement noise uses the functions $D_1\left(x\right)=1-x$ and $D_2\left(x\right)=1-x+x^2$.

Figure 2

Noise dependence of coefficients ${\widehat{d}}_{ij}$ defining functions ${\widehat{D}}_1\left(y\right)$ and ${\widehat{D}}_2\left(y\right)$ [see text and Eq. (6)]. The underlying Langevin time series $x\left(t\right)$ without noise is the same as in Fig. 1.

Figure 3

Conditional moments ${\widehat{M}}_1\left(y_i,\tau \right)$ and ${\widehat{M}}_2\left(y_i,\tau \right)$ as a function of bin $y_i$, for $\tau =0$ and different measurement noise strengths. The asymmetry of ${\widehat{M}}_2$ is due to $d_{21}\ne 0$ [see Eqs. (7)]. The same $x\left(t\right)$ as in Fig. 1 was used.

Figure 4

Conditional moments (a) ${\widehat{M}}_1\left(y_i,\tau \right)$ and (b) ${\widehat{M}}_2\left(y_i,\tau \right)$ as a function of $\tau$, for bin $y_i=0$ and different measurement noise strengths. In (c) one compares the true measurement noise with the approximation ${\sigma }_{app}={\widehat{M}}_2\left(0,0\right)\sim 2{\sigma }^2$ given in Eq. (9). In the inset the corresponding absolute and relative errors are given by ${\zeta }_a=\left|\sigma -{\sigma }_{app}\right|$ and ${\zeta }_r={\zeta }_a/\sigma$, respectively. Errors for ${\widehat{M}}_2$ are negligible. The same $x\left(t\right)$ as in Fig. 1 was used.

Figure 5

Functions ${\widehat{m}}_1$, ${\widehat{m}}_2$, ${\widehat{\gamma }}_1$, and ${\widehat{\gamma }}_2$ (symbols) defining the conditional moments in Eqs. (8b). The underlying Langevin time series $x\left(t\right)$ without noise is characterized by a drift coefficient $D_1\left(x\right)=1-x$ and a diffusion coefficient $D_2\left(x\right)=1-x+x^2$. The measurement noise was fixed at $\sigma =1$. Each hat function is compared with the corresponding integral form in Eqs. (10) using the first estimate of parameters values (dashed lines) and the true values (solid lines).

Figure 6

Function $F$ in Eq. (11) as a function of (a) $d_{10}$, (b) $d_{11}$, (c) $d_{20}$, (d) $d_{21}$, (e) $d_{22}$, and (f) $\sigma$. The same situation as in Fig. 2 is here chosen: $D_1\left(x\right)=1-x$, $D_2\left(x\right)=1-x+x^2$, and $\sigma =1$. Dashed lines indicate the true values used for generating the data series, while the bullet indicates the estimated values of functions $D_1$ and $D_2$ for $\sigma =0$. In each plot while varying one parameter, the remaining ones are fixed at their true values (see text).

Figure 7

Functions $m_1$, $m_2$, ${\gamma }_1$, and ${\gamma }_2$ for the Langevin process with $D_1\left(x\right)=1-x$, $D_2\left(x\right)=1-x+x^2$, and $\sigma =1$. Symbols indicate the functions obtained for the data, dashed line corresponds to the first estimate of the parameters and solid line corresponds to the parameter values obtained from the Levenberg-Marquardt procedure (see text). In this case, for the first estimate one has $F_0=3720$ while the final estimate retrieves $F_{LM}=33.1$. The true minimum is $F_m=29.2$.

Figure 8

Comparison of the optimized parameters values (bullet) with the first estimate and the true values for different input measurement noise strengths ${\sigma }_I$: (a) $2{\sigma }^2$, (b) $d_{10}$, (c) $d_{11}$, (d) $d_{20}$, (e) $d_{21}$, and (f) $d_{22}$. The measurement noise is correctly extracted as well as the parameters defining the drift coefficient $D_1\left(x\right)$ which controls the deterministic part of the underlying evolution equation (see text).

Figure 9

Time-series of the NAO daily index, from 1950 until 1996. The data were extracted from [25]. More details in Ref. [16].

Figure 10

[(a) and (b)] Estimate of the drift and diffusion coefficients $D_1\left(N\right)$ and $D_2\left(N\right)$ of the daily North Atlantic index $N$ [16] ($16801$ data points), together with the corresponding (c) $m_1\left(N\right)$, (d) $m_2\left(N\right)$, (e) ${\gamma }_1\left(N\right)$, and (f) ${\gamma }_2\left(N\right)$. Results for the empirical NAO index are represented with bullets whereas the synthetic data also with $16801$ data points and parameter values given by Table is shown with circles for comparison. The corresponding fits of $D_1$ and $D_2$ and the analytical functions $m_i$ and ${\gamma }_i$ with the best parameter choices are given with solid and dashed lines, respectively.