Improved statistical test for nonstationarity using recurrence time statistics

We have recently introduced a measure for nonstationarity using a recurrence time statistic to assess stationarity. In this paper we propose an extension of this method based on a detailed study of the statistics for the case of stationary systems. We derive a simple scheme that allows us to estimate the effective number of degrees of freedom relevant for this statistic. This substantially improves the statistical significance of the method and can be used to improve the significance of various other nonlinear statistics.

Figure 1

(a) A priori expected frequency distributions ${\varphi }_{N,r,t}\left(l\right)$ and (b) distribution functions ${\Phi }_{N,r,t}\left(l\right)$, respectively, for $N={10}^4$, $r∊\left\{1,2000,7000\right\},$ and $t=100$ under the assumption of a stationary system.

Figure 2

Relation of neighboring vectors of two successive and correlated reference vectors ${\vec{x}}_r$ and ${\vec{x}}_{r+1}$ with $\vec{x}=\left(x_1,x_2\right)$.

Figure 3

Histograms of $\tilde{l}$ for time series $M\left(1\right)$ (left column) and $M\left(10\right)$ (right column) with $k=4$ neighboring vectors for embedding dimensions $m=1$ and $m=10$ and different correction schemes. From top to bottom: (1) without correction, (2) with $\text{RN}$-correction, and (3) with $\text{RN}$ and $\text{NN}$ corrections.

Figure 4

Distance $D\left(L\right)$ between the empirical distribution of the KS test ${\Phi }_{KS}\left(D_{meas}\right)$ and the theoretical distribution function in dependence on the number of degrees of freedom $L$. The distribution function ${\Phi }_{KS}\left(D_{meas}\right)$ from (a) the $M\left(1\right)$ process and (b) the $M\left(10\right)$ process were estimated both with 200 realizations using embedding dimensions $m=1$ and $m=10$, $k=4$ nearest neighbors, and $N={10}^4$ data points leading to $L=4\times {10}^4$.

Figure 5

Empirical frequency distribution ${\varphi }^{KS}\left(D_{meas}\right)$ of 200 realizations of the $M\left(10\right)$ process estimated with $N={10}^4$, $m=10$, and $k=4$ and the theoretical frequency functions $p^{KS}\left(L\right)$, $p^{KS}\left({\widehat{L}}_{\mathit{eff}}\right)$, and $p^{KS}\left(L_{\mathit{min}}\right)$.

Figure 6

Estimated divisor ${\widehat{q}}_{eff}$ (symbols) in dependence on the embedding dimension $m$ for different numbers of nearest neighbors $k∊\{1,2,4\}$ and different processes: (a) $M\left(1\right)$, (b) $M\left(10\right)$, and (c) $M\left(50\right)$. Approximation $q(m,k,t)$ of $q_{eff}$ using Eq. (8) added as different lines, solid $\left(k=1\right)$, dashed $\left(k=2\right)$, and dotted $\left(k=4\right)$.

Figure 7

Distance $D_{meas}$ in dependence on the embedding dimension $m$ with $k=4$ nearest neighbors and $N={10}^5$ data points, for one realization of stationary $\left(B1\right)$ and nonstationary $(B2,B3)$ baker’s map (upper part) and stationary $\left(L1\right)$ and nonstationary $(L2,L3)$ Lorenz system (lower part).

Figure 8

Empirical frequency distribution ${\varphi }^{KS}\left(D_{meas}\right)$ of $500$ realizations of the $M\left(10\right)$ process with $m=10$, the stationary baker’s map $\left(B1\right)$ with $m=6$, and the stationary Lorenz system $\left(L1\right)$ with $m=24$ all estimated using $k=4$ nearest neighbors and different observation times $N={10}^3$, $N={10}^4$, and $N={10}^5$ (steps). The theoretical frequency functions $p^{KS}\left(L_q\right)$ are depicted as different lines.

Figure 9

$D_{meas}$ in dependence on the observation time $N$ (in data points) for (a) stochastic processes $M\left(1\right)$, $M\left(10\right)$, and $M\left(50\right)$; (b) stationary $\left(B1\right)$ and nonstationary baker’s maps $(B2$ and $B3)$, and (c) stationary $\left(L1\right)$ and nonstationary $(L2$ and $L3)$ Lorenz system. Mean of $\varphi \left(D_{meas}\right)$ as points and $2\sigma$ in error bars, estimated from $500$ realizations with embedding parameters $m=10$ (a), $m=6$ (b), and $m=24$ (c) with $k=4$ neighboring vectors. The confidence interval $C_{0.1}\left(L\right)$ of the theoretical distributions $p^{KS}$ for $L_q$ and $L_{\mathit{min}}$ as number of degrees of freedom at a significance level $\alpha =0.1$ for a one-sided test is plotted as lines.

Figure 10

Autocorrelation function $\rho \left(\nu \right)$ of the $x$ component of the Lorenz system using different lengths $N$ of the time series.

Figure 11

Relative statistical error of the nonlinear local prediction error in dependence on the number of data points $N$ for systems $\left(B1\right)$ and $\left(L1\right)$. Errors $\delta$ were calculated using $200$ realizations of a Monte Carlo simulation for each observation time (empty circles). Estimated error $\delta$ with $L_q$ degrees of freedom (solid circles) and with $L_{\mathit{min}}$, respectively, $L$ as number of degrees of freedom (error bars).