Fluctuation of similarity to detect transitions between distinct dynamical regimes in short time series

A method to identify distinct dynamical regimes and transitions between those regimes in a short univariate time series was recently introduced [N. Malik et al., Europhys. Lett. 97, 40009 (2012)], employing the computation of fluctuations in a measure of nonlinear similarity based on local recurrence properties. In this work, we describe the details of the analytical relationships between this newly introduced measure and the well-known concepts of attractor dimensions and Lyapunov exponents. We show that the new measure has linear dependence on the effective dimension of the attractor and it measures the variations in the sum of the Lyapunov spectrum. To illustrate the practical usefulness of the method, we identify various types of dynamical transitions in different nonlinear models. We present testbed examples for the new method's robustness against noise and missing values in the time series. We also use this method to analyze time series of social dynamics, specifically an analysis of the US crime record time series from 1975 to 1993. Using this method, we find that dynamical complexity in robberies was influenced by the unemployment rate until the late 1980s. We have also observed a dynamical transition in homicide and robbery rates in the late 1980s and early 1990s, leading to increase in the dynamical complexity of these rates.

Figure 1

Schematic representation of the ${\epsilon }_j\left(k\right)$ ball neighborhood $U\left({\mathbf{x}}_j\right)$, corresponding to the $k$ nearest neighbors of ${\mathbf{x}}_j$, and its deformation (region within red boundary) due to application of the mapping $\phi$, ${\mathbf{x}}_{j+1}=\phi \left({\mathbf{x}}_j\right)$. Note that the neighborhood of ${\mathbf{x}}_{j+1}$ corresponding to its $k$ nearest neighbors is usually different. Locally at ${\mathbf{x}}_j$ the mapping $\phi$ is approximated by a linear transformation. Expansion of $U\left({\mathbf{x}}_j\right)$ by inclusion of more points rescales $\phi \left[U\left({\mathbf{x}}_j\right)\right]$ by stretching or contraction in different directions. Hence, their radii will scale by the same exponent.

Figure 2

(a), (b) Scaling laws for the Rössler system for two different randomly chosen time points taken over a short time series of length $N=4500$. See Eq. (6) for red lines (lines fitted to the dots) and Eq. (8) for blue lines (lines fitted to the open squares). Note that $\frac{{\beta }_j}{{\alpha }_j}\to 1$ here in the absence of dynamical transitions in the model. The embedding parameters used were $m=10$ and $L=15$. (c), (d) Scaling laws in the logistic map for two different randomly chosen time points over a short time series of length $N=4500$. See Eq. (6) for red lines and Eq. (8) for blue lines. Note again that $\frac{{\beta }_j}{{\alpha }_j}\to 1$ as there are no dynamical transitions. The embedding parameters used were $m=3$ and $L=2$. We also observe the fluctuations in the values of exponents $\alpha$ and $\beta$, possibly caused by the shortness of these time series and numerical error in the evolution of the dynamics.

Figure 3

Convergence of ${\sigma }_S$ under sampling over windows of size $n$. We generate different realizations of ${\sigma }_S$ by bootstrapping, i.e., resampling windows of size $n$ of $S_{j|j+1}$ values with replacement and calculating ${\sigma }_S$ for each of the resampled window. ${\tilde{\sigma }}_S$ is the median of $10000$ such bootstrap realizations and $\epsilon$ gives the corresponding standard error in the estimation of the median over these realizations. (a) Logistic map, (b) Rössler system (in both cases, the total length of the time series is $N=4500$).

Figure 4

An illustration of ${\sigma }_S$'s sensitivity to the changes in the dynamical complexity: identifying transition between chaos and strange nonchaotic attractor (SNA). Chaos exists when both ${\Lambda }_T$ (largest transverse Lyapunov exponent) and ${\Lambda }_y$ (largest Lyapunov exponent) have positive values, whereas SNA exists when ${\Lambda }_y$ is negative and ${\Lambda }_T$ is positive. (a), (c) Largest transverse Lyapunov exponent ${\Lambda }_T$ (blue lines) and largest Lyapunov exponent ${\Lambda }_y$ (red dots) on varying parameter $a$ [see Eqs. (17) and (18)]. Transitions between chaos and SNA are highlighted by gray vertical band in all the panels. (a) Observe ${\Lambda }_y$ becoming negative while $a$ is increased, whereas ${\Lambda }_T$ is positive and continues to increase leading to a transition from chaos to SNA (gray band). In (b) we show ${\sigma }_S$ (black dots) calculated from time series of $x$ variable of Eq. (17). Observe the sharp drop in values of ${\sigma }_S$ as the chaos converts into SNA (gray band). (c) Observe ${\Lambda }_y$ becoming positive while $a$ is increased, whereas ${\Lambda }_T$ is positive and continues to decrease leading to a transition from SNA to chaos (gray band). (d) The value of ${\sigma }_S$ shows a sharp drop as SNA disappears into chaos.

Figure 5

Evolution of the $\epsilon \left(k\right)$ neighborhood of the time point ${\mathbf{x}}_j$ into an ellipsoid by the application of the smooth mapping $\phi$ such that ${\mathbf{x}}_{j+1}=\phi \left({\mathbf{x}}_j\right)$. The expansion or contraction in any direction $i$ is a multiple of $exp\left({\lambda }_i^j\right)$, where ${\lambda }_i^j$ are the eigenvalues of $\mathbf{D}\phi$ at $j$.

Figure 6

Correspondence between ${\mu }_S$ [blue line in (b)] and the Lyapunov exponent $\lambda$ [black line in (a)] for the logistic map. The parameters used for this figure are the same as used for Fig. 2.

Figure 7

Identifying drift in the dynamics of the Baker's map with continuously changing parameter $\beta$. As indicated in Eq. (27), ${\lambda }_1$ is constant with respect to $\beta$. In contrast, ${\lambda }_2=ln\beta$. We observe ${\sigma }_S$ changing from significantly higher values to significantly lower values as the time progresses. A quadratic fit describes this evolution (red curve), indicating the nonlinear change of parameter of the system.

Figure 8

Formation of Lorenz's attractor, when parameter $\gamma$ is varied from $23.5$ to $25.25$ [see Eq. (29)]. When $\gamma <24.06$, transient chaos exists, whereas $24.06<\gamma <24.74$ is a transition zone where multiple attractors coexist and for $\gamma >24.746$ only Lorenz's attractor exists. The ${\sigma }_S$ distinguishes between these three regions, for $\gamma <24.06$ green open squares indicate low complexity dynamics, for $\gamma >24.74$ only orange dots exist indicating higher complexity dynamics. In-between these two regions, we have a state where multiple attractors coexist, indicated by jumps in the values of ${\sigma }_S$ below and above the significance bands (red dotted horizontal lines). The thick black vertical lines are drawn at the times points when $\gamma \approx 24.06$ (crisis) and $24.74$ (subcritical Hopf bifurcation).

Figure 9

Identifying abrupt transition between chaos and intermittent chaos induced by an interior crisis. The iterations of the Ikeda map [Eq. (30)] for $\eta =7.2$ and $7.4$ belong to the precrisis chaos and the postcrisis intermittent chaos, respectively. $y_j$ is the imaginary part of $z_j$. Observe the values of ${\sigma }_S$ jumping between the green color markers (open squares) and the orange color markers (dots) as $\eta$ is abruptly changed between two dynamical regimes. The vertical gray bars indicate the region where transitions should be found and their width is equal to the window size $n$.

Figure 10

Test the effect of presence of observational noise on the method: (a) $x_j$ are values of $x$ variable of Eq. (9) sampled at time step $j$, (b) shows the effect of adding white noise to $x_j$. (c) Shows the variation of control parameter $a$, note the crossing of four gray bands by the parameter $a$. These gray bands represent the four dynamical transitions that take place when $a$ crosses the value close to $0.38$. The width and location of these bands is the same as in the noiseless case presented in [1]. (d)–(f) Show the variation of ${\sigma }_S$ with three different levels of noise added to the signal. For $1\%$ we see the smallest error bars and all the four transitions are clearly visible, i.e., crossing of significant band by values of ${\sigma }_S$ right between the gray bands. In higher noise levels, the transitions are still intact but with increasing error bars.

Figure 11

Testing the effect of missing values on the method: (a) red points represent the time points missing from the dynamics (black points) of the Rössler system at the level of $5\%$ missing values. The transitions highlighted by gray bands are introduced by changing the parameter $a$ in Eq. (9) as described above in Sec. 3d. (c)–(e) Show the variation of ${\sigma }_S$ at different levels of missing values; in all the three we observe the transitions to remain intact. An important point to note is that even on using Chebyshev distance, there is no structural change in evolution of ${\sigma }_S$ compared to the previous example (see Fig. 10).

Figure 12

(a) Black line indicates the monthly robberies in the US between 1975 to 1993, whereas the blue dotted line is the monthly unemployment rate between the same period. (b) Values of ${\sigma }_S$ calculated for monthly robberies time series, represented by green open squares, black plus signs, and orange dots. The blue dotted line is the average value of monthly unemployment rate, calculated exactly with the same window sizes as used for ${\sigma }_S$. Note the change in the values of ${\sigma }_S$ from green open squares to orange dots, representing a transition from one dynamical regime to the other, as also highlighted with the gray band. (c) Continuous color variation shows running windowed linear cross correlation $\rho$ between average values of monthly unemployment rate and values ${\sigma }_S$. Observe the high correlation between the two until 1987.

Figure 13

(a) Black line indicates the monthly homicides in the US between 1975 to 1993, whereas the blue dotted line is the monthly unemployment rate between the same period. (b) Values of ${\sigma }_S$ calculated for monthly homicides time series, represented by green open squares, black plus signs, and orange dots. The blue dotted line is the average value of monthly unemployment rate, calculated exactly with the same window sizes as used for ${\sigma }_S$. Note the change in the values of ${\sigma }_S$ from green open squares to orange dots, representing a transition from one dynamical regime to other, as also highlighted with the gray band. (c) Continuous color variation shows running windowed linear cross correlation $\rho$ between average values of monthly unemployment rate and values ${\sigma }_S$. Observe the low correlation between the two for almost over the whole of time period.

Figure 14

(a) Scatter plot between the level of unemployment and monthly robberies. (b) Scatter plot between the level of unemployment and monthly homicides. $\rho$ is the cross correlation between the plotted variables and the red color line is the linear fit. Note the low correlations between the plotted variables in both (a) and (b).