# A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots

## Abstract

**:**

## 1. Historical Background

## 2. Recurrence Plots in Engineering Research

## 3. About This Tutorial Review

## 4. Consequences of Nonlinearity

#### 4.1. Predictability

#### 4.2. Transitions

#### 4.3. Synchronization

#### 4.4. Characterization

## 5. Dynamical Systems: The Basics

#### 5.1. What Is a Dynamical System?

#### 5.2. Attractors

- Invariance: the attractor should map to itself under $\mathbf{F}$.
- Attractivity: any set of initial conditions in the state space should, for large t i.e., $t\to \infty $, converge to the attractor.
- Irreducibility: the attracting set of states should be connected by one trajectory and it should not be possible to decompose the attractor to subsets of states which have non-overlapping trajectories. In this case, each subset would be an attractor and not their union.
- Persistence: the attractor should be stable under small perturbations, i.e., small deviations from the trajectory on the attractor should return back to the attractor.
- Compactness: the attracting set of states for the dynamic should be compact.

#### 5.3. Bifurcations

## 6. State Space Reconstruction

#### 6.1. The Measurement Paradigm and Time Delay Embedding

#### 6.2. Time Delay Embedding in Practice

- Determining the time delay. The choice of $\tau $ impacts the resulting embedding critically. When $\tau $ is smaller than the desired value, consecutive coordinates of ${\mathbf{y}}_{t}$ are correlated and the attractor is not sufficiently unfolded. When $\tau $ is larger than the desired value, successive coordinates are almost independent, resulting largely uncorrelated cloud of points in ${\mathbb{R}}^{m}$ without much structure. It is important that the fundamental idea in determining the time delay $\tau $ is that each coordinate of the reconstructed m-dimensional vector ${\mathbf{y}}_{t}$ must be functionally independent. In order to achieve this, it is recommended to set $\tau $ to the first zero-crossing of the autocorrelation function. However, the autocorrelation function captures only linear self-interrelations, and it is more preferable to use the first minimum of the self-mutual information function (Figure 8a), as first shown in [94]. For the scalar time series ${\left\{{s}_{t}\right\}}_{t=1}^{N}$, the self-mutual information at lag $\tau $ is,$$I\left(\tau \right)=\underset{\tilde{S}}{\int}\phantom{\rule{0.166667em}{0ex}}\underset{\tilde{S}\left(\tau \right)}{\int}p(\tilde{s},\tilde{s}\left(\tau \right))log\left({\displaystyle \frac{p(\tilde{s},\tilde{s}\left(\tau \right))}{p\left(\tilde{s}\right)\phantom{\rule{0.166667em}{0ex}}p\left(\tilde{s}\left(\tau \right)\right)}}\right),$$$${\tau}_{e}=\underset{\tau}{min}\left\{\tau :{\displaystyle \frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}I\left(\tau \right)}{\mathrm{d}\tau}}=0,{\displaystyle \frac{{\mathrm{d}}^{2}\phantom{\rule{0.166667em}{0ex}}I\left(\tau \right)}{\mathrm{d}{\tau}^{2}}}>0\right\}.$$
- Determining the embedding dimension. The method of false nearest neighbours (FNN) put forward by Kennel, Brown and Abarbanel in 1992 [95] is typically used to determine the embedding dimension, once a time delay $\tau $ is chosen. This approach is based on the geometric reasoning that given an embedding ${\mathbf{y}}_{t}^{m}$ in dimension m, it is possible to differentiate between ‘true’ and ‘false’ neighbours of points on the reconstructed trajectory. In this method, we first choose a reasonable definition of ‘neighbourhood’. Based on this definition, we identify the neighbours of all points on the trajectory in ${\mathbb{R}}^{m}$. Next, we look for the false neighbours, defined as those neighbours which cease to be neighbours in $m+1$ dimensions, i.e., when we consider the trajectory ${\mathbf{y}}_{t}^{m+1}\in {\mathbb{R}}^{m+1}$. The false neighbours were neighbours in the lower dimensional embedding solely because the attractor was not properly unfolded and we were actually looking at a projection of the attractor rather than the attractor itself. As an example, consider the 2D limit cycle trajectory on a circle, where opposite points that are almost on the same vertical line would be seen as neighbours if the same dynamic were projected on to the horizontal axis, i.e., the 1D real line ${\mathbb{R}}^{1}$. Once the attractor is properly unfolded, however, the number of false neighbours would go to zero. In practice, this notion is implemented by the following formula (after Equation (3.8) of [86]),$$FNN\left(r\right)=\frac{{\displaystyle \sum _{i=1}^{N-m\tau}}\mathsf{\Theta}\left({\displaystyle \frac{\left|\right|{\mathbf{y}}_{i}^{m+1}-{\mathbf{y}}_{k(i,\phantom{\rule{0.166667em}{0ex}}m)}^{m+1}\left|\right|}{\left|\right|{\mathbf{y}}_{i}^{m}-{\mathbf{y}}_{k(i,\phantom{\rule{0.166667em}{0ex}}m)}^{m}\left|\right|}}-r\right)\mathsf{\Theta}\left({\displaystyle \frac{\sigma}{r}}-\left|\right|{\mathbf{y}}_{i}^{m}-{\mathbf{y}}_{k(i,\phantom{\rule{0.166667em}{0ex}}m)}^{m}\left|\right|\right)}{{\displaystyle \sum _{i=1}^{N-m\tau}}\mathsf{\Theta}\left({\displaystyle \frac{\sigma}{r}}-\left|\right|{\mathbf{y}}_{i}^{m}-{\mathbf{y}}_{k(i,\phantom{\rule{0.166667em}{0ex}}m)}^{m}\left|\right|\right)},$$

## 7. Recurrence-Based Analysis

#### 7.1. Recurrence Plots

#### 7.2. Recurrence Networks

#### 7.3. Quantification Based on Recurrence Patterns

- Determinism. A prevalent feature found in most recurrence plots are diagonal lines, which show up when there are periods in which trajectories evolve in parallel to each other. A diagonal line of length l occurs when the following condition is satisfied: ${\mathbf{y}}_{i}\approx {\mathbf{y}}_{j}$, ${\mathbf{y}}_{i+1}\approx {\mathbf{y}}_{j+1}$, ${\mathbf{y}}_{i+2}\approx {\mathbf{y}}_{j+2}$, …, ${\mathbf{y}}_{i+l-1}\approx {\mathbf{y}}_{j+l-1}$. This condition can hold only when the two sections of the trajectory—one between ${\mathbf{y}}_{i}$ and ${\mathbf{y}}_{i+l-1}$ and the other between ${\mathbf{y}}_{j}$ and ${\mathbf{y}}_{j+l-1}$ are parallel to each other in the reconstructed state space, which occurs for periodically repeating portions of the trajectory. A higher number of such periodically repeating sections of the trajectory would imply that the state of the system can be predicted on timescales equal to the period of oscillation which, in this example, would be the time difference $\Delta t=|i-j|$. Diagonal lines are thus typically used as an indicator of deterministic behavior, as is also seen in the recurrence plots given in Figure 9. To quantify the extent of determinism contained in the recurrence plot, the recurrence plot-based measure $DET$ is defined as,$$DET={\displaystyle \frac{{\displaystyle \sum _{l={l}_{0}}^{N}}lK\left(l\right)}{{\displaystyle \sum _{l=1}^{N}}lK\left(l\right)}},$$
- Average shortest path length. A ‘path’ between two nodes i and j in a network is defined as a sequence of nodes that needs to be traversed in order to go to node j from node i. In general, there exist many possible paths between any pair of nodes in a network, and there can be even several possible shortest paths between a pair of nodes. However, it is possible to uniquely define a shortest path length ${d}_{ij}$ between two nodes i and j which is the smallest number of nodes that need to be traversed in order to reach j from i. Often the average shortest path length is a characteristic feature of networks that can help distinguish the topology of one network from another. In recurrence networks, shortest path length helps to characterize the topology of nearest-neighbor relationships. Each shortest path ${d}_{ij}$ is the distance between two states i and j of the system measured by laying out ${d}_{ij}$ straight line segments between them such that: (i) each line segment cannot be more than $\epsilon $ units long, and (ii) the ends of each line segment must lie on a measured state, the first and last of which are i and j respectively. Thus, ${d}_{ij}$ is bounded from below by the straight line between the states i and j, i.e., ${d}_{ij}$ is the upper bound for the Euclidean distance between two states on the attractor [115,117], and its average value is an upper bound for the mean separation of states of the attractor [117]. The average shortest path length, $SPL$, is estimated as—$$SPL=\frac{1}{N(N-1)}\sum _{i,j=1}^{N}\phantom{\rule{0.166667em}{0ex}}{d}_{ij},$$

#### 7.4. Inferring Dependencies Using Recurrences

- Correlation of probabilities of recurrence. The determination of the phase from the measured time series of a chaotic oscillation is a challenging task. Especially in the case when the attractor is in a non-phase-coherent dynamical regime, it is nontrivial to determine which particular combination of the state space variables would result in a reliable definition of the ‘phase’ of the motion, in the sense that with every time period, the phase should increase by $2\pi $. In their study, Romano et al. [125] exploit this idea to note that for complex systems, we need to relax the condition $\left|\right|{\mathbf{x}}_{i}-{\mathbf{x}}_{i+\tau}\left|\right|=0$ (which is true for a purely periodic system with a single well defined period $\tau $) to rather have $\left|\right|{\mathbf{x}}_{i}-{\mathbf{x}}_{i+\tau}\left|\right|\approx 0$, i.e., $\left|\right|{\mathbf{x}}_{i}-{\mathbf{x}}_{i+\tau}\left|\right|<\epsilon $, which allows us to define the function,$$q\left(\tau \right)=\frac{1}{N-\tau}\sum _{i=1}^{N-\tau}\mathsf{\Theta}\left(\epsilon -\left|\right|\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}_{i}-{\mathbf{x}}_{i+\tau}\left|\right|\right)=\frac{1}{N-\tau}\sum _{i=1}^{N-\tau}{\mathbf{R}}_{i,\phantom{\rule{0.166667em}{0ex}}i+\tau},$$$$CPR={\displaystyle \frac{1}{N-{\tau}_{0}}}{\displaystyle \sum _{\tilde{\tau}={\tau}_{0}}^{N}}{\displaystyle \frac{\left({q}^{\mathcal{X}}\left(\tau \right)-{\tilde{\mu}}_{q}^{\mathcal{X}}\right)\phantom{\rule{0.166667em}{0ex}}\left({q}^{\mathcal{Y}}\left(\tau \right)-{\tilde{\mu}}_{q}^{\mathcal{Y}}\right)}{{\tilde{\sigma}}_{q}^{\mathcal{X}}\phantom{\rule{0.166667em}{0ex}}{\tilde{\sigma}}_{q}^{\mathcal{Y}}}}$$
- Recurrence-based measure of dependence. Goswami et al. [131] recently proposed a statistically motivated measure of dependence based on recurrence plots. This idea was further developed by Ramos et al. [132] to include conditional dependences as well which helped to identify and remove ‘common driver’ effects in multivariate analyses. The so-called recurrence-based measure of dependence ($RMD$ in Equation (20) below) is the mutual information of the probabilities of recurrence of two dynamical systems $\mathcal{X}$ and $\mathcal{Y}$. Consider the recurrence plot ${\mathbf{R}}^{\mathcal{X}}$ constructed from the measured/embedded series ${\left\{{\mathbf{x}}_{t}\right\}}_{t=1}^{N}$: we can estimate the probability that the system $\mathcal{X}$ recurs to the state at time $t=i$ as,$${p}_{i}^{\mathcal{X}}=\frac{1}{N}\sum _{j=1}^{N}{\mathbf{R}}_{ij}^{\mathcal{X}},$$$${\mathbf{R}}_{ij}^{\mathcal{X}\mathcal{Y}}={\mathbf{R}}_{ij}^{\mathcal{X}}{\mathbf{R}}_{ij}^{\mathcal{Y}},$$$${p}_{i}^{\mathcal{X}\mathcal{Y}}=\frac{1}{N}\sum _{j=1}^{N}{\mathbf{R}}_{ij}^{\mathcal{X}\mathcal{Y}},$$$$RMD=\sum _{i=1}^{N}{p}_{i}^{\mathcal{X}\mathcal{Y}}log\frac{{p}_{i}^{\mathcal{X}\mathcal{Y}}}{{p}_{i}^{\mathcal{X}}\phantom{\rule{0.166667em}{0ex}}{p}_{i}^{\mathcal{Y}}},$$

#### 7.5. Detecting Dynamical Regimes Using Recurrences

## 8. Surrogate-Based Hypothesis Testing

- Estimate the time series analysis quantifier Q from the original time-series, denote it as ${Q}^{orig}$.
- Generate K surrogate time series using an appropriate surrogate generation method.
- Estimate the same quantifier Q from each of the surrogate time series in the exact same manner as was done for the original time series. This results in a sample of K values of Q, which we denote as ${\left\{{Q}_{i}^{surr}\right\}}_{i=1}^{K}$.
- Estimate the probability distribution ${P}^{null}\left(Q\right)$ from the sample ${\left\{{Q}_{i}^{surr}\right\}}_{i=1}^{K}$ using a histogram function or a kernel density estimate. This distribution is known as the ‘null distribution’ as it is the distribution of values Q for the situation the null hypothesis is true, i.e., for whatever characteristic the surrogates preserve.
- Using ${P}^{null}\left(Q\right)$, estimate the so-called ‘p-value’, defined as the total probability of obtaining a value at least as extreme as the observed value ${Q}^{orig}$, i.e.,$$p=\underset{Q>{Q}^{orig}}{\int}\phantom{\rule{0.166667em}{0ex}}{P}^{null}\left(Q\right).$$The p-value encodes how less likely is the observed value ${Q}^{orig}$ to be obtained from the null distribution ${P}^{null}$.
- Based on a chosen confidence level of the test $\alpha $, determine whether ${Q}^{orig}$ is statistically significant at level $\alpha $ by checking whether $p<\alpha $ or not. When $p<\alpha $, the observed value ${Q}^{orig}$ is statistically significant with respect to the chosen null hypothesis, and we fail to accept the null hypothesis, indicating that the observed ${Q}^{orig}$ is caused by characteristics other than what is retained in the surrogates. By convention, $\alpha $ is typically chosen at 5%, i.e., $\alpha =0.05$ or in some cases at 1%, i.e., $\alpha =0.01$. Values of $\alpha $ higher than 5%, such as 10%, is not recommended as the statistical evidence in such cases is rather weak.
- In cases when there is more than one statistical test, we have to take into account the problem of multiple comparisons. This situation commonly arises in a sliding window analysis, where we divide a time series into smaller (often overlapping) sections and estimate the quantifier Q for each section. If $\alpha =0.05$, then 5% of the windows are possibly false positives. To reduce the effect of false positives, ‘correction factors’ such as the Bonferroni correction or the Dunn-Šidák correction are used [144]. In particular, Holm’s method [145] is preferable as it does not require that the different tests be independent. The fundamental idea behind correction factors is to use a corrected value of $\alpha $ which is far lower than the actual reported $\alpha $, thereby reducing the effective number of false positives at the reported level of confidence.

## 9. Application: Climatic Variability in the Equatorial and Northern Pacific

## 10. Summary and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Code Availability

## References

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**Figure 1.**Publications in nonlinear time series analysis and recurrence plots. Based on an online search in the Web of Science Core Collection database, the number of publications per year are shown which contain the terms “nonlinear time series” (

**a**) and “recurrence plots” (

**b**), either in the title (yellow bars) or anywhere in the text (green bars). Studies in nonlinear time series analysis begin to increase from the early 1990s on, coinciding with the increasing ubiquity of more powerful and more compact computers. The field of recurrence plots, however, receive increased attention only about a decade later, after the turn of the century. Moreover, studies that are explicitly based on nonlinear time series techniques peak ca. 2005–2010, whereas explicit recurrence plot studies are still on the rise. The difference between the “All fields” and “In title” counts indicate that both concepts are increasingly seen as methodological tools to be applied to other systems.

**Figure 2.**Predictability of linear and nonlinear systems. Two signals are measured up to time $t=250$ (red vertical line): in (

**a**), the signal is from a linear process where it is essentially a superposition of three sinusoids with frequencies, whereas in (

**b**), the signal comes from a nonlinear system in deterministic chaos. We assume perfect knowledge of all three components of the harmonic signal, and also of the system of differential equations of the chaotic system. Using the knowledge of the governing equations, at $t=500$, we predict the future trajectories for both systems with an initial uncertainty in determining the state of the system of the order of ${10}^{-6}$. For the chaotic system, the 95% confidence interval (i.e., the interval that contains 95% of all initial conditions chosen within initial error bounds) of future projections soon become as large as the entire spread of the signal (yellow shaded area at ca. $t\approx 325$). Exponential divergence of chaotic systems thus imply that even in the situation when we have perfect knowledge about system dynamics, errors in measuring the state of the system result in prediction errors that become as large as the diameter of the system after some time. This is not the case for linear systems (

**a**).

**Figure 3.**Nonlinear transitions in the Rössler system. The Rössler system (Equation (1)) is integrated up to $t=500$ with a sudden change in the parameter a at $t=250$ (red vertical line in (

**a**)). The change in a results in a change in the behavior of the system from spiral-type chaos ($a=0.32$) shown in (

**b**) to screw-type chaos ($a=0.38$) shown in (

**c**). A running mean of a window of size 25 time units (red line in (

**a**)) fails to show any change at the time of the transition. Although the running variance of 25 time units (yellow line in (

**a**)) shows slightly different behaviour for screw-type chaos, it is not possible to uniquely infer the particular dynamic on its basis. Transitions in nonlinear dynamical systems can be of widely different types, and at times can be very subtle and hidden, not easily evident in linear measures estimated from the scalar time series.

**Figure 4.**Phase synchronization of inherently chaotic systems. Two chaotic Rössler systems are coupled bidirectionally via the x-components (

**a**) such that they are in phase synchrony (cf. Equations (21) and (22) and [18,88]). The amplitudes, i.e., the x-components of both systems, are almost uncorrelated (Pearson’s cross-correlation of 0.013 in (

**b**)) while the phases, obtained by a Hilbert transform of the x-components, are almost identical, evidenced by the diagonal line obtained when the phase ${\varphi}_{2}$ of the second Rössler oscillator is plotted against the phase of ${\varphi}_{1}$ of the first (

**c**).

**Figure 5.**Power spectrum is insufficient to describe high dimensional nonlinear dynamics. The power spectrum (

**b**) of two signals shown in (

**a**) do not reflect the stark difference between the two as is evident from a three-dimensional representation of their time evolution in (

**c**). Signal 1 (green) is the x-component of the chaotic Rössler system and signal 2 (yellow) is constructed using an algorithm than randomizes signal 1 such that the power spectrum is (approximately) unchanged. This highlights that linear characteristics are at times insufficient to describe high dimensional nonlinear complexity.

**Figure 6.**Nonlinear dynamical systems: Flows and maps. The Rössler system (

**a**) is an example of a continuous dynamical system whereas the Hénon map (

**b**) is a paradigmatic example of a 2D discrete dynamical system. For the Rössler, trajectories in the 3D state-space are smooth, differentiable paths that settle on the attractor as $t\to \infty $. For the Hénon map however, the trajectories jump from point to point in the state space and the attractor is therefore discontinuous. Here, the parameters for the Rössler model are fixed at $a=0.432,b=2,c=4$ and those for the Hénon are fixed at $a=1.4,b=0.3$, for which both attractors are chaotic. Three different initial conditions (marked as “×”s) all converge to the attractors as time progresses.

**Figure 7.**Bifurcations in nonlinear dynamical systems. Both the Rössler system (

**a**) and the Hénon map (

**b**) undergo qualitative behaviour in the their equilibrium dynamics as the parameter a is varied. For the continuous Rössler system, the bifurcation diagram is obtained by discretizing the dynamic by taking the local extrema of the time series $y\left(t\right)$. Cross-sections such as the $y=0$ plane (known as Poincaré sections) can also be considered, which leads to a different projection of the dynamic. The similar pattern seen in both systems—as they move from a single stable fixed point for low values of a to periodic behavior and finally to chaos—is a universal phenomenon seen in chaotic systems known as the ‘period-doubling’ route to chaos [93]. The vertical red dashed lines indicate the value of a used in Figure 6.

**Figure 8.**State space reconstruction using time delay embedding. The first minima of the self-mutual information (red circle and dashed yellow line in (

**a**) is used to determine the time delay of embedding $\tau $. Using ${\tau}_{e}=137$ chosen from the $I\left(\tau \right)$ curve, the fraction of false neighbors $FNN\left(m\right)$ (Equation (8)) determines the minimum embedding dimension ${m}_{e}$ required to properly unfold the attractor (

**b**). In the example chosen here, the true attractor of the Rössler system is shown in (

**c**). Considering only the x-component as the measured signal, the embedding parameters are determined using (

**a**,

**b**) as ${\tau}_{e}=137$ and ${m}_{e}=3$. The reconstructed attractor (

**d**) is not identical to the one in (

**c**) but the two are topologically equivalent.

**Figure 9.**Typology of recurrence plots. Recurrence plots obtained from: Gaussian white noise (

**a**), superposed harmonics with periods 10, 50, and 75 (

**b**), chaotic Rössler (

**c**), and geometric Brownian motion (

**d**). The differences in the dynamical characteristics lead to drastically different patterns. Pure white noise shows no structure, while harmonic and chaotic behavior show diagonal lines to different degrees, and Brownian motion with drift reveals modular periods interspersed with bottleneck periods. The embedding parameters and the recurrence plot thresholds are given in the title of each subplot. In (

**b**), $\epsilon $ was used as a fixed recurrence rate, while for the rest it was the threshold for the Euclidean norm.

**Figure 10.**Recurrence-based quantification. The determinism $DET$ (green) estimated from the recurrence plot and the average shortest path length shortest path length ($SPL$) (yellow) estimated from the recurrence network quantify dynamical changes for the Hénon map (Equation (2)) on changing the bifurcation parameter a (cf. Figure 7). For comparison, the MLE $\mathsf{\Lambda}$ (red) is shown alongside the recurrence quantification measures to indicate the chaotic and periodic windows. $DET$ ($SPL$) is low (high) for chaotic dynamics (i.e., $\mathsf{\Lambda}>0$) and high (low) for periodic behaviour. Recurrence plots were estimated with $\tau =1,m=2,\epsilon =2$ for the x variable of the Hénon map. The shaded areas denote the interquartile range obtained using 100 randomly chosen initial conditions for each value of a.

**Figure 11.**Inferring interdependencies using recurrences. Estimates of $CPR$ (blue), recurrence-based measure of dependence ($RMD$) (yellow), and $PCC$ (red) for the coupled Rössler system (Equations (21) and (22)) for varying coupling strength $\mu $. The shaded interval denote the interquartile ranges of the measure as obtained from 100 different initial condition choices for the coupled system. The onset of phase synchronization at $\mu \approx 0.042$ is recorded by $CPR$ as it becomes almost one, but it is also recorded by $RMD$ as it suddenly jumps from values close to zero to a plateau at around $RMD\approx 10$. This plateau for $RMD$ ends at $\mu \approx 0.075$; and after $\mu \approx 0.102$, $RMD$ plateaus again, signaling the onset of lag synchronisation [88]. Between $\mu =0.075$ and $\mu =0.102$, $RMD$ has a much larger spread indicating large fluctuations due to intermittent lag synchronisation. $RMD$ shows similar results to that of $JPR$ (cf. Figure 43C of [88]), but is easier to implement and computationally less expensive. Note that $PCC$ dips just before the onset of phase synchronisation and has a continues increase afterwards, but is unable to detect either the precise onset of phase or lag synchronisation.

**Figure 12.**Detecting dynamical regimes using recurrences. The three wells (boundaries indicated by horizontal dashed lines) of a triple-well potential $U\left(x\right)$ ((

**a**), cf. Equations (25)–(27)) identified using the community structure of the recurrence network ($m=1,\tau =1$, fixed recurrence rate = 20%) obtained from the time series (grey line in (

**b**)) sampled every $t=0.25$ units (circles in (

**b**)). The three communities (indicated in (

**b**) with the color of circles) are obtained by optimizing the modularity (Equation (24)) of all possible partitions of the recurrence network. This approach allows us to: (a) infer that the dynamics has three regimes, (b) classify the time series.

**Figure 13.**Dynamics and interrelations of the ENSO and PDO indices. (

**a**) $DET$ for the ENSO Niño 3.4 index (green) and the PDO index (yellow). Dashed horizontal lines indicate inter-quartile ranges obtained from the iAAFT null model. While the two systems are anti-correlated up to ca. 1930, they show a changing lead-lag behavior after that. (

**b**) $SPL$ for the ENSO Niño 3.4 index (green) and the PDO index (yellow). The two $SPL$ series are largely correlated, but after ca. 1960, they show the opposite behavior in terms of being higher or lower than the iAAFT null model. (

**c**) $CPR$ (green) and $RMD$ (yellow) between the ENSO and PDO indices. Statistical evidence for interdependence is rather weak, given the twin surrogate null model, and the confidence level of 5% (horizontal dashed lines) apart for three periods of significant interrelation for $RMD$ (solid yellow markers), centered around the Niño–like conditions of 1897, 1914, 1941, 1956, and 1998. (

**d**,

**e**) Dynamical regimes of the ENSO (

**d**) and the PDO (

**e**) by maximising $MOD$. A 3-community structure is obtained, in which two communities are predominant, demarcating the positive (red markers) and negative phases (green markers) of the two climatic phenomena. Especially for the PDO, we recover the broad periods of warm and cold phases, marked in (

**d**) as “W” and “C” respectively. Recurrence plot parameters used are: (i) ENSO: ${\tau}_{e}=10,{m}_{e}=3$, (ii) PDO: ${\tau}_{e}=16,{m}_{e}=3$. For the calculations of $RMD$ and $CPR$, both indices were embedded with the maximum values of both m and $\tau $.

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Goswami, B. A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots. *Vibration* **2019**, *2*, 332-368.
https://doi.org/10.3390/vibration2040021

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https://doi.org/10.3390/vibration2040021

**Chicago/Turabian Style**

Goswami, Bedartha. 2019. "A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots" *Vibration* 2, no. 4: 332-368.
https://doi.org/10.3390/vibration2040021