Information–Theoretic Analysis of Visibility Graph Properties of Extremes in Time Series Generated by a Nonlinear Langevin Equation

Abstract

In this study, we examined how the nonlinearity $\alpha$ of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex structure was investigated using the visibility graph (VG) method. For both types of series, the information measures of the Shannon entropy measure and Fisher Information Measure were utilized for illustrating the influence of the nonlinearity $\alpha$ on the distribution of the degree, which is the main topological parameter describing the graph constructed by the VG method. The main finding of our study was that the Shannon entropy of the degree of the run lengths and the surplus magnitudes of the extremes is mostly influenced by the nonlinearity, which decreases with with an increase in $\alpha$. This result suggests that the run lengths and surplus magnitudes of extremes are characterized by a sort of order that increases with increases in nonlinearity.

1. Introduction

The behavior of extremes has been extensively investigated [1,2], particularly regarding the influence of long-range correlations [3,4].

To define extremes in a time series, three distinct methods are commonly used: (i) the Block Maxima Series (BMS) approach, which involves dividing the observation period into non-overlapping intervals of equal length and selecting the maximum value within each interval; (ii) the Peak Over Threshold (POT) method, which identifies values exceeding a specified threshold; and (iii) the run theory (or crossing theory) [5,6,7,8], which was utilized in this study. The run theory is particularly effective for estimating the duration of periods exceeding a certain level (such as droughts, floods, or stock market trends), and it offers a robust measure of persistence. In this approach, a run is defined as a sequence of contiguous values above a threshold, and an extreme is characterized by the time in which the run starts, the duration or run length that is the interval from the starting and ending time of the run, and the magnitude that depends on the values of the time series within the run.

Systematic studies on the effect of the nonlinearity of background mechanisms on extremes have not yet garnered significant attention in the broader field. Nonetheless, this issue is significant because it could illuminate the relationship between the dynamics of nonlinear Markov processes and the characteristics of extreme events. In practical applications, this understanding may prove valuable in developing more appropriate nonlinear stochastic models for underlying processes.

As a nonlinear generator of time series, we considered the classical Langevin Equation [9]:

$${\displaystyle \frac{dy}{dt}}=a\left(y\right)+\sqrt{b\left(y\right)}{\zeta }_t,$$

(1)

where $y\left(t\right)$ is a stochastic process, ${\zeta }_t$ is Gaussian white noise, and $a\left(y\right)$ and $b\left(y\right)$ are drift and diffusion functions, respectively. Equation (1) is a general model of scalar, time-homogeneous, and diffusion Markov processes. In the present work, we utilize specific forms of the nonlinear Langevin equation, in which the drift and diffusion functions are expressed through the powers of y (see Equations (3) and (4) in the next section). The power exponents are given by a single parameter $\alpha$, which defines the degree of nonlinearity.

In recent years, methods for transforming time series into networks have been proposed, leading to the development of innovative techniques for analyzing and characterizing these time series [10,11,12,13,14,15,16,17,18]. Other methods consider the “visibility” among the values of a time series, like the visibility graph (VG) method [19], whose versatility is demonstrated by the large variety of applications across many research fields, including economics [20,21,22], weather forecasting [23,24], medicine [25], oceanography [26], and seismology [27,28,29,30,31,32], among others.

Some of the topological characteristics of graphs generated by the VG method may form a good tool for the discrimination of time series features. Topological time series analysis (or, more generally, topological data analysis) is a modern and promising field of knowledge with wide applications in nonlinear dynamical systems [33,34,35,36].

In this study, we used Information Theory quantifiers to examine the influence of the nonlinearity on the distribution of the VG degree of run lengths and surplus magnitudes (defined later) of extremes in time series generated by the nonlinear Langevin equation.

The structure of this work is as follows. In Section 2, the model is introduced. In Section 3, the definition of extremes is presented. Section 4 contains a brief outline of the visibility graph method. The information quantifiers are presented in Section 5. Section 6 discusses the behavior of extreme sizes and magnitudes, and it also presents an informational analysis of the topological properties of graph representations of extreme series. The final section summarizes the findings and concludes the work.

2. The Model

Among stochastic time series models, the Langevin equation [9,37] describes a wide range of Markov processes, and it has the advantage of being associated with the corresponding Fokker–Planck equation for the distribution function. The Langevin equation, originally derived to describe Brownian motion, has proven to be useful in many different research fields, from physics to biology and from economics to social sciences [38,39,40]. In the context of applications of the Langevin equation to natural time series, well-functioning methods have been developed for reconstructing the form of Langevin’s drift and diffusion functions based on time series [38].

In this paper, we assumed the following forward Euler discrete approximation for the Langevin equation [41]:

$$y(t+\Delta t)=y\left(t\right)+a\left(y\left(t\right)\right)\Delta t+\sqrt{b\left(y\right(t\left)\right)\Delta t}{\xi }_t,$$

(2)

where $y\left(t\right)$ is the stochastic process, $\Delta t$ is a time step, and ${\xi }_t$ are independent random variables with normal density. The Euler approximation is the simplest method of discretization; nevertheless, in our context of stationary and very long time series and using a small time step, this method proves to be adequate [42]. The functions $a\left(y\right)$ and $b\left(y\right)$ represent the drift and diffusion functions, respectively, assuming the following forms:

$$b\left(y\right)=y^{\alpha }$$

(3)

and

$$a\left(y\right)=-y^{\alpha +1}+{\displaystyle \frac{1}{2}}\alpha y^{\alpha -1},$$

(4)

which are determined by the parameter $\alpha$ that describes the degree of the nonlinearity. With Equations (3) and (4), Equation (2) generates stationary time series with the same half-Gaussian distribution function

$$p\left(y\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{e}^{-y^2},y\ge 0,$$

(5)

regardless of $\alpha$ (i.e., $p\left(y\right)$ is fixed).

Technically, assuming the form of $b\left(y\right)$ and $p\left(y\right)$, the drift function $a\left(y\right)$ was determined using the general formula for a stationary distribution function [43]:

$$p_{st}\left(y\right)={\displaystyle \frac{c}{b\left(y\right)}}{\int }_{\delta }^y{\displaystyle \frac{a\left(x\right)}{b\left(x\right)}}dx.$$

(6)

Therefore, the nonlinearity was expressed by powers of y, which is a simple and natural extension beyond linear forms. Polynomial forms of drift and diffusion functions are quite common. For instance, they were used in the well-known population models such as Verhulst–Pearl diffusion [44]. It is worth highlighting that our model with $\alpha$ = 2 refers to a cubic-drift diffusion model (where $a\left(y\right)=A{y}^3+By$) which, for example, was applied for modeling the global CO₂ emissions in Spain [45].

Introducing only one nonlinearity parameter will significantly simplify the analysis of how the model’s nonlinearity affects extreme behavior in a generated time series. Similarly, adopting a fixed form of the distribution function will enable us to study the influence of the relationship between drift and diffusion forces without interference from various forms of the distribution function, such as the length of the distribution tails. Moreover, the Gaussian form is common in many research fields. The dynamics of the stochastic process is ruled by the interaction of two forces: the drift force $F_d=a\left(y\right)$ and the stochastic force $F_s={(b\left(y\right)/\Delta t)}^{1/2}{\xi }_t$ [46]; therefore, the model describes the fluctuations of a particle in the potential niche $V\left(y\right)$ (which is given by the drift function, i.e., $dV\left(y\right)/dy=-a\left(y\right)$), where the fluctuations are governed by the diffusion term.

In our analysis, we assumed five values for the nonlinearity parameter $\alpha$, i.e., 1.0, 1.5, 2.0, 2.5, and 3.0. The corresponding forms of potentials are shown in Figure 1. We observed that, with the increase in $\alpha$, the minimum of the potential moves on the right and the right potential wall becomes steeper to balance the increase in the diffusion force.

The model defined by Equations (2)–(4) was used as the generator of time series. For each case of nonlinearity parameter $\alpha$, ten time series with the length n = 1,000,000 and time step $\Delta t=0.01$ were generated for our analysis.

3. Definition of Extremes

In this paper, the extremes were defined using the run theory (or crossing theory) [5,7,8]. According to this approach for discrete data, an extreme $E_i$ is defined as a sequence of $R{L}_i$ contiguous values (or run) that exceed a certain threshold T. Given a time series $y\left(t\right)$, an extreme $E_i$ can be characterized by three components $(t_i,R{L}_i,S{M}_i)$, where $t_i$ indicates the time when the run starts, $R{L}_i$ represents the run’s length or extreme’s duration, and $S{M}_i$ denotes the surplus magnitude defined as follows:

$$S{M}_i=\sum _{j=1}^{R{L}_i}\left(y(t_i+j-1)-T\right).$$

(7)

The surplus magnitude can indicate, for example, the excess wind energy during an extreme weather event, the volume of water collected by floodplains during a flood, the soil moisture level during a drought, the energy of seismic shocks exceeding a non-destructive threshold, the financial impact of stock market spikes, etc. [47,48].

The threshold T is defined as a percentile of the distribution of the series’ values [49]. In our paper, we assumed thresholds above which the extremes are detected to be the $92.5$th, $95.0$th, and $97.5$th percentiles of the distribution of the background series’ values. Percentile-based extremes are frequently used in the investigation of hydrological or climate extremes [49,50,51]. Figure 2 provides, as an example, the graphical representation of two extremes. The Supplementary Materials show some of the realizations of the model with different nonlinearity parameters along with the corresponding sequences of run lengths and surplus magnitudes for different thresholds.

4. The Fisher–Shannon Information Plane

The Fisher–Shannon Information Plane (FSIP), initially introduced by Vignat and Bercher [52] and later employed in various studies [53,54,55], is a plane where the horizontal axis represents Shannon Entropy (S) and the vertical axis represents the Fisher Information Measure ($FIM$), both of which are derived from the probability distribution function (PDF). This tool provides a convenient way to depict both the global and local aspects of the PDFs associated with the studied system on the same information plane.

For a continuous probability distribution function (PDF) $f\left(x\right)$ with $x\in \Delta \subset \mathbb{R}$ and ${\int }_{\Delta }f\left(x\right)dx=1$, its Shannon Entropy is defined as [56]

$$S\left[f\right]=-{\int }_{\Delta }f\left(x\right)lnf\left(x\right)dx,$$

(8)

where a ’global’ measure that is not excessively influenced by significant changes occurring in localized areas of the distribution’s support set is $\Delta$.

On the contrary, the Fisher’s Information Measure (FIM) quantifies the gradient of the PDF, and thus is particularly sensitive even to small localized perturbations [57,58,59]. It is defined as

$$F\left[f\right]={\int }_{\Delta }{\displaystyle \frac{1}{f\left(x\right)}}{\left({\displaystyle \frac{df\left(x\right)}{dx}}\right)}^{2}dx.$$

(9)

There are various interpretations of the FIM, like as a measure of the precision of parameter estimation, as the extent of information retrievable from a set of measurements, and as a metric quantifying the degree of disorder within a system or phenomenon [59]. In the earlier definition of FIM (Equation (9)), the division by $f\left(x\right)$ raises difficulties when $f\left(x\right)$ is too small to be accurately computed. This issue could be avoided using the probability amplitudes $\psi =\sqrt{f}$ [59,60]. Since the gradient operator significantly influences the contribution of subtle local variations to the value of FIM, this quantifier is acknowledged as a ’local’ metric [59].

For discrete probability distributions ($P=\left\{p_j:j=1,\dots ,M\right\}$), its Shannon entropy $S\left[P\right]$ [56] is defined as

$$S\left[P\right]=-\sum _{j=1}^M{p}_j·ln\left(p_j\right).$$

(10)

If it is certain which of the possible outcomes j with probabilities $p_j$ will occur, then $S\left[P\right]=0$, indicating maximal knowledge of the underlying process described by the probability distribution. Conversely, this knowledge is minimal for a uniform distribution. For a given distribution P, the “normalized Shannon entropy” is calculated as

$$S_n\left[P\right]={\displaystyle \frac{S\left[P\right]}{S_{max}}},$$

(11)

where $S_{max}=ln\left(M\right)$.

For the same probability distribution P, when considering the real probability amplitudes (Equation (9)), the FIM is defined as

$$F\left[P\right]=F_0\sum _{i=1}^{M-1}{\left[{\left(p_{i+1}\right)}^{{\displaystyle \frac{1}{2}}}-{\left(p_i\right)}^{{\displaystyle \frac{1}{2}}}\right]}^2,$$

(12)

where the normalization constant $F_0$ is 1 if $p_1=1$ or $p_M=1$; otherwise, it is 1/2.

It has been extensively discussed that this discretization is the best behaved in a discrete environment [61].

In the discrete case, the axes of FSIP are the normalized Shannon Entropy $S_n$ and the FIM F, which is well suited for investigating both the global and local features of the probability distribution under study.

5. The Visibility Graph

The visibility graph (VG) [19] transforms a time series into a graph or network, where each node corresponds to a value in the series, and the connections between two nodes at $t_a$ and $t_b$ adhere to a specific geometric visibility criterion:

$$y_c<y_b+(y_a-y_b){\displaystyle \frac{t_b-t_c}{t_b-t_a}},$$

(13)

where $t_a<t_c<t_b$. In practice, the two values $y_a\left(t_a\right)$ and $y_b\left(t_b\right)$ are visible to each other if any other value $y_c\left(t_c\right)$ fulfills Equation (13).

A key parameter in the VG approach is the degree, representing the number of links originating from a node. Typically, higher values in the series act as hubs within the graph, attracting more links compared to nodes with lower values.

A sequence of extremes can be investigated by means of the VG method if we consider the run lengths (RL) or the surplus magnitudes (SM) as the nodes of the graph and draw the links between two nodes fulfilling the “visibility” rule (Equation (13)), thus converting the extreme sequence into a graph. Figure 3 shows a sketch of the VG method applied to the first 50 values of a run length (Figure 3a) and the surplus magnitude (Figure 3b) of extremes from a time series generated with $\alpha$ = 1.0 and threshold T = 92.5%.

6. Results

Before analyzing the topological properties of extremes, we will first examine the behavior of the surplus magnitude (SM) and run length (RL) in extreme series. This examination will consider the level of nonlinearity, as determined by the parameter $\alpha$ in the time series model, and the percentage levels (i.e., 92.5th, 95.0th, and 97.5th percentiles) used to define the threshold T. Figure 4 shows the relationship between RL and SM for the five different values of the nonlinearity $\alpha$ and three different threshold values. Each point presented on the SM vs. RL plane represents an extremum $E_i$ with the corresponding $R{L}_i$ and $S{M}_i$ values. Ten different colors were used to visualize the results from ten time series with the same parameters ($\alpha$ and T). It is important to note the dispersion of the elements: for example, extremes with the same run length (RL) may have different surplus magnitudes (SM), and, conversely, extremes with the same SM may have different run lengths. However, the clusters of points in Figure 4 tend to align along linear functions, with the slopes of these lines depending on $\alpha$ and T (see Figure 5). Specifically, the slope of these linear relationships decreases as $\alpha$ and T increase. The behavior of the averages $\langle \mathrm{RL}\rangle$ and $\langle \mathrm{SM}\rangle$ is shown in Figure 6 (where each graphic symbol represents the average of the corresponding cluster of points from the diagrams in Figure 4). Different symbols denote different values of the nonlinearity parameter $\alpha$, while the colors of these symbols represent different thresholds. An increase in $\alpha$ causes a linear decrease in $\langle \mathrm{SM}\rangle$ and $\langle \mathrm{RL}\rangle$ on the $\langle \mathrm{SM}\rangle$ vs. $\langle \mathrm{RL}\rangle$ plane for a fixed T. Similarly, an increase in T results in a linear decrease for a fixed $\alpha$.

In summary, the results presented in Figure 4 and Figure 5 demonstrate a clear and systematic (linear) influence of the level of nonlinearity in the Langevin equation on extreme elements such as run length and surplus magnitude. This influence is observed despite the background time series having the same half-Gaussian distribution, irrespective of the value of $\alpha$.

We applied the VG method on the sequences of run lengths {$R{L}_i$} and of surplus magnitudes {$S{M}_i$} generating graphs, whose nodes are given by the values {$R{L}_i$} and {$S{M}_i$}. Among the several parameters describing the topological characteristics of the graphs generated by the VG method, we considered the degree $k_{RL}$ for run length and $k_{SM}$ for surplus magnitude, which is the number of links departing from each node of the graph. We analyzed three properties of the degree (which are, respectively, defined by Equations (11) and (12)): the distribution, the Shannon Entropy (SH), and the Fisher Information Measure (FIM).

The top plots in Figure 7, Figure 8 and Figure 9 display the density of $k_{RL}$ and $k_{SM}$ in log-log scales. To distinguish between the results, five colors were used to represent the data for five different values of the nonlinearity $\alpha$. The variability of these densities with $\alpha$ is practically unnoticeable from these figures as the points overlap. However, the densities share a common feature: the tails of the distributions are much longer than those of an exponential distribution.

The information measures of the Shannon Entropy and the Fisher Information Measure, corresponding to $k_{RL}$ and $k_{SM}$, allow one to identify and quantify the influence of $\alpha$ on changes in the topological properties of the VG representation for {$R{L}_i$} and {$S{M}_i$}. The middle plots in Figure 7, Figure 8 and Figure 9 display the results for SH and FIM presented on the Fisher–Shannon Information Plane. Ten graphic symbols of the same color correspond to ten VGs for the same values of the parameters $\alpha$ and T. The bottom plots show the average values of SH and FIM over these ten results. One can observe a clear effect of the nonlinearity on the behavior of SH as SH decreases with increasing $\alpha$ in both cases of $k_{RL}$ and $k_{SM}$. In the case of $\langle \mathrm{FIM}\rangle$, a decreasing trend was visible for the threshold of 92.5% only. For larger threshold values, we observed a wide spread of results, making it impossible to determine a regular behavior of $\langle \mathrm{FIM}\rangle$ with increasing $\alpha$.

7. Conclusions

The main aim of this work was to investigate the impact of different levels of nonlinearity in the model generating the time series on the behavior of extreme values within these series. To eliminate the influence of the distribution function’s form, which significantly affects extremes, we chose a model characterized by the same half-Gaussian distribution regardless of the level of nonlinearity. We defined the extremes according to run theory, which emphasizes the role of the length of the time period when the values of the time series are above the adopted threshold. In practical applications, the run length corresponds to, for example, the duration of floods, droughts, or stock market runs. The surplus magnitude for a given extreme is related to the run length (see Equation (7)) and is stochastic in nature. However, the average values $\langle \mathrm{SM}\rangle$ and $\langle \mathrm{RL}\rangle$ decrease linearly with the increase in the level of nonlinearity $\alpha$ and with the increase in threshold T. Instead of analyzing the behavior of inter-extreme time and run length series separately (as in [62,63]), we used the VG representation of extreme run length (and magnitude) series. In this approach, the graph topology, characterized by the connectivity distribution function, encapsulates both the temporal and size (or amplitude) structure of a series of extremes with a variable time step. To quantify the features of the connectivity distribution, we used an information approach by presenting the results on the Fisher–Shannon Information Plane.

Our analysis revealed that the Shannon Entropy for both $k_{RL}$ and $k_{SM}$ decreased with increasing $\alpha$. In contrast, the Fisher Information Measure did not exhibit a consistent dependence on the level of nonlinearity for thresholds T greater than 92.5%. The decrease in Shannon Entropy indicates an increase in the ordering of the visibility graph representations of extreme magnitude and run length series.

We demonstrated that the Shannon Entropy (SH) measure, in relation to the graph representation of a series of extremes, is sensitive to changes in the dynamics of the background process caused by variations in $\alpha$. This sensitivity persists even though the distribution function of the process remains unchanged with respect to the level of model nonlinearity, while the drift and diffusion forces do vary. For different values of $\alpha$, the balance between these forces imposes specific constraints on the $y\left(t\right)$ process, leading to varying levels of ordering in the extremes, both in terms of the time and magnitudes (or run lengths).