Introduction

As social sciences and technology advance rapidly, complex systems are assuming an increasingly vital role in human production and daily life. The intricate time series produced by these systems have garnered considerable interest in academia. Investigating the complexity inherent in complex time series holds significant research value, spanning multiple disciplines1,2,3.

In the study of complex time series characteristics, a pivotal concern revolves around the complexity of such series. Various complexity metrics have been proposed by researchers, including Lyapunov exponents4,5 and statistical complexity measures (SCM)6. Lyapunov exponents serve as crucial indicators for evaluating complex systems, denoting the rate of separation between neighboring points. Positive Lyapunov exponents indicate the eventual divergence of points, whereas negative ones signify convergence. Statistical complexity measures (SCM), rooted in information theory, utilize entropy to assess a system's complexity; higher entropy denotes a more complex system with multiple modes.

In nonlinear time series analysis, Bandt and Pompe7 introduced the ordinal methodology in 2002, grounded in information theory. This methodology posits time series as projections of a high-dimensional state space trajectory along the time axis. Initially, the time series undergoes reconstruction into a state space, representing each moment as a trajectory. Subsequently, specific techniques are employed to derive pattern sequences from the state space, transforming trajectory sequences into pattern sequences. The probabilities of different patterns are then statistically analyzed, and information-theoretic metrics like entropy, statistical complexity, and Fisher information are utilized to quantify the distribution's characteristics. The ordinal methodology has sparked considerable interest, giving rise to entropy-based methods such as fuzzy entropy8, Phase entropy9, slope entropy10, bubble entropy11, and attention entropy12, significantly enriching the arsenal of time series analysis techniques.

State space reconstruction serves as the theoretical bedrock of the ordinal methodology, enabling the recovery of chaotic motion trajectories from one-dimensional time series, thereby facilitating a deeper understanding of the system's dynamical properties. Typically, two methods are employed for state space reconstruction: derivative reconstruction and coordinate delay reconstruction13. Derivative reconstruction, relying on time series derivatives, utilizes derivatives to construct state space vectors, thereby capturing richer information to describe the system's dynamics. Coordinate delay reconstruction, a time delay-based approach, constructs high-dimensional state space vectors using different time delays of the one-dimensional time series. In accordance with Takens' embedding theorem14, a state space topologically equivalent to the original dynamical system can be reconstructed from a one-dimensional chaotic time series.

State space reconstruction provides significant advantages; however, it is highly sensitive to outliers and data noise. Noise contamination in a time series can disrupt the reconstructed state space, thus adversely affecting subsequent analyses. Furthermore, selecting appropriate parameters for state space reconstruction presents a challenging task. In real-world scenarios, many time series exhibit long-range dependencies and heavy-tailed distributions, particularly those generated by complex systems. Existing state space reconstruction methods, based on derivative and coordinate delay, lack the capability to capture these temporal memory characteristics, resulting in limited short-term memory capabilities15.

Differential equations have long been used to model natural phenomena, enabling a comprehensive understanding of dynamic processes. However, traditional local differential operators often fall short when simulating complex real-world problems, especially those exhibiting fractal characteristics. Mathematically, fractal derivatives, a non-standard type of derivative where the variable scales according to \({t}^{a}\), were developed to model physical problems where classical principles like Darcy's law, Fourier's law, and Fick's law are ineffective. These classical principles assume Euclidean geometry, making them incompatible with non-integer dimensional media16. Studies have shown that differential equations modeled using fractional operators can more accurately replicate real-world results. For instance, Liu et al. found that fractional differential equations more precisely modeled the variation of diffusion coefficients over time and scale in solute transport within fractal porous media17. Similarly, in 2023, Khan et al. proposed a Covid-19 model involving diabetic patients. This model belongs to a subclass of fractional-fractal (FF) derivatives. When the order of the fractional model reaches 1, the results gradually approach the classical scenario, while all other solutions deviate from the classical situation. Therefore, this fractional-order model provides additional crucial information for case studies18. As research progresses, scholars have increasingly turned to numerical methods to solve fractional differential equations due to the difficulty of obtaining analytical solutions. Kumar et al. converted general fractional differential equations into Volterra-type integral equations using the unique properties of Caputo derivatives and applied existing numerical methods to solve them in 200619. Adomian's method was adapted by Daftardar-Geji et al. to solve fractional nonlinear diffusion-wave equations, achieving notable results20.

With the advancement of fractional calculus theory, its applications have extended to continuous time series. For example, Tarasov and Tarasova21 applied dynamic memory to economics within a fractional calculus-based continuous time framework, incorporating power-law memory into the Harold-Domar model. Ines developed a fractional economic forecasting model to predict the GDP of the G7 countries, demonstrating that fractional-order modeling resulted in smaller prediction errors compared to integer-order models22.

However, the aforementioned fractional theories are model-based, and it is often challenging to fully model real-world complexities. Therefore, this paper attempts to apply fractional calculus theory to data processing from a model-free perspective. We propose a Multi-span Transition Network based on fractional-order state space to analyze the complexity of multivariate time series. This approach effectively addresses the limitations of phase space reconstruction affected by outliers and captures the long-memory properties of time series, offering robust capabilities for analyzing time series complexity.

Inspired by fractional calculus, this paper introduces a multi-span transition network (MSTN) based on fractional-order state space for analyzing the complexity of multivariate time series. Fractional-order state space models effectively address the sensitivity to outliers in state space reconstruction and excel in capturing the long-term memory properties of time series, demonstrating strong capabilities in analyzing time series complexity.

Sequential partition transition networks (OPTNs) have been widely employed for time series analysis1,23,24,25. While OPTNs capture the frequency of transitions between states in the phase space, they focus solely on adjacent transitions. This paper extends the concept by introducing multi-span dimensions to construct a time network, allowing for a more comprehensive representation of transition relationships. When \(\tau = 1\), the transition frequency is measured between patterns \({\pi }_{i}\) and \({\pi }_{i+1}\); for \(\tau = 2\), it's between \({\pi }_{i}\) and \({\pi }_{i+2}\), and so on.

Furthermore, a novel multi-span transition entropy component method (MTECM) is proposed to analyze multivariate time series, examining both spatial relationships among variables and multi-span temporal changes to reveal the complexity of complex time series. By decomposing the complexity of multivariate time series into within-sample and between-sample components, this approach represents a significant departure from existing complexity measures. The MTECM-FOSS algorithm is applied to handle various simulated single and multivariate time series, revealing that time series with distinct memory characteristics exhibit extreme behavior at specific fractional orders. As the spanning dimension increases, the entropy of the MTECM-FOSS algorithm exhibits nonlinear growth. The model is tested on empirical data (including single-channel epilepsy classification, multi-channel epilepsy classification, and fault diagnosis), demonstrating that MTECM-FOSS achieves competitive or even superior classification performance with fewer features compared to state-of-the-art feature extraction methods.

The key innovations of this paper are:

  1. 1.

    The paper presents a novel fractional-order state space model tailored for enhanced complex sequence analysis, demonstrating superior noise resilience compared to its integer-order counterparts. Fractional derivatives' inherent memory and the model's flexible parameter adjustment facilitate a more comprehensive understanding of intricate time series dynamics.

  2. 2.

    This paper introduces a novel multi-span transition entropy component method (MTECM), designed to offer a comprehensive analysis of multivariate time series by simultaneously considering both spatial dependencies and multi-scale temporal patterns.

  3. 3.

    The MTECM-FOSS algorithm is employed to analyze diverse simulated single and multivariate time series, revealing that those with unique memory properties exhibit distinct patterns at specific fractional orders. Notably, the entropy of the MTECM-FOSS exhibits non-linear growth as the spanning dimension increases.

  4. 4.

    The model is tested on empirical data (including single-channel epilepsy classification, multi-channel epilepsy classification, and fault diagnosis), demonstrating that MTECM-FOSS achieves competitive or even superior classification performance with fewer features compared to state-of-the-art feature extraction methods.

The structure of this article is as follows:

In Sect. "Methodology", we present the process of constructing a fractional-order state space for time series and provide a definition and explanation of the multi-span transition entropy component method (MTECM-FOSS). Section “Simulation data validation” tests the superiority of MTECM-FOSS based on fractional-order state space compared to integer-order state spaces using several simulated datasets. Section "Empirical data validation" demonstrates the practical performance of the model on real-world experimental data. Section “Discussion” is dedicated to discussions, and Sect. “Conclusion” concludes the paper.

Methodology

In this section, we will delve into the process of constructing a fractional-order state space for time series, and provide a formal definition of the multi-span transition entropy component method (MTECM). Before introducing the methods, it is necessary to define the symbols used throughout the paper, as shown in Table 1.

Table 1 Symbol definitions and their interpretations.
Full size table

Fractional order state space reconstruction methods

This section offers a detailed explanation of the concept of fractional calculus and the method of fractional-order state space reconstruction. For a complex time series \(X=\{{x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}\}\), integer-order differences can be expressed as:

$$ \nabla x_{i} = \left\{ {\nabla x_{i} |\nabla x_{i} = x_{i} - x_{i - 1} } \right\},i = 2,3,4, \ldots ,n $$
(1)
$$ \begin{array}{*{20}c} {\nabla^{2} x_{i} = \nabla \left( {\nabla X} \right) = \left\{ {x_{i} - 2x_{i - 1} + x_{i - 2} } \right\},i = 2,3,4, \ldots ,n} \\ \end{array} $$
(2)

Integer-order differences demonstrate short-term memory as they rely on adjacent observations, serving as a means to maintain stationarity in time series. To streamline mathematical representation and enable the definition of fractional differences, we introduce the backward shift operator \(B\) and \({B}^{k}\).

$$\nabla {x}_{i}=\left\{{x}_{i}-{x}_{i-1}\right\}=(1-B){x}_{i}$$
$${\nabla }^{2}{x}_{i}=\left\{{x}_{i}-2{x}_{i-1}+{x}_{i-2}\right\}={(1-B)}^{2}{x}_{i}$$

For integer-order derivatives, this can be expressed as:

$$\begin{array}{c}{\nabla }^{\alpha }{x}_{i}={\left(1-B\right)}^{\alpha }{x}_{i}=\left(\sum_{k=0}^{\alpha }\left(\begin{array}{c}\alpha \\ k\end{array}\right){(-B)}^{k}\right){x}_{i}.\end{array}$$
(3)

When extending to fractional derivatives, the integer \(\alpha \) becomes a fraction, and the fractional operator can be written as:

$${\nabla }^{\alpha }{x}_{i}={\left(1-B\right)}^{\alpha }{x}_{i}=(\sum_{k=0}^{\infty }{\left(-B\right)}^{k}\prod_{i=0}^{k-1}\frac{\alpha -i}{k-i}){x}_{i}$$
$$\begin{array}{c}=(1-\alpha B+\frac{\alpha \left(\alpha -1\right)}{2!}{B}^{2}+\dots +\left(-1{)}^{k}\frac{\prod_{i=0}^{k-1}\left(\alpha -i\right)}{k!}{B}^{k}+\dots \right){x}_{i}.\end{array}$$
(4)

Please refer to the Appendix for a detailed explanation of the operators \(B\) and fractional-order derivatives.

From Eq. (4), we observe that fractional differences involve all points preceding the current one, indicating long-term memory. As \(\alpha \) approaches 1, the weight of \({x}_{i-1}\) increases, while for \(\alpha \) close to 0, its weight decreases.

Historically, computational constraints hindered the adoption of fractional models, but modern technology allows for efficient computation. Various methods have been developed to approximate the weights of fractional derivatives, broadening the application of fractional models.

Given a time series \(X=\{{x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}\}\), with embedding dimension m and fractional derivative order α, we can reconstruct it into a new sequence of dimension m and length n, denoted as:

$$\begin{array}{c}X=\left[{X}_{1},{X}_{2}, \dots ,{X}_{m}\right],\end{array}$$
(5)

where \({X}_{i}=\left({\nabla }^{\alpha }{X}_{i-1}\right),{X}_{1}=X,i=\text{2,3},4,\dots m\)

Univariate multi-span statistical complexity

Following the fractional-order state space reconstruction of a univariate time series using fractional calculus, we proceed to compute its statistical complexity. In this study, we utilize the dispersion entropy mapping method26 to transform the state space trajectory into a symbolic sequence. The detailed procedure is outlined below:

Step 1: Mapping time series into symbolic sequences.

Initially, the cumulative distribution function (CDF) of the normal distribution is employed to transform the original sequence \(\mathbf{X}={\{{X}_{k,i}\}}_{i=1,\dots ,m}^{k=1,\dots ,n}\) into a new sequence \(\mathbf{Y}={\{{Y}_{k,i}\}}_{i=1,\dots ,m}^{k=1,\dots ,n}\) with values ranging between 0 and 1:

$$\begin{array}{c}{y}_{k,i}=\frac{1}{{\sigma }_{i}\sqrt{2\pi }}{\int }_{-\infty }^{{x}_{k,i}}{e}^{\frac{-(t-{\mu }_{i}{)}^{2}}{2{\sigma }_{i}^{2}}}dt,\end{array}$$
(6)

where \({\mu }_{i}\) and \({\sigma }_{i}\) represent the mean and standard deviation of each column in the original sequence \(\mathbf{X}\).

The new sequence \(\mathbf{Y}={\{{Y}_{k,i}\}}_{i=1,\dots ,m}^{k=1,\dots ,n}\) is mapped to a discrete integer sequence ranging from 1 to \(c\), as expressed in Eq. (7):

$$\begin{array}{c}{Z}_{k,i}^{c}=round\left({y}_{k,i}\cdot c+0.5\right).\end{array}$$
(7)

The \(round(\cdot )\) function rounds the number to the nearest integer.

Step 2: Build the transition network.

Given an embedding dimension \(m\) and a range of 1 to c for the trajectory \({Z}_{k,i}^{c}\), there are \({m}^{\text{c}}\) possible arrangements after mapping to symbols. Each arrangement, denoted as \({\pi }_{q}\) where \(0\le q\le {m}^{\text{c}}\), contributes to the formation of the transition network, a specialized complex network focusing on transition relationships within the state space. Represented by an adjacency matrix \(\mathbf{A}\), each node signifies an arrangement \({\pi }_{q}\), while edges depict the transition probabilities from one pattern \({\pi }_{x}\) to another pattern \({\pi }_{y}\) as follows:

$$\begin{array}{c}{a}_{{\pi }_{x}\to {\pi }_{y}}=\frac{\#\{{\pi }_{x}\to {\pi }_{y}\}}{\sum_{q=1}^{{m}^{\text{c}}}\#\{{\pi }_{q}\}-1}.\end{array}$$
(8)

Here, \(\#(\cdot )\) denotes the count, and \({\pi }_{x}\to {\pi }_{y}\) represents the transition from pattern \({\pi }_{x}\) at time \(t\) to pattern \({\pi }_{y}\) at time \(t+1\).

Step 3: define the transition entropy under single spanning dimension.

The Single-Span Transition Entropy is defined as in Eq. (9):

$$\begin{array}{c}E\left(Y|X\right)=\sum_{x=1}^{{m}^{\text{c}}}p\left(x\right)\times E\left(y|X=x\right),\end{array}$$
(9)

where \(p\left(x\right)\) and \(E\left(Y|X=x\right)\) are defined in Eqs. (10) and (11):

$$\begin{array}{c}p\left(x\right)=\frac{\sum_{y=1}^{{m}^{\text{c}}}{a}_{xy}}{\sum_{y=1}^{{m}^{\text{c}}}\sum_{x=1}^{{m}^{\text{c}}}{a}_{xy}},\end{array}$$
(10)
$$\begin{array}{c}E\left(Y|X=x\right)=-\sum_{y=1}^{ {m}^{\text{c}}}{a}_{xy}\times \text{log}\left({a}_{xy}\right),\end{array}$$
(11)

Here, \({a}_{xy}\) refers to the value of \({a}_{{\pi }_{x}\to {\pi }_{y}}\) defined in Eq. (8).

Step 3: define the transition entropy under the multi-span dimension.

The multi-span transition entropy is defined to encompass multi-span information in time series, extending beyond traditional adjacent transitions. While traditional transitions analyze the relationship between two consecutive symbols, a more comprehensive relationship, denoted as \({\pi }_{x}\stackrel{\tau }{\to }{\pi }_{y}\), signifies a transition from a pattern \({\pi }_{x}\) at time t to pattern \({\pi }_{y}\) at time \(t+\tau \). This is a time network representing a set of matrices across multiple time scales.

$${\varvec{\Pi}}=\{{A}_{1},{A}_{2},\dots {A}_{\tau },\dots \}$$

where \(\tau \) represents the time factor, and for a given time dimension \(s\), \(1\le \tau \le s\).

Multi-span transition entropy is defined as:

$$\begin{array}{c}{E}_{\tau }\left(Y|X\right)=\sum_{x=1}^{{m}^{\text{c}}}{p}_{\tau }\left(x\right)\times {E}_{\tau }\left(y|X=x\right),1\le \tau \le s\end{array}$$
(12)

where \({p}_{\tau }\left(x\right)\) and \({E}_{\tau }\left(Y|X=x\right)\) are defined in Eqs. (13) and (14):

$$\begin{array}{c}{p}_{\tau }\left(x\right)=\frac{\sum_{y=1}^{{m}^{\text{c}}}{a}_{xy}^{\tau }}{\sum_{y=1}^{{m}^{\text{c}}}\sum_{x=1}^{{m}^{\text{c}}}{a}_{xy}^{\tau }},\end{array}$$
(13)
$$\begin{array}{c}{E}_{\tau }\left(Y|X=x\right)=-\sum_{y=1}^{ {m}^{\text{c}}}{a}_{xy}^{\tau }\times \text{log}\left({a}_{xy}^{\tau }\right)\end{array}$$
(14)

Here, \({a}_{xy}^{\tau }\) refers to the value of \({a}_{{\pi }_{x}\to {\pi }_{y}}^{\tau }\) defined in Eq. (15) for the transition probability at time scale \(\tau \).

$$\begin{array}{c}{a}_{{\pi }_{x}\to {\pi }_{y}}^{\tau }=\frac{\#\{{\pi }_{x}\stackrel{\tau }{\to }{\pi }_{y}\}}{\sum_{q=1}^{{m}^{\text{c}}}\#\{{\pi }_{q}\}-\tau },\end{array}$$
(15)

where \(\#(\cdot )\) represents the count, and \({\pi }_{x}\stackrel{\tau }{\to }{\pi }_{y}\) denotes the transition from pattern \({\pi }_{x}\) at time t to pattern \({\pi }_{y}\) at time \(t+\tau \).

Multivariate multispan transition entropy component method

Traditional transition networks are limited to analyzing univariate complexity, yet real-life scenarios often involve vast amounts of multivariate data, such as high-density electroencephalography (EEG) and stock market prediction. Consequently, there is a pressing need to extend traditional transition networks to encompass the complexity analysis of multivariate time series.

For two time series \(\text{P}\) and \(\text{Q}\), the first and second steps of Sect. “Univariate multi-span statistical complexity” process them individually, yielding the adjacency matrices \(\{{p}_{xy}\}\) and \(\{{q}_{xy}\}\) of their respective transition networks.

The complexity of bivariate time series is defined as the sum of intra-sample and inter-sample complexity.

$$\begin{array}{c}E\left(p,q\right)=0.5*(E\left(p\left(y|x\right)\right)+E\left(q\left(y|x\right)\right))+E\left(p\left(y|x\right)|q\left(y|x\right)\right),\end{array}$$
(16)
$$\begin{array}{c}=0.5*(\sum_{x=1}^{{m}^{\text{c}}}p\left(x\right)\times E\left(p\left(y\right)|X=x\right)+\\ \sum_{x=1}^{{m}^{\text{c}}}q\left(x\right)\times E\left(q\left(y\right)|X=x\right))+\\ \sum_{x=1}^{{m}^{\text{c}}}p\left(x\right)\times (\sum_{y=1}^{{m}^{\text{c}}}p\left(y|x\right)\times \text{log}\left(\frac{p\left(y|x\right)}{q\left(y|x\right)}\right))),\end{array}$$
(17)

where \(p\left(x\right)=\sum_{y=1}^{{m}^{\text{c}}}{p}_{xy},q\left(x\right)=\sum_{y=1}^{{m}^{\text{c}}}{q}_{xy},p\left(y|x\right)=\frac{{p}_{xy}}{p(X=x)},E\left(p(y)|X=x\right)=-\sum_{y=1}^{ {m}^{\text{c}}}\frac{{p}_{xy}}{p(X=x)}\times \text{log}\left(\frac{{p}_{xy}}{p(X=x)}\right)\).

From Eq. (16), the complexity between two-time series variables can be decomposed into within-sample and between-sample complexities. \(E(p(y|x))+E(q(y|x))\) represents inter-sample complexity, while \(E\left(p(y|x)|q(y|x)\right)\) defines intra-sample complexity.

If \(P=\{{p}_{xy}^{1},{p}_{xy}^{2},\dots ,{p}_{xy}^{k}\}\) are the transition networks obtained after the first and second steps for k-dimensional time series data, we use \({T}_{P}\) to denote the inter-sample complexity for multivariate time series, as defined in Eq. (18).

$$\begin{array}{c}{T}_{P}=\frac{1}{k}\sum_{i=1}^{k}E\left({p}^{i}\left(y|x\right)\right).\end{array}$$
(18)

\({W}_{P}\), representing the intra-sample complexity, is defined in Eq. (15).

$$\begin{array}{c}{W}_{P}=\frac{1}{\left(k-1\right)!}\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}E\left({p}^{j}\left(y|x\right)|{p}^{i}\left(y|x\right)\right).\end{array}$$
(19)

Thus, the complexity \(E\left(P\right)\) of the multivariate time series can be expressed in terms of inter-sample complexity \({T}_{P}\) and intra-sample complexity \({W}_{P}\).

$$\begin{array}{c}E\left(P\right)={T}_{P}+{W}_{P}.\end{array}$$
(20)

Equation (20) only represents single-span complexity. To generalize, we can define the multi-span complexity with a time factor \(\tau \).

If \({P}^{\tau }=\{{p}_{xy}^{1,\tau },{p}_{xy}^{2,\tau },\dots ,{p}_{xy}^{k,\tau }\}\) are \(k\) dimensional time series data's transition networks at scale factor τ, the multi-span multivariate time series complexity can be expressed in Eq. (21).

$$\begin{array}{c}E\left({P}^{\tau }\right)={T}_{{P}^{\tau }}+{W}_{{P}^{\tau }}.\end{array}$$
(21)
$${T}_{{P}^{\tau }}=\frac{1}{k}\sum_{i=1}^{k}E\left({p}^{i,\tau }\left(y|x\right)\right),$$
$${W}_{{P}^{\tau }}=\frac{1}{\left(k-1\right)!}\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}E\left({p}^{j,\tau }\left(y|x\right)|{p}^{i,\tau }\left(y|x\right)\right).$$

In summary, the multi-span transition entropy component method based on fractional order state space enhances time series stability and long-term memory by fractional-order state space reconstruction, enabling the study of both time series complexity and structural similarity in multivariate data.

Simulation data validation

Noise experiment

The fractional derivative, unlike integer-order derivatives reliant on past values, aggregates all past outcomes, rendering fractional state space notably adept at attenuating the influence of random noise on signals. For example, when various levels of noise are introduced to a sine signal, as depicted in Eq. (22):

$$\begin{array}{c}x=\text{sin}\left(t\right)+\delta \times \theta \left(t\right),\end{array}$$
(22)

Here, \(\delta \in [\text{0,1}]\) is a proportion factor, and \(\theta (t)\) is a random variable following the uniform distribution \(\widehat{U}(\text{0,1})\).

Figure 1 illustrates the MTECM-FOSS algorithm values for the resulting noisy time series when \(\delta \) is set to 0, 0.1, 0.2, 0.3, 0.5, and 0.8. The algorithm value for the original periodic signal (Fig. 1a) is consistently small, with minimal variation across different spanning dimensions. However, as the signal-to-noise ratio diminishes, the MTECM-FOSS algorithm value escalates. This phenomenon stems from fractional derivatives with \(\alpha \) close to 0 assigning greater weight to past values, signifying long-term memory, while those closer to 1 prioritize recent values, denoting short-term memory. Thus, for smaller α values (e.g. \(\alpha \) = 0.1), historical derivative values contribute to noise cancellation, thereby enhancing the signal-to-noise ratio. Figure 1b demonstrates that when \(\delta \) = 0.1, the MTECM-FOSS algorithm value increases with α, indicating heightened noise influence. Furthermore, as \(\delta \) increases, the overall algorithm value rises; nevertheless, for \(\alpha \) = 0.1, the MTECM-FOSS value remains relatively subdued.

Figure 1
figure 1

Depth plots of the MTECM-FOSS algorithm under varying noise levels, demonstrating the changes with fractional order and spanning dimension.

Full size image

Auto-regressive fractional integrated moving average (ARFIMA) model

The ARFIMA model is widely used for simulating signals with time-series long-range dependence. The general expression for the generation process of data \({y}_{t}\) in an ARFIMA(p,d,q) model, as shown in Eq. (23).

$$\begin{array}{c}{y}_{t}=\sum_{n=1}^{\infty }{a}_{n}(d){y}_{t-n}+{\varepsilon }_{t},\end{array}$$
(23)

where \({\varepsilon }_{t}\) follows a standard normal distribution, \(d\) is a coefficient, \({a}_{n}(d)\) are weights, and \({a}_{n}\left(d\right)=d\times \frac{\Gamma \left(n-d\right)}{\Gamma \left(1-d\right)\Gamma \left(n+1\right)}\) with d belonging to the interval (− 0.5, 0.5), representing the scale parameter. \(\Gamma \left(\cdot \right)\) denotes the gamma function, and \(n\) is the time index.

In the case where \(d=0\), the generated variable \({y}_{t}\) becomes a random variable. For \(0<d<0.5\), the model exhibits long-term memory, characterized by the autocorrelation function values of the time series declining at a hyperbolic rate. Conversely, for \(-0.5<d<0\), the sum of the absolute values of the time series autocorrelation function converges to a constant, indicative of short-term memory. Long-memory processes, unlike short-memory processes, demonstrate slower autocorrelation decay and exhibit stationarity. This study simulates time series generated under six distinct parameter settings: ARFIMA (0, − 0.4, 0), ARFIMA(0, − 0.3, 0), ARFIMA(0, − 0.1, 0), ARFIMA(0, 0.1, 0), ARFIMA(0, 0.3, 0), and ARFIMA(0, 0.4, 0).

Figure 2 depicts the variation of MTECM-FOSS algorithm values across different fractional orders and spanning dimensions within the ARFIMA model. In the range \(-0.5<d<0\), indicating short-term memory, the highest MTECM-FOSS values are typically situated to the right of α \(=0.5\). For instance, when the scale parameter \(d = -0.4\), the peak occurs at α \(=0.6\) or \(0.7\); for \(d = -0.3\), it's at \(\alpha = 0.6\); and for \(d = -0.1\), it's at α \(=0.5\). Conversely, in the case of \(0<d<0.5\), corresponding to long-term memory, the highest MTECM-FOSS values are predominantly observed to the left of \(\alpha = 0.5\), and as the scale parameter \(d\) increases, the peak of the MTECM-FOSS values gradually shifts towards α \(= 0\).

Figure 2
figure 2

Depth plots of the MTECM-FOSS algorithm under varying scale parameter, demonstrating the changes with fractional order and spanning dimension.

Full size image

Fractal Brownian motion

Fractional Brownian motion stands as a valuable mathematical model for time series analysis, extending the foundational concept of Brownian motion and contributing significantly to both mathematical and physical research. The fractal dimension of various trajectory fractional Brownian motions can be delineated by the smoothing coefficient \(H\), where (0 < \(H\)  < 1). As \(H\) approaches 0, the trajectory curve exhibits maximal irregularity, while it attains smoother characteristics with \(H\) approaches 1. The \(H\)-index represents the scaling behavior of the trajectory curve \({T}_{H}(t)\), reflecting its increments.

The increment of the trajectory curve \(T(t)\) is defined as:

$$\Delta {T}_{H}={T}_{H}\left({t}_{2}\right)-{T}_{H}\left({t}_{1}\right).$$

According to the scaling law, we have:\(\Delta {T}_{H}\sim (\Delta t{)}^{H},\Delta t={t}_{2}-{t}_{1}\)

Fractional Brownian motion \(T(t)\) exhibits long-range dependence. Statistically, for any three times \({t}_{1}, t\) and \({t}_{2}\)(\({t}_{1}< t <{t}_{2}\)), the increments \({T}_{H}\left(t\right)-{B}_{H}({t}_{1})\) are related to \({T}_{H}\left({t}_{2}\right)-{T}_{H}(t)\). If \(H>0.5\), the increments of \({T}_{H}\left(t\right)\) are positively correlated or exhibit persistence, meaning that on average, past increasing trends tend to continue increasing in the future. If \(H<0.5\), the increments are negatively correlated or show antipersistence, implying that past increases tend to decrease on average in the future. When \(H=1/2\), the long-range dependence of increments vanishes, and the fractional Brownian motion reduces to ordinary Brownian motion.

Figure 3 depicts contour plots showcasing the MTECM-FOSS algorithm values across varying Smoothness coefficients (\(H\)) and fractional orders (\(\alpha \)), alongside different spanning dimensions. With an increase in the spanning dimension, there is a corresponding rise in the MTECM-FOSS algorithm value, visually denoted by the expanding red regions, which effectively capture more sensitive dynamic changes.

Figure 3
figure 3

Depth plots of the MTECM-FOSS algorithm under varying spanning dimension, demonstrating the changes with smoothing coefficient and fractional order.

Full size image

Figure 4 displays contour plots of the MTECM-FOSS algorithm values concerning different Smoothness coefficients (\(H\)), fractional orders (\(\alpha \)), and spanning dimensions (\(\uptau \)). It illustrates that as the Smoothness coefficient (\(H\)) increases, the MTECM-FOSS algorithm value decreases, as evident from the lightening red regions in the plots. For \(H<0.5\) (Fig. 4a–c), the peak MTECM-FOSS value occurs around \(\alpha =0.2\). Conversely, for \(H>0.5\) (Fig. 4d–f), the algorithm demonstrates a minimum at \(\alpha \approx 0.2\), with the MTECM-FOSS value increasing as \(\alpha \) rises.

Figure 4
figure 4

Depth plots of the MTECM-FOSS algorithm under varying smoothing coefficient, demonstrating the changes with fractional order and spanning dimension.

Full size image

Figure 5 presents the contour plots of the MTECM-FOSS algorithm values under different fractional order (α), Smoothness coefficient (\(H\)), and spanning dimensions (\(\uptau \)). At α = 0.1, smaller Smoothness coefficients (\(H\)) correspond to higher entropy values, while larger \(H\) values have lower entropy, roughly demarcated by \(H \approx 0.5\). Initially, the red region in Fig. 5 increases with α, followed by a decrease, while the blue region continually contracts as α increases.

Figure 5
figure 5

Depth plots of the MTECM-FOSS algorithm under varying fractional order, demonstrating the changes with smoothing coefficient and spanning dimension.

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Multivariate data analysis for logical graph

In this section, we will demonstrate the relationship between the multi-variable time series composed of sequences in a one-dimensional logical graph and how this changes with fractional order and spanning dimension when different system parameters are chosen. The connection between system parameters and Lyapunov exponents in traditional integer-order state spaces and transition networks has been discussed in literature27.

The control equation for a one-dimensional logical graph, as expressed in Eq. (24), is:

$$\begin{array}{c}{x}_{i+1}=r{x}_{i}\left(1-{x}_{i}\right),\end{array}$$
(24)

where \(r\) represents the system parameter. We generated time series for six distinct parameter values, calculating inter-group and intra-group distances to illustrate the MTECM-FOSS algorithm in detail. Initially, we defined three distinct parameters \(r\), specifically 3, 3.7, and 3.9. At \(r\) = 3, the system enters a periodic state, stabilizing into a time series with a period of 2. At \(r\) = 3.7, the system exhibits chaotic behavior, where even minute initial condition variations lead to vastly different long-term dynamics. As \(r\) = 3.9, the system's Lyapunov exponent increases further, and disorder becomes more pronounced. Two different initial conditions, 0.1 and 0.4, were designated as starting points for separate sequences. The combination of three different system parameters and two initial conditions resulted in six-time series sets. The system underwent 5000 iterations, with the last 3000 points selected for analysis, discarding the first 2000 to ensure the stability of the time series. The MTECM-FOSS algorithm was employed to compute distances between the six time series pairs, with parameters set to fractional order \(\alpha =0.1\), reconstructed state space dimension \(m=4\), and spanning dimension \(\tau =10\). The computed results are presented in Table 2.

Table 2 MTECM-FOSS algorithm values for different parameters and initial values.
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When \(r\) = 3, the system is in a periodic state, and the initial value does not affect its convergence value. As shown in Table 2, the within-group complexity is quite low. At \(r\) = 3.7, the system is in a chaotic state, highly sensitive to initial conditions, resulting in entirely different time series for different initial values. For initial values of 0.1 and 0.4, the within-group entropy is 1.44 and 1.45, respectively. Although the initial values differ, their complexity representation is nearly identical. When \(r\) = 3.9, the system's Lyapunov exponent increases further, leading to an increase in within-group energy distance, which is 1.567 and 1.574, respectively. For different initial values, the within-group complexity remains almost the same.

For two groups of time series with the same system parameters but different initial values, their between-group distance is very small, indicating a similar dynamical mechanism. The between-group distance and the complexity of the time series are related to time series with different parameters. The larger the complexity difference between the two time series, the greater the between-group distance. For example, when \(r\) = 3 and \(r\) = 3.9, with an initial value of (\({x}_{0}=0.1\)), the between-group distance between the two time series is 0.673, while for \(r\) = 3 and \(r\) = 3.7, with \({x}_{0}=0.1\), the distance is 0.165. This shows that the system with \(r\) = 3.9 is farther from the periodic system than when \(r\) = 3.7. The ability to quantify the interrelationship between multi-variable time series is precisely the strength of the MTECM-FOSS algorithm. Next, we will test the algorithm's behavior under different spanning dimensions and fractional orders by synthesizing multi-variable time series.

Using the MTECM-FOSS algorithm, we calculate the multi-variable simulated time series composed of time series generated by the logical graph. First, we combine time series produced at different \(r\) values into multi-variable sequences. For instance, we have a two-period multi-variable data set with \({r}_{1}=[\text{3.1,3.2,3.3}]\), a mixed-period multi-variable data set with one variable having a two-period cycle and the other two having a four-period cycle, using \({r}_{2}=[\text{3.1,3.5,3.54}]\), a four-period multi-variable data set with \({r}_{3}=[\text{3.46,3.5,3.54}]\), a multi-variable data set with one variable having a two-period cycle, another a four-period cycle, and a chaotic component using \({r}_{4}=[\text{3.1,3.5,3.7}]\), a multi-variable data set with one variable having a two-period cycle and two chaotic sequences using \({r}_{5}=[\text{3.1,3.7,3.8}]\), and a purely chaotic multi-variable data set using \({r}_{6}=[\text{3.7,3.8,3.9}]\). We process the multi-variable time series generated by different parameter vectors using the MTECM-FOSS algorithm, varying the fractional order \(\alpha \) and the multi-span dimension \(\tau \).

As shown in Fig. 6a–c depict the MTECM-FOSS algorithm's contour plots for \({r}_{1},{r}_{2}\), and \({r}_{3}\) vector parameters, respectively, at different fractional order \(\alpha \) values and spanning dimensions. The entropy of periodic signals is generally low, but it exhibits an increasing trend as the sequence's possible states grow. Moving on to (d), (e), and (f), these represent the MTECM-FOSS algorithm’s contour plots for \({r}_{4},{r}_{5}\), and \({r}_{6}\) vector parameters, respectively. Notably, as the proportion of chaotic data in the multivariate data increases from (d) to (f), the red regions also expand. The multivariate time series data represented by \({r}_{4},{r}_{5}\), and \({r}_{6}\) exhibit a rich variation in entropy with increasing spanning dimension, with entropy increasing linearly. Furthermore, for these parameters, the entropy is relatively small when the fractional order \(\alpha \) is below 0.4, and it gradually increases as \(\alpha \) approaches 1.

Figure 6
figure 6

Depth plots of the MTECM-FOSS algorithm under varying parameter vector, demonstrating the changes with fractional order and spanning dimension.

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Empirical data validation

To assess the MTECM-FOSS algorithm's performance in practical data, we employed single-channel EEG data from epilepsy patients, high-density EEG data from epilepsy, and fault diagnosis data. Due to MTECM-FOSS normalizing the data during the transition from state space to symbol space, amplitude information is lost. To compensate for this, we incorporated the data's variance feature into the classifier. In summary, the classification strategy employed in this study is outlined in Table 3.

Table 3 Data sets and parameter settings.
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Where \(m\) refers to the State space embedding dimension, \(\alpha \) refers to the Fractional order, and \(\tau \) refers to the Spanning dimension.

The following are the definitions of four measurement indicators:

Sensitivity: Sensitivity measures the ratio of true positive test results to all positive results.

$$Sensitivity=\frac{{T}_{P}}{{T}_{P}+{F}_{N}}\times 100\text{\%}.$$

Specificity: Represents the ratio of the number of predicted true negatives to all negatives.

$$Specificity=\frac{{T}_{N}}{{T}_{N}+{F}_{P}}\times 100\text{\%}.$$

Accuracy: Accuracy is one of the most important metrics in classification tasks, representing the proportion of correct predictions made by the model to the total number of predictions.

$$Accuracy=\frac{{T}_{P}+{T}_{N}}{{T}_{P}+{T}_{N}+{F}_{P}+{F}_{N}}\times 100\text{\%}.$$

G-Means: Since some datasets have a huge difference in the quantity of data between different classes, the classification of such data belongs to an imbalanced classification problem. This metric is highly valuable and applicable for evaluating imbalanced classification in this study, defined as:

$$G-Mean=\sqrt{Precision\times Specificity.}$$

\({T}_{P}\): the number of true positives, \({T}_{N}\): the number of true negatives, \({F}_{P}\): the number of false positives, \({F}_{N}\): the number of false negatives.

Dataset selection and introduction

Dataset 1

Dataset 1 was gathered by Swami et al.28 at the Neurology Sleep Center in Hauz Khas, New Delhi. It captures EEG signals from 10 epilepsy patients across three stages: preictal, ictal, and interictal, corresponding to pre-seizure, seizure, and post-seizure states. This dataset is accessible for legal download from the following website: https://www.researchgate.net/publication/308719109_EEG_Epilepsy_Datasets. The EEG signals were recorded using a Grass Telefactor Comet AS40 amplifier with a sampling rate of 200 Hz. Scalp EEG electrodes were placed following the 10–20 system and filtered within the frequency band of 0.5 to 70 Hz to eliminate noise and artifacts. Subsequently, the collected data were segmented into seizure, pre-seizure, and post-seizure periods, and further filtered within the 0.5–70 Hz range. The signals were then partitioned into 5.12-s windows, each containing 1024 samples. Overall, the dataset includes 50 MAT files for each stage, with each file storing a single-channel EEG signal time series data.

Dataset 2

The Bearing Data Center at Case Western Reserve University (CWRU) School of Engineering offers a dataset widely employed in fault diagnosis research. This dataset encompasses normal bearings and bearings with single-point defects at the drive end and fan end. Data were collected at speeds of 12,000 samples per second and 48,000 samples per second, specifically for experiments on drive end bearings. All fan end bearing data were collected at a speed of 12,000 samples per second.

Data files are provided in Matlab format, with each file containing vibration data for both the fan end and drive end, along with motor speed data. The dataset can be downloaded from the Case Western Reserve University (CWRU) School of Engineering website: https://engineering.case.edu/bearingdatacenter/download-data-file. In this study, 12 kHz data was utilized. Given that the official dataset offers only 48 kHz data for "normal" type data, the data used in this study were downsampled to 12 kHz. Additionally, we applied windowing to extract subsequences of length 1024 with a step size of 512, resulting in the data volume outlined in Table 4. Furthermore, we employed five-fold cross-validation and utilized support vector machines to classify the extracted features.

Table 4 Details of the CWRU dataset.
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Dataset 3

Dataset 329 was curated by Boston Children's Hospital, where EEG records were collected from children with refractory epilepsy. These subjects underwent monitoring for several days post cessation of antiepileptic drugs to characterize seizures and assess suitability for surgical intervention. Signals were sampled at 256 Hz with a resolution of 16 bits. Most files contain 23 EEG signals (in some cases, 24 or 26), utilizing the international 10–20 EEG electrode placement and naming system. This dataset can be accessed from the PhysioNet website: https://physionet.org/content/chbmit/1.0.0/.

Given the extensive duration and volume of EEG data spanning several days, we segmented seizure-labeled data for each subject. We then randomly selected non-seizure period data, doubling the duration of seizure periods to achieve a 1:2 ratio in duration. Subsequently, we divided the long-term EEG signals of each channel into non-overlapping sliding windows lasting 4 s (1024 data points). For instance, in 1 h of EEG recording, the data were segmented into 900 non-overlapping sub-sequences, each lasting 4 s. These sub-sequences were labeled as seizure or non-seizure using the proposed method. Finally, each of the 900 time segments in the 1-h EEG recording was assigned its own label, aiding in determining whether the test label matched the true seizure label. Specific details and data selection criteria for Dataset 2 are outlined in Table 5. Additionally, we utilized five-fold cross-validation in this dataset to compute accuracy and other metrics.

Table 5 Details of the CHB-MIT dataset.
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Result and analysis

Analysis of dataset 1

Table 6 shows the processing results for dataset 1. It can be observed that our method achieves a classification accuracy of 100% for distinguishing between seizure and interictal periods. It also achieves a near-perfect accuracy of 96% for distinguishing between preictal and ictal periods. Meanwhile, the classification accuracy is lowest, but still at 88%, for distinguishing between preictal and interictal periods. This is because the EEG signals during the preictal and interictal periods exhibit similar dynamic characteristics, making them difficult to differentiate. Table 6 also presents a brief comparison between our method and previous studies. In scenarios involving classification between interictal and ictal periods, our method achieves a classification accuracy of 100%, outperforming other listed studies. For the task of classifying between ictal and preictal periods, the accuracy reaches 96%, which is better than the studies by Gupta et al.30, Hadiyoso et al.31, and Sharma et al.32. Lastly, for the task of classifying between interictal and preictal periods, our method achieves an accuracy of 88%, surpassing all listed methods and achieving the highest accuracy.

Table 6 A comparative analysis of MTECM-FOSS performance against contemporary techniques for dataset 1.
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Analysis of dataset 2

Dataset 2 contains data with different faults under different operating conditions, allowing classification of different faults under different operating conditions, considered as a ten-classification task. We select parameter scores with fractional order α\( =0.1\), state space reconstruction dimension \(m=4\), and spanning dimension \(\tau =10\). The MTECM-FOSS algorithm is used to calculate entropy features under different spanning dimensions. Then, the "fitcecoc" function in MATLAB is utilized to train a multiclass error-correcting output codes (ECOC) model using support vector machine (SVM) binary learners. The accuracy results of classification are shown in Table 7, while Fig. 7 displays the classification results and confusion matrix of the MTECM-FOSS algorithm in the ten-classification task.

Table 7 A comparative analysis of MTECM-FOSS performance against contemporary techniques for dataset 2.
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Figure 7
figure 7

(a) Classification result, and (b) confusion matrix of the MTECM-FOSS method with fivefold cross-validation.

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In the field of bearing fault diagnosis, convolutional neural networks and entropy methods are common feature extraction methods. This paper lists five methods for classifying time series data into images using neural networks, such as transforming time series data into Gramian Angular Field (GAF) or grayscale images36, or time–frequency images37, and then using convolutional neural networks to build feature extractors and classification modules. Generally, multiscale methods are more competitive than single-scale methods. Therefore, this paper lists common multiscale entropy methods for bearing fault diagnosis classification, such as multiscale entropy (MSE)39, improved multiscale dispersion entropy(MDE)40, RCMDE41, and GGD-RCMDE38.

We can see that the MTECM-FOSS algorithm achieves excellent classification accuracy of 99.5%, whether compared with deep learning methods or commonly used entropy methods. Also, only 200 total samples per category were used in the study of Dhandapani et al.38, not samples from the full dataset. Moreover, because the MTECM-FOSS algorithm only has 14 features when the state space reconstruction dimension m = 4 and spanning dimension \(\uptau \) = 10, it occupies very little memory space and has a relatively fast prediction time.

Analysis of dataset 3

Table 8 presents the processing results for dataset 3. We can see that all classification accuracies exceed 90%, with many reaching 100% for some subjects. In many epilepsy patients, the performance of Specificity is higher than Sensitivity, mostly above 95%. From the last row of Table 8, it can be seen that the average sensitivity and specificity of the MTECM-FOSS algorithm are both greater than 95%. However, the sensitivity of patient 13 is below 90%, which may be due to frequent waveform changes in the EEG data of patient 13 that resemble epileptic EEG waves, such as sudden amplitude changes, and so on.

Table 8 MTECM-FOSS performance for dataset 3.
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Multi-channel EEG data were used for dataset 3, with almost 23 channels for each subject. The MTECM-FOSS algorithm directly processes multi-channel data, calculating the complexity of internal components of multi-channel time series data.

Table 9 provides a comparison between this method and others. Due to the large volume of data in this dataset and the numerous epilepsy patients, most studies only used partial time data or data from some patients in this dataset. Additionally, the division between testing and training sets is inconsistent, limiting the comparability of various result indicators. However, we can still see that MTECM-FOSS remains the most accurate among many classification algorithms, demonstrating strong competitiveness.

Table 9 The table presents the comparison of accuracy between MTECM-FOSS algorithm and other algorithms for dataset 3.
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Discussion

This paper proposes a Novel multi-span Transition Entropy Component Method Based on Fractional Order State Space for multivariate Complex Time series. Fractional order systems are a natural extension of traditional integer order systems. Although there have been extensive efforts by many researchers in fractional-order physical models, fractional-order analysis methods based on data still require further exploration. According to Takens' embedding theorem, we can reconstruct a state space from a one-dimensional chaotic time series that is topologically equivalent to the original dynamical system. There are generally two methods for reconstructing the state space: the derivative reconstruction method and the coordinate delay reconstruction method. How we choose the state variables is quite flexible, as long as the selected state variables are not linearly correlated. Therefore, this paper proposes a state space reconstruction method based on fractional order derivatives. This method is not contradictory to the traditional derivative reconstruction method because integers are a special case of fractions. In addition, this paper also proposes a multi-span Transition Entropy Component Method for analyzing the complexity of multivariate time series. The MTECM-FOSS algorithm increases the universality and flexibility of the method by introducing two flexible variables (fractional order \(\alpha \) and spanning dimension \(\tau \)).

This method has been validated on multiple simulated and empirical datasets. For example, in the noise addition experiment in Sect. “Noise experiment”, different levels of noise produce significantly different results. Although overall, the MTECM-FOSS algorithm value increases as the signal-to-noise ratio decreases. However, at the same noise level, when the fractional order \(\alpha \) is small, the MTECM-FOSS algorithm value always remains relatively small. This is because when the fractional order \(\alpha \) is small, the past values of the time series data often have a relatively large weight, and the sum of past data often offsets random noise to some extent.

In the analysis of the Auto-regressive Fractional Integrated Moving Average (ARFIMA) model in Sect. “Auto-regressive fractional integrated moving average (ARFIMA) model”, we found that when the scale parameter \(d\) is in the range of (− 0.5, 0), most of the peak points of the MTECM-FOSS algorithm value are concentrated to the right of \(\alpha \) = 0.5. When \(d\) is in the range of (0, 0.5), the model is considered to have long-term memory, and most of the peak points of the MTECM-FOSS algorithm value are concentrated to the left of \(\alpha \) = 0.5. Moreover, as the scale parameter d increases, the peak points of the MTECM-FOSS algorithm value gradually shift towards \(\alpha \) = 0.

In the analysis of fractional Brownian motion in Sect. “Fractal Brownian motion”, it was found that as the Smoothness coefficient \(H\) increases, the overall MTECM-FOSS algorithm value decreases. When the Smoothness coefficient \(H\) < 0.5, the maximum value of the MTECM-FOSS algorithm appears around \(\alpha \) = 0.2. When the Smoothness coefficient \(H\) > 0.5, the MTECM-FOSS algorithm value shows a minimum value around \(\alpha \) = 0.2.

In Sect. “Multivariate data analysis for logical graph”, six different parameter vectors were selected to generate various types of multivariate time series data for analysis. The MTECM-FOSS algorithm effectively distinguishes between multivariate time series data composed of different periods and chaotic sequences. Subsequently, the algorithm was validated using empirical datasets, including single-channel epilepsy EEG datasets, single-channel fault diagnosis datasets, and multi-channel (23-channel) children's epilepsy datasets, demonstrating performance comparable to or better than state-of-the-art methods with a minimal number of classification features.

The multi-span transition entropy component method based on fractional order state space for multivariate Complex Time series represents a substantial advancement over integer-order state space and traditional univariate time series analysis transition networks. This algorithm decomposes the complexity of multivariate time series into intra-sample and inter-sample components, a notable improvement compared to existing complexity measurement methods.

However, the MTECM-FOSS algorithm also presents some drawbacks. Firstly, because it relies on fractional order state space calculations, it theoretically requires all historical data for fractional order derivative computation, inevitably increasing computational time, particularly for long sequence data. Secondly, after fractional order state space reconstruction, the trajectory of particles in state space must be mapped to symbol space using the Dispersion Entropy method, necessitating data normalization which results in the loss of amplitude information. Thirdly, the algorithm introduces two variables, fractional order α and spanning dimension \(\uptau \), enhancing method universality and flexibility but potentially causing confusion in parameter selection.

Conclusion

This paper proposes a novel multi-span transition entropy component method based on fractional order state space for multivariate complex time series. We introduce fractional order state space into nonlinear time series analysis and define the multi-span Transition Entropy Component Method for multivariate time series. The fractional order state space adjusts the "perspective" of analyzing time series by adjusting the fractional order \(\alpha \). The fractional order state space method is a natural extension of the traditional integer-order inverse reconstruction method and has more universal properties. Unlike traditional transition networks analyzing univariate time series, the multi-span transition entropy component method decomposes the complexity of multivariate time series into intra-sample and inter-sample components and introduces spanning dimension \(\uptau \) to insightfully examine time series in a universal transition pattern, transforming traditional single-channel transition networks into a special case of this method. This significant innovation is expected to advance research in nonlinear dynamics and find applications in various fields, and its superior performance has been empirically tested and verified.