Multifractal Analysis of Marathon Pacing—Physiological Background and Practical Implications

Abstract

Marathons are one of the ultimate challenges of human endeavor. As a consequence of the growing passion of amateur runners for this discipline, a strong need has been shown for counselling during the preparation and for advice on how to manage their efforts during the race. This monitoring should be based on parameters collected during the race and correctly interpreted. Multifractality parameters, which have proved their relevance in many other areas of signal processing, are natural candidates for this purpose. This paper shows that, due to the extreme irregularity of the data, the previously used multifractal techniques cannot be applied in this context, in contrast with the recently introduced parameters based on the weak scaling exponent, which require no a priori assumptions for their use; these parameters yield new classification parameters in the processing of physiological data captured on marathon runners. The comparison of their values reveals how marathon runners handle variations in the irregularity of their races and therefore gives a new insight on the way that runners of different levels conduct their run; therefore, this study shows that the use of these parameters offers a promising tool in order to give advice on how to improve performances.

1. Introduction

1.1. General Setting of the Study

Marathon running is a complex endurance activity that requires an intricate balance between speed, stamina, and strategic pacing. Indeed, the ability to sustain optimal performance over long distances is influenced by multiple physiological and biomechanical factors, including heart rate variability (HRV), stride dynamics, lactate accumulation, and muscular fatigue [1]. Biomedical signals often display structural characteristics that are visually recognizable but are not adequately described by traditional metrics such as average amplitude. These signals, derived from various physiological processes, exhibit scale-invariant structures. A biomedical signal is considered scale-invariant when its structural patterns recur consistently across different time scales or subintervals. Fractal analysis is employed to estimate the corresponding scaling exponent, thereby identifying and quantifying scale-invariant structures within biomedical signals [2]. Such structures have also been observed in various temporal biomedical data, including inter-spike intervals in neuronal firing patterns, stride intervals in human gait, breathing intervals, and cardiac inter-beat intervals. Crucially, they can distinguish between healthy and pathological states [3,4], as well as differentiate among various pathological conditions [5]. Over the past decade, several studies have indicated that alterations in the fractal structure of biomedical signals may reflect shifts in physiological adaptability [6]. Consequently, fractal analysis has emerged as a promising method for prognosis and diagnosis in biomedical signal processing. Specifically, fractal and multifractal analyses of heart rate variability, cadence, and running speed in marathon runners were shown to have the potential to detect early signs of fatigue. Unlike clinical studies that use heart-rate multifractality mainly for diagnosis, they are used here as a marker of physiological regulation and pacing control in healthy runners. Indeed, whereas prior work has used multifractal features of heart-rate dynamics primarily for clinical assessments, our objective is to interpret them as indicators of control and regulation during marathon running.

These analytical methods could allow us to identify the critical point at which fatigue begins to affect performance, even before there is an observable decline in running speed—a phenomenon commonly referred to as “hitting the marathon wall.” Traditional methods for assessing these physiological responses, such as HRV analysis and detrended fluctuation analysis (DFA), have been used to investigate the scaling behavior in physiological signals like heart rate and speed during prolonged exertion such as climbing to Mont Blanc (4808 m) [7]. Multifractal detrended fluctuation analysis (MF-DFA) has met successes in many fields of applications, including the study of financial time series that exhibit volatility clustering and other irregular behaviors; this approach helps in identifying intricate patterns in market data, which are crucial for risk management and developing trading strategies [8].

1.2. Main Purposes of the Present Study

In a series of studies initiated by V. Billat [9], the physiological responses and pacing strategies of marathon runners were extensively studied, providing a foundation for understanding how different variables interact during a race. The purpose of this new study is to explore the potential of employing multifractal analysis for enhancing the runner’s feedback regarding the optimization of their pacing strategy. To this end, the performances of marathoners who have completed the same marathon are compared to minimize the impact of the profile race. Subsequently, for each runner, the first 21 km and the last 10 km of the marathon are compared. Multifractal parameters are used to enhance the degree of adequate pacing strategy, thereby enabling the runner to achieve the race while maintaining the degree of high pace variation that has been demonstrated to be associated with personal bests, irrespective of the level of the runner (recreational or elite) [10].

Marathon running requires a balance between speed and stamina, and analyzing physiological signals can provide insights into performance, pacing strategy, and fatigue. Therefore, in this study, multifractal techniques based on the weak scaling exponent are shown to provide new insights into how runners of different levels manage their runs, helping to detect fatigue and optimize pacing strategies. Therefore, the method described in this article provides a mathematical framework for characterizing the complex and irregular variations of heart rate data of marathon runners.

1.3. A Short Overview of the Previous Studies

Traditional methods for the analysis of physiological signals often assume the data to be relatively regular or stationary, which is rarely the case under the extreme physiological stress of running marathons, where performance is affected by fluctuating metabolic and biomechanical constraints [11]. To overcome the limitations of traditional signal analysis methods, multifractal analysis has emerged as a robust approach for studying the complex variability inherent in physiological signals. Unlike linear models, which may fail to capture the self-similarity and heterogeneous fluctuations observed in long-duration exercise, multifractal techniques provide a framework for quantifying the complexity and adaptability of an athlete’s physiological regulation. Multifractal detrended fluctuation analysis (MF-DFA) has been particularly useful in detecting fatigue-induced changes in physiological responses, distinguishing between elite and non-elite runners, and identifying individual pacing control mechanisms [12,13].

Some traditional methods have been used to reveal hidden dynamics in the data:

• the analysis of Heart Rate Variability (HRV) to monitor the autonomic nervous system’s response during marathons, analyzing the effects of fatigue and endurance on heart rate patterns [3],
• Detrended Fluctuation Analysis (DFA), which allows us to identify scaling behavior in physiological signals like heart rate and speed, revealing the impacts of prolonged exercise and fatigue [14],
• multifractal analysis, which emerged as a powerful tool to qualitatively assess physiological signals, offering a more comprehensive perspective on individual marathon performance by supplying new classification and model selection parameters based on scaling invariance exponents [3,6].

Multifractal analysis is a mathematical approach that allows for the characterization across scales of complex, irregular patterns in a dataset [15]. In the 1990s, wavelet-based multifractal analysis provided significant insights into the theoretical underpinnings of multifractal analysis [16]; this laid the ground for the application of multifractal methods in various fields; it promoted a variant based on local suprema of wavelet coefficients as a powerful tool in both theoretical and applied mathematics. Indeed, this method has proven effective in various fields due to its ability to capture intricate variability and interdependencies within datasets.

Multifractal analysis has been extensively applied to physiological data, such as in the study of heart rate variability [3,6]. In particular, it has been shown to capture the complex dynamics of physiological signals of marathon runners, offering possible insights into their performances and their health conditions [17,18].

Several variants of multifractal analysis have been challenged on marathon runners’ data. The first one, the Wavelet Transform Modulus Maxima method (WTMM) allows us to estimate the multifractal spectrum (i.e., the fractional dimensions of the singularity sets of a given order) of a signal [19]. It has been applied to heart rate time series, demonstrating that heart rate variability can be described by scaling laws where multifractal analysis is applied to the physiological signals of marathon runners, focusing on the detection of fatigue and the optimization of pacing strategies. Advanced multifractal techniques such as multifractal detrended fluctuation analysis (MF-DFA) and wavelet-based Legendre spectra were used to quantify the multifractality of time series data collected during marathon races [9]. They both quantify the multifractal properties of non-stationary time series. These studies have shown that multifractal techniques can effectively characterize the variability and complexity of physiological signals collected on marathon runners, offering new insights into endurance performance.

1.4. Limitations of the Previous Studies and the New Perspectives Offered by the Weak-Scaling Exponent

Classical signal analysis techniques often assume a minimal level of regularity in data (e.g., stationarity or smoothness), which is not always present in physiological responses during extreme conditions such as marathons; the implications of this question will be detailed in Section 2.1. Previous research has demonstrated that physiological signals, particularly those related to running cadence and stride variability, exhibit complex dynamical behaviors that may not be well captured by traditional analysis methods [12,20]. To address these limitations, recent studies have explored the use of multifractal analysis, particularly via p-exponents, for analyzing variability in cadence and stride length in endurance sports [21,22]. These techniques allow for a more refined characterization of physiological responses that exhibit irregular fluctuations.

The weak-scaling exponent was designed for signals exhibiting negative pointwise regularity, where multifractal analysis via p-exponents struggles due to extreme irregularity [23,24]. The marathon context (extreme physiological conditions) matches the scenarios for which this method was developed [21,22]. It is particularly suited for physiological signals like heart rate variability (HRV) under stress (e.g., during marathons), where classical assumptions of multifractality may not hold [25]. Indeed, heart rate data during marathons are highly irregular and fall outside the range of applicability of MF-DFA or traditional multifractal methods. In this article, weak-scaling leaders are shown to handle such data better, avoiding issues like saturation phenomena.

More recently, we have examined how cadence and stride variability during marathons can be analyzed using multifractal techniques based on alternative regularity exponents such as p-exponents (see Section 2.1 below) [26]. In some cases, even this extended version of multifractal analysis cannot be performed, and the least demanding setting supplied by the weak-scaling pointwise regularity must be used; indeed, it provides a robust framework for analyzing signals without assuming any prior global regularity on the data; a first application of this alternative technique has been performed in [26], where the focus was put on cadence (in this study, the analysis of velocity was not performed due to GPS measurement inaccuracies).

1.5. Main Contributions and Limitations

The key findings of the present work are:

• Initial raw data show significant variations during different phases of the marathon, including warm-ups and breaks. Cleaning the data to remove non-marathon activities requires continuous reconnections to maintain homogeneity; the purpose of the Appendix A is to prove that the procedure which is used does not alter the regularity properties of the data.
• On a methodological level, a clarification is given on the implicit assumptions under which the currently used techniques in order to derive multifractality parameters are valid, see Section 2.1, and, on the applied side, in the case of the extremely irregular data collected on marathon runners, these assumptions are shown not to be met by most of these methods, see Section 2.1. This indicates the need for a multifractal analysis based on the weak-scaling exponent, which requires no a priori assumption on the data, see Section 2.2.
• This study provides insights into how multifractality parameters using the weak-scaling exponent characterize changes in physiological signals due to fatigue, particularly around the 30th kilometer mark, where perceived exertion is significantly increased.

Note, however, that this study is exploratory, and the too small number of data does not allow us to infer firmly based statistical conclusions regarding the coaching of marathon runners, and in particular the use of confidence intervals or statistical tests would not be relevant in order to confirm these preliminary results. Our purpose in the present paper is more at the level of a proof of concept: multifractality parameters derived from the weak-scaling exponent characterize different types of races and therefore should prove pertinent quantities in future comprehensive statistical studies.

Multifractal parameters are used to enhance the degree of adequate pacing strategy, thereby enabling the runner to achieve the race while maintaining the degree of high pace variation that has been demonstrated to be associated with personal bests, irrespective of the level of the runner (recreational or elite) [10]. Marathon running requires a balance between speed and stamina, and analyzing physiological signals can provide insights into performance, pacing strategy, and fatigue. Traditional methods often assume signals to be relatively regular or stationary, which is rarely the case under the extreme physiological stress of running marathons. The multifractal analysis method based on the weak scaling exponent provides a mathematical framework for characterizing the complex and irregular variations of heart rate data of marathon runners, providing new insights into how runners of different levels manage their runs, helping to detect fatigue and optimize pacing strategies.

1.6. Structure of This Paper

Section 2 deals with the mathematical foundations of multifractal analysis and the numerical methods that are derived: Section 2.1 recalls the classical notions of pointwise regularity and their respective ranges of applicability, and Section 2.2 deals with the alternative based on the weak-scaling exponent recently considered for applications and which is relevant for the analysis of extremely irregular data, such as the ones considered in this paper. Section 2.3 presents the different variants of multifractal analysis, which are used in applications, focusing on their respective ranges of validity, and Section 2.4 introduces the weak-scaling multifractality parameters that will be estimated and interpreted in the next section.

Section 3 deals with the applications on marathon runners’ heart-beat analysis. A technical description of the data is performed in Section 3.1. The runners’ individual characteristics are exposed in Section 3.2. In Section 3.3, a physiological interpretation of multifractality parameters is given, putting in light their relevance for detecting fatigue during a marathon. The multifractal characteristics of the data are derived and discussed in Section 3.4, and a focus on discussing Joyners’ model is the purpose of Section 4.

The conclusion in Section 5 discusses the advances and limitations of the results obtained in this paper and proposes several directions for future research which would allow to support of or complement the conclusions of the present exploratory work by appropriate statistical backing.

Finally, Appendix A is a mathematical appendix showing that the reconnection technique used in order to eliminate corrupted or irrelevant parts of the data does not introduce spurious isolated singularities and therefore does not alter the results of their multifractal analysis.

2. Materials and Methods

In this section, the main techniques used in order to derive multifractality classification parameters are recalled, and their respective limitations are discussed.

2.1. Pointwise Regularity Exponents

Multifractal analysis deals with the analysis and classification of everywhere irregular signals. Its purpose is to estimate the fractional dimensions of the sets of points where a pointwise regularity exponent takes a given value H. These dimensions, considered as a function of H, are referred to as the multifractal spectrum, see (5). It follows that a prerequisite of the method is to determine which notion of pointwise regularity is relevant for a given signal. Indeed, several possible definitions have been introduced, each one making sense in a particular functional setting. The most widely used ones are the pointwise Hölder exponent, and the p-exponent, see [16,23,27].

Definition 1.

Let $f\in L_{\mathrm{loc}}^p\left(\mathbb{R}\right)$ with $p\ge 1$. Let $x_0\in \mathbb{R}$; $f\in T_{\alpha }^p\left(x_0\right)$ if there exists a polynomial $P_{f,x_0}$ of degree less than α and $C>0$ such that, for r small enough,

$${\left({\int }_{x_0-r}^{x_0+r}{|f\left(x\right)-P_{f,x_0}(x-x_0)|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}\le C{r}^{\alpha +1/p}.$$

(1)

The p-exponent of f at $x_0$ is

$$h_f^p\left(x_0\right)=sup\{\alpha :f\in T_{\alpha }^p\left(x_0\right)\}.$$

If $p=+\infty$, then the local $L^p$ norm in (1) is replaced by a local $L^{\infty }$ norm and the ∞-exponent is referred to as the Hölder exponent and simply denoted as $h_f\left(x_0\right)$ instead of $h_f^{\infty }\left(x_0\right)$; see [23,24] for the use of p-exponents in the context of multifractal analysis and [17,28] for their use in the context of physiological data.

Note that p-exponents can take negative values down to $-1/p$, which gives a mathematical formalization of the notion of singularities of a negative exponent; a typical example is supplied by the cusp $f\left(x\right)=|x-x_0{|}^{-H}$, which locally belongs to $L^p$ if $p<1/H$, in which case it satisfies $h_f^p\left(x_0\right)=-H$.

It is important to note that Definition 1 makes sense only if the data can be modeled by functions that locally belong to $L^p$; otherwise, the corresponding exponent is not defined. Therefore, a prerequisite for the use of p-exponents is to determine if such is the case. This can be performed using the wavelet decomposition of f, which is now recalled.

Let $\psi \left(x\right)$ be a wavelet, i.e., a well-localized, smooth function such that the

$${\psi }_{j,k}\left(x\right):=2^{j/2}\psi (2^{j}x-k),j\in \mathbf{Z},k\in \mathbf{Z},$$

form an orthonormal basis of $L^2\left(\mathbb{R}\right)$. The wavelet coefficients of f are defined by

$$c_{j,k}=2^j{\int }_{\mathbf{R}}f\left(t\right)\psi (2^{j}t-k)dt.$$

(2)

A numerically robust way to determine if a signal can be modeled by a locally bounded function consists of estimating the value taken by its uniform Hölder exponent, which is defined through a log–log plot regression as

$$H_{min}=\underset{j\to +\infty }{lim\; sup}\left({\displaystyle \frac{log\left({sup}_k\left|c_{j,k}\right|\right)}{log\left(2^{-j}\right)}}\right).$$

(3)

This parameter also presents an interest by itself since it is also used for classification, see Section 3.3. If $H_{min}>0$, then the data can be modeled by a locally bounded function, and a multifractal analysis based on the pointwise Hölder exponent can be performed [16]. However, as will be shown in Section 3.3, for physiological data, $H_{min}$ often is found to be negative, see Figure 1, which shows a corresponding log–log plot regression.

Under less restrictive conditions, one can use a p-exponent for $p>\infty$: it is the case if data can be modeled by locally $L^p$ functions. This assumption can be verified numerically as follows: let $p>0$; the wavelet structure functions are defined as

$${\displaystyle \forall j\ge 0,S_f^w(j,p)=2^{-j}\sum _k{\left|c_{j,k}\right|}^p;}$$

and the wavelet scaling function is derived through a log–log plot regression as

$${\displaystyle {\zeta }_f\left(p\right)=\underset{j\to +\infty }{lim\; inf}\frac{log\left(S_{\mu }^w(j,p)\right)}{log\left(2^{-j}\right)},}$$

(4)

see Figure 2, where a log–log plot regression for the estimation of ${\zeta }_f\left(2\right)$ is shown.

The definition of ${\zeta }_f$ does not require any a priori assumption on the data (provided that the wavelet used is smoother than the maximal regularity met in the data). If ${\zeta }_f\left(p\right)>0$, then $f\in L^p$, and p-exponents can be defined, whereas it is no longer the case if ${\zeta }_f\left(p\right)<0$, see [23,24]. However, physiological data can prove so irregular that $\forall p>0$, ${\zeta }_f\left(p\right)<0$ and no p-exponent can be used, as illustrated in Figure 3 (see [26] for data recorded on marathon runners, and [18] for MEG data). It is well documented that, if these techniques are run out of their range of validity, the smallest scales dominate in the computation of multiresolution quantities, leading to a saturation phenomenon in the log–log plot regressions, and their outputs lead to misleading conclusions [24]; this is illustrated in Figure 4, which shows the estimation of Legendre spectra of fractionally derived multifractal random walks. For illustrative purposes, an example of a realization of a multifractal random walk is shown in Figure 5. This is an interesting toy example for which the theoretical multifractal spectra are simply shifted by the order of the fractional derivation (when the corresponding exponent is well defined), see [15,29]. Figure 4 shows that the Hölder and p-spectra are correctly estimated only when the data satisfy the admissibility conditions (respectively, $H_{min}>0$ and ${\zeta }_f\left(p\right)<0$), but they fail to be correctly estimated otherwise.

Let us also mention additional classical interpretations of the parameter ${\zeta }_f\left(2\right)$; the first one is in terms of long-range dependence, a notion usually associated to a power-law decay of the autocovariance function of the form $1/{\left|k\right|}^{1+\alpha }$, which also amounts to a power-law scaling of the Fourier spectrum of the form $C{\nu }^{-\alpha }$ when $\nu \to 0$; a wavelet reformulation of these scaling laws yields that $\alpha ={\zeta }_f\left(2\right)$; this also yields the Hurst parameter $H=(1+\alpha )/2$, see [30]. This parameter has an important interpretation in terms of correlations of increments: following the intuition supplied by fBm, one interprets $H<1/2$ as anti-correlated increments and $H>1/2$ as positively correlated increments. The estimation of ${\zeta }_f\left(2\right)$ on the data considered in the present article leads to values of the Hurst exponent below $1/2$, pointing towards anti-correlated increments of the data.

2.2. Weak-Scaling Multifractal Analysis (WS-MFA)

The pitfall mentioned when no p-exponent can be used can be turned in two ways: first, one can perform a preprocessing, which consists of a fractional integration of the data; this question will be addressed in Section 2.3. Alternatively, if one does not want to alter the nature of the singularities by this smoothing procedure, then one can use another regularity exponent, the weak scaling exponent introduced by Y. Meyer [31] as a substitute for the previously used pointwise regularity exponents when they cannot be used; in contrast with p-exponents, the definition of weak scaling exponents does not require a priori assumption on the data, see [26,31].

Definition 2.

Let $f:\mathbb{R}\to \mathbb{R}$ denote a tempered Schwartz distribution. $f\in C^{s,s^{\prime }}\left(x_0\right)$, if its wavelet coefficients verify

$$\exists C\forall j,k,|c_{j,k}|\le C{2}^{-sj}(1+|2^j{x}_0-k{\left|\right)}^{-s^{\prime }};$$

$f\in {\mathsf{\Gamma }}^s\left(x_0\right)$ if $\exists s^{\prime }>0$ such that $f\in C^{s,-s^{\prime }}\left(x_0\right)$. The weak scaling exponent of f is defined by

$$h_f^{ws}\left(x_0\right)=sup\{s:f\in {\mathsf{\Gamma }}^s\left(x_0\right)\}.$$

This regularity exponent formalizes the heuristic idea that the wavelet coefficients decay like $2^{-h_f^{ws}\left(x_0\right)j}$ in the “cone of influence” $|x_0-k{2}^{-j}|\le C{2}^{-j}$, which can be interpreted as a local scaling invariance of the data corresponding to a Hurst exponent equal to $h_f^{ws}\left(x_0\right)$. Note that, in sharp contrast with p-exponents that can take negative values only above the threshold $H=-1/p$, the weak-scaling exponent can take any negative value.

2.3. Practical Multifractal Analysis

Initially introduced by U. Frisch and G. Parisi [32] in the framework of fully developed turbulence, multifractal analysis provides a global description of the sets of pointwise singularities of f. These sets can have complex geometric structures, and their sizes are characterized by fractional dimensions: the multifractal spectrum is

$$D_f\left(H\right)={dim}_H\left(\{x:h_f\left(x\right)=H\}\right),$$

(5)

where dim stands for the Hausdorff dimension. A multifractal spectrum encapsulates information concerning how the irregularity of the data evolves with time: Simple random processes, such as fractional Brownian motion, display everywhere the same regularity, which leads to a degenerate multifractal spectrum supported by one point. On the other hand, the spectra derived in the present paper are supported by a full interval, indicating a rich structure in terms of variability of the sizes of oscillations, and our purpose will be to analyze how this variability has to be controlled along the race in order to improve the performance.

Legendre spectra allow us to estimate these quantities in a numerically stable way. This approach is based on multiresolution quantities $d_{j,k}$ associated with the chosen pointwise regularity exponent. Mathematically, this requirement means that the exponent can be recovered at each point $x_0$ by local log–log plot regressions of the $d_{j,k}$ (where j and k are chosen such that $x_0\in {\lambda }_{j,k}:=[k{2}^{-j},(k+1)2^{-j}]$) vs. the scale $2^{-j}$, see [16]. Multiresolution quantities commonly used are:

• local $L^2$ norms of $|f-l|$ (where l is a linear term which consists of a local estimation of the Taylor polynomial of degree 1 of f in ${\lambda }_{j,k}$),
• local maxima of the continuous wavelet transform,
• local suprema of wavelet coefficients,
• local $l^p$ norms of wavelet coefficients.

The first possibility leads to multifractal detrended fluctuations analysis. By construction, it is associated with the 2-exponent (see [24]).

The second possibility is referred to as the wavelet transform maxima method (WTMM) and is fitted to the estimation of the weak-scaling exponent, see [31]. However, it requires the local maxima to be finite so that a renormalization of the continuous wavelet transform (which amounts to performing a fractional integration of a large enough order) is first performed before the computation of the maxima.

The third method (wavelet leaders) requires the function to be locally bounded, i.e., $H_{min}>0$.

The last method (p-leaders) requires that, for the value of p selected, the data can be modeled by $L^p$ functions, i.e., that ${\zeta }_f\left(p\right)>0$.

We now discuss the relevance of the different numerical methods which have been proposed in order to reveal the scaling properties of the data and the information that they yield on the pointwise singularities of these data. A critical problem for the analysis of marathon runners’ physiological data is addressed: the extreme conditions which they meet imply an extra irregularity for the data which is not met in other circumstances, and this pushes these methods out of their range of validity. A special emphasis will therefore be put on the specific assumptions that each method requires, how to verify them in practice, and which pitfalls can be expected if these conditions are not met.

Since the data considered satisfy $H_{min}<0$ and $\forall p>0$, $\zeta \left(p\right)<0$ (see Figure 1 and Figure 3), it follows that none of the previously discussed methods can be used directly on the data; their implementation out of their domain of validity leads to the computation of quantities that do not have any interpretations in terms of local singularities of the data (the pitfalls following such a wrong use are well documented, see, e.g., [23,24]). This explains why they are not tested on the data considered in this paper. The second to fourth methods have been implemented after a fractional integration of the data of a sufficiently large order is performed in order to smooth them. However, this track will not be followed for the following reasons:

• A fractional integration adds an additional parameter, and the shape of the corresponding spectrum thus obtained may depend on the order of fractional integration used, see [23,24].
• This adds an artificial degree of liberty in the analysis, which makes the interpretation of the results more complex, see [18], where this problem is addressed.
• Since the characteristic of the data being analyzed is their unusually strong irregularity, it is more meaningful not to artificially smooth them before the analysis. Indeed, this smoothing may hide some important information contained in the data.

Let us also mention that the perspective of real-time monitoring calls for signal analysis algorithms which are fast, which is the case for those based on orthonormal wavelet bases, where fast algorithms exist (in $O\left(n\right)$ operations) both for the computation of the wavelet coefficients and for the estimation of multifractality parameters.

These reasons explain why, though the previously mentioned methods do yield relevant classification information on smoother data, the analysis will be restricted to the use of the weak scaling exponent (WS-MFA).

2.4. WS-MFA Parameters

Weak-scaling leaders are now introduced; they are multiresolution quantities associated with the weak-scaling exponent, see [26,31] for the motivation and justification of this definition and for a discussion on more general choices for the couple $(\theta ,\omega )$.

Definition 3.

Let f be a tempered Schwartz distribution, and denote its wavelet coefficients by $\left(c_{j,k}\right)$; let ${\theta \left(j\right)=(logj))}^a$ and $\omega \left(k\right)=j^b$ for $a,b>0$. The $(\theta ,\omega )$-neighborhood of the couple $(j,k)$ is the set of indices $(j^{\prime },k^{\prime })$ satisfying

$$j\le j^{\prime }\le j+\theta \left(j\right)and\left|{\displaystyle \frac{k}{2^j}}-{\displaystyle \frac{k^{\prime }}{2^{j^{\prime }}}}\right|\le {\displaystyle \frac{\omega \left(j^{\prime }\right)}{2^j}};$$

in the following, this neighborhood will be denoted by $V_{(\theta ,\omega )}(j,k)$. The weak-scaling leaders of f are

$$d_{j,k}=\underset{(j^{\prime },k^{\prime })\in V_{(\theta ,\omega )}(j,k)}{sup}\left|c_{j^{\prime },k^{\prime }}\right|.$$

(6)

For a given scale $2^{-j}$, let

$${\displaystyle \forall p\in \mathbb{R}{S}_f(p,j)=2^j\sum _k{\left|d_{j,k}\right|}^p.}$$

In practice, in the limit of small scales $2^{-j}\to 0$, these structure functions often exhibit power law behaviors with respect to the analysis scale $2^{-j}$: $S_f(p,j)\sim 2^{-j{\eta }_f\left(q\right)}$. More precisely, the weak-scaling function ${\eta }_f\left(q\right)$ is

$${\displaystyle {\eta }_f\left(p\right)=\underset{j\to +\infty }{lim\; inf}\frac{log\left(S_f(p,j)\right)}{log\left(2^{-j}\right)}.}$$

Its Legendre transform

$$\mathcal{L}\left(H\right)=\underset{p\in \mathbb{R}}{inf}\left(Hp-{\eta }_f\left(p\right)+1\right)$$

is referred to as the (weak scaling) Legendre spectrum, see Figure 6, where these Legendre spectra are computed for the heart rates of two marathon runners, putting in light their evolution between the first half and the last quarter of the races.

Mathematically, one can prove that the Legendre spectrum provides an upper bound for the weak-scaling multifractal spectrum which can be estimated in practice: $\forall H$, $D_f^{ws}\left(H\right)\le \mathcal{L}\left(H\right),$ see [18,26]. Therefore, in applications, the multifractal spectrum is estimated by the Legendre spectrum. As regards physiological data, multifractal analysis based on the weak-scaling exponent has provided a successful alternative to previous wavelet-based methods in several situations where no p-exponent could be used, see [18,26].

Practical multifractal analysis only retains a few characteristic parameters for the full Legendre spectrum $\mathcal{L}\left(H\right)$. The first one is the exponent $H_{min}$, which can be interpreted as the minimal value taken by the pointwise exponent $h\left(x\right)$, i.e., characterizes the strongest singularities that are met in the data.

Two other multifractality parameters can be derived from the expansion of the scaling function ${\zeta }_X\left(q\right)$ around $q=0$ as

$${\zeta }_X\left(q\right)=\sum _{m\ge 1}{c}_m{\displaystyle \frac{q^m}{m!}}.$$

(7)

The log-cumulants $c_m$ are estimated using log–log plot regressions, see [33]; $c_1^{ws}$ gives the value of the pointwise exponent $h\left(x\right)$, which is mostly met in the data, and $c_2^{ws}$ estimates the width of the multifractal spectrum and therefore indicates the range of values taken by the exponent $h\left(x\right)$.

We use for these analyses the Multifractal Analysis Toolbox, which has been developed by P. Abry, H. Wendt, and their collaborators, see [24,34]; it provides a robust framework for performing multifractal analysis of time series. Wavelet-based techniques are implemented and allow us to estimate multifractal spectra, scaling functions, and other key multifractal parameters; it constitutes a valuable tool for the study of complex, scale-invariant structures in signals. It is available on open access at: https://www.irit.fr/~Herwig.Wendt/software.html (accessed on 11 December 2025).

2.5. Why WS-MFA?

Multifractal detrended fluctuation analysis (MF-DFA) has demonstrated promising results in specific populations of marathon runners. Notably, its effectiveness is particularly evident in runners who adopt a conservative pacing strategy, where the primary objective is to maintain a constant target pace throughout the race. These athletes often choose to operate at an intensity level below their physiological limits to minimize the risk of encountering the phenomenon known as the “marathon wall” (i.e., a sudden onset of severe fatigue typically around the 30th to 35th kilometer) [11,12,20,21,22]. In this context, MF-DFA is well suited because:

• MF-DFA assumes a certain degree of regularity (e.g., stationarity or weak non-stationarity) in the time series, which is often met with these runners’ heart rate or speed data, given their emphasis on steady-state effort and risk-averse strategies.
• These athletes exhibit moderate heart rate variability (HRV) fluctuations as they avoid overexertion phases or tactical surges (e.g., breakaways or late accelerations).
• The avoidance of “the wall” is, in this scenario, a byproduct of a submaximal performance strategy, where athletes deliberately sacrifice potential performance peaks in favor of pacing regularity and fatigue avoidance. Consequently, the MF-DFA method can effectively detect and quantify this controlled behavior, as it captures scaling properties of heart rate signals that remain relatively stable across time scales.

However, traditional methods such as MF-DFA or wavelet leader-based techniques present significant limitations when analyzing physiological data acquired under extreme conditions. Marathon runners, especially those pushing their physiological limits or employing aggressive pacing strategies, exhibit highly irregular and non-stationary signals, with sudden shifts in heart rate variability (HRV) due to fatigue onset, metabolic disturbances, or biomechanical adaptations (e.g., changes in stride frequency or ground contact time). In these contexts:

• MF-DFA and similar techniques assume a minimal level of data regularity, such as local stationarity or weak correlations, which often do not hold under marathon-specific stress conditions.
• The irregularity present in such datasets leads to saturation effects in log–log regressions, where the smallest scales dominate the multifractal estimates, resulting in misleading or unstable outputs that lack physiological interpretability.

In contrast, WS-MFA developed in this study:

• Requires no prior assumption about the local boundedness or $L^p$-integrability of the data, making it robust against extreme irregularities often observed in marathon HRV signals.
• Captures the full range of local singularities without necessitating fractional integration or smoothing preprocessing steps that could mask critical fluctuations related to fatigue and pacing breakdown.
• Provides a stable and physically meaningful characterization of the most irregular segments of the data, which are often the most relevant for detecting early signs of maladaptation to exertion (e.g., pre-collapse HRV signatures).

Thus, WS-MFA emerges as a superior framework for analyzing complex physiological signals under non-ideal and highly variable conditions, such as those encountered during prolonged endurance events. In doing so, it enables the extraction of actionable insights into individual pacing strategies and physiological resilience, even in the presence of extreme signal variability and fatigue-induced perturbations.

3. Results

3.1. Description of the Data

The heart rates of marathon runners were recorded using Garmin Forerunner 630 watches, see Figure 7, which shows one of these data before and after applying the denoising procedure, which is described in Appendix A. Each participant wore the device securely on their wrist to ensure accurate heart rate measurements. The watches were configured to capture real-time physiological data throughout the marathon. To ensure precise timing, runners manually started the recording upon crossing the start line and stopped it immediately after finishing. Following the race, the recorded data were synchronized with the Garmin Connect platform for further processing and analysis. Each heart rate time series consisted of approximately $10,000$ to $15,000$ sampled values per runner. This standardized procedure ensured the collection of reliable physiological and performance metrics. The dataset used is available at: A link will be provided in the final version of this article.

3.2. The Subjects’ Characteristics and Their Performance

Table 1. Characteristics and performance metrics of marathon runners, including time performance, marathon name, age, weight, and height. Runner M6 is the only female participant, and runners M4 and M5 represent the same individual with two different performances. Rankings are based on the percentage of time relative to the record for each gender and age category. Runners highlighted in blue are those who participated in the Paris Marathon.

	M1	M2	M3	M4	M5
Time	4:07:06	3:45:37	3:05:07	2:52:24	2:47:50
Marathon	Paris	Tokyo	Montpellier	Paris	Paris
Rank	10	7	5	1	2
Age	44	41	37	50	48
Weight (kg)	79	72	79	65	67
Height (cm)	180	173	185	174	174
	M6 Women	M7	M8	M9	M10
Time	4:06:19	4:13:35	4:09:04	3:22:19	0.4399
Marathon	Sully sur Loire	Paris	Paris	Paris	La Rochelle
Rank	4	8	9	6	3
Age	55	53	48	32	52
Weight (kg)	53	83	78	80	75
Height (cm)	170	178	171	181	180

provides information about the subjects’ characteristics and their performance, offering context and perspective on the results obtained. It includes data on the time performance, marathon name, age, weight, and height of experimented marathon runners, where all participants are men except for M6. Additionally, runners M4 and M5 represent the same person but with two different performances. The rank is determined by calculating the percentage of time taken in the race relative to the record of each category to which the marathon runner belongs based on gender and age (a lower percentage of the record corresponds to a higher ranking for the marathon runner).

Multifractality parameters are estimated using log–log regressions of quantities based on $(\theta ,\omega )$-leaders, namely the log-cumulants, see [26]. Numerical estimations are performed using linear regressions over the range $j=5$ to $j=8$ (i.e., between 26 s and 3 min 25 s), corresponding to relevant physiological scales, see [35] and refs. therein. Based on numerical studies regarding the selection of $\theta$ and $\omega$ parameters, see [26], and to ensure consistency, the same parameters $\theta \left(j\right)=j^{0.25}$ and $\omega \left(j\right)=j$ are adopted for all data considered; indeed, they provide an optimal parameterization, yielding accurate estimates of the theoretical spectrum for several models. It is important to note that, as part of future studies, a theoretical statistical investigation is necessary to robustly determine the most appropriate parameter selection across various types of signals. We present in

Table 2. Representation of different multifractal parameters $(H_{min},c_1^{ws},c_2^{ws})$ of the heart rate signals for each marathon runner. Runners shown in blue participated in the Paris Marathon, while runners shown in black participated in other marathon events.

	M1	M2	M3	M4	M5
$H_{min}$	0.18	0.14	0.31	0.29	0.35
$c_1^{ws}$	0.41	0.47	0.43	0.34	0.46
$c_2^{ws}$	−0.05	−0.01	−0.007	−0.05	−0.03
	M6 Women	M7	M8	M9	M10
$H_{min}$	0.36	0.26	0.24	0.29	0.25
$c_1^{ws}$	0.67	0.60	0.59	0.44	0.44
$c_2^{ws}$	−0.14	−0.11	−0.07	−0.004	0.44

the three multifractal parameters linked to the multifractal spectrum of ten heart rate signals (in beats per second). In addition, these parameters are compared in Figure 8 and Figure 9.

3.3. Physiological Interpretation of Multifractality Parameters

We now explore the relevance of the three multifractality parameters for detecting fatigue during a marathon. In this study, three multifractal parameters are introduced:

$H_{min}$ (minimal regularity): This parameter represents the strongest singularities (the sharpest and most abrupt changes) found within the physiological data. A higher value of $H_{min}$ corresponds to more uniformity in the physiological signals, suggesting better management of physiological stress and less severe singularities, often linked to higher-ranked runners who maintain more consistent pacing throughout the marathon. In sports terms, higher values of $H_{min}$ mean that the runner maintains smoother, steadier physiological regulation (e.g., fewer abrupt changes in heart rate variability), indicating better pacing strategy and resistance to fatigue.

$c_1^{ws}$ (most common level of regularity): It can be considered as the typical regularity of physiological responses for a runner during the marathon. In sports terms, lower-performing runners usually have higher values of $c_1^{ws}$, suggesting that their heart rate variability fluctuates more and usually is less stable or controlled. On other hand, higher-performing runners exhibit lower $c_1^{ws}$, corresponding to more consistent physiological control and stable pacing strategies.

$c_2^{ws}$ (multifractal spectrum’s width): This essentially reflects how much the regularity (or physiological responses) varies throughout the run. Positive or close to zero values indicate a narrower spectrum, meaning fewer fluctuations—i.e., a more stable physiological state during the marathon; negative values further from zero indicate broader, more pronounced multifractal spectra, meaning greater variability and complexity in physiological responses, often correlating with increased fatigue and less efficient pacing.

3.4. Results and Discussion

Figure 8 shows that the better the ranking, the larger the $H_{min}$ value. Conversely, when examining the figure for $c_1^{ws}$, which represents the regularity almost everywhere versus the record, the opposite trend is observed: marathoner runners with lower performance levels have a higher $c_1^{ws}$ value compared to those at the top ranks. These two observations suggest that the better the ranking, the narrower the multifractal spectrum of marathon runners. In other words, higher-performing runners exhibit more uniform regularity, indicating less “fractality”, i.e., the regularity exponent $h\left(x\right)$ varies over a small interval.

In addition, Figure 9, and particularly the $c_1^{ws}-H_{min}$ representation, confirms our conclusion: the smaller the gap, the better the ranking of marathon runners. When discussing the parameter $c_2^{ws}$, a focus is put on its influence on the spectrum’s concavity. As $c_2^{ws}$ approaches zero, the spectrum becomes less concave and narrower, indicating a more monofractal nature, where the pointwise regularity parameter remains constant. Conversely, as $c_2^{ws}$ moves away from zero in the negative direction, the spectrum becomes more concave and broader, thus indicating higher multifractality. The analysis of the $c_2^{ws}$ graph in relation to the record supports our conclusions: athletes with lower performance exhibit more negative $c_2^{ws}$ values, corresponding to a higher degree of multifractality.

We enhanced the previous analysis by examining variations in multifractal parameters throughout the marathon. Indeed, the multifractal analysis of stationary data yields local spectrums that do not evolve with time. In contrast, spectrums that evolve with time are the signature of a non-stationarity in the data. In such cases, it is important to analyze this evolution by doing a local multifractal analysis. The global spectrums computed on the whole length of the data available yield the concave hull of the different local spectrums and therefore supply mixed information, which is hard to interpret; this is in sharp distinction to the derivation of the local spectra, which gives relevant information on the evolution of the multifractality parameters during the race. However, in practice, such an analysis can be made difficult if the windows considered are too short to derive reliable log–log plot regression, and this is why the data are only split into two parts.

Note that, in all generality, when data do not present stationarity properties, their multifractal characteristics may vary with time, and the output of a global multifractal analysis mixes these characteristics so that it is hard to infer from it pertinent information. Therefore, a preprocessing consisting of splitting the signal into homogeneous pieces is required; see [36] for such procedures and their use on finance data. However, here, this splitting can be performed accurately by hand because of the a priori knowledge on the data. Indeed, it is well established that during the final 12 km, many runners experience a significant increase in difficulty, with Borg rate of perceived exertion (RPE) scores exceeding 15/20, indicating strenuous effort. An important issue is to explore how these changes impact multifractality parameters. To account for these physiological changes, the race was segmented into the first half and the last quarter. The first half corresponds to a period of relatively steady pacing with moderate fatigue, whereas the last quarter captures the phase where fatigue typically intensifies, around the so-called “marathon wall”. This segmentation ensures that the multifractality parameters computed for each segment meaningfully reflect the effects of increased exertion and fatigue, as shown by the evolution of $H_{min}$ and $c_1^{ws}$ between these segments in Figure 10, highlighting the differences in physiological responses to fatigue beyond the 30th kilometer. Note that these two parameters do not necessarily exhibit the same type of variation: the variations of $H_{min}$ and $c_1^{ws}$ for marathoner M1, who finished last, and M4, who finished first, are very similar. This indicates that both the first and the last runners experienced the same variation in spectrum, meaning the same change in regularity. Despite finishing last, M1 managed to adjust and pace his race similarly to M4. It means that, despite his lower performance level, he was able to adequately self-pace his race, which is an important factor of performance.

M4 and M5 correspond to the same runners in different instances. Figure 9 indicates that his rates of change behave in a very different way; this can be interpreted as showing that the runner’s shape and training state have a strong influence on the pacing strategy, and this can be captured by multifractal analysis.

Runner M6, being the only female participant, presents particularly interesting multifractal characteristic (with notably high $H_{min}$ and $c_1^{ws}$, and a minimal $c_2^{ws}$). However, drawing definitive conclusions or generalizations from a single female participant is not scientifically realistic or robust. At this point, one can only say that the results of the only female runner indicate an interesting direction for future investigation. Additional female physiological data during marathon races are essential for a valid comparative analysis of multifractal characteristics between genders. This is a critical limitation of our study and suggests that future research should explicitly include a larger sample of female runners to meaningfully explore potential gender-based differences in physiological multifractality.

Our study suggests that multifractal analysis can provide valuable feedback for optimizing pacing strategies. Indeed, the analysis reveals that multifractal parameters detect changes in physiological signals due to fatigue, especially around the 30th kilometer mark. This study also shows that better-ranked runners had more uniform regularity in their physiological signals, confirming the results derived in [37,38]. This indicates that amateur runners should strive to control the variability of their pacing so that it is evenly distributed along the marathon, rather than simply reduce this variability, which is commonly met advice.

4. Additional Considerations

Joyner’s contributions to this field established a conceptual framework for understanding the limits of human endurance performance [39]. His model has been employed to investigate the theoretical potential for marathon times under optimal conditions. Nevertheless, this approach did not consider the optimization of pacing strategy, as the constant speed was assumed to be optimal. However, a recent study analyzing the best performance in marathons showed that marathon performance depends on pacing oscillations between asymmetric extreme values [40]. The variation of parameters between the two parts of the race provides insight into the evolution of the corresponding quantity. However, to assess the significance of this change, it must be compared to its initial value. Therefore, in the perspective of self-improving the runner’s performance in the next marathon, we propose the examination of the rate of change of the multifractal parameters $(H_{min}$, $c_1^{ws}$, $c_1^{ws})$ in Figure 11, as a biofeedback for improving the pace management that could constitute the fourth dimension of the marathon performance.

Practical Use: From WS-MFA Parameters to Coachable Decisions

The variation in WS-MFA parameters between two race segments provides insight into the evolution of physiological regulation under fatigue. However, for practical use by runners and coaches, absolute values are less informative than individualized changes relative to an early-race reference. Accordingly, we propose to monitor (i) the relative change of the parameters $H_{min}$, $c_1^{ws}$, and $c_2^{ws}$ with respect to a stable baseline segment and (ii) their rate of change along the race. These two quantities can be used as biofeedback to detect an approaching loss of control of pacing and regulation (often observed around the 30th kilometer) and to guide real-time adjustments. A sustained decrease in $H_{min}$ and/or a shift of $c_2^{ws}$ toward more negative values indicates increased irregularity and a loss of controlled regulation under fatigue. Coaches can implement simple, individualized alerts based on relative change (e.g., $H_{min}$ drop $>10\%$ for two consecutive windows = yellow alert; >20% = red alert). Thresholds should be calibrated on repeated long runs and races for each athlete. In the post-race learning phase, coaches can compare parameter trajectories across races and key long runs to identify the distance/time at which $H_{min}$ starts to fall rapidly (individual break-point).

5. Conclusions

In this study, a recently introduced technique, WS-MFA, has been employed to quantify the multifractality of time series collected during marathons. Its advantage over previously introduced techniques has been demonstrated for the data that are considered in the present article: it is the consequence of their extreme irregularity, a challenge that was not met previously in the domains of applications of other multifractal methods and which was put in evidence by the (global and pointwise) function space modeling approach that was developed. The theoretical study worked out in the Appendix A shows that the reconnection procedure used allows us to eliminate spurious artefacts without adding new singularities in the data and therefore provides an acceptable denoising procedure.

We have shown that WS-MFA provides a comprehensive view of the individual performance of marathon runners, identifying alterations in physiological signals resulting from fatigue and optimizing race pace strategies. Therefore, it yields a comprehensive perspective on individual performance offering a new way to understand the complex dynamics of physiological signals during marathons. This approach can help optimize training and pacing strategies to improve performance.

We examined the multifractality characteristics of marathons achieved by a diverse group of recreational runners, including individuals of varying age, gender, and performance levels. Considering the aforementioned diversity, the performance was standardized in percentage of the world performance for each individual category.

Our results show a possible use of multifractality parameters for fatigue detection: around the 30th kilometer (the typical onset point of heightened fatigue), changes in these parameters indicate increased physiological stress and altered pacing behavior. They constitute sensitive indicators of when fatigue significantly alters physiological signals:

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Changes in $H_{min}$, $c_1^{ws}$, and $c_2^{ws}$ from the first to the last segments of the race clearly highlight the physiological impact of fatigue on the runner. In particular, marked decreases in $H_{min}$ over the course of a marathon (particularly around the 30 km mark) suggest increased physiological stress and onset of fatigue.

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Early detection through multifractal analysis could help runners and coaches adjust pacing strategies proactively, thereby optimizing performance. Note however that automatic segmentation methods (based on change-point detection or entropy-based clustering) could yield a way to refine the a priori segmentation used, and we intend to explore these techniques in future works.

Limitations

A major limitation of this study is the small sample size and the limited amount of data analyzed. A larger and more diverse dataset is necessary to confirm the robustness and generalizability of the present observations, including additional signals such as speed/pace and cadence, and involving enough runners to support inferential statistics (e.g., confidence intervals, effect sizes, and hypothesis testing). Accordingly, the present results should be interpreted as exploratory. In particular, any statement related to sex/gender differences must be considered preliminary, as the current dataset is underpowered to draw reliable conclusions on this point. From a practical perspective, progress in this area requires broader data sharing under open and reproducible conditions. We therefore encourage runners and practitioners to contribute anonymized race and training data to open-access repositories, which would facilitate independent validation and benchmarking. Such collective datasets would also enable the development and calibration of algorithms providing individualized indices of pacing regulation and race optimization, with the applied goal of reducing late-race performance collapse (often referred to as the “marathon wall”) through earlier detection of a loss of physiological control.

Methodologically, WS-MFA is a recent technique, and systematic studies are still needed to optimize the design of the $(\theta ,\omega )$ boxes and to assess the sensitivity of the estimates to these choices. This line of work should lead to more precise and stable multifractality parameter estimation. Finally, an important next step will be to investigate the origin of the observed multifractality by comparing the estimated parameters with those obtained from shuffled and surrogate data (see, e.g., [41,42]) in order to disentangle genuine physiological structure from potential confounds related to sampling, preprocessing, or measurement artefacts.