Physica A 388 (2009) 3798–3808

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Physica A journal homepage: www.elsevier.com/locate/physa

Detection of changes in the fractal scaling of heart rate and speed in a marathon race Véronique L. Billat a,∗ , Laurence Mille-Hamard a , Yves Meyer b , Eva Wesfreid b,c a

Unité INSERM 902, Genopole-University d’Evry Val d’Essonne, France

b

CMLA, École Normale Supérieure de Cachan, 94235 Cachan, France

c

LMPA, Université du Littoral, Côte d’Opale, 62228 Calais, France

article

info

Article history: Received 19 August 2008 Received in revised form 2 April 2009 Available online 31 May 2009 PACS: 02.50.Ey 05.40.-a 87.19.Hh 87.80.Vt 89.75.Da Keywords: Fractal analysis Detrended Fluctuation Analysis (DFA) Wavelet Heart rate, running

abstract The aim of this study was to detect changes in the fractal scaling behavior of heart rate and speed fluctuations when the average runner’s speed decreased with fatigue. Scaling analysis in heart rate (HR) and speed (S) dynamics of marathon runners was performed using the detrended fluctuation analysis (DFA) and the wavelet based structure function. We considered both: the short-range (α1 ) and the long-range (α2 ) scaling exponents for the DFA method separated by a change-point, n0 = 64 = 5.3 min (box length), the same for all the races. The variability of HR and S decreased in the second part of the marathon race, while the cardiac cost time series (i.e. the number of cardiac beats per meter) increased due to the decreasing speed behavior. The scaling exponents α1 and α2 of HR and α1 of S, increased during the race (p < 0.01) as did the HR wavelet scaling exponent (τ ). These findings provide evidence of the significant effect of fatigue induced by long exercise on the heart rate and speed variability. Published by Elsevier B.V.

1. Introduction In the last 20 years, marathon running has been a social and athletic phenomenon [1] with 60,000 runners applying to take part in the race each year. There are around 35,000 participants in Paris, New York or London. The marathon race is no longer reserved to elite runners, and the average performance is 3 h 50 min, and the average marathon age is 45 ± 4 years (both for the females and males) for the greatest marathons (Boston, New York, Paris, London). The marathon race is run at a high percentage (80%–100%) of maximal heart rate reserve and oxygen uptake [2,3]. The incidence of sudden death (cardiac arrest) happening during or following long exhaustive exercise as marathon, increases [4]. Indeed, myocardial injury and ventricular dysfunction related to training levels among non elite participants in the Boston marathon has recently been reported [4]. Two other studies identified a ‘‘cardiac fatigue’’ in sub elite participants after triathlon and cross-country skiing races. The ‘‘cardiac fatigue’’ is defined by abnormalities in resting left ventricular diastolic and systolic functions by echocardiography [5] or by a modification of the cardiac autonomy regulation associated with a reduced vagal. Cardio respiratory dynamics have revealed fractal scaling behavior of heart rate, blood pressure and respiration rate [6– 13]. It has been demonstrated that a scaling analysis of heartbeat time series allowed detection of changes of the long range power-law correlations in patients with congestive heart failure [14–18]. Therefore, the cardiac fatigue induced by prolonged

∗

Corresponding author. Tel.: +33 6 89 87 75 76; fax: +33 1 42 39 20 83. E-mail address: [email protected] (V.L. Billat).

0378-4371/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.physa.2009.05.029

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Table 1 The subjects’ anthropometry and performance data. Runners

Age (yrs)

Weight (kg)

Height

Marathon time

v VO2max

N signals

1 2 3 4 5 6 7 8 9 10 Mean SD

37 33 34 30 40 32 44 30 29 37 34.6 4.9

71 71 67 68 67 71 73 78 69 72 70.7 3.3

174 179 175 177 176 180 181 188 174 181 179 4

03:56:36 03:34:33 03:27:31 05:08:45 03:22:32 03:11:48 03:44:03 03:11:48 04:29:20 02:41:15 03:40:49 00:42:20

15.0 16.0 17.5 12.0 17.0 17.5 15.5 17.5 12.5 19.0 16.0 2.3

2839 2575 2490 3705 2430 2302 2689 2302 3232 1935 2650 508

endurance exercise could be associated to changes in the fractal scaling of heart rate and speed fluctuations throughout the race. Nowadays, most of the marathon runners can wear a cardio frequency meter which allows to get both heart rate (HR) and speed (S) signals during whole the race. The distribution of the HR behavior in healthy subjects has been described by a DFA function over a wide range of timescales which indicates the presence of long range (power law) correlations [19,20,7,21]. The detrended long memory process of a time series is estimated by a scaling behavior exponent (α ) which is the slope of the linear regression of a scale dependent quantity in a log–log plot. The long-range HR correlation behavior is shown to break down in older persons or in persons with severe congestive heart failure [22,7,8,13,23]. A multifractal analysis of heart rate time series during 10 km races and marathon was presented in an attempt to unveil the scaling law behavior using the Wavelet Transform Modulus Maxima (WTMM) [24]. This method was proposed by Arneodo et al. [25]. This prior work performed by our group demonstrated that the distribution of the variations in the beatto-beat intervals in healthy subjects was described by a structure function over a wide range of scales, and that fatigue did not significantly alter the scaling behavior of the HR time series. However, this prior study did not apply the DFA method, which has been reported to be a particularly robust estimator [26] for small series and for signals with superimposed linear trends, as reported in the heart rate time series (the so-called ‘‘cardiovascular drift’’) reported in long and intense exercise [27–29]. Furthermore, no studies have yet analyzed the speed time series in marathon races, as most of the races are stochastic exercises where the speed is not imposed. Given that HR response depends on the exercise intensity, it is therefore necessary, for understanding HR variations, to also analyze those of the speed. For non elite marathon runners, the speed decreases with the progressive glycogen depletion and the increasing neuromuscular disorders [30]. This speed decrease which alters the final performance (i.e. the time for achieving the given distance), is considered as a global fatigue criterion [5,31], but we still do not know how the fatigue affects the speed variability during an exhaustive race. The hypothesis is that sensory feedback from the fatiguing muscle would modulate central motor output [29]. Indeed, some studies have suggested that humans might regulate their speed during competition, based on feedback from a variety of receptors that monitor the physiological responses to the demands of the activity as the heart rate [32–36,29]. However, heart rate time series have still never been examined with those of the speed. Therefore, the present study aimed to test the hypothesis that there are changes in the fractal scaling of heart rate and speed fluctuations during a marathon race, and that modification is associated with the speed decrease. By applying two techniques of scaling analysis (i.e. detrended fluctuation analysis (DFA) and wavelet structure function) we calculated the scaling exponents of heart rate, speed and cardiac cost (heart rate/speed) time series in non elite runners during a marathon race. We then compared the time series of the second vs. the first half marathon to test the hypothesis that the scaling exponents of heartbeat and speed time series were modified when the race is in progress. 2. Methods 2.1. Subjects The subjects included ten non elite marathon runners (Table 1). Before participation, all subjects were informed of the risks and stresses associated with the protocol, and gave their written voluntary informed consent. The present study conformed to the standards set by the Declaration of Helsinki and its procedures were approved by the local ethics committee of the Saint Louis Hospital of Paris. 2.2. Experimental design Speed and heart rate signals were recorded simultaneously during the marathon race of Paris in April 2008. The race commenced at 9 am and a temperature of 17 ◦ C without wind (< 2 m/s, anemometer, Windwatch, ALBA, Silva, Sweden) was observed. Sequential speed and heart rate signals were recorded simultaneously using a cardio-frequency meter coupled

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Speed - mean(Speed) signal

Integrated signal

4 2 0 -2 -4 -6 -8

500

1000

1500

2000

2500

Integrated time series and trend segments in each box of size 500 300 200 100 0 500

1000

1500

2000

2500

Integrated signal

Time boxes of size 500 Integrated time series and trend segments in each box of size 250 300 200 100 0 500

1000

1500

2000

2500

Time boxes of size 250 Fig. 1. Athlete’s speed signal (at top) and its integrated time series with local linear regression for boxes of size n = 500 (41.66 min) (at middle) and size n = 250 (20.83 min) (at bottom).

with an accelerometer (Polar RS 800, FI) or a GPS system (Feed Forward, FI). Two weeks prior to the race, the subjects performed a test to determine individual v VO2max and maximal heart rate. 2.3. Procedures Speed and Heart rate measurements The athlete’s physiological time series was recorded every 5 s due to the speed sampling which does not allow the analysis of beat by beat data in marathon races. The following physiological variables were recorded: - Heart rate (heart beats per minute), - Speed (km per hour). The heart rate/speed ratio (HRS) named the cardiac cost is expressed as the number of heart beats per meter. 2.4. DFA algorithm Each marathon time series f = (f [n])1≤n≤N was analyzed using the detrended fluctuation analysis (DFA) which estimates the scaling exponents of the detrended integrated time series. Description:

• The time series is integrated after subtracting its average value f : y[k] =

n X [f [i] − f ].

(1)

i=1

• The integrated time series y = (y[k])1≤k≤N is divided into M (n) boxes of equal length n, M (n) is the greatest multiple of n, less then N. In each box of length n, a segment line yn is fitted to the integrated time series, therefore, yn = (yn [k])1≤k≤M (n) is a curve of length M (n), linear over each box, which represents a local linear trend (Fig. 1) of the integrated time series. • The integrated time series (y[k])1≤k≤N is detrended by subtracting the local linear trend curve yn = (yn [k])1≤k≤M (n) . • The root-mean-square of this integrated and detrended time series is the DFA function: v u (n) u 1 M X [y[k] − yn [k]]2 . F [ n] = t (2) M (n) k=1 • The DFA function F [n] is computed for a broad range of time-scales (box size n) to provide a relationship between F [n] and the box size n.

• A linear behavior of log(F [n]) vs. log[n], over a time-scale range [n1 , n2 ] indicates the presence of a time-scale power law: F [n] ≈ cnα .

(3)

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Speed time series 15

10

5 500

1000

1500

2000

2500

Short range scale(α1) and long range scale (α2)

log(F[n])

1 0.5

α2 = 0.54

α1 = 1.28

0 -0.5 log(n)

Fig. 2. Athlete’s speed time series (at top) and its DFA function, in a log–log plot, with regression lines (at bottom).

Indeed, Eq. (3) is equivalent to the following linear equation: log(F [n]) ≈ log(cnα ) ≈ α log(n) + log(c )

(4)

c is a constant, α is the linear slope. Therefore, the DFA function can be characterized by a scaling exponent α , estimated as the slope of the least square regression line which approaches the graph of log(F [n]) vs. log[n] for some time-scale range [n1 , n2 ] (for boxes of size between n1 and n2 ). 2.5. Application Let us show the local linear trend of an athlete’s speed time series recorded every 5 s during 3.47 h. This signal has 2500 recorded points. Its integrated time series was divided into boxes of length n; each box has n/12 min length. Fig. 1 shows at top this signal after its mean subtraction, at middle (resp. at bottom) its integrated time series y = (y[k])1≤k≤n with the linear trends over boxes of size n equal to 500 (resp. 250), of 41.66 min (resp. of 20.83 min) length. Figs. 2–4 show at top, speed (S), heart rate (HR) and heart cost (HRS) time series and, at bottom, log(F [n]) vs. log[n] with its linear regression over two distinct regions separated by a change-point, n0 = log10 (64), with the corresponding shortrange (α1 ) and long-range (α2 ) scaling exponents. This break point corresponds to a box length of 64/12 = 5.33 min. It was the same for all analyzed time series and all athletes. Most athletes were slowing down at the energy stations situated at the end of the race, but Figs. 1–4 show HR, S and HRS time series of a particular runner which was slowing down at each station. As is shown in Figs. 2–4, DFA can be used without removing the slowing down dips at energy stations. Dips modify the scaling exponent’s values α1 , α2 but global results obtained in this paper are unchanged: α1 significantly higher than α2 and the short scaling exponent α1 of HR and S greater in the second half of the race. 3. Wavelet analysis Each marathon time series f = (f [n])1≤n≤N was also analyzed using the wavelet leader based structure function. Sf (j, q) =

n(j) 1 X

n(j) k=1

Lf (j, k)q ≈ cq (2j )τ (q)

where:

P (j) q • Sf (j, q) = n1(j) nk= 1 Lf (j, k) is the structure function [25,45] • Lf (j, k) are the wavelet leaders coefficients [57]

(5)

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Heart Rate time series 185 180 175 170 165 160 500

1000

1500

2000

2500

Short range scale(α1) and long range scale (α2) 2

log(F[n])

1.5 1

α1 = 1.50

α2 = 0.84

0.5 0

log(n) Fig. 3. Athlete’s heart rate time series (at top) and its DFA function, in a log–log plot, with regression lines (at bottom).

Heart Cost time series 3 2.5 2 1.5 1 500

1000

1500

2000

2500

Short range scale(α1) and long range scale (α2) 0.5

log(F[n])

0

α1 = 1.26 α2 = 0.50

-0.5 -1 -1.5 log(n)

Fig. 4. Athlete’s cardiac cost time series (at top) and its DFA function, in a log–log plot, with regression lines (at bottom).

• n(j) ≈ 2Nj is the number of such coefficients available at each scale 2j , • N denotes the signal length, • τ (q) is the scaling exponent for some range of q values (q ∈ [q1 , q2 ]) which can be estimated as the slope of log2 (Sf (j, q)), indeed, Eq. (6) is equivalent to the following linear equation: log2 (Sf (j, q)) ≈ τ (q)j + log2 (cq ) for each value of q.

(6)

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Table 2 Mean and variation of the heart rate, speed and cardiac cost in marathon race. Variablesa

Marathon time

Heart rate (beats min−1 ) CV (%) Speed mean (km h−1 ) CV (%) Cardiac cost (beats m−1 ) CV (%)

0%–25% 162 ± 12 7±3 11.8 ± 1.5 28.1 ± 6.1 0.23 ± 0.39 1.3 ± 0.2

a

P 25%–50% 162 ± 12 4±5 12.1 ± 2.1 16.7 ± 12.8 0.23 ± 0.03 0.75 ± 0.397

50%–75% 163 ± 11 3±2 11.8 ± 2.2 13.4 ± 6.7 0.52 ± 0.05 0.52 ± 0.44

75%–100% 162 ± 11 3±2 10.2 ± 2.2 14.5 ± 5.9 0.250 ± 0.05 0.54 ± 0.44

0.95 0.001 0.02 < 0.001 < 0.001 < 0.0005

Mean, Coefficient of variation (CV %).

Table 3 Mean and variation of the scaling exponents of the heart rate, speed and cardiac cost, in marathon race. Part of the signal

Small scale [0.60–1.77] Heart rate

Whole race First part time of the marathon race Second part time of the marathon race P value between the first and second half race

1.36

±0.11 1.32

±0.13 1.34

Speed 1.12

±0.11 1.06

±0.10 1.17

[Large scale 1.80–2.78] Cardiac cost

Heart rate

Speed

1.14

0.98 ±0.20 0.83 ±0.25 1.07 ±0.24 0.03

0.74 ±0.12 0.67 ±0.12 0.753 ±0.14 0.07

±0.08 1.06

±0.10 1.15

±0.14

±0.13

±0.10

0.0044

0.0006

0.005

Cardiac cost 0.70

±0.12 0.63

±0.11 0.72

±0.17 0.06

3.1. Time-scales vs. scales (DFA function vs. structure function) The scaling power law was assessed using:

• the detrended fluctuation analysis (DFA) with time-scales (boxes length) and • The wavelet leader based structure function (WLSF) with scales (which behaves as the inverse of the frequencies) used to dilate or contract wavelets. We estimated the short and long range scaling exponents, α1 , α2 (the scaling exponents τ (q)) as the slope of the segment lines which fit the graph of the DFA (resp. WLSF) function in a log–log plot. 3.2. Statistical methods Values were expressed as means ±SD. The change in each value with fatigue between two (0%–50%, 50%–100% racing time) or four equal parts of the marathon race (0%–25%, 25%–50%, 50%–75%, 75%–100% racing time) were analyzed using a t-test of student for paired data or an analysis of variance (ANOVA) for repeated measures and a Newman–Keuls post hoc test. For DFA, we analyzed the scale and fatigue effect using a two-way ANOVA. We analyzed the effect of fatigue using the wavelet analysis by comparing the scaling exponents τ1 (q), τ2 (q) of the two race parts by a t-test for paired data. The level of significance was set at P < 0.05. 3.3. Results Average and coefficient of variation of HR, Speed and HRS As the race progressed, the speed decreased (p < 0.002) but the average HR remained at 81 ± 7% of the maximal heart rate reserve (p = 0.95). Consequently, the cardiac cost was shown to increase (p < 0.001) (Table 2). The variability of all these physiological signals measured by the coefficient of variation decreased according to the time run (Table 2). Racing scale invariance in HR, S and HRS time series estimated by the DFA function As well as heart rate, speed and cardiac cost, time series are also shown to fluctuate in a complex, apparently erratic manner (Figs. 2–4). Two distinct regions of scale invariance ([0.60 − 1.77] and [1.80 − 2.78]) were identified in the races which correspond to short and long time scaling exponents α1 , α2 (Figs. 2–4). The estimated short-range and long-range scaling exponents α1 and α2 were not significantly different between the speed, heart rate, and cardiac cost time series (F < 4, p < 0.001). The short scaling exponent α1 was significantly higher than the long scaling exponent α2 (p < 0.0001) for all the signals (HR, S and HRS), (Table 3). The exponent α1 , but not α2 , increased significantly between the two marathon halves for the speed and cardiac cost time series (Table 2, Fig. 5). The HR scaling exponent, α1 (short-range scale) and α2 (longrange scale) were greater in the second half of the race (Table 3, Fig. 6). Racing HR, S and HRS time series, estimated by the wavelet scaling law behavior The marathon heart rate, speed and cardiac cost time series were also analyzed using the wavelet leader based structure function (WLSF). The corresponding scaling exponents τ1 (q) and τ2 (q) of the two racing halves are shown in Table 4 for q = 2. Heart rate scaling exponents values belong to [0.55, 2.19] while the speed and cardiac cost scaling exponents values belong, respectively, to [0.30, 0.84] and to [0.38, 1.07]. Analysis revealed that the exponents (τ1 (2)) and (τ2 (2)) for HR were

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Short range speed scaling exponents

Short range heart cost scaling exponents 1.6

1.6 first half second half

second half

1.4

α1

1.4

α1

first half

1.2

1.2

1

1

0.8 2

4

6

8 Athletes

10

12

0.8

14

2

4

6 8 Athletes

10

12

Long range heart cost scaling exponents

Long range speed scaling exponents first half

first half

second half

second half

1

α2

α2

1

0.8

0.8

0.6 0.6

2

4

6

8 Athletes

10

12

0.4

14

2

4

6 8 Athletes

10

12

Fig. 5. Short and long ranges scaling exponents (α1 and α2 ) for speed and cardiac cost time series of the first vs. second parts of the race. Table 4 Two parts of the marathon scaling exponents τ1 (2) and τ2 (2) estimated by the WLSF. Heart rate (HR) (b min−1 ) (p = 0.007)

Athlete 1 Athlete 2 Athlete 3 Athlete 4 Athlete 5 Athlete 6 Athlete 7 Athlete 8 Athlete 9 Athlete 10 Mean SD

Speed (S) (m m−1 ) (p = 0.38)

Cardiac cost (HRS) (b m−1 ) (p = 0.73)

τ1 (2)

τ2 (2)

τ1 (2)

τ2 (2)

τ1 (2)

τ2 (2)

0.98 1.04 1.20 0.64 0.93 0.91 0.80 0.56 0.55 0.78 0.84 0.21

1.40 0.90 2.04 1.31 1.61 1.83 0.84 0.65 0.81 0.97 1.24 0.48

0.30 0.77 0.39 0.84 0.34 0.71 0.65 0.54 0.53 0.38 0.55 0.20

0.32 0.72 1.07 0.49 0.49 0.72 0.53 0.74 0.62 0.44 0.32 0.50

0.52 0.63 0.38 0.56 0.47 0.75 0.72 0.60 0.64 0.69 0.60 0.12

0.41 0.75 1.12 0.58 0.43 0.65 0.57 0.56 0.64 0.55 0.63 0.20

significantly higher that those of the speed and cardiac cost (p < 0.01) (Table 4). The HR’s scaling exponent τ2 (2) was higher than τ1 (2) for 9 out of 10 athletes (Figs. 7 and 8) (p < 0.01, Table 4). 4. Discussion and conclusion In accordance with our hypothesis, the marathon race affects the fractal short scaling exponent of heart rate in the second vs. the first half of the race. This was shown in short range scales with DFA methods for S and HR and, less significantly (only for HR), by the wavelet analysis.

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Short range heart rate scaling exponents 1.8 first half second half

α1

1.6

1.4

1.2

1

1

2

3

4

5

6

7

8

9

10

11

9

10

11

Athletes Long range heart rate scaling exponents 1.4 first half second half

α2

1.2

1

0.8

1

2

3

4

5

6

7

8

Athletes Fig. 6. Short and long ranges scaling exponents (α1 and α2 ) for heart rate time series of the first vs. second parts of the race.

Speed

Heart rate 170 15

Kilometer per hour

Heart beats per minute

160 150 140 130

10

5 120

0

500

1000

1500

0

2000

500

Scaling exponent: τ(q).

1000

1500

2000

Scaling exponent: τ(q).

4

2

2

τ(q)

τ(q)

0 0

-2

first half

-2

first half lasthalf

lasthalf -4 -4

-2

0 q

2

4

-4 -4

-2

0 q

2

Fig. 7. Heart rate and speed scaling exponents τ1 (q) and τ2 (q) of the two race half parts (Athlete3).

4

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Heart beats per meter

Heart cost 3

2

1

0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Scaling exponent: τ(q). 2

τ(q)

0

-2

first part last part

-4 -3

-2

-1

0 q

1

2

3

Fig. 8. Cardiac cost scaling exponents τ1 (q) and τ2 (q) of the two race half parts (Athlete3).

The scaling exponents of heartbeat, speed and cardiac cost time series get larger when the marathon race is in progress. This result suggests that stronger correlations appear in HR, S and HRS time series after the first half of the marathon race (> 21 km) with both techniques of scaling analysis. The result of HR time series is in accordance with a recent study published in the present journal [39]. The Jun Zhuang et al. [39] paper was one of the first to have analyzed the alteration in scaling behavior of short-term heartbeat time series for professional shooting athletes from rest to exercise. In our present study, we compared HR time series with the exercise duration in the first vs. the second half marathon when the runner started to have a glycogen depletion, which is a major cause of speed decrease at the end of the race. There was a change point in the fractal scaling of heart rate and speed fluctuations during the marathon race. Indeed, the present study reported a change point of scaling in heart rate, speed and cardiac energy cost (the heart rate/speed ratio) dynamics of runners in the marathon race with both the two methods applied. The variability of heart rate and speed decreased in the second part of the race. We chose to analyze the heart rate and speed variability with the most common nonparametric estimators of scaling law exponents based on the Detrended Fluctuation Analysis (DFA) and wavelet structure functions. The DFA is a version, for time series with trend, of the aggregated variance method that is frequently used in physiological data processing, and notably in heart beat signals recorded in healthy or sick subjects [40,15,16,41,42,38]. It involves: (1) aggregating the signal process by windows with fixed length, (2) detrending the signal from a linear regression in each windows, (3) computing the standard error of the residual errors (the DFA function) for all data, (4) estimating the exponent of the power law from a log–log regression of the DFA function on the length of the chosen window. After the first stage, the process is supposed to behave like a self-similar process with stationary increments added with a trend. The second stage is supposed to remove the trend. Finally, the third and fourth stages are the same as those of the aggregated method [43]. The detrended phase is a major advantage of this DFA method for heart beat signals in long exercises of more than 10 minutes of prolonged moderateintensity (e.g. 50%–75% VO2max ) which is increasing with the time. Audit et al., [44] showed that the best non parametric estimator in minimizing the mean squared error is the DFA estimator which has been initially introduced by Peng et al. [26] and then employed by Taqqu et al. [18] for characterizing long-range dependence properties of DNA sequences. This method has been reported to be sensitive enough to demonstrate the fatigue effect on a sample including less than 3000 data [26,18]. The second method used in that present study is the wavelet leaders based structure function scaling behavior. Waveletbased estimators have been used successfully in very different contexts for estimating scaling behavior [15,16,45,46,37]. Indeed, since wavelets can be made orthogonal to low-order polynomials behavior, these estimators are blind to eventual superimposed polynomial trends [45]. Through observations that the best performances in humans are obtained during variable pace running, our work has previously focused on the heart rate signal by successfully using the wavelet method to demonstrate the effect of an imposed constant vs. free pace on a 10,000 m performed at the same average pace [24]. In this previous study, we showed that a constant pace led to an increase in heart rate and oxygen uptake throughout the exercise in contrast to a freely chosen pace. We considered the short-range and long-range scales to analyze both, the alteration in speed and HR brought about through exhaustive exercise as the marathon. There are two distinct scaling exponents of the time series, which are both modified with the race progress. Therefore, this study showed that there are two distinct scaling exponents: the short range

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α1 , for n less than n0 (n0 = 64 = 5.3 min) and the long range α2 , for n greater than n0 . The same phenomenon arises in posture control — see Collins and De Luca [47] and Meyer et al. [48]. However, Collins and De Luca [47] derive this result from the study of the variogram, without mathematical justification. This approach could be made rigorous by using DFA or wavelet analysis (see Bardet et al., [49]) and has led to the introduction of multiscale fractional Brownian motion in Bardet and Bertrand [50]. Multiscale fractional Brownian motion is a fractional process with different Hurst indexes corresponding to the different ranges of scales. However, both off-line and on-line detection of change on the Hurst indexes on one (or two) of the different ranges of scale is a challenging problem. A huge amount of literature has been devoted by the statisticians to change-point analysis [2, 51–53]. The HR and Speed scaling exponents, for small scales, increased whereas this was not the case for HRS. This finding might suggest that, even if its average values increased, the fractal scaling of HRS is not modified with fatigue short-range scales and can be seen as a regularity Hölder exponent. We still do not know how HRS (i.e. the parameter we named the ‘‘cardiac cost’’) could be a signal for perceived exertion, but we can suggest that the variability, more than the average tendency of a physiological signal, could allow rapid speed adjustments. Other studies have suggested that speed is controlled by physiological (such as average heart rate) or psycho-physiological markers [32,54–56,17]. Preliminary investigations performed in sub elite runners achieving the marathon of Paris, showed a speed and heart rate power scale relationship and a fractal scaling which increased with fatigue (i.e. average speed decrease in the second part of the race). 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