Comparative Analysis of Generalized Multiscale Entropy Methods for

Coarse-Grained Time Series Construction in Assessing Autonomic

Balance in Peripheral Arterial Disease Patients

O. Barquero-P

erez

1 a

, R. Goya-Esteban

1 b

, E. Sarabia-Cachadi

2 c

and J. Naranjo-Orellana

3 d

Dept. Signal Theory and Communications, Universidad Rey Juan Carlos, Fuenlabrada, Madrid, Spain

Centro de Estudios Universitarios Cardenal Sp

ınola CEU, Sevilla, Spain

Departamento Deporte e Inform

atica, Universidad Pablo de Olavide, Sevilla, Spain

{oscar.barquero, rebeca.goyaesteban}@urjc.es, esarabia@ceu.es, jnarore@upo.es

Keywords:

Autonomic Nervous Systems, Heart Rate Variability, Generalized Multiscale Entropy, Peripheral Arterial

Disease.

Abstract:

Peripheral Arterial Disease (PAD) is a chronic condition that signiﬁcantly impacts autonomic balance, as

reﬂected in Heart Rate Variability (HRV). However, the characterization of autonomic balance in PAD pa-

tients using HRV is still unclear. Generalized Multiscale Entropy (GMSE) is a nonlinear method capable of

characterizing the complexity of HRV across multiple time scales, offering a more nuanced understanding

of autonomic dysfunction in PAD patients. 14 healthy male subjects (60±5 years) and 14 male intermittent

claudication patients (64±6 years) underwent 10 minutes of ECG recording from which RR interval time se-

ries were obtained. This study provides a comparative analysis of different GMSE methods for constructing

coarse-grained time series, speciﬁcally using the mean, mean absolute deviation (MAD), standard deviation

(σ), and variance (σ

) approaches. By applying these methods, we investigate their efﬁcacy in differentiat-

ing between healthy individuals and PAD patients. Our results demonstrate that the variance coarse-grained

method offers superior discriminatory power, revealing statistically signiﬁcant differences. These ﬁndings

suggest that the variance-based GMSE method is the most effective approach for assessing autonomic imbal-

ance in PAD patients, with potential applications in improving diagnostic tools and treatment strategies.

1 INTRODUCTION

Peripheral Arterial Disease (PAD) is an atheroscle-

rotic condition characterized by the occlusion of ar-

teries located distal to the aortic bifurcation (Ramos

et al., 2009). This arterial blockage reduces the oxy-

gen supply to the lower limb muscles during physi-

cal activity, leading to pain and forcing individuals to

stop walking. This condition, known as intermittent

claudication, often results in signiﬁcant limitations in

daily physical activities and negatively impacts the

health-related quality of life of those patients (Fein-

glass et al., 1996; Crowther et al., 2007; Celis et al.,

2009). PAD is a chronic and progressive disorder that

can severely restrict an individual’s mobility and inde-

pendence, making it an important public health con-

https://orcid.org/0000-0002-7235-3986

https://orcid.org/0000-0002-0402-8487

https://orcid.org/0000-0002-4444-6755

https://orcid.org/0000-0001-9180-1732

cern, particularly among the elderly population.

Atherosclerosis, and consequently PAD, is partic-

ularly prevalent among the elderly population. This

condition is closely associated with several risk fac-

tors, including diabetes mellitus, hypertension, el-

evated body mass index, and dyslipidemia (Ramos

et al., 2009; Diehm et al., 2009). Furthermore, the

risk of developing PAD is further increased by both

current and past smoking habits (Diehm et al., 2009).

Heart Rate Variability (HRV) is a ﬂuctuation in

the time intervals between consecutive heartbeats. It

allows to non-invasively measure the autonomic ner-

vous system regulation of the cardiovascular system.

It provides valuable insight into the balance between

sympathetic and parasympathetic nervous activity,

making it a crucial tool for assessing autonomic func-

tion (Camm et al., 1996; Shaffer and Ginsberg, 2017).

HRV analysis can reveal subtle abnormalities in the

autonomic regulation of the heart, which may have

important implications for understanding the physio-

Barquero-Pérez, O., Goya-Esteban, R., Sarabia-Cachadiña, E. and Naranjo-Orellana, J.

Comparative Analysis of Generalized Multiscale Entropy Methods for Coarse-Grained Time Series Construction in Assessing Autonomic Balance in Peripheral Arterial Disease Patients.

DOI: 10.5220/0013165000003911

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 18th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2025) - Volume 1, pages 893-898

ISBN: 978-989-758-731-3; ISSN: 2184-4305

893

logical mechanisms underlying various cardiovascu-

lar disorders, including peripheral arterial disease.

While HRV has been extensively studied in var-

ious cardiovascular diseases, and some studies have

examined HRV in peripheral arterial disease patients,

the role of nonlinear characterization of HRV in pre-

dicting intermittent claudication in PAD patients re-

mains underexplored. Traditional HRV analysis of-

ten focuses on linear methods, which may not fully

capture the complex, nonlinear dynamics of the HRV.

Therefore, a nonlinear approach may be required to

reveal more subtle autonomic dysfunctions in PAD

patients and potentially improve our understanding of

the relationship between HRV and symptoms like in-

termittent claudication.

Some previous studies have not found a signif-

icant relationship between heart rate variability and

improvements in walking ability in peripheral arte-

rial disease patients (Leicht et al., 2011; Sandercock

et al., 2007), other research has revealed a positive

association between HRV indices and maximal walk-

ing distance, though not with claudication distance,

in symptomatic PAD individuals (Lima et al., 2016).

These mixed ﬁndings suggest that further investiga-

tion is needed to better understand the capacity of

HRV to characterize autonomic dysfunction in PAD

patients and its potential relationship to clinical out-

comes like walking performance.

The primary objective of this study is to in-

vestigate whether resting HRV differs between pa-

tients with intermittent claudication and healthy con-

trol subjects. To achieve this, we will employ Gen-

eralized Multiscale Entropy (GMSE), a nonlinear

analysis method that characterizes the complexity

of physiological time series over multiple temporal

scales (Costa and Goldberger, 2015). We hypothe-

size that GMSE will detect signiﬁcant differences in

HRV patterns between PAD patients with intermittent

claudication and healthy individuals. This could po-

tentially offer a new strategy for understanding au-

tonomic imbalance in this patient population, which

may have important implications for managing and

monitoring this chronic and debilitating condition.

The structure of the paper is as follows. In Sec-

tion 2, the dataset is explained. In Section 3, GMSE

is explained. In Section 4, the statistical analysis is

explained. In Section 5, results are reported. Finally,

in Section 6, conclusions are presented.

2 DATASET

Fourteen male control individuals (60±5 years old,

90±12 kg, 174±7 cm) and 14 male PAD patients

(64±6 years old, 83±17 kg, 168±7 cm) exhibiting in-

termittent claudication were recruited from two hos-

pitals in Seville, Spain. The control subjects were se-

lected based on the following criteria: absence of car-

diovascular disease, no ongoing medical treatment,

and an ankle-brachial index greater than 1. In con-

trast, the PAD patient group consisted of individuals

referred by the vascular surgery departments of the

participating hospitals, with a conﬁrmed PAD diag-

nosis, no prior surgical interventions, and an ankle-

brachial index less than 0.9 (Schroll and Munck,

1981; Aboyans et al., 2012). All participants, in-

cluding both the control and PAD groups, were non-

smokers and had not taken any cardiovascular-related

medications for at least three months prior to the

study. The participants reported to the laboratory in

the morning, two hours after breakfast, having ab-

stained from caffeine consumption and exercise for

24 hours before the data recording. The RR interval

time series were recorded for 15 minutes at rest in the

supine position using a Firstbeat Bodyguard recorder.

The initial ﬁve minutes of each recording were ex-

cluded to allow for participant relaxation (Chidean

et al., 2018; Cachadi

na et al., 2018).

3 GENERALIZED MULTISCALE

ENTROPY FOR HEART RATE

VARIABILITY

GMSE is an advanced nonlinear method used to quan-

tify the complexity of physiological time series across

multiple temporal scales. Traditional entropy mea-

sures, such as Sample Entropy (SampEn), quantify

the unpredictability or irregularity of a time series on

a single scale (Richman and Moorman, 2000). Al-

though effective, these measures may not fully cap-

ture the multiscale nature of physiological signals.

Multiscale Entropy (MSE) extends this approach by

evaluating entropy over a range of scales, providing

a more comprehensive characterization of the under-

lying dynamics (Costa et al., 2005). GMSE further

enhances MSE by allowing for ﬂexible scaling pa-

rameters, which can be adjusted to ﬁt speciﬁc types

of data or clinical contexts (Costa and Goldberger,

2015). This type of methods has been used with suc-

cess to characterize long-range correlated cardiovas-

cular an respiratory signals (Martins et al., 2020; Silva

et al., 2015)

In MSE deﬁnition, {y

(τ)

} where each coarse-

grained time series at scale τ is constructed by av-

eraging non-overlapping blocks of τ consecutive data

points from the original time series {x

}. The GMSE

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894

modiﬁcations use different statistical moments to

build the coarse-grained time-series. In this work we

are going to compare the following four approaches:

• Averaging: Each coarse-grained time series is

constructed by averaging non-overlapping blocks

of τ consecutive data points from the original time

series {x

(τ)

jτ

∑

i=( j−1)τ+1

, j = 1, 2, . . . ,





• Mean Absolute Deviation (MAD): The coarse-

grained time series can also be computed using

the mean absolute deviation of τ consecutive data

points:

(τ)

jτ

∑

i=( j−1)τ+1

− ˆµ|

where ˆµ is the sample mean of the τ data points in

each block.

• Standard Deviation, σ: Alternatively, each block

of τ data points can be used to compute the stan-

dard deviation:

(τ)

jτ

∑

i=( j−1)τ+1

− ˆµ)

• Variance, σ

: Finally, the coarse-grained series

can be based on the variance of the data points

within each block:

(τ)

jτ

∑

i=( j−1)τ+1

− ˆµ)

Where N is the length of the original time series,

and τ is the scale factor.

Once the coarse-grained time series are generated,

the entropy of each is calculated using a method such

as Sample Entropy (SampEn):

SampEn(m, r, N) = − ln



A(m + 1)

B(m)



where m is the embedding dimension, r is the toler-

ance (typically a percentage of the time series stan-

dard deviation), and A(m +1) and B(m) represent the

number of matching template vectors of length m + 1

and m, respectively.

GMSE is deﬁned as the entropy measure com-

puted across all scales τ, producing a proﬁle that re-

ﬂects the time series complexity at various temporal

resolutions. This approach is particularly effective in

HRV analysis, where entropy at different scales can

reveal the balance between sympathetic and parasym-

pathetic inﬂuences on the heart (Costa and Gold-

berger, 2015).

4 STATISTICAL ANALYSIS

To statistically analyze the differences between the

control and PAD patients groups, we will employ a

combination of bootstrap resampling and exponential

curve ﬁtting to the GMSE curve. This approach will

allow us model the complexity and autonomic bal-

ance of heart rate variability measured by GMSE.

First, we will perform bootstrap resampling with

replacement on the GMSE curves for both the con-

trol and PAD groups. This will generate multiple re-

sampled GMSE curves for each group, allowing us

to compute the average GMSE curve and assess the

variability within each group.

For each average GMSE curve from the bootstrap

resampling, we will then ﬁt an exponential curve us-

ing the equation:

GMSE(τ) = C · e

−κτ

where C and κ are constants of the model deter-

mined by optimization methods. Whereas τ repre-

sents the different time scales.

This exponential curve ﬁtting will allow us to ex-

tract the values of C

and C

pad

as well as κ

and

pad

for the control and PAD patient groups, respec-

tively, from the bootstrap resampling process. We can

then assess whether there are signiﬁcant differences

in these exponential model parameters between the

two groups by comparing the distributions of the ﬁtted

model parameters obtained from the bootstrap resam-

pling procedure.

Speciﬁcally, this statistical analysis will provide

insights into the underlying differences in the com-

plexity and autonomic balance of heart rate variabil-

ity between the healthy control subjects and the PAD

patients. The exponential curve ﬁtting and compari-

son of the model parameters between the groups can

shed light on the speciﬁc alterations in the cardiovas-

cular autonomic regulation associated with peripheral

arterial disease, potentially leading to a better under-

standing of the disease and improved diagnostic or

prognostic tools.

5 RESULTS

Figure 1 shows the average value and standard de-

viation of the GMSE curve obtained from the boot-

strap resamplings for the different methods to build

the coarse-grained time series, in particular, (a) us-

ing the average MSE

, (b) using the MAD MSE

MAD

, and (d) using

the variance MSE

. From scale two, the curves are

statistically different, showing that GMSE, with vari-

ance, is able to distinguish control subjects from PAD

Comparative Analysis of Generalized Multiscale Entropy Methods for Coarse-Grained Time Series Construction in Assessing Autonomic

Balance in Peripheral Arterial Disease Patients

895

(a) GMSE average values for healthy subjects and PAD

patients, using average to build coarse-grained time se-

ries.

(b) GMSE average values for healthy subjects and PAD

patients, using MAD to build coarse-grained time se-

ries.

patients, using σ to build coarse-grained time series.

(d) GMSE average values for healthy subjects and PAD

patients, using σ

to build coarse-grained time series.

Figure 1: GMSE comparison of methods to build the coarse-grained time series for healthy subjects (dots) and PAD patients

(squares) with mean and standard deviation curves obtained from the bootstrap resampling.

patients. It is clear, that using the variance to create

the coarse-grained time series allows to better distin-

guish healthy subjects and PAD patients.

Figure 2 shows the mean and standard deviation

for the C parameter of the exponential model. Among

the methods evaluated for constructing coarse-grained

time series, only the σ

approach demonstrates suf-

ﬁcient discriminatory power to reliably differentiate

between healthy individuals and PAD patients, yield-

ing statistically signiﬁcant differences.

6 CONCLUSIONS

The ﬁndings of this study provide signiﬁcant insights

into the autonomic dysfunction associated with PAD

and its manifestation in patients with intermittent

claudication. By employing GMSE analysis of HRV,

we were able to discern notable differences in auto-

nomic regulation between PAD patients and healthy

controls. This study underscores the utility of non-

linear methods in capturing the complexity of physio-

logical signals, which traditional linear methods may

overlook. In fact, previous studies using linear meth-

ods (both time and frequency domain) and nonlin-

ear (SampEn, DFA) did not show strong differences

in HRV (Cachadi

na et al., 2018; Leicht et al., 2011;

Sandercock et al., 2007; Lima et al., 2016).

Our results indicate that PAD patients exhibit a

reduced complexity in HRV across multiple time

scales, as evidenced by lower GMSE values com-

pared to healthy subjects. This reduction in com-

plexity suggests a diminished adaptability of the au-

tonomic nervous system in PAD patients, potentially

due to chronic ischemic conditions affecting the car-

diovascular system. Our study uniquely contributes

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896

Figure 2: Mean and standard deviation for the C parameter of the exponential model for the different methods to build the

coarse-grained time series in GMSE.

by utilizing a MSE approach to provide a more strong

proof of these alterations. Moreover, among all the

possible ways to build the coarse-grained time series

in GMSE, the best option to be able to distinguish

between healthy and PAD patients is using the σ

The results indicated a reduction in the complexity of

HRV time series in PAD patients, which is in agree-

ment with previous results.

The clinical implications of these ﬁndings are

substantial. The ability to distinguish PAD patients

from healthy individuals based on HRV complexity

could help to the search for new diagnostic tools and

therapeutic strategies aimed at improving autonomic

function in this population. Furthermore, the use of

GMSE as a diagnostic marker could enhance the mon-

itoring of disease progression and the efﬁcacy of in-

terventions aimed at restoring autonomic balance.

However, this study is not without limitations.

The sample size was relatively small, and the study

population was limited to male subjects, which may

affect the results. Future research should aim to in-

clude a larger and more diverse cohort to validate

these results and explore the potential sex-based dif-

ferences in the autonomic dysfunction associated with

PAD. As a future work, it would be worth to com-

pare these results with other multiscale entropy mea-

sures (Bari et al., 2014; Valencia et al., 2009)

In conclusion, this study demonstrates that GMSE

analysis, using σ

of HRV is a powerful tool for as-

sessing autonomic dysfunction in patients with Pe-

ripheral Arterial Disease. The signiﬁcant differ-

ences in HRV complexity between PAD patients and

healthy controls highlight the potential of GMSE as a

diagnostic and monitoring tool in clinical settings.

ACKNOWLEDGEMENTS

This work was partially funded by Grant

PID2022-136887NB-I00 by MCIN/AEI/10.13039/

501100011033.

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