Abstract—The Detrended Fluctuation Analysis (DFA) is a
popular method for quantifying the self-similarity of the heart
rate that may reveal complexity aspects in cardiovascular
regulation. However, the self-similarity coefficients provided by
DFA may be affected by an overestimation error associated
with the shortest scales. Recently, the DFA has been extended
to calculate the multifractal-multiscale self-similarity and some
evidence suggests that overestimation errors may affect
different multifractal orders. If this is the case, the error might
alter substantially the multifractal-multiscale representation of
the cardiovascular self-similarity. The aim of this work is 1) to
describe how this error depends on the multifractal orders and
scales and 2) to propose a way to mitigate this error applicable
to real cardiovascular series.
Clinical Relevance— The proposed correction method may
extend the multifractal analysis at the shortest scales, thus
allowing to better assess complexity alterations in the cardiac
autonomic regulation and to increase the clinical value of DFA.
I. INTRODUCTION
The interest in the fractal analysis of the heart rate has
been progressively growing aimed at describing the complex
cardiovascular regulation, its adaptations to external stimuli
and behavioral conditions, and its interactions with the
autonomic control and the respiratory system. In this context,
the Detrended Fluctuation Analysis (DFA) is a powerful tool
for quantifying the fractal structure of cardiovascular series
[1] based on the estimation of a coefficient, α, related to the
Hurst's exponent. The DFA calculates the 2
nd
order moment
of the fluctuations of the integrated series after polynomial
detrending on consecutive blocks of n beats, F(n). For fractal
series, α is estimated as the slope of log F(n) vs. log n [2].
Since the heart rate shows a multiscale behavior, it has
been proposed to estimate α as a function of n, α(n), by
calculating "local" slopes around n [3], [4]. However, the
DFA may overestimate the local slopes at the shorter scales,
likely because of the overfitting of the detrending polynomial
(the overestimation increases with the polynomial order) [5].
A proposed correction method assumes the same α at all the
scales [5] and thus cannot be applied on real cardiovascular
series because of their multiscale nature. Alternatively, it was
suggested to use high-order polynomials at the larger scales,
where the overfitting is negligible, and the 1
st
order
polynomial, which causes the lowest overfitting, at the
shortest scales [6]. However, also the 1
st
order detrending
P. Castiglioni is with IRCCS Fondazione Don Carlo Gnocchi, Milan,
Italy (email: pcastiglioni@dongnocchi.it ). G. Parati and A. Faini are with
Istituto Auxologico Italiano, IRCCS, Dept. of Cardiovascular, Neural and
Metabolic Sciences Milan, Italy (email: a.faini@auxologico.it).
polynomial may cause overestimations that limit the shortest
analyzable n.
Recently, the multiscale DFA was modified to evaluate
the multifractality [7] calculating the q
th
moment order of the
residual variances, F
q
(n), and the multifractal multiscale self-
similarity coefficients, α(q,n), as the local slopes of log F
q
(n)
vs. log n. It is possible that the estimation bias affecting the
monofractal α also influences α(q,n) at q≠2, but whether the
error depends on both q and n and if it alters the
quantification of multifractality has never been described
systematically. Therefore, this work aims to describe the
estimation bias in the multifractal multiscale DFA and to
propose a method applicable to real cardiovascular series for
mitigating this error.
II. DFA ESTIMATION BIAS
A. The Multifractal Multiscale DFA
The DFA evaluates the multifractality of a time series S
i
of N samples (i=1,...,N), with mean μ and standard deviation
σ, by calculating the summation y
i
=∑
σ
(1),
by splitting the y
i
series into M blocks of n samples, and by
evaluating the variability function F
q
(n)
=
∑
! " ≠ 0
= %
&
’
∑ ()*
+
’
,-&
! " = 0
(2)
with σ
2
n
(k) the variance of the residuals of y
i
in each block k
after polynomial detrending [8]. In our work, n increased
linearly on a log scale from 6 up to N/4 samples. Moreover,
we considered integer q between -5 and +5; maximally
overlapped blocks, which means that M=N-n+1; and linear
detrending. The local slopes α(q,n) were calculated as the
derivative of log F
q
(n) vs. log n [6].
The cumulative functions of the α(q,n) squared
increments were calculated separately for positive q
α
/0
1
= ∑ 23", − 3" − 1, 7
8
(3)
and negative q
α
/0
= ∑ 23", − 3" − 1, 7
9
81
(4).
The α
/0
1
) and α
/0
functions represent scale-by-scale
measures of the degree of multifractality [9] .
Multifractal and Multiscale Detrended Fluctuation Analysis of
Cardiovascular Signals: how the Estimation Bias Affects Short-
Term Coefficients and a Way to mitigate this Error
Paolo Castiglioni, Gianfranco Parati, and Andrea Faini
2021 43rd Annual International Conference of the
IEEE Engineering in Medicine & Biology Society (EMBC)
Oct 31 - Nov 4, 2021. Virtual Conference
978-1-7281-1178-0/21/$31.00 ©2021 IEEE 257