Non-linear time series analysis: methods and applications to atrial fibrillation.

Abstract

We apply methods from non-linear statistical time series analysis to characterize electrograms of atrial fibrillation. These are based on concepts originating from the theory of non-linear dynamical systems and use the empirical reconstruction density in reconstructed phase space. Application of these methods is not restricted to deterministic chaos but is valid in a general time series context. We illustrate this by applying three recently proposed non-linear time series methods to fibrillation electrograms: 1) a test for time reversibility in atrial electrograms during paroxysmal atrial fibrillation in patients; 2) a test to detect differences in the dynamical behaviour during the pharmacological conversion of sustained atrial fibrillation in instrumented conscious goats; 3) a test for general Granger causality to identify couplings and information transport in the atria during fibrillation. We conclude that a characterization of the dynamics via the reconstruction density offers a useful framework for the non-linear analysis of electrograms of atrial fibrillation.

Full text

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Non–linear Time Series Analysis:
Methods and Applications to Atrial Fibrillation
B.P.T. Hoekstra1, C.G.H. Diks2, M.A. Allessie3and J. DeGoede4
Published in:
Analysis and Processing of Cardiac Electrograms in Atrial Fibrillation
Annali dell’Istituto Superiore di Sanit` a, Roma, 2001
Edited by:
V.Barbaro, P. Bartolini, G. Calcagnini, G. Boriani
Reference:
Ann. Ist. Super. Sanit` a, vol. 37, no. 3 (2001), pp. 325–333
1Division of Image Processing, Department of Radiology, Leiden University Medical
Center, The Netherlands
2Center for Nonlinear Dynamics in Economics and Finance, Department of Economics
and Econometrics, University of Amsterdam, The Netherlands
3Cardiovascular Research Institute, Department of Physiology, Maastricht University,
The Netherlands
4Department of Physiology, University of Leiden, The Netherlands
Address: B.P.T. Hoekstra, Leiden University Medical Center, Division of Image Processing,
Department of Radiology, P.O. Box 9600, 2300 RC Leiden, The Netherlands. Tel: +31–71–526 2137;
Fax: +31–71–526 6801; Email: B.P.T.Hoekstra@LUMC.NL

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Summary
We apply methods from nonlinear statistical time series analysis to characterize elec-
trograms of atrial fibrillation. These are based on concepts originating from the theory of
nonlinear dynamical systems and use the empirical reconstruction density in reconstructed
phase space. Application of these methods is not restricted to deterministic chaos but is
valid in a general time series context. We illustrate this by applying three recently proposed
nonlinear time series methods to fibrillation electrograms: 1) a test for time reversibility in
atrial electrograms during paroxysmal atrial fibrillation in patients; 2) a test to detect differ-
ences in the dynamical behaviour during the pharmacological conversion of sustained atrial
fibrillation in instrumented conscious goats; 3) a test for general Granger causality to iden-
tify couplings and information transport in the atria during fibrillation. We conclude that a
characterization of the dynamics via the reconstruction density offers a useful framework for
the nonlinear analysis of electrograms of atrial fibrillation.
Keywords: atrial fibrillation, Granger causality, nonlinear time series analysis, reconstruction density,
time reversibility
Introduction
Physiological time series typically are short, nonlinear and noisy. Such time series usually cannot
be studied satisfactorily by linear time series analysis. Although linear techniques such as Fourier
analysis are useful to study characteristic oscillations in detail, these methods fail to detect any
nonlinear correlations present and cannot provide a complete characterization of the underlying
dynamics.
Over the lasts two decades many nonlinear time series methods have been developed in the
theory of nonlinear dynamics, commonly known as chaos theory. These methods are suited to
characterize the dynamics in noise free, low–dimensional deterministic systems and have proven
highly successful in characterizing irregular (chaotic) time series from mathematical models and
well–controlled physical experiments.
Biological systems are subjected to changes in their environment triggered both by stochastic
sources and feedback control mechanisms. Hence, time series recorded from the natural world
consist of a mixture of random and deterministic features in which chaos in a strong sense can
hardly be expected. Although much initial work was inspired by the idea that chaos might
be omnipresent in nature and focused on whether observed irregularity was chaos or noise, it
has become increasingly clear that such a sharp distinction between deterministic chaos and
randomness cannot be made [1].
Here we adopt a different point of view on nonlinear time series analysis and demonstrate
that concepts originating from the theory of nonlinear dynamics can be used to characterize
the underlying dynamics of physiological time series. In particular, we apply recently developed
statistical nonlinear time series methods which are based on the reconstruction density.
We present a test for time reversibility [2] which is applied to discriminate and classify atrial
electrograms of acute atrial fibrillation (AF) in patients undergoing open–chest surgery. The
test has been previously applied to characterize the dynamics of EEG epochs recorded before
and during an epileptical seizure [3].
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We also present a test to detect differences between reconstruction densities [4] which is
applied to electrograms of sustained AF in instrumented conscious goats to characterize changes
in the dynamics during the pharmacological conversion of sustained AF by the class IC anti–
arrhythmic agent cibenzoline. The test is also used to investigate if the fibrillation electrograms
are stationary by comparing different segments of the recorded time series.
We previously studied linear and nonlinear association lengths to characterize spatial or-
ganization during AF [5, 6]. However, the association lengths do not provide the direction of
couplings and the flow of information between different atrial sites. In 1969, Granger [7] in-
troduced an operational definition of causality between two variables. This so–called Granger
causality is well suited to identify the relevant space–time couplings in the activation pattern of
AF. The testing for Granger causality has mostly been performed in a linear framework. Here
we propose to test for general Granger causality, i.e. linear as well as nonlinear causality, as a
first step in the characterization of the space–time pattern of electrical heart activity.
The nonlinear time series methods presented here have in common that they are based on
the reconstruction density in reconstructed phase space. A characterization of the dynamics via
the reconstruction density offers a useful framework for the analysis of electrograms of atrial
fibrillation in particular and time series in general.
Methods
Reconstructed Phase Space and Reconstruction Density
The reconstruction theorem proved by Takens [8] is the starting point for many nonlinear time
series methods. Given a scalar–valued time series {xi}∞
system can be represented in a reconstructed phase space by defining m–dimensional delay
vectors
? xi= (xi, xi−τ,..., xi−(m−1)τ),
where τ is the reconstruction delay and m is the embedding dimension.
We will only be concerned with stationary time series. We will follow the methodology of
Takens [9] in order to give a definition of stationarity which can be applied to a single time
series rather than to an ensemble of time series. In this methodology a time series by definition
is called stationary if its reconstruction measure exists, defined by
i=1, the state of a deterministic dynamical
(1)
ρ(? x) = lim
L→∞
1
L
L
?
i=1
δ(? x − ? xi),
(2)
where δ(·) is the Dirac delta function. In practice where we have time series of finite length we
approximate the reconstruction density by the empirical reconstruction density, given by
ρ(? x) =1
L
L
?
i=1
δ(? x − ? xi),
(3)
where L = N − (m − 1)τ, and N is the length of the time series.
We can study the reconstruction density without proposing a model for the time series.
For example, it can be examined if the reconstruction density has a certain symmetry. In the
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following section a test is presented which explores the symmetry with respect to time reversal.
Next, we elaborate on a test which was designed to examine if two given time series have the
same reconstruction density. This test can also be used to investigate (strict) stationarity by
comparing different segments of an observed time series. Finally, we consider a test for general
Granger causility to study the direction of couplings and the information transport between
different spatial positions on the atria.
Test for Time Reversibility
A time series is (time) reversible if its statistical properties are invariant under reversal of time
direction. This means that the time series “looks the same” read forward or backward in time.
Otherwise the time series is said to be (time) irreversible.
If a time series is generated by a linear Gaussian random process, the result is a reversible time
series. Furthermore, all static transformations of the time series preserve reversibility. Therefore,
if reversibility can be ruled out, the null hypothesis that the time series is a realization of a static
transformation of a linear Gaussian random process can be rejected.
We will briefly describe a test for reversibility proposed by Diks et al. [2]. We want to test
the null hypothesis that the reconstruction density ρ of the delay vectors is invariant under time
reversal for all embedding dimensions m and reconstruction delays τ, i.e.
H0:
ρ(? x) = ρ(P? x),
where P denotes the time reversal operator which is defined by
P(xi, xi−τ, ···, xi−(m−1)τ) = (xi−(m−1)τ, ···, xi−τ, xi).
(4)
A squared distance Qrbetween the time forward and time reversed reconstruction densities is
defined as
?d√π?m?
in which ρ?(? x) is defined as the convolution
?
where κ(·) is a Gaussian distribution defined by
?2
with | · | denoting the Euclidean norm in Rm. The parameter d > 0 is a bandwidth, which is
the length scale at which the reconstruction densities ρ(? x) and ρ(P? x) are compared.
Qrcan be estimated unbiasedly from the set of delay vectors {? xi}N
from ρ(? x). The result is [2]
Qr
=
1
2
d? x
?ρ?(? x) − ρ?(P? x)
?2,
(5)
ρ?(? x) =d? r ρ(? r) κ(? x −? r),
(6)
κ(? x) =
πd2
?m/2
exp(−2|? x|2/d2),
(7)
i=1sampled independently
?
Qr
=
2
N (N − 1)
?
i<j
wij,
(8)
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where
wij= exp(−|? xi− ? xj|2/d2) − exp(−|? xi− P? xj|2/d2).
Under the H0the expected value of?Qris equal to zero and its standard deviation σris given
(9)
by
σr
=
2
N (N − 1)

?
i<j
w2
ij


1/2
.
(10)
The test statistic Sris now defined as
Sr=
?Qr
σr.
(11)
Under the H0, Sr has expected value zero and unit standard deviation. We remark that the
asymptotic distribution of Srmay not be normal. However, assuming unimodality, reversibility
can be rejected at a rejection probability of at most 0.05 when the test statistic gets larger than
three [10].
In the derivation of the test statistic Srwe have assumed that the delay vectors are sampled
independently from the reconstruction density ρ(? x). However, this is an oversimplification,
since the reconstruction procedure by itself introduces dependence among the delay vectors.
Furthermore, dynamical structure in the time series will introduce additional dependences. In
order to reduce their effects, we excluded all pairs (? xi, ? xj) which are closer in time than some lag
W as proposed by Theiler [11]. Furthermore, higher order dependences between delay vectors
were suppressed by comparing non–overlapping blocks consisting of ? consecutive delay vectors,
instead of single pairs of delay vectors [4]. The parameter ? is preferably chosen large with
respect to the characteristic time scale of the time series.
Insight into the presence of irreversibility in the time series may already be obtained from
a visual inspection of a two–dimensional phase plot. By interchanging the two components of
the delay vectors, which is equivalent to mirroring the phase plot in the main diagonal, it can
be visually inspected if the plot is symmetric with respect to this operation. If not, it can be
concluded that the time series is irreversible.
For example, Fig. 1 shows a right atrial electrogram (RA) of AF and a surrogate electrogram
constructed from the measured electrogram by the procedure of phase randomization in the
frequency domain [12]. From the lack of symmetry in the phase plot of the measured electrogram
(there are more points above the main diagonal than below), it may be concluded that the
electrogram is irreversible.In contrast, the distribution of points in the phase plot of the
surrogate electrogram is symmetric and therefore suggests that the surrogate time series is
reversible. This is indeed the case, since the surrogate electrogram is by construction a realization
of a linear Gaussian random process.
Finally, we remark that a test for time reversibility is not a test for linearity, since there exist
stationary nonlinear processes that are time reversible. On the other hand, time irreversibility
indicates nonlinearity or non–Gaussianity.
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