Biomedical Engineering Reference
In-Depth Information
is analyzed in respect to local trends in data windows. The method allows to detect
long-range correlations embedded in a non-stationary time series. The procedure
relies on the conversion of a bounded time series
x
t
(
t
∈
N)
into an unbound process:
X
t
:
t
i
=
1
(
x
i
−
x
i
)
X
t
=
(2.134)
where
X
t
is called a cumulative sum and
is the average in the window
t
. Then the
integrated time series is divided into boxes of equal length
L
, and a local straight line
(with slope and intercept parameters
a
and
b
)isfitted to the data by the least squares
method:
x
i
L
i
=
1
(
X
i
−
a
i
−
b
)
E
2
2
=
(2.135)
Next, the fluctuation—the root-mean-square deviation from the trend is calculated
over every window at every time scale:
1
L
L
i
=
1
(
X
i
−
a
i
−
b
)
F
(
L
)=
2
(2.136)
This detrending procedure followed by the fluctuation measurement process is
repeated over the whole signal over all time scales (different box sizes
L
). Next, a
log
is constructed.
A straight line on this graph indicates statistical self-affinity, expressed as
F
−
log graph of
L
against
F
(
L
)
(
)
∝
L
α
. The scaling exponent α is calculated as the slope of a straight line fittothe
log
L
−
(
)
.
The fluctuation exponent α has different values depending on the character of the
data. For uncorrelated white noise it has a value close to
log graph of
L
against
F
L
1
2
, for correlated process
1
2
, for a pink noise α
3
2
α
>
=
1 (power decaying as 1
/
f
), α
=
corresponds to
Brownian noise.
In the case of power-law decaying autocorrelations, the correlation function de-
cays with an exponentγ:
R
L
−
γ
and the power spectrum decays as
S
f
−
β
.
(
L
)
∼
(
f
)
∼
The three exponents are related by relations: γ
β.
As with most methods that depend upon line fitting, it is always possible to find a
numberα by the DFA method, but this does not necessarily mean that the time series
is self-similar. Self-similarity requires that the points on the log
=
2
−
2α
,
β
=
2α
−
1, and γ
=
1
−
log graph are suffi-
ciently collinear across a very wide range of window sizes. Detrended fluctuation is
used in HRV time series analysis and its properties will be further discussed in Sect.
4.2.2.4.3.
−
2.5.4 Recurrence plots
Usually the dimension of a phase space is higher than two or three which makes
its visualization difficult. Recurrence plot [Eckmann et al., 1987] enables us to in-
vestigate the
m
-dimensional phase space trajectory through a two-dimensional repre-
sentation of its recurrences; namely recurrence plot (RP) reveals all the times when
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