Biomedical Engineering Reference
In-Depth Information
5. repeat steps 3 and 4 many times to construct the distribution of the statistics;
6. use that distribution to compute the probability of observing the original or
more extreme value of statistics for processes conforming to the null hypothe-
sis.
Two approaches to surrogate data construction will be described below. The sim-
plest question one would like to answer is whether there is evidence for any dynamics
at all in the time series, i.e., if there are any relations between the consecutive sam-
ples. The null hypothesis in this case is that the observed data is fully described by a
series in which each sample is an independent random variable taken from identical
distribution. The surrogate data can be generated by shuffling the time-order of the
original time series. The surrogate data will have the same amplitude distribution as
the original data, but any temporal correlations that may have been in the original
data are destroyed.
The test for linearity corresponds to the null hypothesis that all the structure in the
time series is given by the autocorrelation function, or equivalently, by the Fourier
power spectrum. The test may be performed by fitting autoregressive model to the
series and examination of the residuals, or by randomizing the phases of the Fourier
transform. The second approach is recommended, since it is more stable numerically.
The main steps of the procedure are the following:
1. compute the Fourier transform
X
(
f
)
of the original data
x
(
t
)
;
e
i
φ
(
f
)
2. generate a randomized version of the signal
Y
(
f
)=
X
(
f
)
by multiply-
by
e
i
φ
(
f
)
,whereφ
ing each complex amplitude
X
is a random value,
independently drawn for each frequency
f
, from the interval
(
f
)
(
f
)
[
0
,
2π
]
;
3. in order to get real components from the inverse Fourier transform symmetrize
the phases, so that φ
(
f
)=
−
φ
(
−
f
)
;
4. perform the inverse transform.
(
)
In this way we obtain surrogate data
y
t
characterized by the same power spectrum
as the original time series
x
, but with the correlation structure destroyed. Then the
statistical test is performed comparing the estimators (i.e., statistics) obtained from
original and the distribution of that estimator obtained from surrogate data.
In case of multivariate time series the surrogates with randomized phases may be
used to test for the presence of the phase dependencies between the signals. The re-
jection of the null hypothesis assuming independence between the time series does
not mean the presence of the non-linearities. In biological time series analysis the
Occham's razor approach should be used: we should seek the simplest model con-
sistent with the data and the surrogate data tests are helpful in this respect.
(
t
)
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