Biomedical Engineering Reference
In-Depth Information
et al., 2004, Zygierewicz et al., 2005] a massive univariate approach with the mul-
tiple comparisons problem controlled with FDR (Sect.1.5.3.3) was proposed as an
efficient solution to that question, and will be described below.
For the null hypothesis of no changes at the given time-frequency coordinates,
we have to reduce the resolution to time-frequency boxes
(
Δ
t
×
Δ
f
)
.Therearetwo
reasons for decreasing the time-frequency resolution:
•
The time and frequency resolution are bounded by the uncertainty principle,
which for the frequencies defined as the inverse of the period (Hz) gives:
Δ
t
1
×
Δ
f
≥
4π
. The lower bound is only reached by the Gabor functions
•
In real data there are big variations of energy estimates due to different kinds of
noise. Increasing the product Δ
t
×
Δ
f
reduces the variance of energy estimate
Results from [Durka et al., 2004], suggest that Δ
t
2 gives robust results.
This introduces a discretization of the time-frequency plane into resels (resolution
elements)
r
×
Δ
f
=
1
/
(
i
,
j
)
. In order to obtain
E
n
(
i
,
j
)
— energy in resels
r
n
(
i
,
j
)
(subscript
n
denotes the trial) — we integrate
7
energy density
E
n
(
t
,
f
)
:
Z
(
i
+
1
)
·
Δ
t
Z
(
j
+
1
)
·
Δ
f
E
n
(
i
,
j
)=
E
n
(
t
,
f
)
dtd f
(4.24)
i
·
Δ
t
j
·
Δ
f
At this point, we may proceed to testing the null hypothesis of no significant
changes in
E
n
. The distribution of energy density for a given frequency is not
normal. However, in many practical cases it can be transformed to an approximately
normal one using an appropriate Box-Cox transformation [Box and Cox, 1964]. The
Box-Cox transformations are the family of power transformations:
(
i
,
j
)
x
λ
−
1
if
λ
=
0
BC
(
x
,
λ
)=
(4.25)
λ
log
(
x
)
if
λ
=
0
For each frequency
j
the λ parameter is optimized by maximization of the log-
likelihood function (LLF) [Hyde, 1999] in the reference period:
k
=
1
log
x
m
m
2
λ
opt
=
logσ
BC
(
x
,
λ
)
+(
ma
λ
{
LLF
(
λ
)
}
=
max
λ
−
λ
−
1
)
(4.26)
. The optimal λ
opt
is then used to transform all the resels in frequency
j
. Standard parametric tests
can be applied to the normalized data. However, we cannot a priori assume equal
variances in the two tested groups. The known solution to the problem of variance
where
m
is the length of data
x
,
x
∈{
E
n
(
i
,
j
)
:
i
∈
t
b
,
n
=
1
,...,
N
}
7
In the spectrogram equation (2.84) and scalogram equation (2.95), the energy is primarily computed on
the finest possible grid and then the integration is approximated by a discrete summation. In case of MP
the integration of equation (2.117) can be strict. The procedure was described in detail in [Durka et al.,
2004].
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