IEEE Systems, Man and Cybernetics Magazine - January 2022 - 21
can be solved using [34] and [35]. In the literature, the
information theoretic filter extraction algorithm is the
most commonly used technique for selecting filters
obtained in matrix W [36].
FTDCSSP
Fuzzy time-delayed filters are used, generating the extended
state space model
2
ZW X*(xx x
kk),
x=0
. nd
|
where
d XX()() ()
xx=
kkis
the delay operator across the signal state space, nx
the fuzzy membership value for the variable x, and W()x
Z WWW @
k =6 012
() () ()
ces WW,()
R
T
S
S
S
01
n X()
n
n
11
22
X
X
k
k
-
-
k
()
()
V
X
W
W
W
(7)
is
is
the optimized fuzzy CSSP weights matrix. Further, the
terms in (6) can be simplified to obtain
.
(8)
+
The fuzzy CSSP filters, which are the rows in the matri()
and W ,()2
maximize the fuzzy mutual
information criterion [5, eq. (22)], [37]. Each of the
matrices WW,,() ()
n
01 and W()2
-
apply to X ,()k
n0
n
11 X ,()k
-
select at least two filters (pertaining to the largest and
smallest variances) for every class. In this manner,
FK26
== is chosen in the experiments for K = 3. In (8),
estimating three spatial transforms consists of calculating
32 K 18
# = row vectors.
Stationarity-Based Approaches
Nonstationarities are very frequent and can arise at different
time instances. They are mainly caused by eye
blinking, head/body movements, and drowsiness during
the course of a trial. Between sessions, they can be triggered
by different calibration settings and by constantly
changing the positions of electrodes. In addition, subjects
have physiological differences, leading to various signal
probability distributions. These result in time-varying
feature vectors. In fact, several traditional methods,
including the fuzzy CSP algorithm, produce poor results
from this feature space. We discuss approaches to deal
with the problem of nonstationarity in EEG regression
machine learning problems. One of them is the divergence-based
technique. We generalize the notion of divergence-based
CSP for regression through the concept of
fuzzy sets.
F-DivCSP-WS
We propose a cost function for regression by deploying
fuzzy covariance matrices. We begin by using two fuzzy
classes and later generalize for multiple classes (K > 2). The
conditional probability of every fuzzy class is normal, i.e.,
- d,
and X ,()k22 respectively. In the CSP method [38], we +-2d].
-1
= [(() (
Tr((
2
1
Here, W()F
Tr
WW WW
WW WW
<<
<<
RR
RR
-1
1
2 )(
2
1
))
)),
(10)
denotes the regression cost function and
represents the regression approach's forecasting strength.
The symmetric KL divergence [39] ()sD ppkl 12
<
sD pp 2
1
kl 12
() ;
<
R
=+ R
ww
ww
<
R
R
2
1
<< -
<
ww
ww
1
2
2E,
conditional probabilities of two fuzzy classes, after spatial
filtering, can be written as
(11)
which is synonymous with the CSP objective function
given by (13):
WW WW W),
sklkl 12
W
= RR<
<
argmin sD (
W* =
WW
argmaxWW.
<
<
W
R
R
2
1
<
(12)
(13)
In this article, we address an EEG-based driving scenario
in which we examine the stationarity within sessions for
every subject. The regularization function W()G
is designed
to optimize stationarity covering every fuzzy class:
WW WW W
G() (),
= NN 1 j=
12 1
+
where Ni
i=
class. In (14), R and i
ji,
captures the number of trials in the ith fuzzy
R indicate the trial and class fuzzy
covariances. We therefore put together a mixed objective
function that conjointly optimizes coupled prediction and
stationarity objectives:
January 2022 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 21
ji kl
,, i
D
ji
<
<
1 ||n RR (14)
<
2
Ni
linking the
FsDp p
Dp pD pp
W =
kl
=+
=
kl
2
1
[
+ Tr(( <<
<
2
1
log
log
WW WW
WW
det
det
+ Tr(( RR )) ,
WW
WW WW
RR
R
()
()
1 )(
R
-1
2
<
<
2
1
,
<< F
-1
1 )(
2
det
det
12
12 21
<
<<
R
kl
()
()
WW
WW
<
<
R
1
2
,
))],
() (),
() (),
() ()
(6)
N (, )0 1
R and (, )N 0 2
( 1R and 2
00
R for two fuzzy classes, respectively
R denote the fuzzy class covariances). The Kullback-Leibler
(KL) divergence across two D variate gaussians
~ (, )Np1
Dp ptrace(),RR0
IEEE Systems, Man and Cybernetics Magazine - January 2022
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