Biomedical Engineering Reference
In-Depth Information
where
e
k
is the instantaneous squared error of the
k
th predictor. The memory term,
λ
, is identical
for all experts, but can be manually adjusted during training. The gating function, which moderates
the learning rate of the predictors, is determined by the distance from the winning predictor,
k
*, to
the other predictors.
2
d
k k
∗
( )
,
g
k
( )
=
exp
(3.54)
2
σ
2
( )
where
k
is the predictor to be updated,
k
* is the winning predictor,
d
k,k
*
is the neighborhood distance
between them, and σ is an annealing parameter that controls the neighborhood width. This results
in the model shown in Figure
3.17
. The signal flow graph is shown for only one predictor.
In exact analogy with training a Kohonen map, both the neighborhood width and the global
learning rate are annealed during training according to an exponentially decreasing schedule
-
----
i
-
----
i
σ
( )
σ
=
0
e
µ ( )
=
µ
0
e
(3.55)
where
τ
is the annealing rate and
i
is the iteration number. The overall learning rate for the
k
th
predictor at the
i
ith iteration is
η
k
( )
=
µ ( )
g
k
⋅
( )
(3.56)
3.2.3.2 gated Competitive Experts in BMIs.
Let us see how the hard competition of gated ex-
perts can be applied to BMIs. Hypothesizing that the neural activity will demonstrate varying
characteristics for different localities in the space of the hand trajectories, we expect the multiple
model approach, in which each linear model specializes in a local region, to provide a better overall
input-output mapping. However, the BMI problem is slightly different from the signal segmenta-
tion because the goal is to segment the joint input/desired signal space.
The training of the multiple linear models is accomplished by competitively (hard or soft
competition) updating their weights in accordance with previous approaches using the NLMS al-
gorithm. The winning model is determined by comparing the (leaky) integrated squared errors of
all competing models and selecting the model that exhibits the least integrated error for the cor-
responding input [
48
]. The leaky integrated squared error for the
i
ith model is given by
(
)
(
)
(
1
2
(
),
1
,
,
(3.57)
ε
n
=
−
µ
ε
n
−
+
µ
e
n
i
=
L
M
i
i
i
where
M
is the number of models and
µ
is the time constant of the leaky integrator. Then, the
j
th
model wins competition if
ε
j
(
n
) <
ε
i
(
n
) for all
i
≠
j
. If hard competition is employed, only the weight
