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=
k
10), where the hyperparameters space is explored according to a grid search
procedure (Anguita et al.
2009
).
5.2.2 Generalization to Multiclass SVM with Probability
Estimates
In SVMs, the output of the FFP is either positive or negative. Its sign represents if
the new sample is either classified as a given class or not. But the magnitude of its
value, just as it is, does not represent a quantity directly comparable against other
SVMs. Ideally, for a given sample in a multiclass problem, only one of the binary
classifiers should be positive and the rest negative but this is not always the case.
Therefore, output normalization methods are required to have comparable SVMs.
We opted to compute probability estimates
p
c
(
1] which represent how
probable it is for a new sample pattern to be classified as a given class
c
.Thisis
appropriate when used in OVA classification as each binary classifier is associated
with a particular class (or, in this case, human activity). For a given number of classes
m
and a test sample
x
, the probability output of each SVM (
p
c
(
x
)
∈
[0
,
)
is compared against the others to find the class
c
∗
with the Maximum A Posteriori
Probability (MAP). Assuming that all the classes have the same a priori distribution
then:
c
∗
=
x
)
∀
c
∈ {
1
, ...,
m
}
.
The probability estimation is implemented using the approach presented in (Platt
1999
) in which the output of the SVM FFP (
f
argmax
c
p
c
(
x
)
(
x
)
) obtained from the training set is
fit to a sigmoid function of the following form:
1
p
(
x
)
=
e
(
f
(
x
)
+
)
,
(5.7)
1
+
where
and
are function parameters whose optimal values can be found using the
f
(
x
i
)
values and the targets of the training samples (modified as
t
i
=
(
y
i
+
1
) /
2)
in the following error minimization function:
n
argmin
,
−
t
i
log
(
p
(
x
i
))
+
(
1
−
t
i
)
log
(
1
−
p
(
x
i
)),
(5.8)
i
=
1
Considering the fixed-point arithmetic limitation, the sigmoid function, which
works also with real numbers, cannot be directly used for estimating
p
(
x
)
.Thisis
solved by means of Look-Up-Tables (LUTs) which link
f
through a
simple indexing operation. For this, a fixed number of bits must be defined in order
to map the probability estimates
p
(
x
)
with
p
(
x
)
without the need of floating-point arithmetic.
The completeMC-HF-SVMprocess for the recognition of 6 activities is illustrated
in Fig.
5.2
. It depicts the binary classifiers for each activity along with their associated
LUT. The notation allows to understand how an input sample
x
is processed until
the most likely activity
c
∗
is selected.
(
x
)
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