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the likelihood of an outcome, while fuzzy logic deals with the degree of ambi-
guity, that is expressed as membership value. A probability of 1 indicates that
the event is certain to occur. The membership function value, on the contrary,
measures the degree to which objects satisfy imprecisely defined properties. An
important difference w.r.t. classical set theory is that the sum of fuzzy member-
ship functions can be different from one, in contrast with the sum of mutually
independent probabilities of a system in classical set theory that must add to
one. An important advantage in fuzzy logic, is that fuzzy membership functions
can be developed using a wide range of techniques including probability den-
sity functions. Fuzzy logic provides a simple way to define a precise conclusion
based upon vague, ambiguous, imprecise, noisy, or missing input information.
The input information and the output decision can be combined by means of
a rule-based approach, even based on linguistic relationships. This approach to
control problems mimics how humans make decisions. Thanks to the ability of
fuzzy logic to deal with vague and noisy input, it has been widely used in the
olfactory signal analysis. In the following a brief overview of its main application
is provided.
-
Fuzzy
k
-NN
: Fuzzy
k
-NN, a variation of the classic
k
-NN based on a fuzzy
logic approach, assigns a fuzzy class membership to each sample and pro-
vides an output in a fuzzy form [15]. In particular, the membership value of
unlabeled sample
x
to
i
th
class is influenced by the inverse of the distances
from neighbors and their class memberships:
μ
i
(
x
)=
j
=1
μ
ij
(
)
−
2
x
−
x
j
m
−
1
(6)
j
=1
(
)
−
2
x
−
x
j
m
−
1
where
μ
ij
represents the membership of labeled sample
x
j
to the
i
th
class.
This value can be crisp or it can be calculated according to a particular
fuzzy rule: in our work we defined a Gaussian membership function with
maximum value at the average of the class and standard deviation related
to the minimum and maximum values of it. In this way, the closer the sample
j
is to the average point of class
i
, the closer its membership value
μ
ij
will
be to 1 and vice-versa. Of course other types of membership function may be
defined, e.g., according to the lung cancer stage or to the confidence assigned
to the target labels. The parameter
m
determines how heavily the distance
is weighted when calculating each neighbor contribution to the membership
value; we chose
m
= 2, but almost the same error rates have been obtained
on these data over a wide range of values of
m
.
-
Fuzzy Adaptive Resonance Map (FAM)
: FAM is a specific case of Adaptive
Resonance Theory (ART) networks, devised by Grossberg [16]. Adaptive
resonant theory networks were introduced as a theory of human cognition in
information processing, and are systems capable of adapting autonomously
in real time to changes in the environment. Their basic idea takes inspiration
from the fact that a human brain can learn new events without necessarily
forgetting events learnt in the past. They have indeed been designed to solve
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