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Assume that a very large number of measurements is available; if the null
hypothesis is valid, the numerator of Z is very small since the minimization
of the cost function gives values equal to zero to the “useless” parameters of
the complete model, hence g Q and g Q−q are very close; if the null hypothesis
is not valid, the two models cannot be very similar, even if the amount of
data is large, since the submodel does not have the appropriate complexity
for accounting for the data. That explains why the realization of Z must be
small if the null hypothesis is valid.
Thus, Fisher's test consists in choosing a risk α , and computing, from the
Fisher distribution, the value z α such that Pr( z<z α )= α . Then the quantity
y p
g Q−q w Q−q
mc
y p
g Q w mc
2
2
z = N
Q
1
w mc
2
q
y p
g Q
(realization of Z with the available data) is computed, and the null hypothesis
is accepted if and only if z<z α .
2.10.4.2 Computation of the Cumulative Distribution Function of
the Rank of the Probe Feature
In the present section, we discuss the computation of the probability for the
probe feature to have a higher rank (rank 1 being the highest rank), at a given
step of the selection procedure, than one of the features selected during the
previous steps. The complete computation can be found in [Stoppiglia 1998].
We denote by H k− 1 the probability for the probe vector to be ranked higher
than one of the k
1 features selected at previous steps. The probability for the
probe feature to have a lower rank than the first k
1 features is therefore 1
H k− 1 . The probability for the probe feature to be ranked higher than the k
1
first features but lower than the k th feature is thus P N−k (cos 2 ( θ k ))[1
H k− 1 ],
where P N−k (cos 2 ( θ k )) is the probability for the angle of the projection of the
feature k under consideration, onto the null subspace of the previously selected
features, and the projection of the process output on the same subspace, to
be smaller than θ k . Therefore, the probability H k for the probe feature to
be more significant than one of the k selected features is given by: H k
=
H k− 1 + P N−k (cos 2 θ k )(1
H k− 1 ). Thus, H k can be computed recursively, with
H 0 = 0. That requires the computation of P N−k (cos 2 θ k ), which is given by
the following relations:
P n (cos 2 θ )=1
fr n (cos 2 θ ) (n positive integer), with
for n even: fr n ( x )=2 [sin 1 x + x (1 −x ) P ( n/ 2) 2 ( x )], where, for
n
6 ,P ( n/ 2) 2 ( x )=1+ ( n/ 2) 2
k =1
[2 k ( k !) / ((2 k + 1)!!)(1
x ) k ]; for n =
4: P 0 =1;for n =2: P 1 =0;
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