Training the Classifiers - Design and Analysis of Learning Classifier Systems - page 82

D X from which
w can be recovered by w = Λ − 1 ( Λ w ). Even though the initial Λ might be
singular and therefore cannot be inverted to calculate w , it can still be updated
by (5.54) until it is non-singular and can be inverted. This allows for using the
non-informative prior Λ 0 = 0 that cannot be used when applying the covariance
form of the Kalman filter.

Thus, w is updated indirectly through the vector ( Λ w ) ∈ R

Minimum Model Error Philosophy

For deriving the Kalman filter update equations we have assumed knowledge
of the measurement noise variances τ − 1

1

,τ − 1
2

. In our application of the
Kalman filter that is not the case, and so we have find a method that allows us
to estimate the variances at the same time as the system state.
Assuming a different measurement noise variance for each observation makes
estimating these prohibitive, as it would require estimating more parameters
than there are observations. To reduce the degrees of freedom of the model it
will be assumed that τ is constant for all observations, that is τ 1 = τ 2 =

{ ,... }

= τ .
In addition, we adopt the Minimum Model Error (MME) philosophy [170] that
aims at finding the model parameters that minimises the model error, which
is determined by the noise variance τ . The MME is based on the Covariance
Constraint condition, which states that the observation-minus-estimate error
variance must match the observation-minus-truth error variance, that is

···

w T x n ) 2 ( m ( x n ) τ ) − 1 . ( y n − ≈ (5.58)

Given that constraint and the assumption of not having any process noise, the
model error for the n th observation is given by weighting the left-hand side of
(5.58) by the inverted right-hand side, which, for N observations results in

m ( x n ) w T x n − y n 2

N

τ . (5.59)

n =1

Minimising the above is independent of τ and therefore equivalent to (5.5). Thus,
assuming a constant measurement noise variance has led us back to minimising
the error that we originally intended to minimise.

Relation to Recursive Least Squares

The Kalman filter update equations are very similar but not quite the same
as the RLS update equations. Maybe the most obvious match is the inverse
covariance update (5.54) of the Kalman filter, and (5.31) of the RLS algorithm,
only differing by the additional term τ N +1 in (5.54). Similarly, (5.56) and (5.34)
differ by the same term.
In fact, if all Λ in the RLS update equations are substituted by τ − 1 Λ ,in
addition to assuming τ 1 = τ 2 = = τ for the Kalman filter, these equations

become equivalent. More specifically, the covariance form of the Kalman filter

···