NONLINEAR SEQUENTIAL STATE ESTIMATION FOR SOLVING PATTERN CLASSIFICATION PROBLEMS - Adaptive Signal Processing: Next Generation Solutions - page 341

1. System (state) model , which is described by the first-order autoregressive
equation

w nþ 1 ¼ w n þv n : (6 : 4)

The dynamic noise v n is white Gaussian noise of zero mean and covariance
matrix Q n which is purposely included in the system model to anneal the
supervised training of the MLP over time. In the early stages of the training
session, Q n is large in order to encourage the supervised-learning algorithm
to escape local minima,

then it is gradually reduced to some finite but

small value.
2. Measurement model , which is described by the equation

d n ¼ b ( w n , u n ) þv n (6 : 5)

where the new entities are defined as follows:
† d n is the vector denoting observables
† u n is the vector denoting the input signal applied to the network
† v n is the vector denoting the measurement noise process of zero mean and

diagonal covariance matrix R n . The source of this noise is attributed to the
way in which d n is actually obtained.

The vector-valued measurement function b (.,.) in (6.5) accounts for the overall non-
linearity of the MLP from the input to the output layer; it is the only source of non-
linearity in the state-space model of the MLP. Here, the notion of state refers to an
externally adjustable state, which manifests itself in adjustments applied applied to
the MLP's weights through supervised training—hence the inclusion of the weight
vector w n in the state-space model described by both (6.4) and (6.5).

6.4 THE EXTENDED KALMAN FILTER

Given the training sample { u n , d n } n¼ 1 , the issue of interest is how to undertake the
supervised training of the MLP by means of a sequential state estimator. Since
the MLP is nonlinear by virtue of the nonlinear measurement model of (6.5),
the sequential state estimator would have to be correspondingly nonlinear. With this
requirement in mind, we begin the discussion by considering how the extended
Kalman filter (EKF) can be used to fulfil this rule.
For the purpose of our present discussion, the relevant equations of the EKF algor-
ithm summarized in Table 6.1 are the following two, using the terminology of the
state-space model of (6.4) and (6.5).

1. The innovations process , defined by

a n ¼ d n b ( w njn 1 , u n )

(6 : 6)

where the desired response d n plays the role of the observable for the EKF.