Household Electrical Consumptions Modeling and Management Through Neural Networks and Fuzzy Logic Approaches - Complex System Modelling and Control Through Intelligent Soft Computations - page 447

The update equation is given by:

w ð n Þ ¼ w ð n 1 Þþ k ð n Þ e ð n Þ ð 18 Þ

where the gain vector k ð n Þ is expressed by:

1

T

k ð n Þ ¼ P ð n 1 Þ a ð n Þ r ð n Þþ a ð n Þ P ð n 1 Þ a ð n Þ ð 19 Þ

a ð

Þ ð ð ÞÞ

with n the gradient vector of the function f x n with respect to the parameter

ð Þ

vector w Kadirkamanathan and Niranjan ( 1993 ), Sundararajan et al. ( 2002 ),

r(n) is the variance of the measurement noise and P ð n 1 Þ is the error covariance
matrix which is updated by:

n 1

P

T

P ð n Þ ¼ I k ð n Þ a ð n Þ ð n 1 Þþ Q ð n 1 Þ ð 20 Þ

where Q is introduced to avoid that the rapid convergence of the EKF

algorithm prevents the model from adapting to future data Kadirkamanathan and
Niranjan ( 1993 ), Sundararajan et al. ( 2002 ). The z

ð n 1 Þ

× ð Þ

z matrix P n is positive

de nite symmetric and z is the number of parameters to be adjusted. When a new

hidden neuron is allocated, the dimension of P ð n Þ increases to:

P ð n 1 Þ 0

P ð n Þ ¼ ð 21 Þ

0 p 0 I z 1 z 1

where p 0 is an estimate of the uncertainty in the initial values assigned to the
parameters and z 1 is the number of new parameters introduced by adding the new
hidden neuron. As stated in Sundararajan et al. ( 2002 ), Yingwei et al. ( 1998 )to
keep the RBF network in a minimal size a pruning strategy removes those hidden
units that contribute little to the overall network output over a number of consec-
utive observations. To carry out this pruning strategy, for every observation

ð x ð n Þ; y ð n ÞÞ the hidden unit outputs are computed:

o i ð n Þ ¼k i / ð k x ð n Þ c i k Þ; i ¼ 1 ; ...; K ð 22 Þ

and normalized with respect to the highest output:

o i ð n Þ

o i ð n Þ ¼

Þg ;

i ¼ 1 ; ...; K ð 23 Þ

max f o i ð n

The hidden units for which the normalized output ( 23 ) is less than a threshold ʴ for

ʾ consecutive observations are removed and the dimensionality of all the related

matrices are adjusted to suit

the reduced network (Sundararajan et al. 2002 ;

Yingwei et al. ( 1998 ).