Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 38

It is also sensitive to the initial phase of sinusoidal signals, particularly for
short digital samples resulting in a frequency bias. Nevertheless, these prob-
lems have been addressed with modifications such as a weighting sequence on
the squared forward and backward errors or the use of window methods.
An important issue in using the AR model is the selection of the optimal
order, p . If the model order is too low, we obtain a smoother representation
of the signal and also a highly smoothed spectrum. Conversely, if the model
order is too high, we potentially introduce spurious low-level peaks in the
spectrum. There have been several criteria to select the optimal AR order;
the simplest method being to use the mean squared value of the residual
error, which decreases as model order increases and is dependant on the esti-
mation method used. One can then monitor the rate of decrease and use the
order where the rate of decrease is no longer significant. Akaike (1969, 1974)
proposed two criteria, namely the final prediction criterion (FPE) and the
Akaike information criterion (AIC), which could be used to select the optimal
AR model order. The FPE selects the optimal order based on minimization
of the following performance index
FPE( p )= σ wp N + p +1
N

(2.88)

− p − 1

where σ wp is the estimated variance of the linear prediction error. The AIC
is more often used and is based on selecting the order that minimizes the
following:

AIC( p ) = log e σ wp +2 p
N

(2.89)

Other criteria include Rissanen's (1983) criterion based on the order that
minimizes the description length (MDL), where MDL is defined as

MDL( p )= N log e σ wp + p ln N (2.90)

Parzen (1974) had also proposed a criterion autoregressive transfer function
where the optimal model order was found by minimizing the function.

2.4.3.2 Moving Average Model

In the MA model, the coecients a k are set to zero and the difference equation
for the input-output relationship is given as

q

x ( n )= b k w ( n − k ) (2.91)

k =0

The noise whitening filter for the MA process is regarded as an all-pole filter.
The AIC can also be used to estimate the best MA model and has a simi-
lar form to that of Equation 2.89. This estimation method is approximately