Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 42

and compute the coefficients a .
For m = 0wehave:

p
i = 1 a i R ( − i )+ σ 2 ε

R ( 0 )= (2.51)

which allows us to compute σ 2 ε
Methods for estimation of AR model parameters implemented in MATLAB Signal
Processing Toolbox [Orfanidis, 1988, Kay, 1988] are:

•

arburg Estimate AR model parameters using Burg method

•

arcov Estimate AR model parameters using covariance method

•

armcov Estimate AR model parameters using modified covariance method

•

aryule Estimate AR model parameters using Yule-Walker method

2.3.2.2.2 Choice of the AR model order When fitting the AR model to the sig-
nal we have to make an assumption that the signal can be described by the autore-
gressive process. The correct order of the process can be assessed by finding the min-
imum of the Akaike information criterion (AIC), which is a function of the model
order p :

2 p
N +

AIC ( p )= log V (2.52)

where p is the model order (the number of free parameters in the model), N - number
of signal samples used for model parameters estimation, V - residual noise variance.
The higher the model order, the less variance remains unaccounted; however an ex-
cessive number of model parameters increases the statistical uncertainty of their esti-
mates. The AIC is a sort of cost function that we seek to minimize. The first element
of that function expresses punishment for using high order model and the second
element expresses a reward for reducing the unexplained variance.
AIC is the one mostly applied in practice, but other criteria may be commonly
found in the literature, e.g., minimum description length MDL [Marple, 1987]:

p log N
N

MDL ( p )= + log V (2.53)

MDL is said to be statistically consistent, since p log N increases with N faster than
with p . The criteria for model order determination work well and give similar results
when the data are reasonably well modeled by AR process. An example of an AIC
function is shown in Figure 2.8.

2.3.2.2.3 AR model power spectrum Once the model is properly fittedtothe
signal all the signal's properties are contained in the model coefficients. Especially,
we can derive an analytical formula for AR model power spectrum. First we rewrite
the model equation (2.46):