Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 150

6.2.4. Order selection criteria

Choosing a model system is a problem which is as important as the choice of the
model itself. Selecting too small an order means smoothing the obtained spectrum,
while choosing too large an order introduces secondary spurious peaks.

There is a large number of order selection criteria which are for most cases based
on the statistic properties of the signal (maximum likelihood estimation: MLE).
Others, simpler and less efficient are based on the comparison of the eigenvalues of
the correlation matrix to some threshold correlation matrix [KON 88].

A large number of order selection criteria use the prediction error power
decrease when the order increases. When the theoretic order is reached, this power
remains constant. However, a criterion based only on the prediction error power
shape does not make it possible to take the estimated spectrum variance increase
into account when the order is overestimated. That is why the criteria integrate these
two phenomena. One of the first criteria proposed by Akaike [AKA 70] was the
FPE (Final Prediction Error): the estimated error corresponds to the value that
minimizes:

Nk +

()

ˆ k

FPE k =

Nk ρ

[6.32]

−

where:

k

()

+ ∑

()

ργ ˆ = ˆ 0 a ˆ γ ˆ l [6.33]

k xx l xx

l = 1

is the power of the prediction error that decreases with k while the term Nk
Nk

+

−
increases with k (to take the estimated spectrum variance augmentation into account
when k increases). The AR parameters are estimated through Yule-Walker equations
with the biased estimator of the correlation. The most well known criterion proposed
by Akaike is the AIC (Akaike Information Criterion) [AKA 74]:

() ()

ˆ

k

[6.34]

AIC k = N 1n ρ + 2

k

This criterion is more general than FPE and it can be applied by determining the
order of an MA part of an ARMA model. Asymptotically (

)

→ N FPE and AIC
are equivalent, but for a small number of samples AIC is better. It was proved that
AIC is inconsistent and it tends to overestimate the order [KAS 80]. [RIS 83]
proposed to modify AIC by replacing the term 2k by a term, which increases more