[a,v,k] = arburg(x,poles,criterion)

Calculate coefficients of an autoregressive (AR) model of complex data
"x" using the whitening lattice-filter method of Burg (1968).  The inverse
of the model is a moving-average filter which reduces "x" to white noise.
The power spectrum of the AR model is an estimate of the maximum
entropy power spectrum of the data.  The function "ar_psd" calculates the
power spectrum of the AR model.

ARGUMENTS:
x         %% [vector] sampled data

poles     %% [integer scalar] number of poles in the AR model or
limit to the number of poles if a
valid "stop_crit" is provided.

criterion %% [optional string arg]  model-selection criterion.  Limits
the number of poles so that spurious poles are not
added when the whitened data has no more information
in it (see Kay & Marple, 1981). Recognised values are
'AKICc' -- approximate corrected Kullback information
criterion (recommended),
'KIC'  -- Kullback information criterion
'AICc' -- corrected Akaike information criterion
'AIC'  -- Akaike information criterion
'FPE'  -- final prediction error" criterion
The default is to NOT use a model-selection criterion

RETURNED VALUES:
a         %% [polynomial/vector] list of (P+1) autoregression coeffic-
ients; for data input x(n) and  white noise e(n),
the model is
P+1
x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
k=1

v         %% [real scalar] mean square of residual noise from the
whitening operation of the Burg lattice filter.

k         %% [column vector] reflection coefficients defining the
lattice-filter embodiment of the model

HINTS:
(1) arburg does not remove the mean from the data.  You should remove
the mean from the data if you want a power spectrum.  A non-zero mean
can produce large errors in a power-spectrum estimate.  See
"help detrend".
(2) If you don't know what the value of "poles" should be, choose the
largest (reasonable) value you could want and use the recommended
value, criterion='AKICc', so that arburg can find it.
E.g. arburg(x,64,'AKICc')
The AKICc has the least bias and best resolution of the available
model-selection criteria.
(3) arburg runs in octave and matlab, does not depend on octave forge
or signal-processing-toolbox functions.
(4) Autoregressive and moving-average filters are stored as polynomials
which, in matlab, are row vectors.

NOTE ON SELECTION CRITERION
AIC, AICc, KIC and AKICc are based on information theory.  They  attempt
to balance the complexity (or length) of the model against how well the
model fits the data.  AIC and KIC are biassed estimates of the asymmetric
and the symmetric Kullback-Leibler divergence respectively.  AICc and
AKICc attempt to correct the bias. See reference [4].


REFERENCES
[1] John Parker Burg (1968)
"A new analysis technique for time series data",
NATO advanced study Institute on Signal Processing with Emphasis on
Underwater Acoustics, Enschede, Netherlands, Aug. 12-23, 1968.

[2] Steven M. Kay and Stanley Lawrence Marple Jr.:
"Spectrum analysis -- a modern perspective",
Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981

[3] William H. Press and Saul A. Teukolsky and William T. Vetterling and
Brian P. Flannery
"Numerical recipes in C, The art of scientific computing", 2nd edition,
Cambridge University Press, 2002 --- Section 13.7.

[4] Abd-Krim Seghouane and Maiza Bekara
"A small sample model selection criterion based on Kullback's symmetric
divergence", IEEE Transactions on Signal Processing,
Vol. 52(12), pp 3314-3323, Dec. 2004