Fit Autoregression Using Burg's Algorithm
DESCRIPTION:
Fits a model of the form x(t) = a(1)x(t-1) + ... + a(p)x(t-p) + e(t) using Burg's algorithm.
USAGE: 
ar.burg(x, aic=T, order.max=<<see below>>)
REQUIRED ARGUMENTS:
x:
a univariate or multivariate time series, or a vector, or a matrix with columns representing univariate components of a multivariate time series. Missing values are allowed only at the beginning or end of series.
OPTIONAL ARGUMENTS:
aic:
logical flag: if TRUE, use the Akaike information criterion to choose the best order not greater than order.max. If FALSE, order.max will be the order of the fitted model.
order.max:
the maximum order of autoregression to fit to the time series. The default value is 10 * log10(n.used/ncol(x)).
VALUE:
a list with the following components:
order:
the order of the autoregression fitted. If aic=TRUE, then this is the order less than or equal to order.max which minimizes the AIC, otherwise, it is order.max.
ar:
the autoregressive coefficients. These are in an order by "nser" by "nser" array, where "nser" is the number of univariate components of x (1 or ncol(x)). If order is 0, ar will have dimensions 1 by "nser" by "nser" and will be filled with zeros. The first level of the first dimension corresponds to one observation back in time, the second level corresponds to two observations back, etc. The second dimension corresponds to the predicted series, and the third corresponds to the predicting series.
var.pred:
the estimated prediction variance ("nser" by "nser" matrix) of the process with AR coefficients ar.
aic:
vector of the values of Akaike's information criterion for orders 0 through order.max. These have the minimum value subtracted from all of them so the minimum is always zero.
n.used:
the length of the (nonmissing part of the) input time series.
order.max:
the input or default value of order.max.
resid:
a time series representing the estimate of e(t) for the model above. The estimates contained in ar are used in the forward direction on the series with mean(s) removed.
method:
the string "burg".
series:
the name used for the input x.
DETAILS:
The a(i) coefficients i=1,...,order.max are estimated by minimizing a sum of backward and forward prediction squared errors (Burg's algorithm). The output may be used in spec.ar to estimate the spectrum of the process.

The estimation is performed using the sample mean of each univariate series as the estimate of the mean. Remember that the coefficients in ar are for the series with the mean(s) removed.

REFERENCE:
Burg, J. P. (1967). Maximum Entropy Spectral Analysis. Paper presented at the 37th Annual International S.E.G. Meeting, Oklahoma City, Oklahoma.
BUGS:
Estimation for multivariate series appears to be incorrect (it gives slightly different answers when the series are permuted).
SEE ALSO:
acf.plot , ar , ar.yw , arima.mle , spec.ar .
EXAMPLES: 
a <- ar.burg(log(lynx))
tsplot(a$aic)
# Fit an AR(11) to this time series
llynx.ar.fit <- ar.burg(log(lynx), aic=F, order=11)