mulbar {timsac}R Documentation

Multivariate Bayesian Method of AR Model Fitting

Description

Determine multivariate autoregressive models by a Bayesian procedure. The basic least squares estimates of the parameters are obtained by the householder transformation.

Usage

  mulbar(y, max.order=NULL, plot=FALSE)

Arguments

y a multivariate time series.
max.order upper limit of the order of AR model. Default is 2*sqrt(n), where n is the length of the time series y.
plot logical. If TRUE daic is plotted.

Details

The statistic AIC is defined by

AIC = nlog(det(v)) + 2k,

where n is the number of data, v is the estimate of innovation variance matrix, det is the determinant and k is the number of free parameters.

Bayesian weight of the m-th order model is defined by

W(n) = const * C(m) / (m+1),

where const is the normalizing constant and C(m)=exp(-0.5AIC(m)).

The Bayesian estimates of partial autoregression coefficient matrices of forward and backward models are obtained by (m = 1,...,lag)

G(m) = G(m) D(m),

H(m) = H(m) D(m),

where the original G(m) and H(m) are the (conditional) maximum likelihood estimates of the highest order coefficient matrices of forward and backward AR models of order m and D(m) is defined by

D(m) = W(m) + ... + W(lag).

The equivalent number of parameters for the Bayesian model is defined by

ek = (D(1)^2 + ... + D(lag)^2) id + id (id+1)/2

where id denotes dimension of the process.

Value

mean mean.
var variance.
v innovation variance.
aic AIC(m), (m = 0,...,max.order).
aicmin minimum AIC.
daic AIC(m)-aicmin (m = 0,...,max.order).
order.maice order of minimum AIC.
v.maice innovation variance attained at m=order.maice.
bweight Bayesian weights.
integra.bweight integrated Bayesian Weights.
arcoef.for AR coefficients (forward model). arcoef.for[i,j,k] shows the value of i-th row, j-th column, k-th order.
arcoef.back AR coefficients (backward model). arcoef.back[i,j,k] shows the value of i-th row, j-th column, k-th order.
pacoef.for partial autoregression coefficients (forward model).
pacoef.back partial autoregression coefficients (backward model).
v.bay innovation variance of the Bayesian model.
aic.bay equivalent AIC of the Bayesian (forward) model.

References

H.Akaike (1978) A Bayesian Extension of The Minimum AIC Procedure of Autoregressive Model Fitting. Research Memo. NO.126, The Institute of Statistical Mathematics.

G.Kiagawa and H.Akaike (1978) A Procedure for The Modeling of Non-stationary Time Series. Ann. Inst. Statist. Math., 30, B, 351–363.

H.Akaike, G.Kitagawa, E.Arahata and F.Tada (1979) Computer Science Monograph, No.11, Timsac78. The Institute of Statistical Mathematics.

Examples

  data(Powerplant)
  z <- mulbar(Powerplant, max.order=10)
  z$pacoef.for
  z$pacoef.back

[Package timsac version 1.2.1 Index]