timsac (version 1.3.0)

unibar: Univariate Bayesian Method of AR Model Fitting

Description

This program fits an autoregressive model by a Bayesian procedure. The least squares estimates of the parameters are obtained by the householder transformation.

Usage

unibar(y, ar.order=NULL, plot=TRUE)

Arguments

y

a univariate time series.

ar.order

order of the AR model. Default is $2 \sqrt{n}$, where $n$ is the length of the time series y.

plot

logical. If TRUE (default) daic, pacoef and pspec are plotted.

Value

• mean
  mean.
• var
  variance.
• v
  innovation variance.
• aic
  AIC.
• aicmin
  minimum AIC.
• daic
  AIC-aicmin.
• order.maice
  order of minimum AIC.
• v.maice
  innovation variance attained at m=order.maice.
• pacoef
  partial autocorrelation coefficients (least squares estimate).
• bweight
  Bayesian Weight.
• integra.bweight
  integrated Bayesian weights.
• v.bay
  innovation variance of Bayesian model.
• aic.bay
  AIC of Bayesian model.
• np
  equivalent number of parameters.
• pacoef.bay
  partial autocorrelation coefficients of Bayesian model.
• arcoef
  AR coefficients of Bayesian model.
• pspec
  power spectrum.

Details

The AR model is given by $$y(t) = a(1)y(t-1) + \ldots + a(p)y(t-p) + u(t),$$ where $p$ is AR order and $u(t)$ is Gaussian white noise with mean $0$ and variance $v(p)$. The basic statistic AIC is defined by $$AIC = n\log(det(v)) + 2m,$$ where $n$ is the length of data, $v$ is the estimate of innovation variance, and $m$ is the order of the model. Bayesian weight of the $m$-th order model is defined by $$W(m) = CONST \times C(m) / (m+1),$$ where $CONST$ is the normalizing constant and $C(m)=\exp(-0.5AIC(m))$. The equivalent number of free parameter for the Bayesian model is defined by $$ek = D(1)^2 + \ldots + D(k)^2 +1,$$ where $D(j)$ is defined by $D(j)=W(j) + \ldots + W(k)$. $m$ in the definition of AIC is replaced by $ek$ to be define an equivalent AIC for a Bayesian 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.Kitagawa 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(Canadianlynx)
  z <- unibar(Canadianlynx, ar.order=20)
  z$arcoef