timsac (version 1.3.8)

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)

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.

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.

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 $$W(m) = CONST \times \frac{C(m)}{m+1},$$ 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