timsac (version 1.3.8-4)

mulmar: Multivariate Case of Minimum AIC Method of AR Model Fitting

Description

Fit a multivariate autoregressive model by the minimum AIC procedure. Only the possibilities of zero coefficients at the beginning and end of the model are considered. The least squares estimates of the parameters are obtained by the householder transformation.

Usage

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

Value

mean

mean.

var

variance.

v

innovation variance.

aic

AIC.

aicmin

minimum AIC.

daic

AIC-aicmin.

order.maice

order of minimum AIC.

v.maice

MAICE innovation variance.

np

number of parameters.

jnd

specification of \(i\)-th regressor.

subregcoef

subset regression coefficients.

rvar

residual variance.

aicf

final estimate of AIC (\(=n\log\)(rvar)\(+2\)np).

respns

instantaneous response.

regcoef

regression coefficients matrix.

matv

innovation variance matrix.

morder

order of the MAICE model.

arcoef

AR coefficients. arcoef[i,j,k] shows the value of \(i\)-th row, \(j\)-th column, \(k-\)th order.

aicsum

the sum of aicf.

Arguments

y

a multivariate time series.

max.order

upper limit of the order of AR model, less than or equal to \(n/2d\) where \(n\) is the length and \(d\) is the dimension of the time series y. Default is \(min(2 \sqrt{n}, n/2d)\).

plot

logical. If TRUE, daic[[1]]\(, \ldots , \)daic[[d]] are plotted.

Details

Multivariate autoregressive model is defined by $$y(t) = A(1)y(t-1) + A(2)y(t-2) +\ldots+ A(p)y(t-p) + u(t),$$ where \(p\) is order of the model and \(u(t)\) is Gaussian white noise with mean \(0\) and variance matrix matv. AIC is defined by $$AIC = n \log(det(v)) + 2k,$$ 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.

References

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

# Example 1
data(Powerplant)
z <- mulmar(Powerplant, max.order = 10)
z$arcoef

# Example 2
ar <- array(0, dim = c(3,3,2))
ar[, , 1] <- matrix(c(0.4,  0,   0.3,
                      0.2, -0.1, -0.5,
                      0.3,  0.1, 0), nrow = 3, ncol = 3, byrow = TRUE)
ar[, , 2] <- matrix(c(0,  -0.3,  0.5,
                      0.7, -0.4,  1,
                      0,   -0.5,  0.3), nrow = 3, ncol = 3,byrow = TRUE)
x <- matrix(rnorm(200*3), nrow = 200, ncol = 3)
y <- mfilter(x, ar, "recursive")
z <- mulmar(y, max.order = 10)
z$arcoef