Procedure loglikef computes the logarithm of likelihood function from the PARMA
sequence x for matrices
phi (of size \(T \times p\)) and theta (of size \(T \times (q+1)\))
inputed in their Fourier representation as a and b, respectively.
loglikef(ab, x, conpars)negative value of the logarithm of likelihood function: y.
matrix [a,b] taken as a vector, where
a is Fourier representation of phi
(use phi=ab2phth(a) to recover phi),
b is Fourier representation of theta
(use del=ab2phth(b[,1]) to recover del and
theta=ab2phth(b[,2:q+1]) to recover theta).
Vector ab contains only non-zero coefficients form a and b.
input time series.
vector of parameters [T,p,q,naf,nbf,del_mask,iaf,ibf,stype]:
T_t period of PC-T structure,
p, q maximum PAR and PMA order, respectively,
naf, nbf total active coefficients in a and b, respectively,
del_mask vector of length T (it will be used in the future, so far the user should set del_mask=matrix(1,T,1)),
iaf, ibf linear indexes of active coefficients in a and b, respectively,
stype numeric parameter connected with covariance matrix computation, so far should be equal to 0 to use procedure
R_w_ma (see R_w_ma description). In the future also other values of stype will be available for
full covariance matrix computation.
Harry Hurd
This method of computation of logarithm of likelihood function makes use of the representation of the periodically varying parameters by Fourier series.
This alternative parametrization of PARMA system, introduced by
Jones and Breslford, can sometimes substantially reduce the number of parameters required to represent PARMA system. Mapping between phi and theta coefficients
and a and b coefficients is one-to-one, so first
logarithm of likelihood is computed for transformed coefficients and then these coefficients are transformed to phi and theta.
Fourier series parametrization permits us
to reduce the total number of parameters by constraining some frequencies to have zero amplitude. Then the code includes
the computation of covariance matrix from the parameters phi, del, theta.
Since the inverse of the computed covariance is needed for computing the likelihood, and it is sometimes ill conditioned (or even singular), the condition is improved by removing rows and columns corresponding to very small eigenvalues. This corresponds to removing input data that is highly linearly dependent on the remaining input data. The procedure contains a threshold ZTHRS (which current value is 10*eps) that governs the discarding of rows and column corresponding to small eigenvalues (these are determined by a Cholesky decomposition). Any eigenvalue smaller than the threshold has its row and column deleted from the matrix. Then the
inverse and the likelihood are computed from the reduced rank covariance matrix.
Box, G. E. P., Jenkins, G. M., Reinsel, G. (1994), Time Series Analysis, 3rd Ed., Prentice-Hall, Englewood Cliffs, NJ.
Brockwell, P. J., Davis, R. A., (1991), Time Series: Theory and Methods, 2nd Ed., Springer: New York.
Jones, R., Brelsford, W., (1967), Time series with periodic structure, Biometrika 54, 403-408.
Makagon, A., (1999), Theoretical prediction of periodically correlated sequences, Probability and Mathematical Statistics 19 (2), 287-322.
Sakai, H., (1989), On the spectral density matrix of a periodic ARMA process, J. Time Series Analysis, v. 12, no. 2, pp. 73-82.
Vecchia, A., (1985), Maximum Likelihood Estimation for Periodic Autoregressive Moving Average Models, Technometrics, v. 27, pp.375-384.
R_w_ma, parmaresid, parmaf