| pseudo.spectrum {tsdecomp} | R Documentation |
Pseudo-Spectrum of an ARIMA Model
Description
Compute the polynomials in the numerators of a partial fraction
decomposition of the pseudo-spectrum in an ARIMA model.
The polynomials are in terms of the variable 2\cos\omega,
with \omega\in [0, 2\pi].
Usage
pseudo.spectrum(mod, ar)
## S3 method for class 'tsdecPSP'
print(x, ...)
Arguments
mod |
an object of class |
ar |
an object of class |
x |
an object of class |
... |
further arguments to be passed to |
Details
The coefficients of the ARIMA models for each component (e.g., trend, seasonal) are obtained from the following relationship.
\sigma^2\frac{\theta(B)\theta(F)}{\phi(B)\phi(F)} =
\sigma^2_a\frac{\theta_T(B)\theta_T(F)}{\phi_T(B)\phi_T(F)} +
\sigma^2_b\frac{\theta_S(B)\theta_S(F)}{\phi_S(B)\phi_S(F)} + \sigma^2_e \,,
where B is the backshift operator and F=B^{-1} is the
forward operator. Each term in the right-hand-side is related to the ARIMA models
of each one of the unobserved components.
pseudo.spectrum computes the symmetric polynomials of the type
\varphi(B)\varphi(F) for the polynomials in
the left-hand-side LHS (based on the fitted model) and for the polynomials
in the denominators of the right-hand-side RHS
(based on the allocation of roots of the fitted AR polynomial,
roots.allocation).
Then coefficients in the numerators of the RHS are obtained
by means of partial.fraction .To do so
the terms in the RHS are multiplied by the denominator in the LHS;
then, the coefficients of the numerators in the RHS are obtained
by equating the coefficients of the same order on both sides of
the relationship (the orders of the unknown polynomials are set
to one degree lower than those polynomials of the corresponding denominator).
Value
A list of class tsdecPSP containing:
the quotient of the polynomial division (if the degree of the
numerator in the LHS is equal or higher than the degree of the denominator);
the coefficients of total polynomials (numerator and denominator in the LHS)
and the denominators in the RHS.
References
Burman, J. P. (1980) ‘Seasonal Adjustment by Signal Extraction’. Journal of the Royal Statistical Society. Series A (General), 143(3), pp. 321-337. doi: 10.2307/2982132
Hillmer, S. C. and Tiao, G. C. (1982) ‘An ARIMA-Model-Based Approach to Seasonal Adjustment’. Journal of the American Statistical Association, 77(377), pp. 63-70. doi: 10.1080/01621459.1982.10477767
See Also
arima, partial.fraction,
roots.allocation.