# canonical-decomposition: Canonical Decomposition In tsdecomp: Decomposition of Time Series Data

## Description

Given the partial fraction decomposition of the pseudo-spectrum, the canonical decomposition allocates the variance of each component so that the variance of the irregular is maximised. Then, the coefficients of the numerators in the pseudo-spectrum (relationship given in `pseudo.spectrum`) are converted into the MA coefficients of the model for each component by means of `acgf2poly`.

## Usage

 ```1 2 3 4``` ```canonical.decomposition(num.trend, den.trend, num.trans, den.trans, num.seas, den.seas, quotient, optim.tol = 1e-04, ...) ## S3 method for class 'tsdecCanDec' print(x, units = c("radians", "degrees", "pi"), digits = 4, ...) ```

## Arguments

 `num.trend` numeric vector, the coefficients of the MA polynomial related to the trend component in the relationship given in `pseudo.spectrum`. `den.trend` numeric vector, the coefficients of the AR polynomial related to the trend component in the relationship given in `pseudo.spectrum`. `num.trans` numeric vector, the coefficients of the MA polynomial related to the transitory component in the relationship given in `pseudo.spectrum`. `den.trans` numeric vector, the coefficients of the AR polynomial related to the transitory component in the relationship given in `pseudo.spectrum`. `num.seas` numeric vector, the coefficients of the MA polynomial related to the seasonal component in the relationship given in `pseudo.spectrum`. `den.seas` numeric vector, the coefficients of the AR polynomial related to the seasonal component in the relationship given in `pseudo.spectrum`. `quotient` numeric vector, the quotient of the polynomial division of the polynomials in the LHS of the relationship given in `pseudo.spectrum`. (Different from zero only when the degree of the MA polynomial is equal or higher than the degree of the AR polynomial in the fitted model). `optim.tol` numeric, the convergence tolerance to be used by `optimize`. `units` character, the units in which the argument of the roots are printed. `units="pi"` prints the argument in radians as multiples of pi. `x` an object of class `tsdecCanDec` returned by `canonical.decomposition`. `digits` numeric, the number of significant digits to be used by `print`. `...` further arguments to be passed to `poly2acgf` or `print`.

## Value

An object of class `tsdecCanDec` containing the MA coefficients of the ARIMA models obtained for the unobserved components (e.g., trend, seasonal) and the variance of the corresponding disturbance terms.

## 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.

`acgf2poly`, `pseudo.spectrum`, `optimize`.