perARMA (version 1.6)

Bcoeff, Bcoeffa: Fourier representation of covariance function

Description

The procedure Bcoeff computes the complex estimators $ B_{k}(\tau) = \frac{1}{T} \sum_{t=0}^{T-1}R(t+\tau,t)\exp(-i 2 \pi t /T)$ as Fourier coefficients in covariance function representation. The procedure first computes the periodic mean (with missing values ignored) and subtracts it from the series. For each specified lag $\tau$, the values of $\hat{B}_{k}(\tau)$ are computed only for $ k= 0, 1, \ldots,\left\lfloor T/2 \right\rfloor$, because for real series $ \hat{B}_{k}(\tau)= \overline{\hat{B}_{T-k}(\tau)}$. Also the p-values for the test $ B_{k}(\tau)=0$ are returned. The function Bcoeffa calculates the estimator of the real coefficients $ a_{k}(\tau)$ which satisfy $ R(t+\tau,t) = B(t,\tau) = a_1(\tau) + \sum (a_{2k}(\tau) \cos(2 \pi k t/ T)+a_{2k+1}(\tau) \sin(2 \pi k t/ T)) $, for all required lags $ \tau$ and $k$.

Usage

Bcoeff(x, T, tau, missval, datastr,...)
Bcoeffa(x, T, tau, missval, datastr,...)

Arguments

x

input time series.

T

period length of PC-T structure.

tau

vector of lag values on which estimation of $B_k(\tau)$ is performed.

missval

notation for missing values.

datastr

string name of data for printing.

...

other arguments: printflg should be a positive parameter to print, meth is a parameter connected to the amount of frequencies used in estimation, if meth=0 all possible frequencies are used in estimation else if meth > 0 then $ \left\lfloor n/2\right\rfloor$ frequencies on either side of the Fourier frequencies $ 2\pi k/T$ are used. By default parameters are fixed to printflg=1, meth=0.

Value

procedures (for positive printflg parameter) print a table containing the following columns:

k

index number of the coefficient $B_k(\tau)$.

reB_k,imB_k/ahat_k

real and imaginary parts of estimated coefficient $B_k(\tau)$ (Bcoeff procedure) / real coefficients in representation of coefficient $B_k(\tau)$ (Bcoeffa procedure).

n1

degrees of freedom associated to the estimator $SS1/n1$ of the variance at frequency $2\pi k /T$.

n2

degrees of freedom associated to the "background" variance estimation $SS2/n2$ based on frequencies in the neighborhood of the frequency $2\pi k /T$.

Fratio

the statistic $(SS1/n1)/(SS2/n2)$, which under the null hypothesis has $F(n1,n2)$ distribution.

pv

p-values for test of $\left|B_k(\tau)\right|^2=0$, based on F distribution.

If printflg is set to be equal to 0, above values are returned just as matrices.

Details

This procedure can be applied to the original series to obtain estimators of $ B_{k}(\tau)$ in covariance function representation or to the normalized series (series after periodic mean removal and division by standard deviations) to obtain correlation coefficients. The p-values for test of $|B_k(\tau)|^2=0$ are based on the ratio of magnitude squares of amplitudes of a high resolution Fourier decompositions. Magnitudes for the frequency corresponding to index $k$ are compared to the magnitudes of neighboring frequencies (via the F distribution) (see Hurd, H. L., Miamee, A. G., 2007, Periodically Correlated Random Sequences, pp. 272-282, 288-292).

References

Dehay, D., Hurd, H. L., (1994), Representation and Estimation for Periodically and Almost Periodically Correlated Random Processes in W. A. Gardner (ed.), Cyclostationarity in Communications and Signal Processing, IEEE Press.

Hurd, H. L., Miamee, A. G., (2007), Periodically Correlated Random Sequences: Spectral Theory and Practice, Wiley InterScience.

Examples

data(volumes)
Bcoeff(volumes,24,seq(0,12),NaN,'volumes')
Bcoeffa(volumes,24,seq(0,12),NaN,'volumes')