Bcoeff, Bcoeffa {perARMA}R Documentation

Fourier representation of covariance function

Description

The procedure Bcoeff computes the complex estimators Bk(τ)=1Tt=0T1R(t+τ,t)exp(i2πt/T) 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 τ\tau, the values of B^k(τ)\hat{B}_{k}(\tau) are computed only for k=0,1,,T/2 k= 0, 1, \ldots,\left\lfloor T/2 \right\rfloor, because for real series B^k(τ)=B^Tk(τ) \hat{B}_{k}(\tau)= \overline{\hat{B}_{T-k}(\tau)}. Also the p-values for the test Bk(τ)=0 B_{k}(\tau)=0 are returned.
The function Bcoeffa calculates the estimator of the real coefficients ak(τ) a_{k}(\tau) which satisfy R(t+τ,t)=B(t,τ)=a1(τ)+(a2k(τ)cos(2πkt/T)+a2k+1(τ)sin(2πkt/T)) 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 τ \tau and kk.

Usage

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

Arguments

x

input time series.

T_t

period length of PC-T structure.

tau

vector of lag values on which estimation of Bk(τ)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 n/2 \left\lfloor n/2\right\rfloor frequencies on either side of the Fourier frequencies 2πk/T 2\pi k/T are used.
By default parameters are fixed to printflg=1, meth=0.

Details

This procedure can be applied to the original series to obtain estimators of Bk(τ) 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 Bk(τ)2=0|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 kk 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).

Value

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

k

index number of the coefficient Bk(τ)B_k(\tau).

reB_k, imB_k/ahat_k

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

n1

degrees of freedom associated to the estimator SS1/n1SS1/n1 of the variance at frequency 2πk/T2\pi k /T.

n2

degrees of freedom associated to the "background" variance estimation SS2/n2SS2/n2 based on frequencies in the neighborhood of the frequency 2πk/T2\pi k /T.

Fratio

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

pv

p-values for test of Bk(τ)2=0\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.

Author(s)

Harry Hurd

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')

[Package perARMA version 1.7 Index]