Bcoeff, Bcoeffa {perARMA} | R Documentation |
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_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 |
missval |
notation for missing values. |
datastr |
string name of data for printing. |
... |
other arguments: |
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).
Value
procedures (for positive printflg
parameter) print a table containing the following columns:
k |
index number of the coefficient |
reB_k , imB_k/ahat_k |
real and imaginary parts of estimated coefficient |
n1 |
degrees of freedom associated to the estimator
|
n2 |
degrees of freedom associated to the
"background" variance estimation |
Fratio |
the statistic |
pv |
p-values for test of |
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')