Function peracf, given an input time series and a specified period T, computes the periodic correlation coefficients for which
\( \rho(t+\tau,t)=\rho(t,\tau)\), where \(t = 1,\ldots, T\)
are seasons and \(\tau\) is lag. For each
possible pair of \( t\) and \(\tau \) confidence limits for
\(\rho(t,\tau)\) are also computed using Fisher
transformation. Procedure peracf
provides also two important tests: \( \rho(t+\tau,t) \equiv \rho(\tau)\) and \(\rho(t+\tau,t) \equiv 0\).
peracf(x, T_t, tau, missval, datastr,...)tables of values for each specified lag \(\tau\):
estimated correlation coefficients.
lower bands of confidence intervals.
upper bands of confidence intervals.
number of samples used in each estimation.
Above values are also returned as matrices.
input time series, at the begining missing values
in x will be treat as zeros and periodic mean will be computed,
then missing values will be replaced by periodic mean.
period of PC-T structure.
vector of lag values for which estimation is made.
notation for missing values (denoted as NaN).
string name of data for printing.
other arguments, that are connected with the plots:
prttaus, plottaus, cialpha, typeci, typerho, pchci, pchrho, colci, colrho, where
prttaus is a set of lags for which correlation coefficients are printed; it is a subset of tau,
plottaus is a set of lags for plotting the correlation coefficients
(one plot per lag); it is a subset of tau,
cialpha threshold for confidence interval,
typeci/ typerho, pchci/ pchrho, colci/colrho
define the type, plot character and colors of confidence intervals/periodic correlation values.
By default these parameters are fixed to prttaus = seq(1,T/2), plottaus = seq(1,T/2), cialpha = 0.05, typeci = "b", typerho = "b", pchci = 10, pchrho = 15, colci = "blue", colrho = "red".
Harry Hurd
Function peracf uses three separate procedures:
rhoci() returns the upper and lower bands defining a \(1 - \alpha\) confidence interval for the true values of
\( \rho(t, \tau)\),
rho.zero.test() tests whether the estimated correlation coefficients are equal to zeros, \( \rho(t+\tau,t) \equiv 0\).
rho.equal.test() tests whether the estimated correlation coefficients are equal to each other for all seasons in the period,
\( \rho(t+\tau,t) \equiv \rho(\tau)\).
In the test \( \rho(t+\tau,t) \equiv \rho(\tau)\), rejection for some \(\tau
> 0\) indicates
that series is properly PC and is not just an amplitude modulated stationary
sequence. In other words, there exists a nonzero
lag \(\tau\) for which \( \rho(t+\tau,t)\) is
properly periodic in \(t\).
In the test \(\rho(t+\tau,t) \equiv 0\), the
rejection for some \( \tau \neq 0\) indicates the sequence is not PC white noise.
Hurd, H. L., Miamee, A. G., (2007), Periodically Correlated Random Sequences: Spectral Theory and Practice, Wiley InterScience.
Bcoeff, perpacf
data(volumes)
dev.set(which=1)
peracf(t(volumes),24,seq(1,12),NaN,'volumes')
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