meanCoeff, acfCoeff {boodd}R Documentation

Fourier Coefficients Estimation of the Mean and Autocovariance Functions.

Description

For both periodically (PC) and almost periodically correlated (APC) data, the functions calculate the Fourier coefficients of the mean and autocovariance functions. The function can also be used for bootstrap samples obtained with the EMBB, CEMBB, GSBB, CGSBB.

Usage

    meanCoeff(x, period, freq, ...)
    acfCoeff(x, tau, period, freq, ...)

    ## Default S3 method:
meanCoeff(x, period, freq, ...)
    ## Default S3 method:
acfCoeff(x, tau, period, freq, ...)

    ## S3 method for class 'ts'
meanCoeff(x, period=frequency(x), freq, ...)
    ## S3 method for class 'ts'
acfCoeff(x, tau, period=frequency(x), freq, ...)

Arguments

x

A vector or time series representing a periodically or almost periodically correlated time series.

period

A positive integer; the period length. By default it is frequency(x).

tau

A vector of integers; a single lag or vector of lags.

freq

A numeric vector of frequencies.

...

Optional additional arguments for the function.

Details

If the freq argument is not specified, the Fourier frequencies are used: 2*k*pi/period for k=0,1,...,period, where period is the frequency of the time series.

The meanCoeff function implements the estimator of the Fourier coefficient of the mean at frequency \gamma:

\widehat{b}(\gamma) = \frac{1}{n}\sum_{t=1}^n X_t e^{-i\gamma t}.

The acfCoeff function implements the estimator of the Fourier coefficient of the autocovariance for given lag tau at frequency \lambda:

\widehat{a}(\lambda,\tau) = \frac{1}{n}\sum_{t=1-\min\{\tau,0\}}^{n-\max\{\tau,0\}} (X_{t+\tau}-\widehat{\mu}_n(t+\tau)) (X_{t}-\widehat{\mu}_n(t))e^{-i\lambda t}.

Value

meanCoeff returns a vector of the same length as freq.

acfCoeff returns either a vector of length length(freq) if a single lag tau is specified, or a matrix with length(tau) rows and length(freq) columns if tau is a vector.

References

Bertail, P. and Dudek, A. (2025). Bootstrap for Dependent Data, with an R package (by Bernard Desgraupes and Karolina Marek) - submitted..

Dudek, A.E. (2015). Circular block bootstrap for coefficients of autocovariance function of almost periodically correlated time series. Metrika, 78, 313-335.

Dudek, A.E. Maiz, S. and Elbadaoui, M. (2014). Generalized Seasonal Block Bootstrap in frequency analysis of cyclostationary signals. Signal Process., 104C, 358-368.

See Also

seasonalMean, seasonalVar, seasonalACF

Examples

# Fourier frequencies for the data nottem (temperatures at Nottingham Castle)
meanCoeff(nottem)
acfCoeff(nottem, 5)

# Given frequencies
freq <- 2 * (0:5) * pi / 12
meanCoeff(nottem, freq = freq)

[Package boodd version 0.1 Index]