tacvfARFIMA {arfima}

The theoretical autocovariance function of a long memory process.

Description

Calculates the tacvf of a mixed long memory-ARMA (with posible seasonal components). Combines long memory and ARMA (and non-seasonal and seasonal) parts via convolution.

Usage

tacvfARFIMA(
  phi = numeric(0),
  theta = numeric(0),
  dfrac = numeric(0),
  phiseas = numeric(0),
  thetaseas = numeric(0),
  dfs = numeric(0),
  H = numeric(0),
  Hs = numeric(0),
  alpha = numeric(0),
  alphas = numeric(0),
  period = 0,
  maxlag,
  useCt = T,
  sigma2 = 1
)

Arguments

phi

The autoregressive parameters in vector form.

theta

The moving average parameters in vector form. See Details for differences from arima.

dfrac

The fractional differencing parameter.

phiseas

The seasonal autoregressive parameters in vector form.

thetaseas

The seasonal moving average parameters in vector form. See Details for differences from arima.

dfs

The seasonal fractional differencing parameter.

H

The Hurst parameter for fractional Gaussian noise (FGN). Should not be mixed with dfrac or alpha: see "Details".

Hs

The Hurst parameter for seasonal fractional Gaussian noise (FGN). Should not be mixed with dfs or alphas: see "Details".

alpha

The decay parameter for power-law autocovariance (PLA) noise. Should not be mixed with dfrac or H: see "Details".

alphas

The decay parameter for seasonal power-law autocovariance (PLA) noise. Should not be mixed with dfs or Hs: see "Details".

period

The periodicity of the seasonal components. Must be >= 2.

maxlag

The number of terms to compute: technically the output sequence is from lags 0 to maxlag, so there are maxlag + 1 terms.

useCt

Whether or not to use C to compute the (parts of the) tacvf.

sigma2

Used in arfima.sim: determines the value of the innovation variance. The tacvf sequence is multiplied by this value.

Details

The log-likelihood is computed for the given series z and the parameters. If two or more of dfrac, H or alpha are present and/or two or more of dfs, Hs or alphas are present, an error will be thrown, as otherwise there is redundancy in the model. Note that non-seasonal and seasonal components can be of different types: for example, there can be seasonal FGN with FDWN at the non-seasonal level.

The moving average parameters are in the Box-Jenkins convention: they are the negative of the parameters given by arima.

Value

A sequence of length maxlag + 1 (lags 0 to maxlag) of the tacvf of the given process.

Author(s)

JQ (Justin) Veenstra and A. I. McLeod

References

Veenstra, J.Q. Persistence and Antipersistence: Theory and Software (PhD Thesis)

P. Borwein (1995) An efficient algorithm for Riemann Zeta function Canadian Math. Soc. Conf. Proc., 27, pp. 29-34.

Examples

t1 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, dfrac = 0.3, maxlag = 30)
t2 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, H = 0.8, maxlag = 30)
t3 <- tacvfARFIMA(phi = c(0.2, 0.1), theta = 0.4, alpha = 0.4, maxlag = 30)
plot(t1, type = "o", col = "blue", pch = 20)
lines(t2, type = "o", col = "red", pch = 20)
lines(t3, type = "o", col = "purple", pch = 20)  #they decay at about the same rate