spectral.density {LSTS}

Spectral Density

Description

Returns theoretical spectral density evaluated in ARMA and ARFIMA processes.

Usage

spectral.density(ar = numeric(), ma = numeric(), d = 0, sd = 1, lambda = NULL)

Arguments

ar

(type: numeric) AR vector. If the time serie doesn't have AR term then omit it. For more details see the examples.

ma

(type: numeric) MA vector. If the time serie doesn't have MA term then omit it. For more details see the examples.

d

(type: numeric) Long-memory parameter. If d is zero, then the process is ARMA(p,q).

sd

(type: numeric) Noise scale factor, by default is 1.

lambda

(type: numeric) \lambda parameter on which the spectral density is calculated/computed. If lambda=NULL then it is considered a sequence between 0 and \pi.

Details

The spectral density of an ARFIMA(p,d,q) processes is

f(\lambda) = \frac{\sigma^2}{2\pi} \cdot \bigg(2\, \sin(\lambda/2)\bigg)^{-2d} \cdot \frac{\bigg|\theta\bigg(\exp\bigg(-i\lambda\bigg)\bigg)\bigg|^2} {\bigg|\phi\bigg(\exp\bigg(-i\lambda\bigg)\bigg)\bigg|^2}

With -\pi \le \lambda \le \pi and -1 < d < 1/2. |x| is the Mod of x. LSTS_sd returns the values corresponding to f(\lambda). When d is zero, the spectral density corresponds to an ARMA(p,q).

Value

An unnamed vector of numeric class.

References

For more information on theoretical foundations and estimation methods see Brockwell PJ, Davis RA, Calder MV (2002). Introduction to time series and forecasting, volume 2. Springer. Palma W (2007). Long-memory time series: theory and methods, volume 662. John Wiley \& Sons.

Examples

# Spectral Density AR(1)
require(ggplot2)
f <- spectral.density(ar = 0.5, lambda = malleco)
ggplot(data.frame(x = malleco, y = f)) +
  geom_line(aes(x = as.numeric(x), y = as.numeric(y))) +
  labs(x = "Frequency", y = "Spectral Density") +
  theme_minimal()