pl: Stationary Power-Law process ('time_series_model')

In gmwmx2: Estimate Functional and Stochastic Parameters of Linear Models with Correlated Residuals and Missing Data

Stationary Power-Law process (time_series_model)

Description

Constructs a time_series_model representing a stationary power-law process with parameters kappa and sigma2. In the frequency domain, a power-law process is often described by a spectrum P(f) = P_0 f^{\kappa} (Bos et al., 2008), where f is the frequency, P_0 is a constant and \kappa is the spectral index. Note that we use the convention that the power spectral density satisfies P(f) \propto |f|^{\kappa}, where \kappa > -1 ensures second-order stationarity. This corresponds to the alternative notation P(f) \propto |f|^{-\alpha} with \alpha = -\kappa. The autocovariance \gamma(h) = \mathrm{cov}(X_t, X_{t+h}) used here (Hosking, 1981) is \gamma(0) = \sigma^{2} \frac{\Gamma(1+\kappa)}{\Gamma\left(1+\kappa/2\right)^2}, and for h > 0 \gamma(h) = \frac{-\kappa/2 + h - 1}{\kappa/2 + h}\,\gamma(h-1).

Usage

pl(kappa = NULL, sigma2 = NULL)

Arguments

kappa	Power-law parameter in (-1, 1).
sigma2	Process variance (> 0).

Value

A time_series_model object.

References

Bos MS, Fernandes RMS, Williams SDP, Bastos L (2008). "Fast error analysis of continuous GPS observations." Journal of Geodesy, 82, 157-166.

Hosking JRM (1981). "Fractional differencing." Biometrika, 68(1), 165-176.

Examples

mod <- pl(kappa = -0.5, sigma2 = 2)
mod

gmwmx2 documentation built on June 10, 2026, 5:06 p.m.