arma11_to_wv: ARMA(1,1) to WV

In SMAC-Group/gmwm: Generalized Method of Wavelet Moments

ARMA(1,1) to WV

Description

This function computes the WV (haar) of an Autoregressive Order 1 - Moving Average Order 1 (ARMA(1,1)) process.

Usage

arma11_to_wv(phi, theta, sigma2, tau)

Arguments

phi	A double corresponding to the autoregressive term.
theta	A double corresponding to the moving average term.
sigma2	A double the variance of the process.
tau	A vec containing the scales e.g. 2^{\tau}

Details

This function is significantly faster than its generalized counter part arma_to_wv

Value

A vec containing the wavelet variance of the ARMA(1,1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order 1 and Moving Average Order 1 (ARMA(1,1)) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = - \frac{{2{\sigma ^2}\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)}}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}

Haar Wavelet Derivation Information

For more information, please see: Supported Haar Wavelet Formulae (Internet Connection Required).

See Also

arma_to_wv

Examples

ntau = 7
tau = 2^(1:ntau)
wv.theo = arma11_to_wv(0.3, 0.1, 1, tau)

SMAC-Group/gmwm documentation built on June 10, 2025, 6:10 a.m.