deriv_arma11: Analytic D matrix for ARMA(1,1) process

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

Description

Obtain the first derivative of the ARMA(1,1) process.

Usage

deriv_arma11(phi, theta, sigma2, tau)

Arguments

phi	A double corresponding to the phi coefficient of an ARMA(1,1) process.
theta	A double corresponding to the theta coefficient of an ARMA(1,1) process.
sigma2	A double corresponding to the error term of an ARMA(1,1) process.
tau	A vec containing the scales e.g. 2^tau

Value

A matrix with:

• The first column containing the partial derivative with respect to phi;

• The second column containing the partial derivative with respect to theta;

• The third column contains the partial derivative with respect to sigma^2.

Process Haar WV First Derivative

Taking the derivative with respect to phi yields:

(1/((-1 + phi)^4*(1 + phi)^2*tau[j]^2))*2*sigma2*((-(3 - 4*phi^(tau[j]/2) + phi^tau[j]))*(1 + phi*(2 + 3*phi) + theta^2*(1 + phi*(2 + 3*phi)) + 2*theta*(1 + phi*(3 + phi + phi^2))) + ((-(1 + theta)^2)*(-1 + phi)*(1 + phi)^2 - 2*phi^(tau[j]/2 - 1)*(theta + phi)*(1 + theta*phi)*(-1 + phi^2) + phi^(tau[j] - 1)*(theta + phi)*(1 + theta*phi)*(-1 + phi^2))*tau[j])

Taking the derivative with respect to theta yields:

(2*sigma2*((1 + 2*theta*phi + phi^2)*(3 - 4*phi^(tau[j]/2) + phi^tau[j]) +(1 + theta)*(-1 + phi^2)*tau[j])) / ((-1 + phi)^3*(1 + phi)*tau[j]^2)

Taking the derivative with respect to sigma^2 yields:

((-2*((-(theta + phi))*(1 + theta*phi)*(3 - 4*phi^(tau[j]/2) + phi^tau[j]) - (1/2)*(1 + theta)^2*(-1 + phi^2)*tau[j]))/((-1 + phi)^3*(1 + phi)*tau[j]^2))

Author(s)

James Joseph Balamuta (JJB)

Examples

deriv_arma11(.3, .4, 1, 2^(1:5))

SMAC-Group/gmwm documentation built on Sept. 11, 2021, 10:06 a.m.