deriv_2nd_arma11: Analytic D matrix for ARMA(1,1) process

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

Description

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

Usage

deriv_2nd_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 second partial derivative with respect to phi;

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

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

• The fourth column contains the partial derivative with respect to phi and theta.

• The fiveth column contains the partial derivative with respect to sigma^2 and phi.

• The sixth column contains the partial derivative with respect to sigma^2 and theta.

Process Haar WV Second Derivative

Taking the second derivative with respect to phi yields:

d^2/dphi^2 nu[j]^2(phi, theta, sigma2) = (1/((phi - 1)^5*(phi + 1)^3*tau[j]^2))*(2*sigma2*((phi - 1)^2* ((phi + 1)^2*(theta^2*phi + theta*phi^2 + theta + phi)*tau[j]^2* (phi^(tau[j]/2) - 1)*phi^(tau[j]/2 - 2) + (phi^2 - 1)*(theta^2*(-phi) + theta*(phi^2 + 4*phi + 1) - phi)*tau[j]*(phi^(tau[j]/2) - 2)*phi^(tau[j]/2 - 2) - 2*(theta - 1)^2*(phi^tau[j] - 4*phi^(tau[j]/2) + 3)) - 12*(phi + 1)^2*(-((1/2)*(theta + 1)^2*(phi^2 - 1)*tau[j]) - (theta + phi)*(theta*phi + 1)* (phi^tau[j] - 4*phi^(tau[j]/2) + 3)) + 6*(phi + 1)*(phi - 1)*((1/2)*(theta + 1)^2*(phi^2 - 1)*tau[j] + (theta + phi)*(theta*phi + 1)*(phi^tau[j] - 4*phi^(tau[j]/2) + 3) + (phi + 1)*(-((theta + phi)*(theta*phi + 1)*tau[j]*(phi^(tau[j]/2) - 2)*phi^(tau[j]/2 - 1)) - theta*(theta + phi)*(phi^tau[j] - 4*phi^(tau[j]/2) + 3) - (theta*phi + 1)*(phi^tau[j] - 4*phi^(tau[j]/2) + 3) - (theta + 1)^2*phi*tau[j]))))

Taking the second derivative with respect to theta yields:

d^2/dtheta^2 nu[j]^2(phi, theta, sigma2) = (2*sigma2*(2*phi*(3 - 4*phi^(tau[j]/2) + phi^tau[j]) + (-1 + phi^2)*tau[j]))/((-1 + phi)^3*(1 + phi)*tau[j]^2)

Taking the second derivative with respect to sigma^2 yields:

d^2/dsigma2^2 nu[j]^2(phi, theta, sigma2) = 0

Taking the derivative with respect to sigma^2 and theta yields:

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

Taking the derivative with respect to sigma^2 and phi yields:

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

Taking the derivative with respect to phi and theta yields:

d/dphi * d/dtheta nu[j]^2(phi, theta, sigma2) =(-(1/((-1 + phi)^4*(1 + phi)^2*tau[j]^2)))*2*sigma2*(2*(3 - 4*phi^(tau[j]/2) + phi^tau[j])* (1 + phi*(3 + phi + phi^2) + theta*(1 + phi*(2 + 3*phi))) + (2*(1 + theta)*(-1 + phi)*(1 + phi)^2 + 2*phi^(tau[j]/2 - 1)*(-1 + phi^2)* (1 + 2*theta*phi + phi^2) - phi^(tau[j] - 1)*(-1 + phi^2)* (1 + 2*theta*phi + phi^2))*tau[j])

Author(s)

James Joseph Balamuta (JJB)

Examples

deriv_2nd_arma11(.3, .4, 1, 2^(1:5))

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