deriv_ma1: Analytic D matrix for MA(1) process
In SMAC-Group/simts: Time Series Analysis Tools

deriv_ma1

R Documentation

Analytic D matrix for MA(1) process

Description

Obtain the first derivative of the MA(1) process.

Usage

deriv_ma1(theta, sigma2, tau)

Arguments

`theta`	A `double` corresponding to the theta coefficient of an MA(1) process.
`sigma2`	A `double` corresponding to the error term of an MA(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 \theta and the second column contains the partial derivative with respect to \sigma ^2

Process Haar WV First Derivative

Taking the derivative with respect to \theta yields:

\frac{\partial }{{\partial \theta }}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{\sigma ^2}\left( {2\left( {\theta + 1} \right){\tau _j} - 6} \right)}}{{\tau _j^2}}

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

\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{{\left( {\theta + 1} \right)}^2}{\tau _j} - 6\theta }}{{\tau _j^2}}