deriv_2nd_arma11: Analytic D matrix for ARMA(1,1) process
In SMAC-Group/simts: Time Series Analysis Tools

deriv_2nd_arma11

R Documentation

Analytic D matrix for ARMA(1,1) process

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:

\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{{(\phi - 1)}^5}{{(\phi + 1)}^3}\tau _j^2}}\left( \begin{array}{cc} &{(\phi - 1)^2}\left( {{{(\phi + 1)}^2}\left( {{\theta ^2}\phi + \theta {\phi ^2} + \theta + \phi } \right)\tau _j^2\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 1} \right){\phi ^{\frac{{{\tau _j}}}{2} - 2}} + \left( {{\phi ^2} - 1} \right)\left( {{\theta ^2}( - \phi ) + \theta \left( {{\phi ^2} + 4\phi + 1} \right) - \phi } \right){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 2}} - 2{{(\theta - 1)}^2}\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right) \\ &- 12{(\phi + 1)^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) \\ &+ 6(\phi + 1)(\phi - 1)\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) + (\phi + 1)\left( { - (\theta + \phi )(\theta \phi + 1){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} - \theta (\theta + \phi )\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) - (\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) - {{(\theta + 1)}^2}\phi {\tau _j}} \right)} \right) \\ \end{array} \right)

Taking the second derivative with respect to \theta yields:

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

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

\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = 0

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

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

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

\frac{\partial }{{\partial \phi }}\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{2}{{{{(\phi - 1)}^4}{{(\phi + 1)}^2}\tau _j^2}}\left( \begin{array}{ll} &- (\phi - 1)(\phi + 1)\left( \begin{array}{ll} &- (\theta + \phi )(\theta \phi + 1){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} \\ &- \theta (\theta + \phi )\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ &- (\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ &- {(\theta + 1)^2}\phi {\tau _j} \\ \end{array} \right) \\ &+ (\phi - 1)\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) \\ &+ 3(\phi + 1)\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) \\ \end{array} \right)

Taking the derivative with respect to \phi and \theta yields:

\frac{\partial }{{\partial \theta }}\frac{\partial }{{\partial \phi }}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = - \frac{{2{\sigma ^2}}}{{{{(\phi - 1)}^4}{{(\phi + 1)}^2}\tau _j^2}}\left( \begin{array}{cc} &{\tau _j}\left( \begin{array}{cc} &2(\theta + 1)(\phi - 1){(\phi + 1)^2} \\ &+ 2\left( {{\phi ^2} - 1} \right)\left( {2\theta \phi + {\phi ^2} + 1} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} \\ &- \left( {{\phi ^2} - 1} \right)\left( {2\theta \phi + {\phi ^2} + 1} \right){\phi ^{{\tau _j} - 1}} \\ \end{array} \right) \\ &+ 2\left( {\theta (\phi (3\phi + 2) + 1) + \phi \left( {{\phi ^2} + \phi + 3} \right) + 1} \right)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ \end{array} \right)