deriv_2nd_ar1: Analytic second derivative matrix for AR(1) process
In simts: Time Series Analysis Tools

deriv_2nd_ar1

R Documentation

Analytic second derivative matrix for AR(1) process

Description

Calculates the second derivative for the AR(1) process and places it into a matrix form. The matrix form in this case is for convenience of the calculation.

Usage

deriv_2nd_ar1(phi, sigma2, tau)

Arguments

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

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, \sigma ^2 \right) = \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j \left(2 (\phi (7 \phi +4)+1) \phi ^{\frac{\tau _j}{2}-1}-(\phi (7 \phi +4)+1) \phi ^{\tau _j-1}+3 (\phi +1)^2\right)+\left(\phi ^2-1\right)^2 \tau _j^2 \left(\phi ^{\frac{\tau _j}{2}}-1\right) \phi ^{\frac{\tau _j}{2}-1}+4 (3 \phi +1) \left(\phi ^2+\phi +1\right) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^5 (\phi +1)^3 \tau _j^2}

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

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

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

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