# deriv_2nd_ar1: Analytic second derivative matrix for AR(1) process In SMAC-Group/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}

### Author(s)

James Joseph Balamuta (JJB)

SMAC-Group/simts documentation built on Sept. 4, 2023, 5:25 a.m.