Process to Haar Wavelet Variance Formulae

May 23, 2016 / by James Balamuta / In computing /

Process to Haar Wavelet Variance Formulae

The following equations are derivations used within the package as they relate to the Haar Wavelet Variance (WV) theoretical quantities. The initial WV formula, $\nu _j^2$ , are used to calculate process to wavelet variance. The later are used within the asymptotic model selection calculations.

The initial equations, marked by $AVa{r_n}\left[ {X\left( t \right)} \right]$ , come from Allan variance of time series models for measurement data by Nien Fan Zhang published in Metrologia and Analysis and Modeling of Inertial Sensors Using Allan Variance by El-Sheimy, et. al. in IEEE Transactions on Instrumentation and Measurement. That is, these equations are derived using the Allan Variance (AV). The relationship between the Allan Variance to the Wavelet Variance is $\frac{1}{2} AVa{r_n}\left[ {X\left( t \right)} \right] = \nu _j^2$ . Note, the $n$ used in the Allan Variance is equivalent to $\frac{\tau_j}{2}$ .

The derivations below were done using Mathematica. The derivation file is available at: http://smac-group.com/assets/supporting_docs/haar_analytical_derivatives_complete.nb

If you notice one of the derivations as being incorrected, please let us know via an issue at https://github.com/smac-group/gmwm/issues.

White Noise

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \frac{ {\sigma _X^2} }{n} \\ \nu _j^2\left( { {\sigma ^2} } \right) &= \frac{1}{2}\frac{ { {\sigma ^2} } }{ {\left( {\frac{ { {\tau _j} } }{2} } \right)} } \\ &= \frac{ { {\sigma ^2} } }{ { {\tau _j} } } \\ \frac{\partial }{ {\partial {\sigma ^2} } }\nu _j^2\left( { {\sigma ^2} } \right) &= \frac{1}{ { {\tau _j} } } \\ \frac{ { {\partial ^2} } }{ {\partial {\sigma ^4} } }\nu _j^2\left( { {\sigma ^2} } \right) &= 0 \\ \end{aligned}$

Random Walk

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \frac{ {2{n^2} + 1} }{ {6n} }\sigma _a^2 \\ \nu _j^2\left( { {\gamma ^2} } \right) &= \frac{1}{2}\frac{ {\left( {2{ {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2} + 1} \right){\gamma ^2} } }{ {6\left( {\frac{ { {\tau _j} } }{2} } \right)} } \\ &= \frac{ {\left( {\tau _j^2 + 2} \right){\gamma ^2} } }{ {12{\tau _j} } } \\ \frac{\partial }{ {\partial \gamma ^2} }\nu _j^2\left( { {\gamma ^2} } \right) &= \frac{ {\left( { {\tau _j^2} + 2} \right)} }{ {12\tau _j} } \\ \frac{\partial ^2 }{ {\partial \gamma ^4} }\nu _j^2\left( { {\gamma ^2} } \right) &= 0 \\ \end{aligned}$

Drift Process

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \frac{ { {n^2}{\omega ^2} } }{2} \\ \nu _j^2\left( \omega \right) &= \frac{1}{2}\frac{ { { {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2}{\omega ^2} } }{2} \\ &= \frac{ {\tau _j^2{\omega ^2} } }{ {16} } \\ \frac{\partial }{ {\partial \omega } }\nu _j^2\left( \omega \right) &= \frac{ {\tau _j^2\omega } }{8} \\ \frac{ { {\partial ^2} } }{ {\partial {\omega ^2} } }\nu _j^2\left( \omega \right) &= \frac{ {\tau _j^2} }{8} \end{aligned}$

Quantization Noise (QN)

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \frac{ {3{Q^2} } }{ { {n^2} } } \\ \nu _j^2\left( Q^2 \right) &= \frac{1}{2}\frac{ {3{Q^2} } }{ { { {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2} } } \\ &= \frac{ {6{Q^2} } }{ {\tau _j^2} } \\ \frac{\partial }{ {\partial Q^2} }{\nu _j^2}\left( Q^2 \right) &= \frac{6}{ { {\tau _j^2} } } \\ \frac{ { {\partial ^2} } }{ {\partial Q^4} }{\nu _j^2}\left( Q^2 \right) &= 0 \end{aligned}$

AR 1 Process

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \frac{ {n - 3{\phi} - n\phi^2 + 4\phi^{n + 1} - \phi^{2n + 1} } }{ { {n^2}{ {\left( {1 - {\phi} } \right)}^2}\left( {1 - \phi^2} \right)} }\sigma _a^2 \\ \nu _j^2\left( { {\phi},{\sigma ^2} } \right) &= \frac{1}{2}\frac{ {\left( {\left( {\frac{ { {\tau _j} } }{2} } \right) - 3{\phi} - \left( {\frac{ { {\tau _j} } }{2} } \right)\phi^2 + 4\phi^{\left( {\frac{ { {\tau _j} } }{2} } \right) + 1} - \phi^{2\left( {\frac{ { {\tau _j} } }{2} } \right) + 1} } \right){\sigma ^2} } }{ { { {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2}{ {\left( {1 - {\phi} } \right)}^2}\left( {1 - \phi^2} \right)} } \\ &= \frac{\sigma ^2 \left(\left(\phi ^2-1\right) \tau _j+2 \phi \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2} }+3\right)\right)}{(\phi -1)^3 (\phi +1) \tau _j^2} \end{aligned}$

Derivatives w.r.t. $\phi$

$\begin{aligned} \frac{\partial }{ {\partial {\phi} } }\nu _j^2\left( { {\phi},{\sigma ^2} } \right) &= \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j \left(\phi ^{\tau _j}-2 \phi ^{\frac{\tau _j}{2} }-\phi -1\right)-((3 \phi +2) \phi +1) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2} }+3\right)\right)}{(\phi -1)^4 (\phi +1)^2 \tau _j^2} \\ \frac{\partial }{ {\partial \phi^2} }\nu _j^2\left( { {\phi},{\sigma ^2} } \right) &= \frac{ {2{\sigma ^2} } }{ { { {(\phi - 1)}^5}{ {(\phi + 1)}^3}\tau _j^2} }\left( \begin{aligned} &\left( { {\phi ^2} - 1} \right){\tau _j}\left( {2((7\phi + 4)\phi + 1){\phi ^{\frac{ { {\tau _j} } }{2} - 1} } - ((7\phi + 4)\phi + 1){\phi ^{ {\tau _j} - 1} } + 3{ {(\phi + 1)}^2} } \right) + \\ &{\left( { {\phi ^2} - 1} \right)^2}\tau _j^2\left( { {\phi ^{\frac{ { {\tau _j} } }{2} } } - 1} \right){\phi ^{\frac{ { {\tau _j} } }{2} - 1} } + \\ &4\left( { {\phi ^2} + \phi + 1} \right)(3\phi + 1)\left( { {\phi ^{ {\tau _j} } } - 4{\phi ^{\frac{ { {\tau _j} } }{2} } } + 3} \right) \\ \end{aligned} \right) \end{aligned}$

Derivatives w.r.t. $\sigma ^2$

$\begin{aligned} \frac{\partial }{ {\partial {\sigma ^2} } }{\nu ^2}(\tau ) &= \frac{\left(\phi ^2-1\right) \tau _j+2 \phi \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2} }+3\right)}{(\phi -1)^3 (\phi +1) \tau _j^2} \\ \frac{\partial }{ {\partial {\sigma ^4} } }{\nu ^2}(\tau ) &= 0 \end{aligned}$

Derivative w.r.t both $\sigma ^2$ and $\phi$

Here we opted to take the derivative w.r.t to $\sigma^2$ first and then $\phi$. The order of derivatives do not matter due to Clairaut’s Theorem.

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

MA 1 Process

> NOTE For the MA(1) process listed in Zhang on Page 552, there is a sign error between equations (21) and (22). This has been corrected here.

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= \left( {1 + \theta^2} \right)\frac{ {\left( {n + \left( {2n - 3} \right)\frac{ { {\theta} } }{ {1 + \theta^2} } } \right)} }{ { {n^2} } }\sigma _a^2 \\ \nu _j^2\left( { {\theta},{\sigma ^2} } \right) &= \frac{1}{2}\left( {1 + \theta^2} \right)\frac{ {\left( {\left( {\frac{ { {\tau _j} } }{2} } \right) + \left( {2\left( {\frac{ { {\tau _j} } }{2} } \right) - 3} \right)\frac{ { {\theta} } }{ {1 + \theta^2} } } \right)} }{ { { {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2} } }{\sigma ^2} \\ &= \frac{ {\left( { { {\left( { {\theta} + 1} \right)}^2}{\tau _j} - 6{\theta} } \right){\sigma ^2} } }{ {\tau _j^2} } \end{aligned}$

Derivatives w.r.t $\theta$

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

Derivatives w.r.t. $\sigma ^2$

$\begin{aligned} \frac{\partial }{ {\partial {\sigma ^2} } }{\nu _j^2\left( { {\theta},{\sigma ^2} } \right)} &= \frac{ {\left( { { {\left( { {\theta} + 1} \right)}^2}{\tau _j} - 6{\theta} } \right)} }{ {\tau _j^2} } \\ \frac{\partial }{ {\partial {\sigma ^4} } }{\nu _j^2\left( { {\theta},{\sigma ^2} } \right)} &= 0 \end{aligned}$

Derivative w.r.t both $\sigma ^2$ and $\theta$

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

ARMA(1,1)

> NOTE For the ARMA(1,1) process listed in Zhang on Page 553, he references Time Series Analysis: Forecasting and Control by Box G E P and Jenkins G M 1976 that contains an error when describing both the process variance and autocorrelation function (ACF).

In this case, the ARMA(1,1) process variance, $\gamma \left( 0 \right)$, and first autocovariance,$\gamma \left( 1 \right)$, is given by:

[\begin{aligned} Var\left( {X\left( t \right)} \right) &= \gamma \left( 0 \right)
&= {\sigma ^2}\frac{ {\left( {1 + 2\theta \phi + {\theta ^2} } \right)} }{ {\left( {1 - {\phi ^2} } \right)} }
\gamma \left( 1 \right) &= {\sigma ^2}\frac{ {\left( {1 + \theta \phi } \right)\left( {\phi + \theta } \right)} }{ {\left( {1 - {\phi ^2} } \right)} } \end{aligned} ]

And the ARMA(1,1)’s autocorrelation function (ACF) is given by:

[\begin{aligned} \rho \left( 1 \right) &= \frac{ {\gamma \left( 1 \right)} }{ {\gamma \left( 0 \right)} }
& = \frac{ {\left( {\phi + \theta } \right)\left( {1 + \phi \theta } \right)} }{ {1 + 2\phi \theta + {\theta ^2} } }
\rho \left( k \right) &= {\phi ^{k - 1} }\rho \left( 1 \right) \end{aligned} ]

for $k \ge 1$.

With this in mind, we rederive the Allan Variance for an ARMA(1,1) using Equation 11 on page 551.

ARMA(1,1) Derivation

We begin by stating Equation 11 on page 551:

$AVa{r_n}\left[ {X\left( t \right)} \right] = \frac{ {n\left( {1 - \rho \left( n \right)} \right) + \sum\limits_{i = 1}^{n - 1} {i\left( {2\rho \left( {n - i} \right) - \rho \left( i \right) - \rho \left( {2n - i} \right)} \right)} } }{ { {n^2} } }\sigma _X^2$

Aside: To continue, we need to solve the series formulation using the recursive properties of ARMA(1,1)’s ACF.

$\begin{aligned} \sum\limits_{i = 1}^{n - 1} {i\left( {2\rho \left( {n - i} \right) - \rho \left( i \right) - \rho \left( {2n - i} \right)} \right)} &= \rho \left( 1 \right)\sum\limits_{i = 1}^{n - 1} {i\left( {2{\phi ^{n - i - 1} } - {\phi ^{i - 1} } - {\phi ^{2n - i - 1} } } \right)} \\ &= \rho \left( 1 \right)\left( {\frac{ { - {\phi ^{2n + 1} } + \left( {n{ {(\phi - 1)}^2} + 4\phi } \right){\phi ^n} + \phi ( - 2n(\phi - 1) - 3)} }{ { { {(\phi - 1)}^2}\phi } } } \right)\\ &= \frac{ {(\phi \theta + 1)(\theta + \phi )\left( { - {\phi ^{2n + 1} } + \left( {n{ {(\phi - 1)}^2} + 4\phi } \right){\phi ^n} + \phi ( - 2n(\phi - 1) - 3)} \right)} }{ { { {(\phi - 1)}^2}\phi \left( { {\theta ^2} + 2\phi \theta + 1} \right)} } \end{aligned}$

Returning: We substitute in to the first equation to obtain the Allan Variance for the ARMA(1,1) process.

$\begin{aligned} &= \frac{ {n\left( {1 - {\phi ^{n - 1} }\frac{ {\left( {\phi + \theta } \right)\left( {1 + \phi \theta } \right)} }{ {1 + 2\phi \theta + {\theta ^2} } } } \right) + \frac{ {(\phi \theta + 1)(\theta + \phi )\left( { - {\phi ^{2n + 1} } + \left( {n{ {(\phi - 1)}^2} + 4\phi } \right){\phi ^n} + \phi ( - 2n(\phi - 1) - 3)} \right)} }{ { { {(\phi - 1)}^2}\phi \left( { {\theta ^2} + 2\phi \theta + 1} \right)} } } }{ { {n^2} } }\left( {\frac{ { {\sigma ^2}\left( { {\theta ^2} + 2\theta \phi + 1} \right)} }{ {1 - {\phi ^2} } } } \right)\\ &= - \frac{ { {\sigma ^2}\left( {(\theta + 1)n(\phi - 1)(\theta (\phi - 1) - \phi - 2\phi \theta - 1) - (\phi \theta + 1)(\theta + \phi )\left( { {\phi ^n} - 3} \right)\left( { {\phi ^n} - 1} \right)} \right)} }{ { {n^2}{ {(\phi - 1)}^3}(\phi + 1)} } \end{aligned}$

ARMA(1,1) Process

$\begin{aligned} AVa{r_n}\left[ {X\left( t \right)} \right] &= - \frac{ { {\sigma ^2}\left( {(\theta + 1)n(\phi - 1)(\theta (\phi - 1) - \phi - 2\phi \theta - 1) - (\phi \theta + 1)(\theta + \phi )\left( { {\phi ^n} - 3} \right)\left( { {\phi ^n} - 1} \right)} \right)} }{ { {n^2}{ {(\phi - 1)}^3}(\phi + 1)} } \\ \nu _j^2\left( {\phi ,\theta ,{\sigma ^2} } \right) &= - \frac{ { {\sigma ^2}\left( {(\theta + 1)\left( {\frac{ { {\tau _j} } }{2} } \right)(\phi - 1)(\theta (\phi - 1) - \phi - 2\phi \theta - 1) - (\phi \theta + 1)(\theta + \phi )\left( { {\phi ^{\left( {\frac{ { {\tau _j} } }{2} } \right)} } - 3} \right)\left( { {\phi ^{\left( {\frac{ { {\tau _j} } }{2} } \right)} } - 1} \right)} \right)} }{ { { {\left( {\frac{ { {\tau _j} } }{2} } \right)}^2}{ {(\phi - 1)}^3}(\phi + 1)} }\left( {\frac{1}{2} } \right) \\ &= - \frac{ {2{\sigma ^2}\left( { - \frac{1}{2}{ {(\theta + 1)}^2}\left( { {\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( { {\phi ^{ {\tau _j} } } - 4{\phi ^{\frac{ { {\tau _j} } }{2} } } + 3} \right)} \right)} }{ { { {(\phi - 1)}^3}(\phi + 1)\tau _j^2} } \\ \end{aligned}$

Derivative w.r.t $\phi$:

$\begin{aligned} \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{aligned} &{\tau _j}\left( { - { {(\theta + 1)}^2}(\phi - 1){ {(\phi + 1)}^2} - 2\left( { {\phi ^2} - 1} \right)(\theta + \phi )(\theta \phi + 1){\phi ^{\frac{ { {\tau _j} } }{2} - 1} } + \left( { {\phi ^2} - 1} \right)(\theta \phi + 1)(\theta + \phi ){\phi ^{ {\tau _j} - 1} } } \right) \\ &- \left( { {\theta ^2}((3\phi + 2)\phi + 1) + 2\theta \left( {\left( { {\phi ^2} + \phi + 3} \right)\phi + 1} \right) + (3\phi + 2)\phi + 1} \right)\left( { {\phi ^{ {\tau _j} } } - 4{\phi ^{\frac{ { {\tau _j} } }{2} } } + 3} \right) \\ \end{aligned} \right) \\ \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{aligned} &{(\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{aligned} \right) \end{aligned}$

Derivative w.r.t $\theta$:

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

Derivative w.r.t. $\sigma^2$:

$\begin{aligned} \frac{\partial }{ {\partial \sigma ^2 } }\nu _j^2\left( {\phi ,\theta ,{\sigma ^2} } \right) &= \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j+2 \phi \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2} }+3\right)\right)}{(\phi -1)^3 (\phi +1) \tau _j^2} \\ \frac{ { {\partial ^2} } }{ {\partial {\sigma ^4} } }\nu _j^2\left( {\phi ,\theta ,{\sigma ^2} } \right) &= 0 \end{aligned}$

Derivative w.r.t. $\phi$ and $\sigma^2$:

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

Derivative w.r.t. $\theta$ and $\sigma^2$:

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

Derivative w.r.t. $\theta$ and $\phi$:

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