In SMAC-Group/gmwm: Generalized Method of Wavelet Moments

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 $\phi$:

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

SMAC-Group/gmwm documentation built on Sept. 11, 2021, 10:06 a.m.