deriv_2nd_arma11 {simts}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. 2τ2^{\tau}

Value

A matrix with:

Process Haar WV Second Derivative

Taking the second derivative with respect to ϕ\phi yields:

2ϕ2νj2(ϕ,θ,σ2)=2σ2(ϕ1)5(ϕ+1)3τj2((ϕ1)2((ϕ+1)2(θ2ϕ+θϕ2+θ+ϕ)τj2(ϕτj21)ϕτj22+(ϕ21)(θ2(ϕ)+θ(ϕ2+4ϕ+1)ϕ)τj(ϕτj22)ϕτj222(θ1)2(ϕτj4ϕτj2+3))12(ϕ+1)2(12(θ+1)2(ϕ21)τj(θ+ϕ)(θϕ+1)(ϕτj4ϕτj2+3))+6(ϕ+1)(ϕ1)(12(θ+1)2(ϕ21)τj+(θ+ϕ)(θϕ+1)(ϕτj4ϕτj2+3)+(ϕ+1)((θ+ϕ)(θϕ+1)τj(ϕτj22)ϕτj21θ(θ+ϕ)(ϕτj4ϕτj2+3)(θϕ+1)(ϕτj4ϕτj2+3)(θ+1)2ϕτj)))\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 θ\theta yields:

2θ2νj2(ϕ,θ,σ2)=2σ2((ϕ21)τj+2ϕ(ϕτj4ϕτj2+3))(ϕ1)3(ϕ+1)τj2\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 σ2\sigma ^2 yields:

2σ4νj2(ϕ,θ,σ2)=0\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = 0

Taking the derivative with respect to σ2\sigma^2 and θ\theta yields:

θσ2νj2(ϕ,θ,σ2)=2(ϕ1)3(ϕ+1)τj2((θ+1)(ϕ21)τj+(2θϕ+ϕ2+1)(ϕτj4ϕτj2+3))\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 σ2\sigma^2 and ϕ\phi yields:

ϕσ2νj2(ϕ,θ,σ2)=2(ϕ1)4(ϕ+1)2τj2((ϕ1)(ϕ+1)((θ+ϕ)(θϕ+1)τj(ϕτj22)ϕτj21θ(θ+ϕ)(ϕτj4ϕτj2+3)(θϕ+1)(ϕτj4ϕτj2+3)(θ+1)2ϕτj)+(ϕ1)(12(θ+1)2(ϕ21)τj(θ+ϕ)(θϕ+1)(ϕτj4ϕτj2+3))+3(ϕ+1)(12(θ+1)2(ϕ21)τj(θ+ϕ)(θϕ+1)(ϕτj4ϕτj2+3)))\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 ϕ\phi and θ\theta yields:

θϕνj2(ϕ,θ,σ2)=2σ2(ϕ1)4(ϕ+1)2τj2(τj(2(θ+1)(ϕ1)(ϕ+1)2+2(ϕ21)(2θϕ+ϕ2+1)ϕτj21(ϕ21)(2θϕ+ϕ2+1)ϕτj1)+2(θ(ϕ(3ϕ+2)+1)+ϕ(ϕ2+ϕ+3)+1)(ϕτj4ϕτj2+3))\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)

Author(s)

James Joseph Balamuta (JJB)


[Package simts version 0.2.2 Index]