deriv_2nd_ar1 {simts} | 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. 2τ
|
Value
A matrix
with the first column containing the
second partial derivative with respect to ϕ
and
the second column contains the second partial derivative with
respect to σ2
Process Haar WV Second Derivative
Taking the second derivative with respect to ϕ
yields:
∂ϕ2∂2νj2(ϕ,σ2)=(ϕ−1)5(ϕ+1)3τj22σ2((ϕ2−1)τj(2(ϕ(7ϕ+4)+1)ϕ2τj−1−(ϕ(7ϕ+4)+1)ϕτj−1+3(ϕ+1)2)+(ϕ2−1)2τj2(ϕ2τj−1)ϕ2τj−1+4(3ϕ+1)(ϕ2+ϕ+1)(ϕτj−4ϕ2τj+3))
Taking the second derivative with respect to σ2
yields:
∂σ4∂2νj2(σ2)=0
Taking the derivative with respect to ϕ
and σ2
yields:
∂ϕ∂σ2∂2νj2(ϕ,σ2)=(ϕ−1)4(ϕ+1)2τj22((ϕ2−1)τj(ϕτj−2ϕ2τj−ϕ−1)−(ϕ(3ϕ+2)+1)(ϕτj−4ϕ2τj+3))
Author(s)
James Joseph Balamuta (JJB)
[Package
simts version 0.2.2
Index]