man-roxygen/deriv_wv/2nd/deriv2_ma1.R

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

#' @section Process Haar WV Second Derivative:
#' Taking the second derivative with respect to \eqn{\theta}{theta} yields:
#' \deqn{\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{\tau _j}}}}{d^2/dtheta^2 nu[j]^2 (theta, sigma2) = (2*sigma2)/tau[j]}
#'
#' Taking the second derivative with respect to \eqn{\sigma^2}{sigma^2} yields:
#' \deqn{\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = 0}{d^2/dsigma2^2 nu[j]^2 (theta, sigma2) = 0}
#'
#' Taking the first derivative with respect to \eqn{\theta}{theta} and \eqn{\sigma^2}{sigma^2} yields:
#' \deqn{\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}}}{d/dtheta * d/dsigma2 nu[j]^2 (theta, sigma2) = (-6 + 2*(1 + theta)*tau[j])/tau[j]^2}