This function computes the (haar) WV of an ARMA process
arma_to_wv_app(ar, ma, sigma2, tau, alpha = 0.9999)A vec containing the coefficients of the AR process
A vec containing the coefficients of the MA process
A double containing the residual variance
A vec containing the scales e.g. \(2^{\tau}\)
A double indicating the cutoff.
A vec containing the wavelet variance of the ARMA process.
The Autoregressive Order \(p\) and Moving Average Order \(q\) (ARMA(\(p\),\(q\))) process has a Haar Wavelet Variance given by: $$\frac{{{\tau _j}\left[ {1 - \rho \left( {\frac{{{\tau _j}}}{2}} \right)} \right] + 2\sum\limits_{i = 1}^{\frac{{{\tau _j}}}{2} - 1} {i\left[ {2\rho \left( {\frac{{{\tau _j}}}{2} - i} \right) - \rho \left( i \right) - \rho \left( {{\tau _j} - i} \right)} \right]} }}{{\tau _j^2}}\sigma _X^2$$ where \(\sigma _X^2\) is given by the variance of the ARMA process. Furthermore, this assumes that stationarity has been achieved as it directly
This function provides an approximation to the arma_to_wv as computation times
were previously a concern. However, this is no longer the case and, thus, this has been left
in for the curious soul to discover...