This function computes the Haar Wavelet Variance of an ARMA process

arma_to_wv(ar, ma, sigma2, tau)

## Arguments

ar |
A `vec` containing the coefficients of the AR process |

ma |
A `vec` containing the coefficients of the MA process |

sigma2 |
A `double` containing the residual variance |

tau |
A `vec` containing the scales e.g. \(2^{\tau}\) |

## Value

A `vec`

containing the wavelet variance of the ARMA process.

## Details

The function is a generic implementation that requires a stationary theoretical autocorrelation function (ACF)
and the ability to transform an ARMA(\(p\),\(q\)) process into an MA(\(\infty\)) (e.g. infinite MA 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

## See also