arma_to_wv_app

This function computes the (haar) WV of an ARMA process

internal

A system contains easy-to-use tools as a support for time series analysis courses. In particular, it incorporates a technique called Generalized Method of Wavelet Moments (GMWM) as well as its robust implementation for fast and robust parameter estimation of time series models which is described, for example, in Guerrier et al. (2013) <doi: 10.1080/01621459.2013.799920>. More details can also be found in the paper linked to via the URL below.

St<c3><a9>phane Guerrier

simts

Time Series Analysis Tools

Stéphane Guerrier

James Balamuta

Roberto Molinari

Justin Lee

Lionel Voirol

Yuming Zhang

Wenchao Yang

Nathanael Claussen

Yunxiang Zhang

Christian Gunning

Romain Francois

Ross Ihaka

R Core Team 

arma_to_wv_app function

<dl><dt>ar</dt>
<dd>A <code>vec</code> containing the coefficients of the AR process</dd>
<dt>ma</dt>
<dd>A <code>vec</code> containing the coefficients of the MA process</dd>
<dt>sigma2</dt>
<dd>A <code>double</code> containing the residual variance</dd>
<dt>tau</dt>
<dd>A <code>vec</code> containing the scales e.g. $2^{\tau}$</dd>
<dt>alpha</dt>
<dd>A <code>double</code> indicating the cutoff.</dd></dl>

Arguments

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

Process Haar Wavelet Variance Formula

ARMA process to WV Approximation — arma_to_wv_app

<dl>

<dt>ar</dt>
<dd>A <code>vec</code> containing the coefficients of the AR process</dd>


<dt>ma</dt>
<dd>A <code>vec</code> containing the coefficients of the MA process</dd>


<dt>sigma2</dt>
<dd>A <code>double</code> containing the residual variance</dd>


<dt>tau</dt>
<dd>A <code>vec</code> containing the scales e.g. $2^{\tau}$</dd>


<dt>alpha</dt>
<dd>A <code>double</code> indicating the cutoff.</dd>

</dl>

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

arma_to_wv_app: ARMA process to WV Approximation

Description

Usage

Value

Arguments

Process Haar Wavelet Variance Formula

Details