May 23, 2016 / by James Balamuta / In computing /

Process to Haar Wavelet Variance Formulae

The following equations are derivations used within the package as they relate to the Haar Wavelet Variance (WV) theoretical quantities. The initial WV formula, , are used to calculate process to wavelet variance. The later are used within the asymptotic model selection calculations.

The initial equations, marked by , come from Allan variance of time series models for measurement data by Nien Fan Zhang published in Metrologia and Analysis and Modeling of Inertial Sensors Using Allan Variance by El-Sheimy, et. al. in IEEE Transactions on Instrumentation and Measurement. That is, these equations are derived using the Allan Variance (AV). The relationship between the Allan Variance to the Wavelet Variance is . Note, the used in the Allan Variance is equivalent to .

The derivations below were done using Mathematica. The derivation file is available at: http://smac-group.com/assets/supporting_docs/haar_analytical_derivatives_complete.nb

If you notice one of the derivations as being incorrected, please let us know via an issue at https://github.com/smac-group/gmwm/issues.

White Noise
Random Walk
Drift Process
Quantization Noise (QN)
AR 1 Process

Derivatives w.r.t.

Derivatives w.r.t.

Derivative w.r.t both and

Here we opted to take the derivative w.r.t to first and then . The order of derivatives do not matter due to Clairaut’s Theorem.

MA 1 Process

> NOTE For the MA(1) process listed in Zhang on Page 552, there is a sign error between equations (21) and (22). This has been corrected here.

Derivatives w.r.t

Derivatives w.r.t.

Derivative w.r.t both and

ARMA(1,1)

NOTE For the ARMA(1,1) process listed in Zhang on Page 553, he references Time Series Analysis: Forecasting and Control by Box G E P and Jenkins G M 1976 that contains an error when describing both the process variance and autocorrelation function (ACF).

In this case, the ARMA(1,1) process variance, , and first autocovariance,, is given by:

And the ARMA(1,1)’s autocorrelation function (ACF) is given by:

for .

With this in mind, we rederive the Allan Variance for an ARMA(1,1) using Equation 11 on page 551.

ARMA(1,1) Derivation

We begin by stating Equation 11 on page 551:

Aside: To continue, we need to solve the series formulation using the recursive properties of ARMA(1,1)’s ACF.

Returning: We substitute in to the first equation to obtain the Allan Variance for the ARMA(1,1) process.

ARMA(1,1) Process

Derivative w.r.t :

Derivative w.r.t :

Derivative w.r.t. :

Derivative w.r.t. and :

Derivative w.r.t. and :

Derivative w.r.t. and :