I have a time serie that I want to analyse through a wavelet decomposition.
I am using the R package WaveThres.
I am interested in the wavelet autocorrelation, but I struggle to understand what does it mean precisely.
I have from the book Wavelet Methods in Statistics with R the following formula
$\Psi_j(\tau)=\sum_{k}\phi_{j,k}(0)\phi_{j,k}(\tau)$
$\tau\in\mathbb{Z}$ being the lag of the autocorrelation
and
$\left \{\phi_{j,k}(t)=\phi_{j,k-t} \right \}_{j,k}$ a set of non decimated wavelets
I would really appreciate to understand the meaning of this formula, and (why/if) it is different from performing a multi resolution analysis (MRA) and computing the Pearson autocorrelation coefficient on a detail.
fRed
Best Answer
The correlation coefficient of two sets of values is one number.
The auto-correlation of one set of values is a function (see e.g. http://en.wikipedia.org/wiki/Autocorrelation ). Let's call the argument of the function
t(looks like it's the $\tau$ in your question), then the value of the auto-correlation function attis the correlation coefficient of the set of values and the set of values shifted byt(I might be ignoring normalization factors here).