Wavelet bicoherence was given by Van Milligen1995, which used to analyze turbulence. And the normalized squared wavelet bicoherence (usually called wavelet bicoherence) is shown below. $$ WBC(a_1,a_2)=\frac{|\int_TW_f(a_1,\tau)W_f(a_2,\tau)W_g^*(a,\tau)\mathrm{d}\tau|^2}{\int_T|W_f(a_1,\tau)W_f(a_2,\tau)|^2\mathrm{d}\tau\int_T|W_g(a,\tau)|^2\mathrm{d}\tau} $$
Van Milligen1995 proposed a concept called statistical noise level which puzzled me. The content of statistical noise is as follows
However, the wavelet coefficients are not all statistically independent, since the chosen wavelet family is not orthogonal. Each coefficient is calculated by evaluating Eq. (4), integrating over the range $- \infty < t< +\infty$. Due to the periodicity $a$ of the wavelets of scale $a$, two statistically independent estimates of the wavelet coefficients are separated by a time $a/2$, or by a number of points $M(a)=a\omega_{samp}/(4\pi)$ ($w_{samp}$ being the sampling frequency). Thus, the summation done in the evaluation of the bicoherence $WBC(a_1,a_2)$ is not really carried out over $N$ points, but only over $N/max[M(a)]$, where the maximum is taken over the values of a that come into play for the evaluation of a specific value of the squared bicoherence. An estimate for the statistical noise level in $WBC(a_1,a_2)$ is, therefore $$ \epsilon(WBC)\approx \Big[ \frac{\omega_{samp}}{2\cdot \mathrm{Min}(|w_1|,|w_2|,|w_1+w_2|)}\frac{1}{N}\Big] $$ where $w=2\pi/a$.
My questions are
- We all know that the wavelet functions are well-localized. So why did Milligen say that "the periodicity $a$ of the wavelets of scale $a$"? And why the wavelet coefficients separated by a time $a/2$ are statistically independent?
- Why take the maximum value of $M(a)$ in $N/max[M(a)]$?
- Milligen didn't show the formal derivation of the statistical noise level, i.e. $\epsilon(WBC)$. So I want to know how to relate the formula of $\epsilon(WBC)$ with $N/max[M(a)]$.