Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as

COI is the region of the wavelet spectrum in which edge effects become important
and is defined here as the e-folding time for the autocorrelation of wavelet
power at each scale.


As an example: We have a vector with length 1001 and then compress it using the Mexican Hat Wavelet. As a result we get the following power spectrum plot:

Then using this tool we obtain the same power spectrum, but with the COI added (cross-hatched region on plot $b$).

The question is how can I ... describe the COI so I can easily add it to my plots given the e-folding time for the Mexican Hat Wavelet which is $\sqrt{2}s$. In other words: Are there any equations/inequalities that model the COI ?

• Not that I am aware of. Of course, you could always come up with your own definition. Just be explicit as to what it means so that the next guy is not in the position you currently are. Sep 17, 2013 at 21:16
• @AnonSubmitter85 Found another definition - Usually the coefficients of a CWT are presented in the timescale {b, a} half plane with linear scale on time b axis, pointing to the right, and logarithmic scale a axis, facing downward with increasing octave. To resolve localized signals, the analyzing wavelet ψ(t) is chosen so that it vanishes outside some interval (t_min, t_max). In this case the domain in the {b,a} half plane that can be influenced by a point (b_0, a_0) mainly lies within the cone of influence defined by Abs[b - b_0] = a Sqrt[2] Sep 19, 2013 at 10:14
• This python library plots them explicitly I haven't looked into it yet: github.com/alsauve/scaleogram . Also there is a good discussion over at mathworks: mathworks.com/help/wavelet/ug/… Jul 2, 2020 at 17:15

Mallat and Hwang provide a succint definition for the cone of influence of a wavelet in their 1992 seminal paper: Singularity Detection and Processing with Wavelets

The definition found in section V-B is as follows:

Let us suppose that the wavelet $$\psi(x)$$ has a symmetrical support equal to $$[-K , K]$$. We call cone of influence of $$x_0$$ in the scale-space plane, the set of points $$(s, x)$$ that satisfy
$$| x - x_0 | < Ks$$.

Note that $$s$$ refers to the CWT scale. In computers, the smallest scale would be $$s=1$$, twice the scale would be $$s=2$$, and so on.

Tangentially, but important for the quantification of singularities, note that the paper follows with a relevant statement: This cone is not generally sufficient to characterize the neighbourhood of $$x_0$$:

In order to characterize the regularity of $$f(x)$$ at a point $$x_0$$, one might think that it is sufficient to measure the decay of the wavelet transform within the cone of influence of $$x_0$$. Theorem 2 proves that this is wrong in general and that one must also measure the decay of the wavelet transform below this cone of influence. This is due to oscillations that can create a singularity at $$x_0$$.

This said, we can use this cone if $$f(x)$$ doesn't oscillate "too much" around $$x_0$$ (the paper ellaborates on this as well).