# Why wavelet power spectrum is a measure of the local variance of a time series

The integral wavelet transform is the integral transform defined as $$\left[W_{\psi }f\right](a,b)={\frac {1}{\sqrt {|a|}}}\int _{-\infty }^{\infty }{\overline {\psi \left({\frac {x-b}{a}}\right)}}f(x)dx$$ I read on this paper that the wavelet power spectrum, defined as $$|{W(a,b)}|^2$$ is a measure of the local variance (of the time series) but I can't understand why or how it becomes a similar measure for a time series (or a function).