# Wavelet-domain gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be specified by multipliers in wavelet domain. A realization of one such GP is: $$s(t) = M_h m(b,a) W_g \eta(t)\, ,$$ where $\eta(t)$ is white noise, $W_g$ is the continuous wavelet transform with respect to wavelet $g$, $m(b,a)$ is the multiplier (kinda like a Fourier coefficient) with scale $a$ and time $b$, and $M_h$ is the inverse wavelet transform with reconstruction wavelet $h$.

One key result of the paper is that if the multipliers $m(b,a)$ only change slowly, then the realization themselves are only "weakly" dependent on the actual choices of $g$ and $h$. Thus $m(b,a)$ specifies the process. They go on to create some significant tests to help infer the wavelet multipliers based on realizations.

Two questions:

1. How do we evaluate the standard GP likelihood which is $p(D) = \mathcal{N}(0,K)$?

I'd guess we're effectively doing a change of coordinates so $K^{-1} = W^T M^{-1} W$ where $W$ are the wavelets and $M$ is the (diagonal?) matrix of wavelet coefficients $m(a,b)$. However, they use a non-orthonormal CWT so I don't know if this is correct.

2. How can this wavelet domain GP be related to a real-space GP? Specifically, can we calculate a real-space (non-stationary) kernel $k$ from $m(a,b)$?

For comparison, the kernel of a stationary Gaussian processes is the Fourier dual of its spectral density (Bochner's theorem, see Rasmussen chapter 4) - which gives an easy way to switch between a real space GP and a frequency space one. Here I'm asking if there's such a relationship in the wavelet domain.

• Did you get anywhere with this. I'm not sure the change of variables is right as that would contradict when they say $K_{g,h}(b−b/a,a/a)=W_{g,h}(b−b/a)$ is called the reproducing kernel? – tdc Nov 10 '17 at 10:30