# 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

If the multipliers $$m(b,a)$$ only change slowly, then the realizations 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
Commented Nov 10, 2017 at 10:30