Wavelet-domain Gaussian processes: what is the covariance?

Question

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.

Answer 1

The driving process, white noise η(t), is independent of the choice of basis. In a CWT (unlike DWT jumping in octaves) there is some redundancy, narrow wavebands do overlap. The "feature" being tested for significance is a variance (power) observed in a narrow frequency over a short time. This clearly does depend mathematically on the chosen wavelet but not very much - narrower bandwidth can detect more slowly changing features with greater sensitivity, wider bandwidth is more responsive but has noisier background and is less specific.

1. As this measures wavelet space it's integrated over wavelet's duration, the transform you've written would be for any "point in time". Generally one needs phase information to invert the CWT. Maraun's test is essentially Chi-squared in power.

2. No. Maraun depends on signal to noise in a frequency band over a time range, this could have many different realisations in noise space and is phase independent. It is sensitive to an AR(1) signal in wavelet domain at a specific frequency, i.e oscillation sustained over time, e.g. CWT domain will tend to suppress an isolated spike in broadband noise.