Warping of stationary stochastic processes

September 16, 2019 — January 21, 2021

Figure 1

Transforming stationary processes into non-stationary ones by transforming their inputs (Sampson and Guttorp 1992; Genton 2001; Genton and Perrin 2004; Perrin and Senoussi 1999, 2000).

This is of interest in the context of composing kernels to have known desirable properties by known transforms, and also learning (somewhat) arbitrary transforms to attain stationarity.

One might consider instead processes that are stationary upon a manifold.

1 Stationary reducible kernels

Figure 2

The main idea is to find a new feature space where stationarity (Sampson and Guttorp 1992) or local stationarity (Perrin and Senoussi 1999, 2000; Genton and Perrin 2004) can be achieved.

Genton (2001) summarises:

We say that a nonstationary kernel \(K(\mathbf{x}, \mathbf{z})\) is stationary reducible if there exist a bijective deformation \(\Phi\) such that: \[ K(\mathbf{x}, \mathbf{z})=K_{S}^{*}(\mathbf{\Phi}(\mathbf{x})-\mathbf{\Phi}(\mathbf{z})) \] where \(K_{S}^{*}\) is a stationary kernel.

2 Classic deformations

2.1 MacKay warping

2.2 As a function of input

Invented apparently by Gibbs (1998) and generalised in Paciorek and Schervish (2003).

Let \(k_S\) be some stationary kernel on \(\mathbb{R}^D.\) Let \(\Sigma(\mathbf{x})\) be a \(D \times D\) matrix-valued function which is positive definite for all \(\mathbf{x},\) and let \(\Sigma_{i} \triangleq \Sigma\left(\mathbf{x}_{i}\right) .\) ) Then define \[ Q_{i j}=\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)^{\top}\left(\left(\Sigma_{i}+\Sigma_{j}\right) / 2\right)^{-1}\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right) \] Then \[ k_{\mathrm{NS}}\left(\mathbf{x}_{i}, \mathbf{x}_{j}\right)=2^{D / 2}\left|\Sigma_{i}\right|^{1 / 4}\left|\Sigma_{j}\right|^{1 / 4}\left|\Sigma_{i}+\Sigma_{j}\right|^{-1 / 2} k_{\mathrm{S}}\left(\sqrt{Q_{i j}}\right) \] is a valid non-stationary covariance function.

Homework question: Is this a product of convolutional gaussian processes.

3 Learning transforms

Figure 3

4 References

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———. 2000. Reducing Non-Stationary Random Fields to Stationarity and Isotropy Using a Space Deformation.” Statistics & Probability Letters.
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