Warping of stationary stochastic processes



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.

Stationary reducible kernels

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.

Classic deformations

MacKay warping

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.

Learning transforms

References

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