Use of Input Deformations with Brownian Motion Filters for Discontinuous Regression

Abstract

Bayesian Gaussian processes are known as ‘smoothing devices’ and in the case of n data points they require O(n ²)... O(n ³) number of multiplications in order to perform a regression analysis. In this work we consider one-dimensional regression with Wiener-Lévy (Brownian motion) covariance functions. We indicate that they require only O(n) number of multiplications and show how one can utilize input deformations in order to define a much broader class of efficient covariance functions suitable for discontinuity-preserving filtering. An example of the selective smoothing is presented which shows that regression with Brownian motion filters outperforms or improves nonlinear diffusion filtering especially when observations are contaminated with noise of larger variance.