Probabilistic-based approach to optimal filtering
The signal-to-noise ratio maximizing approach in optimal filtering provides a robust tool to detect signals in the presence of colored noise. The method fails, however, when the data present a regimelike behavior. An approach is developed in this manuscript to recover local (in phase space) behavior in an intermittent regimelike behaving system. The method is first formulated in its general form within a Gaussian framework, given an estimate of the noise covariance, and demands that the signal corresponds to minimizing the noise probability distribution for any given value, i.e., on isosurfaces, of the data probability distribution. The extension to the non-Gaussian case is provided through the use of finite mixture models for data that show regimelike behavior. The method yields the correct signal when applied in a simplified manner to synthetic time series with and without regimes, compared to the signal-to-noise ratio approach, and helps identify the right frequency of the oscillation spells in the classical and variants of the Lorenz system.