Dual Signal Decomposition of Stochastic Time Series

The decomposition of a stochastic time series into three component series representing a dual signal - namely, the mean and dispersion - while isolating noise is presented. The decomposition is performed by applying machine learning techniques to fit the dual signal. Machine learning minimizes the loss function which compromises between fitting the original time series and penalizing irregularities of the dual signal. The latter includes terms based on the first and second order derivatives along time. To preserve special patterns, weighting of the regularization components of the loss function has been introduced based on Statistical Process Control methodology. The proposed decomposition can be applied as a smoothing algorithm against the mean and dispersion of the time series. By isolating noise, the proposed decomposition can be seen as a denoising algorithm. Two approaches of the learning process have been considered: sequential and jointly. The former approach learns the mean signal first and then dispersion. The latter approach fits the dual signal jointly. Jointly learning can uncover complex relationships for the time series with heteroskedasticity. Learning has been set by solving the direct non-linear unconstrained optimization problem or by applying neural networks that have sequential or twin output architectures. Tuning of the loss function hyperparameters focuses on the isolated noise to be a stationary stochastic process without autocorrelation properties. Depending on the applications, the hyperparameters of the learning can be tuned towards either the discrete states by stepped signal or smoothed series. The decomposed dual signal can be represented on the 2D space and used to learn inherent structures, to forecast both mean and dispersion, or to analyze cross effects in case of multiple time series.