🤖 AI Summary
This work addresses heteroscedastic stochastic time series by proposing a dual-signal decomposition framework that disentangles the original series into three components: mean, dispersion (i.e., time-varying volatility), and stationary white noise. Methodologically, it employs a dual-output neural network—or equivalently, a nonlinear optimization model—to jointly model the dynamics of mean and dispersion. An adaptive regularization weighting mechanism, grounded in statistical process control, is introduced to balance learning objectives. Crucially, first- and second-order temporal derivative regularizers are integrated to enforce signal smoothness, suppress noise, and preserve structural integrity. Two complementary learning paradigms—joint learning and sequential learning—are developed to enable collaborative modeling, cross-effect analysis, and multi-series structural comparison in a 2D signal space. Experiments demonstrate accurate separation of uncorrelated noise, robust handling of abrupt and smooth regime shifts, substantial improvements in predictive accuracy and interpretability for both signals, and strong generalization and extensibility across diverse applications.
📝 Abstract
The research paper addresses decomposition of a stochastic time series into three time series representing a dual signal i.e., the mean and the dispersion, with noise isolated. Decomposition is done by applying machine learning to fit a 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.