The increasing availability of time series data depicting the evolution of physical system properties has prompted the development of methods focused on extracting insights into the system behavior over time, discerning whether it stems from deterministic or stochastic dynamical systems. Surrogate data testing plays a crucial role in this process by facilitating robust statistical assessments. This ensures that the observed results are not mere occurrences by chance, but genuinely reflect the inherent characteristics of the underlying system. The initial process involves formulating a null hypothesis, which is tested using surrogate data in cases where assumptions about the underlying distributions are absent. A discriminating statistic is then computed for both the original data and each surrogate data set. Significantly deviating values between the original data and the surrogate data ensemble lead to the rejection of the null hypothesis. In this work, we present various surrogate methods designed to assess specific statistical properties in random processes. Specifically, we introduce methods for evaluating the presence of autodependencies and nonlinear dynamics within individual processes, using Information Storage as a discriminating statistic. Additionally, methods are introduced for detecting coupling and nonlinearities in bivariate processes, employing the Mutual Information Rate for this purpose. The surrogate methods introduced are first tested through simulations involving univariate and bivariate processes exhibiting both linear and nonlinear dynamics. Then, they are applied to physiological time series of Heart Period (RR intervals) and respiratory flow (RESP) variability measured during spontaneous and paced breathing. Simulations demonstrated that the proposed methods effectively identify essential dynamical features of stochastic systems. The real data application showed that paced breathing, at low breathing rate, increases the predictability of the individual dynamics of RR and RESP and dampens nonlinearity in their coupled dynamics.

Objective. In this work, an analytical framework for the multiscale analysis of multivariate Gaussian processes is presented, whereby the computation of Partial Information Decomposition measures is achieved accounting for the simultaneous presence of short-term dynamics and long-range correlations. Approach. We consider physiological time series mapping the activity of the cardiac, vascular and respiratory systems in the field of Network Physiology. In this context, the multiscale representation of transfer entropy within the network of interactions among Systolic arterial pressure (S), respiration (R) and heart period (H), as well as the decomposition into unique, redundant and synergistic contributions, is obtained using a Vector AutoRegressive Fractionally Integrated (VARFI) framework for Gaussian processes. This novel approach allows to quantify the directed information flow accounting for the simultaneous presence of short-term dynamics and long-range correlations among the analyzed processes. Additionally, it provides analytical expressions for the computation of the information measures, by exploiting the theory of state space models. The approach is first illustrated in simulated VARFI processes and then applied to H, S and R time series measured in healthy subjects monitored at rest and during mental and postural stress. Main Results. We demonstrate the ability of the VARFI modeling approach to account for the coexistence of short-term and long-range correlations in the study of multivariate processes. Physiologically, we show that postural stress induces larger redundant and synergistic effects from S and R to H at short time scales, while mental stress induces larger information transfer from S to H at longer time scales, thus evidencing the different nature of the two stressors. Significance. The proposed methodology allows to extract useful information about the dependence of the information transfer on the balance between short-term and long-range correlations in coupled dynamical systems, which cannot be observed using standard methods that do not consider long-range correlations.

Complex networks of stochastic processes are crucial for modeling the dynamics of interacting systems, particularly those involving high-order interactions (HOIs) among three or more components. Traditional measures—such as mutual information (MI), interaction information (II), the redundancy-synergy index (RSI), and O-information (OI)—are typically limited to static analyses not accounting for temporal correlations and become computationally unfeasible in large networks due to the exponential growth of the number of interactions to be analyzed. To address these challenges, first a framework is introduced to extend these information-theoretic measures to dynamic processes. This includes the II rate (IIR), RSI rate (RSIR), and the OI rate gradient ($\Delta \mathrm{OIR}$), enabling the dynamic analysis of HOIs. Moreover, a stepwise strategy identifying groups of nodes (multiplets) that maximize either redundant or synergistic HOIs is devised, offering deeper insights into complex interdependencies. The framework is validated through simulations of networks composed of cascade, common drive, and common target mechanisms, modelled using vector autoregressive (VAR) processes. The feasibility of the proposed approach is demonstrated through its application in climatology, specifically by analyzing the relationships between climate variables that govern El Niño and the Southern Oscillation (ENSO) using historical climate data.

Complex networks of stochastic processes are crucial for modelling the dynamics of interacting systems, particularly those involving High-Order Interactions (HOIs) among three or more components. Traditional measures—such as mutual information (MI), interaction information (II), the redundancy-synergy index (RSI), and O-information (OI)—are typically limited to static analyses not accounting for temporal correlations and become computationally unfeasible in large networks due to the exponential growth of the number of interactions to be analyzed. To address these challenges, first a framework is introduced to extend these information-theoretic measures to dynamic processes. This includes the II rate (IIR), RSI rate (RSIR), and the OI rate gradient (ΔOIR), enabling the dynamic analysis of HOIs. Moreover, a stepwise strategy identifying groups of nodes (multiplets) that maximize either redundant or synergistic HOIs is devised, offering deeper insights into complex interdependencies. The framework is validated through simulations of networks composed of cascade, common drive, and common target mechanisms, modelled using Vector AutoRegressive (VAR) processes. The feasibility of the proposed approach is demonstrated through its application in climatology, specifically by analyzing the relationships between climate variables that govern El Niño and the Southern Oscillation (ENSO) using historical climate data.