arXiv subjects

X. San Liang

Publications and source records attributed to X. San Liang.

16 recordsLinked to original sources

arXiv2025

Identifying the net information flow direction pattern in mutually coupled non-identical chaotic oscillators

This paper focuses on a fundamental inquiry in a coupled oscillator model framework. It specifically addresses the direction of net information flow in mutually coupled non-identical chaotic oscillators. Adopting a specific form of conditional mutual information as a model-free and asymmetric index, we establish that if the magnitude of the maximum Lyapunov exponent can be defined as the 'degree of chaos' of a given isolated chaotic system, a predominant net information transfer exists from the oscillator exhibiting a higher degree of chaos to the other while they are coupled. We incorporate two distinct categories of coupled 'non-identical' oscillators to strengthen our claim. In the first category, both oscillators share identical functional forms, differing solely in one parameter value. We also adopt another measure, the Liang-Kleeman information flow, to support the generality of our results. The functional forms of the interacting oscillators are entirely different in the second category. We further extend our study to the coupled oscillator models, where the interacting oscillators possess different dimensions in phase space. These comprehensive analyses support the broad applicability of our results.

nlin.CD↗

arXiv2024

Quantitative causality, causality-guided scientific discovery, and causal machine learning

It has been said, arguably, that causality analysis should pave a promising way to interpretable deep learning and generalization. Incorporation of causality into artificial intelligence (AI) algorithms, however, is challenged with its vagueness, non-quantitiveness, computational inefficiency, etc. During the past 18 years, these challenges have been essentially resolved, with the establishment of a rigorous formalism of causality analysis initially motivated from atmospheric predictability. This not only opens a new field in the atmosphere-ocean science, namely, information flow, but also has led to scientific discoveries in other disciplines, such as quantum mechanics, neuroscience, financial economics, etc., through various applications. This note provides a brief review of the decade-long effort, including a list of major theoretical results, a sketch of the causal deep learning framework, and some representative real-world applications in geoscience pertaining to this journal, such as those on the anthropogenic cause of global warming, the decadal prediction of El Niño Modoki, the forecasting of an extreme drought in China, among others.

cs.AI↗

arXiv2023

Quantitative causality analysis with coarsely sampled time series

The information flow-based quantitative causality analysis has been widely applied in different disciplines because of its origin from first principles, its concise form, and its computational efficiency. So far the algorithm for its estimation is based on differential dynamical systems, which, however, may make an issue for coarsely sampled time series. Here, we show that for linear systems, this is fine at least qualitatively; but for highly nonlinear systems, the bias increases significantly as the sampling frequency is reduced. This paper provides a partial solution to this problem, showing how causality analysis is assured faithful with coarsely sampled series when, of course, the statistics is sufficient. An explicit and concise formula has been obtained, with only sample covariances involved. It has been successfully applied to a system comprising of a pair of coupled Rössler oscillators. Particularly remarkable is the success when the two oscillators are nearly synchronized.

nlin.AO↗

arXiv2021

An overview of the quantitative causality analysis and causal graph reconstruction based on a rigorous formalism of information flow

Inference of causal relations from data now has become an important field in artificial intelligence. During the past 16 years, causality analysis (in a quantitative sense) has been developed independently in physics from first principles. This short note is a brief summary of this line of work, including part of the theory and several representative applications.

eess.SY↗

arXiv2021

The causal interaction between the subnetworks of a complex network

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system, in order to investigate problems such as the effective connectivity between two brain regions, each with millions of neurons involved. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems (e.g., the human brain), and is expected to have applications in many real world problems in different disciplines such as neuroscience, climate science, fluid dynamics, financial economics, to name a few.

q-bio.NC↗

arXiv2021

Normalized multivariate time series causality analysis and causal graph reconstruction

Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as real physical notion so as to formulate it from first principles, however, seems to go unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized, and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a huge challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.

cs.AI↗

arXiv2021

El Niño Modoki thus far can be mostly predicted more than 10 years ahead of time

The 2014-2015 "Monster"/"Super" El Niño failed to be predicted one year earlier due to the growing importance of a new type of El Niño, El Niño Modoki, which reportedly has much lower forecast skill with the classical models. In this study, we show that, so far as of today, this new El Niño actually can be mostly predicted at a lead time of more than 10 years. This is achieved through tracing the predictability source with an information flow-based causality analysis, which is rigorously established from first principles in the past decade. We show that the information flowing from the solar activity 45 years ago to the sea surface temperature results in a causal structure resembling the El Niño Modoki mode. Based on this, a multidimensional system is constructed out of the sunspot number series with time delays of 22-50 years. The first 25 principal components are then taken as the predictors to fulfill the prediction, which through information flow-based causal deep learning reproduces rather accurately the events 12 years in advance.

physics.ao-ph↗

arXiv2021

Measuring the importance of individual units in producing the collective behavior of a complex network

A quantitative evaluation of the contribution of individual units in producing the collective behavior of a complex network can allow us to understand the potential damage to the structure integrity due to the failure of local nodes. Given time series for the units, a natural way to do this is to find the information flowing from the unit of concern to the rest of the network. In this study, we show that this flow can be rigorously derived in the setting of a continuous-time dynamical system. With a linear assumption, a maximum likelihood estimator can be obtained, allowing us to estimate it in an easy way. As expected, this "cumulative information flow" does not equal to the sum of the information flows to other individual units, reflecting the collective phenomenon that a group is not the addition of the individual members. For the purpose of demonstration and validation, we have examined a network made of Stuart-Landau oscillators. Depending on the topology, the computed information flow may differ. In some situations, the most crucial nodes for the network are not the hubs; they may have low degrees, and, if depressed or attacked, will cause the failure of the entire network.

physics.soc-ph↗

arXiv2020

A note on causation versus correlation

Recently, it has been shown that the causality and information flow between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event $A$ taking a harmonic form (sine/cosine), and it generates through some process another event $B$ so that $B$ always lags $A$ by a phase of $π/2$. Here the causality is obviously seen, while by computation the correlation is, however, zero. This seemingly contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincaré section; it does not add information. That is to say, though the absolute information flow from $A$ to $B$ is zero, i.e., $T_{A\to B}=0$, the total information increase of $B$ is also zero, so the normalized $T_{A\to B}$, denoted as $τ_{A\to B}$, takes the form of $\frac 0 0$. By slightly perturbating the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that $τ_{A\to B}$ approaches 100\%, just as one would have expected.

physics.data-an↗

arXiv2016

Canonical transfer and multiscale energetics for primitive and quasi-geostrophic atmospheres

The past years have seen the success of a novel multiscale energetic formalism in a variety of ocean and engineering fluid applications. In a self-contained way, this study introduces it to the atmospheric dynamical diagnostics, with important theoretical updates. Multiscale energy equations are derived using a new analysis apparatus, namely, multiscale window transform, with respect to both the primitive equation and quasi-geostrophic models. A reconstruction of the "atomic" energy fluxes on the multiple scale windows allows for a natural and unique separation of the in-scale transports and cross-scale transfers from the intertwined nonlinear processes. The resulting energy transfers bear a Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian mechanics, we hence would call them "canonical". A canonical transfer process is a mere redistribution of energy among scale windows, without generating or destroying energy as a whole. By classification, a multiscale energetic cycle comprises of available potential energy (APE) transport, kinetic energy (KE) transport, pressure work, buoyancy conversion, work done by external forcing and friction, and the cross-scale canonical transfers of APE and KE which correspond respectively to the baroclinic and barotropic instabilities, among others, in geophysical fluid dynamics. A buoyancy conversion takes place in an individual window only, bridging the two types of energy namely KE and APE, it does not involve any processes among different scale windows, and is hence basically not related to instabilities. This formalism is exemplified with a preliminary application to the Madden-Julian Oscillation (MJO) study.

physics.ao-ph↗

arXiv2015

Information flow and causality as rigorous notions ab initio

Information flow (or information transfer as may be called) the widely applicable general physics notion can be rigorously derived from first principles, rather than axiomatically proposed as an ansatz. Its logical association with causality and, particularly, the most stringent one-way causality, if existing, is firmly substantiated and stated as a fact in proved theorems. Established in this study are the information flows among the components of time-discrete mappings and time-continuous dynamical systems of arbitrary dimensionality, both deterministic and stochastic. They have been obtained explicitly in closed form, and all possess the property of causality, which reads: if a component, say $x_i$, has an evolutionary law independent of $x_j$, then the information flow from $x_j$ to $x_i$ vanishes. These results have been put to applications with benchmark systems, such as the Kaplan-Yorke map, the Rössler system, the baker transformation, the Hénon map, and a stochastic potential flow. Besides recovering the properties as expected from the respective systems, some of the applications show that the information flow structure underlying a complex trajectory pattern could be tractable. For linear systems, the resulting remarkably concise formula asserts analytically that causation implies correlation, while correlation does not imply causation, resolving unambiguously the long-standing debate over causation versus correlation.

nlin.CD↗

arXiv2015

Normalizing the causality between time series

Recently, a rigorous yet concise formula has been derived to evaluate the information flow, and hence the causality in a quantitative sense, between time series. To assess the importance of a resulting causality, it needs to be normalized. The normalization is achieved through distinguishing three types of fundamental mechanisms that govern the marginal entropy change of the flow recipient. A normalized or relative flow measures its importance relative to other mechanisms. In analyzing realistic series, both absolute and relative information flows need to be taken into account, since the normalizers for a pair of reverse flows belong to two different entropy balances; it is quite normal that two identical flows may differ a lot in relative importance in their respective balances. We have reproduced these results with several autoregressive models. We have also shown applications to a climate change problem and a financial analysis problem. For the former, reconfirmed is the role of the Indian Ocean Dipole as an uncertainty source to the El Niño prediction. This might partly account for the unpredictability of certain aspects of El Niño that has led to the recent portentous but spurious forecasts of the 2014 "Monster El Niño". For the latter, an unusually strong one-way causality has been identified from IBM (International Business Machines Corporation) to GE (General Electric Company) in their early era, revealing to us an old story, which has almost gone to oblivion, about "Seven Dwarfs" competing a giant for the mainframe computer market.

physics.data-an↗

arXiv2014

Causality between time series

Given two time series, can one tell, in a rigorous and quantitative way, the cause and effect between them? Based on a recently rigorized physical notion namely information flow, we arrive at a concise formula and give this challenging question, which is of wide concern in different disciplines, a positive answer. Here causality is measured by the time rate of change of information flowing from one series, say, X2, to another, X1. The measure is asymmetric between the two parties and, particularly, if the process underlying X1 does not depend on X2, then the resulting causality from X2 to X1 vanishes. The formula is tight in form, involving only the commonly used statistics, sample covariances. It has been validated with touchstone series purportedly generated with one-way causality. It has also been applied to the investigation of real world problems; an example presented here is the cause-effect relation between two climate modes, El Niño and Indian Ocean Dipole, which have been linked to the hazards in far flung regions of the globe, with important results that would otherwise be difficult, if not impossible, to obtain.

stat.ME↗

arXiv2010

Remarkable evolutionary laws of absolute and relative entropies with dynamical systems

The evolution of entropy is derived with respect to dynamical systems. For a stochastic system, its relative entropy $D$ evolves in accordance with the second law of thermodynamics; its absolute entropy $H$ may also be so, provided that the stochastic perturbation is additive and the flow of the vector field is nondivergent. For a deterministic system, $dH/dt$ is equal to the mathematical expectation of the divergence of the flow (a result obtained before), and, remarkably, $dD/dt = 0$. That is to say, relative entropy is always conserved. So, for a nonlinear system, though the trajectories of the state variables, say $\ve x$, may appear chaotic in the phase space, say $Ω$, those of the density function $ρ(\ve x)$ in the new ``phase space'' $L^1(Ω)$ are not; the corresponding Lyapunov exponent is always zero. This result is expected to have important implications for the ensemble predictions in many applied fields, and may help to analyze chaotic data sets.

nlin.CD↗

arXiv2007

Information flow within stochastic dynamical systems

Information flow or information transfer is an important concept in dynamical systems which has applications in a wide variety of scientific disciplines. In this study, we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system. The resulting measure of of information transfer possesses a property of transfer asymmetry and, when the stochastic perturbation to the receiving component does not rely on the giving component, has a form same as that for the corresponding deterministic system. An application with a two-dimensional system is presented, and the resulting transfers are just as expected. A remarkable observation is that, for two highly correlated time series, there could be no information transfer from one certain series, say $x_2$, to the other ($x_1$). That is to say, the evolution of $x_1$ may have nothing to do with $x_2$, even though $x_1$ and $x_2$ are highly correlated. Information transfer analysis thus extends the traditional notion of correlation analysis by providing a quantitative measure of causality between time series.

nlin.CD↗

arXiv2007

Perfect energy transfer and mean-eddy interaction in incompressible fluid flows

The mean-eddy interaction in a flow is investigated in terms of the energy transfer between its mean and eddy processes. In a Reynolds decomposition framework, energy transfer can be defined from the decomposed energy equations, but the definition is subjective with the classical energetics formalism. This paper shows that a faithful formalism can be rigorously established within the same framework. The resulting transfer sums to zero everywhere over the decomposed subspaces, representing a mere redistribution of energy between the mean and eddy processes, without generating or destroying energy as a whole. For this reason, it has been referred to as perfect transfer in distinction from other energy transfers one may have encountered. The perfect transfer has a simple form in expression, with the introduction of an eddy flow coupled with the mean and eddy processes of the field under concern. The eddy or turbulence production is then totally determined by the divergence of this flow. This formalism has been validated with a well-known barotropic instability model, and a wake control problem. It is expected to be useful for the harnessing of turbulence, in that it may help to identify the best locations to place passive controls, or design the performance functional for active controls.

physics.flu-dyn↗