Normalizing the causality between time series

Liang, X. San

doi:10.1103/physreve.92.022126

Cited by 103 publications

(100 citation statements)

References 20 publications

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“…From Liang (, ), the rate of IF from X 2 (e.g., SM or sounding wind speeds) to X 1 (e.g., T2) can be defined as

T_{2 \to 1} = \frac{C_{11} C_{12} C_{2, d 1} - {C_{12}}^{2} C_{1, d 1}}{{C_{11}}^{2} C_{22} - C_{11} {C_{12}}^{2}}

where C ij is the sample covariance between X i and X j and C i , dj is the covariance between X i and

\overset{X_{j}}{true}; \overset{X_{j}}{true}

is the difference approximation of d X j /d t using the Euler forward scheme as defined in equation

C_{ij} = \overset{((), X_{i} - \overset{X_{i}}{true}) ((), X_{j} - \overset{X_{j}}{true})}{true}

C_{i, italicdj} = \overset{((), X_{i} - \overset{X_{i}}{true}) ((), {\overset{\cdot}{X}}_{j} - \overset{{\overset{\cdot}{X}}_{j}}{true})}{true}

\overset{X_{j}}{true} = \frac{X_{j, n + 1} - X_{j, n}}{∆ t}

…”

Section: Data and Analysis Methodsmentioning

confidence: 99%

“…It needs to be normalized in order to have its importance assessed. According to Liang (), the variation of X 1 is usually caused by three terms: the IF from X 2 to X 1 ( T 2 → 1 ), variation within itself (

\frac{{normald H}_{1}^{*}}{normald t}

), and the stochastic effect of noise (

\frac{{normald H}_{1}^{noise}}{normald t}

), where H 1 is the evolution of the marginal entropy of X 1 (see details in Liang, and 2014). To evaluate the importance of IF relative to the other two processes, τ 2 → 1 is proposed as

τ_{2 \to 1} = \frac{T_{2 \to 1}}{Z_{2 \to 1}}

where

Z_{2 \to 1} = (||, T_{2 \to 1}) + (||, \frac{{d H}_{1}^{*}}{d t}) + (||, \frac{{d H}_{1}^{noise}}{d t})

while

\frac{normald {H_{1}}^{*}}{normald t} = p

\frac{normald {H_{1}}^{noise}}{normald t} = \frac{∆ t}{2 C_{11}} ((), C_{d 1, d 1} + p^{2} C_{11} + q^{2} C_{22} - 2 {pC}_{d 1, 1} - 2 {qC}_{d 1, 2} + 2 italicpq C_{12})

p = C_{22} C_{1, italicdt}

…”

Section: Data and Analysis Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Does Soil Moisture Have an Influence on Near‐Surface Temperature?

Liu

2019

JGR Atmospheres

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The relationship between soil moisture (SM) and temperature at 2‐m height (T2) is examined with long‐term meteorological and soil observations and a single‐column model. With the information flow (IF) and correlation analysis methods, results show that the distribution of SM depends on land use and land cover, while an increase in vegetation fraction corresponds to an increase in SM, and this dependence decreases with soil depth. There is causality between T2 and SM at all soil levels. Compared with those in deeper soil, higher IF values and correlations appear in the top soil levels, implying a stronger interaction between the top soil layer and the atmosphere. Meanwhile, most correlation coefficient values rank weak or moderate, and negative feedback is predominant with low IF values, indicating that the causality between them is relatively weak and also proving that SM is not the sole factor that influences changes of near‐surface temperature. The single‐column model demonstrates that an increase (decrease) in SM will result in cooler (warmer) T2 through the redistribution of surface heat flux, while changes in SM in the top soil layers have a stronger influence on T2 compared to those in deep soil layers. The duration of the atmospheric temperature inversion layer is also prolonged (reduced) with an increase (decrease) in SM. Impacts of SM on T2 vary with land use or vegetation fraction and also are more significant in warm months and the daytime than in cold months and nighttime. Moreover, large‐scale and local circulations have steady influences on T2 in different weather conditions.

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“…From Liang (, ), the rate of IF from X 2 (e.g., SM or sounding wind speeds) to X 1 (e.g., T2) can be defined as

T_{2 \to 1} = \frac{C_{11} C_{12} C_{2, d 1} - {C_{12}}^{2} C_{1, d 1}}{{C_{11}}^{2} C_{22} - C_{11} {C_{12}}^{2}}

where C ij is the sample covariance between X i and X j and C i , dj is the covariance between X i and

\overset{X_{j}}{true}; \overset{X_{j}}{true}

is the difference approximation of d X j /d t using the Euler forward scheme as defined in equation

C_{ij} = \overset{((), X_{i} - \overset{X_{i}}{true}) ((), X_{j} - \overset{X_{j}}{true})}{true}

C_{i, italicdj} = \overset{((), X_{i} - \overset{X_{i}}{true}) ((), {\overset{\cdot}{X}}_{j} - \overset{{\overset{\cdot}{X}}_{j}}{true})}{true}

\overset{X_{j}}{true} = \frac{X_{j, n + 1} - X_{j, n}}{∆ t}

…”

Section: Data and Analysis Methodsmentioning

confidence: 99%

\frac{{normald H}_{1}^{*}}{normald t}

), and the stochastic effect of noise (

\frac{{normald H}_{1}^{noise}}{normald t}

), where H 1 is the evolution of the marginal entropy of X 1 (see details in Liang, and 2014). To evaluate the importance of IF relative to the other two processes, τ 2 → 1 is proposed as

τ_{2 \to 1} = \frac{T_{2 \to 1}}{Z_{2 \to 1}}

where

Z_{2 \to 1} = (||, T_{2 \to 1}) + (||, \frac{{d H}_{1}^{*}}{d t}) + (||, \frac{{d H}_{1}^{noise}}{d t})

while

\frac{normald {H_{1}}^{*}}{normald t} = p

\frac{normald {H_{1}}^{noise}}{normald t} = \frac{∆ t}{2 C_{11}} ((), C_{d 1, d 1} + p^{2} C_{11} + q^{2} C_{22} - 2 {pC}_{d 1, 1} - 2 {qC}_{d 1, 2} + 2 italicpq C_{12})

p = C_{22} C_{1, italicdt}

…”

Section: Data and Analysis Methodsmentioning

confidence: 99%

Does Soil Moisture Have an Influence on Near‐Surface Temperature?

Liu

2019

JGR Atmospheres

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“…[37,38] another decomposition is proposed to detect redundant and synergetic contributions of driving variables. Liang [39,40] presents a rigorous approach based on the underlying Langevin description of a system to define the contributions of internal and external driving to the evolution of the entropy of a subprocess Y . This approach is, however, based on the knowledge of the deterministic-stochastic equations of the system, but in principle it can also be estimated from time series alone involving numerical optimization problems.…”

Section: A Quantifying Causal Information Transfermentioning

confidence: 99%

Quantifying information transfer and mediation along causal pathways in complex systems

Runge

2015

Phys. Rev. E

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Measures of information transfer have become a popular approach to analyze interactions in complex systems such as the Earth or the human brain from measured time series. Recent work has focused on causal definitions of information transfer aimed at decompositions of predictive information about a target variable, while excluding effects of common drivers and indirect influences. While common drivers clearly constitute a spurious causality, the aim of the present article is to develop measures quantifying different notions of the strength of information transfer along indirect causal paths, based on first reconstructing the multivariate causal network. Another class of novel measures quantifies to what extent different intermediate processes on causal paths contribute to an interaction mechanism to determine pathways of causal information transfer. The proposed framework complements predictive decomposition schemes by focusing more on the interaction mechanism between multiple processes. A rigorous mathematical framework allows for a clear information-theoretic interpretation that can also be related to the underlying dynamics as proven for certain classes of processes. Generally, however, estimates of information transfer remain hard to interpret for nonlinearly intertwined complex systems. But if experiments or mathematical models are not available, then measuring pathways of information transfer within the causal dependency structure allows at least for an abstraction of the dynamics. The measures are illustrated on a climatological example to disentangle pathways of atmospheric flow over Europe.

show abstract

“…We overcame such limitation by using causality analyses to quantify the connectivity between variables. Such causality or information transfer has been defined within the framework of information theory [69,[84][85][86]. Three basic tenets of the information transfer are the following: (1) Causality implies correlation but correlation does not imply causality, (2) Causality implies directionality, which means that the transfer of information detects the direction of information transfer between two systems, and (3) Asymmetry is a basic property of information transfer.…”

Section: Appendix a Correlation And Causalitymentioning

confidence: 99%

“…A measure of relative flow of information [86], in terms of the marginal entropy, can be given as, Additionally, the two components of entropy (eqn. A7), estimated in terms of p and q are [86] p dt …”

Section: Appendix a Correlation And Causalitymentioning

confidence: 99%

<strong>New insights on land surface-atmosphere feedbacks over tropical South America at interannual timescales</strong>

Bedoya-Soto

Poveda

2017

Proceedings of First International Electronic Conference on the Hydrological Cycle

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Abstract:Using monthly data for the period 1979-2010, we study the dynamics and strength of land surface-atmosphere feedbacks (LAFs) among variables involved in the heat and moisture fluxes, at interannual timescales in Tropical South America (TropSA). The variables include precipitation, surface air temperature, specific humidity at 925 hPa, evaporation, and estimates of volumetric soil water content. We use a Maximum Covariance Analysis (MCA) to reduce the dimensionality and to rank the relative contributions to LAFs and group the time series into Maximum Covariance States (MCS) with common mechanisms among variables. We estimate linear (Pearson correlations) and non-linear association (information transfer or causality) metrics among pairs of variables to configure the structure of linkages. The main MCS associated with LAFs over TropSA are strongly influenced by ENSO, and the meridional and equatorial SSTs modes over the Atlantic and Indian Oceans. ENSO favors a unimodal behavior, with center of action in the Amazon River basin, while the SSTs mode over the Tropical North Atlantic (TNA) results in a dipole between northern and southern TropSA. Our results show that soil moisture plays a leading role in regulating heat and water anomalies, and provides the memory of the LAFs and their subsequent influence. Thus, volumetric soil water becomes a fundamental variable leading up to 9 month-lags. ENSO enhances the interannual connectivity and memory inside LAF mechanisms with respect to other modes. Within the identified multivariate structure, evaporation and soil moisture enhance the interannual connectivity of the whole set of variables since both variables exhibit more frequent two-way feedbacks with the remaining variables.

show abstract

Normalizing the causality between time series

Cited by 103 publications

References 20 publications

Does Soil Moisture Have an Influence on Near‐Surface Temperature?

Does Soil Moisture Have an Influence on Near‐Surface Temperature?

Quantifying information transfer and mediation along causal pathways in complex systems

<strong>New insights on land surface-atmosphere feedbacks over tropical South America at interannual timescales</strong>

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