DOCSLIB.ORG

Discovery of Causal Time Intervals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Discovery of Causal Time Intervals Discovery of Causal Time Intervals Zhenhui Li∗ Guanjie Zheng∗ Amal Agarwaly Lingzhou Xuey Thomas Lauvauxz Abstract Granger test X causes Y: rejected Causality analysis, beyond \mere" correlations, has be- X[t0:t1] causes Y[t0:t1]: accepted come increasingly important for scientific discoveries X and policy decisions. Many of these real-world appli- cations involve time series data. A key observation is Y that the causality between time series could vary signif- icantly over time. For example, a rain could cause severe t0 t1 time traffic jams during the rush hours, but has little impact on the traffic at midnight. However, previous studies Figure 1: An example illustrating the causality in mostly look at the whole time series when determining partial time series. The original Granger test fails to the causal relationship between them. Instead, we pro- detect the causal relationship, since X causes Y only pose to detect the partial time intervals with causality. during the time interval [t0 : t1]. Our goal is to find As it is time consuming to enumerate all time intervals such time intervals. and test causality for each interval, we further propose an efficient algorithm that can avoid unnecessary com- causes Y by Granger test if the full model is significantly putations based on the bounds of F -test in the Granger better than the reduced model (i.e., it is critical to use causality test. We use both synthetic datasets and real the values of X to predict Y ). datasets to demonstrate the efficiency of our pruning However, in real-world scenarios, causal relation- techniques and that our method can effectively discover ships may exist only in partial time series. This is interesting causal intervals in the time series data. illustrated in Figure 1, where X causes Y only during the time interval [t0 : t1]. In such cases, if we apply the 1 Introduction traditional Granger test (i.e., fitting the models using In recent years, driven by a wide range of real-world the whole time series), we may not detect any causal- applications and scientific discoveries, detecting causal- ity between X and Y . However, if we only test on the ity from the data has gained increasing attention in partial time series in interval [t0 : t1], the causality can the data mining community. Time series data is a be discovered. Therefore, given two time series X and type of data frequently seen in many applications such Y , our goal in this paper is to find the causal time in- as stock, traffic, and climate. Various methods have tervals in which X causes Y . This is highly motivated been developed to determine the causality between two by a number of real world applications. Consider the time series. Among all causality test methods, Granger following two examples. causality test [10] is one of the most popular meth- ods [25, 2, 20, 18, 17]. EXAMPLE 1.1. Suppose that a local event (e.g., a In short, the idea of Granger test is based on football game) attracts many people, generating a large volume of traffic at the venue (e.g., a football stadium) prediction: Given a pair of time series X = x1x2 : : : xT during the event hours. In such scenario, time series and Y = y1y2 : : : yT , two different predictive models, called the full model and the reduced model, are fitted X could be the frequency of geo-tagged tweets at the using the whole time series (i.e., all data points). The venue (as a proxy of the popularity of local events), and full model uses the past values of X and the past values time series Y could be the taxi pick-up volume at the of Y to predict Y , whereas the reduced model uses past same location. Intuitively, we can say that the tweet values of Y only to predict Y . Then, we say that X frequency X (which represents the people attending the event) causes traffic volume Y during the time of the event. However, for the rest of the time, tweet frequency ∗College of Information Sciences and Technology, Pennsylvania State University. Email: [email protected] does not necessarily cause the pick-up volume. In fact, yDepartment of Statistics, Pennsylvania State University. during the normal days, the pick-ups are mostly due zDepartment of Meteorology, Pennsylvania State University. to human routine behaviors (e.g., commuting between home and work) and may not have any correlation with computations for the Granger test over similar time the frequency of geo-tagged tweets. intervals. • We conduct experiments on both synthetic data EXAMPLE 1.2. Nowadays, environmental pollution and real data to demonstrate the effectiveness and is a big concern in people's daily life. For example, efficiency of the proposed method. poor air quality in big cities in China [27, 28] and po- tential air/water contamination from shale gas devel- Before proceeding, we emphasize that our work opment [15, 22] in the United States have drawn a lot is built upon the Granger test. We do not claim to of attention lately. Here, one biggest scientific question improve the Granger test itself, which is known to have is whether certain type of emissions (e.g., CO2) gener- some weaknesses (more details in Section 2). Instead, ated by industrial, vehicular, or other human activities we focus on an important yet unaddressed problem in causes the environmental change. But the causal rela- practice, that is, discovering partial time intervals with tionship between an emission source and the air quality Granger causality. may vary over time. For example, vehicle emissions re- act in the presence of sunlight and form ground level 2 Related Work ozone, a primary ingredient of smog, whereas industrial Dependency/causality discovery on time series. emissions have a bigger impact on the air quality in cool Many methods have been proposed to discover the tem- and humid days. poral dependency between time series, such as autocor- relation, cross-correlation [7], transfer entropy [6], ran- To detect all the casual time intervals, a naive way domization test [16], phase slope index [21]. However, is to enumerate all the time intervals of the time series temporal correlation or dependency does not necessar- and conduct Granger test for each interval. But such an ily indicate causality. Granger causality test [10] was enumeration could be very time consuming in practice. first introduced in the area of econometrics for time se- To address this challenge, in this paper we propose an ries analysis and has since gained tremendous success efficient algorithm for causal time interval discovery. across many domains due to its simplicity and robust- Here, our key insight is that similar time intervals (i.e., ness [12, 13, 3, 8]. intervals with significant overlaps) are likely to have Weakness of the Granger test. A well-known diffi- similar Granger test results. In fact, given the fitted culty in causality reasoning (including the Granger test) models of a time interval, we can derive useful upper and is the confounding issue, that is, spurious causality may lower bounds for the statistical significance of Granger be detected if both processes X and Y are impacted by causality for similar time intervals. Such bounds can a third (possibly unknown) process (i.e., a confounder). be used to quickly identify intervals that will definitely In [14], variables which mediate the impact of an ac- pass or fail the test without fitting the actual models. tion on the outcome are identified to reduce the bias We have conducted experiments to verify the ef- in assessing the causal effect for online advertising. [5] fectiveness and efficiency of our method on two inter- exploits confounder path delays in an attempt to cancel esting real-world datasets. First, we study the causal- out the spurious confounder effect. However, a princi- ity relationship between local event (represented by the pled solution to this problem remains elusive. frequency of geo-tagged tweets at the venue) and the traffic volume (represented by the taxi drop-offs/pick- Variations of the Granger test. While the original ups). We have observed that, when there is an event, Granger causality test was designed for two time series, taxi drop-offs cause geo-tagged tweets, and geo-tagged several methods [2, 18, 17] have been proposed to an- tweets cause pick-ups. Second, we analyze variables of alyze time series data involving many features and to climate and their causal relationship with temperature. learn a causal graph structure. Following the work [2], We have observed that the causality varies over time [20] detects causality of spatial time series, [18] pro- and is different for different variables. poses to use hidden Markov Random Field method, [17] handles extreme values in time series, [4] detects In summary, our key contributions are as follows: Granger causality from irregular time series, and [5] • To the best of our knowledge, we are the first to presents Copula-Granger method to efficiently capture study the problem of detecting partial time inter- non-linearity in the data. Learning temporal causal vals in which a causal relationship exists between graph has been applied to biology applications [25], cli- two time series based on the Granger test. mate analysis [9], microbiology [19], fMRI data analy- sis [24], anomaly detection [23], and longitudinal anal- • We propose an efficient algorithm which utilizes ysis [26]. However, none of these work address the is- the bounds of regression error to avoid unnecessary sues that causality could change over time, and that causality only exists in partial time intervals. Previ- (i.e., the number of independent variables) in the models ous study [1] uses a fixed-length sliding window to test Mf and Mr, respectively, and T is the length of the time Granger causality, but they cannot detect all the causal series.
Load more

Recommended publications
  • Neural Granger Causality
    1 Neural Granger Causality Alex Tank*, Ian Covert*, Nick Foti, Ali Shojaie, Emily B. Fox Abstract—While most classical approaches to Granger causality detection assume linear dynamics, many interactions in real-world applications, like neuroscience and genomics, are inherently nonlinear. In these cases, using linear models may lead to inconsistent estimation of Granger causal interactions. We propose a class of nonlinear methods by applying structured multilayer perceptrons (MLPs) or recurrent neural networks (RNNs) combined with sparsity-inducing penalties on the weights. By encouraging specific sets of weights to be zero—in particular, through the use of convex group-lasso penalties—we can extract the Granger causal structure. To further contrast with traditional approaches, our framework naturally enables us to efficiently capture long-range dependencies between series either via our RNNs or through an automatic lag selection in the MLP. We show that our neural Granger causality methods outperform state-of-the-art nonlinear Granger causality methods on the DREAM3 challenge data. This data consists of nonlinear gene expression and regulation time courses with only a limited number of time points. The successes we show in this challenging dataset provide a powerful example of how deep learning can be useful in cases that go beyond prediction on large datasets. We likewise illustrate our methods in detecting nonlinear interactions in a human motion capture dataset. Index Terms—time series, Granger causality, neural networks, structured sparsity, interpretability F 1 INTRODUCTION In many scientific applications of multivariate time se- Most classical model-based methods assume linear time ries, it is important to go beyond prediction and forecast- series dynamics and use the popular vector autoregressive ing and instead interpret the structure within time series.
    [Show full text]
  • The Granger-Causality Between Transportation and GDP: a Panel Data Approach
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Beyzatlar, Mehmet Aldonat; Karacal, Müge; Yetkiner, I. Hakan Working Paper The Granger-Causality between Transportation and GDP: A Panel Data Approach Working Papers in Economics, No. 12/03 Provided in Cooperation with: Department of Economics, Izmir University of Economics Suggested Citation: Beyzatlar, Mehmet Aldonat; Karacal, Müge; Yetkiner, I. Hakan (2012) : The Granger-Causality between Transportation and GDP: A Panel Data Approach, Working Papers in Economics, No. 12/03, Izmir University of Economics, Department of Economics, Izmir This Version is available at: http://hdl.handle.net/10419/175921 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Working Papers in Economics The Granger-Causality between Transportation and GDP: A Panel Data Approach Mehmet Aldonat Beyzatlar, Dokuz Eylül University Müge Karacal, Izmir University of Economics I.
    [Show full text]
  • GRANGER CAUSALITY and U.S. CROP and LIVESTOCK PRICES Rod F
    SOUTHERN JOURNAL OF AGRICULTURAL ECONOMICS JULY, 1984 GRANGER CAUSALITY AND U.S. CROP AND LIVESTOCK PRICES Rod F. Ziemer and Glenn S. Collins Abstract then discussed. Finally, some concluding re- Agricultural economists have recently been marks are offered. attracted to procedures suggested by Granger CAUSALITY: TESTABLE DEFINITIONS and others which allow observed data to reveal causal relationships. Results of this study in- Can observed correlations be used to suggest dicate that "causality" tests can be ambiguous or infer causality? This question lies at the heart in identifying behavioral relationships between of the recent causality literature. Jacobs et al. agricultural price variables. Caution is sug- contend that the null hypothesis commonly gested when using such procedures for model tested is necessary but not sufficient to imply choice. that one variable "causes" another. Further- more, the authors show that exogenity is not Key words: Granger causality, causality tests, empirically testable and that "informativeness" autoregressive processes is the only testable definition of causality. This Economists have long been concerned with testable definition is commonly known as caus- the issue of causality versus correlation. Even ality in the Granger sense. Although Granger beginning economics students are constantly (1969) never suggested that this testable defi- reminded that economic variables can be cor- nition of causality could be used to infer ex- related without being causally related. Re- ogenity, many researchers think of exogenity cently, testable definitions of causality have been when seeing the common parlance "test for suggested by suggestedGranger by(1969,(196, 1980), Sims andand casuality."Granger (1980) and Zellner provide others. These causality tests have been applied further discussion of definitions of "causality" by agricultural economists in livestock markets and Engle et al.
    [Show full text]
  • A Brief Introduction to Temporality and Causality
    A Brief Introduction to Temporality and Causality Kamran Karimi D-Wave Systems Inc. Burnaby, BC, Canada [email protected] Abstract Causality is a non-obvious concept that is often considered to be related to temporality. In this paper we present a number of past and present approaches to the definition of temporality and causality from philosophical, physical, and computational points of view. We note that time is an important ingredient in many relationships and phenomena. The topic is then divided into the two main areas of temporal discovery, which is concerned with finding relations that are stretched over time, and causal discovery, where a claim is made as to the causal influence of certain events on others. We present a number of computational tools used for attempting to automatically discover temporal and causal relations in data. 1. Introduction Time and causality are sources of mystery and sometimes treated as philosophical curiosities. However, there is much benefit in being able to automatically extract temporal and causal relations in practical fields such as Artificial Intelligence or Data Mining. Doing so requires a rigorous treatment of temporality and causality. In this paper we present causality from very different view points and list a number of methods for automatic extraction of temporal and causal relations. The arrow of time is a unidirectional part of the space-time continuum, as verified subjectively by the observation that we can remember the past, but not the future. In this regard time is very different from the spatial dimensions, because one can obviously move back and forth spatially.
    [Show full text]
  • Vargranger — Pairwise Granger Causality Tests After Var Or Svar
    Title stata.com vargranger — Pairwise Granger causality tests after var or svar Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description vargranger performs a set of Granger causality tests for each equation in a VAR, providing a convenient alternative to test; see[ R] test. Quick start Perform a Granger causality test after var or svar vargranger Perform a Granger causality test on vector autoregression estimation results stored as myest vargranger, estimates(myest) Menu Statistics > Multivariate time series > VAR diagnostics and tests > Granger causality tests 1 2 vargranger — Pairwise Granger causality tests after var or svar Syntax vargranger , estimates(estname) separator(#) vargranger can be used only after var or svar; see [TS] var and [TS] var svar. collect is allowed; see [U] 11.1.10 Prefix commands. Options estimates(estname) requests that vargranger use the previously obtained set of var or svar estimates stored as estname. By default, vargranger uses the active results. See[ R] estimates for information on manipulating estimation results. separator(#) specifies how often separator lines should be drawn between rows. By default, separator lines appear every K lines, where K is the number of equations in the VAR under analysis. For example, separator(1) would draw a line between each row, separator(2) between every other row, and so on. separator(0) specifies that lines not appear in the table. Remarks and examples stata.com After fitting a VAR, we may want to know whether one variable “Granger-causes” another (Granger 1969). A variable x is said to Granger-cause a variable y if, given the past values of y, past values of x are useful for predicting y.
    [Show full text]
  • A Non-Linear Granger Causality Framework to Investigate Climate–Vegetation Dynamics Christina Papagiannopoulou1, Diego G
    Geosci. Model Dev. Discuss., doi:10.5194/gmd-2016-266, 2016 Manuscript under review for journal Geosci. Model Dev. Published: 16 November 2016 c Author(s) 2016. CC-BY 3.0 License. A non-linear Granger causality framework to investigate climate–vegetation dynamics Christina Papagiannopoulou1, Diego G. Miralles2,3, Niko E. C. Verhoest3, Wouter A. Dorigo4, and Willem Waegeman1 1Depart. of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Belgium, 2Depart. of Earth Sciences, VU University Amsterdam, the Netherlands 3Laboratory of Hydrology and Water Management, Ghent University, Belgium 4Depart. of Geodesy and Geo-Information, Vienna University of Technology, Austria Correspondence to: Christina Papagiannopoulou ([email protected]) Abstract. Satellite Earth observation has led to the creation of global climate data records of many important environmental and climatic variables. These take the form of multivariate time series with different spatial and temporal resolutions. Data of this kind provide new means to unravel the influence of climate on vegetation dynamics. However, as advocated in this article, existing statistical methods are often too simplistic to represent complex climate-vegetation relationships due to the 5 assumption of linearity of these relationships. Therefore, as an extension of linear Granger causality analysis, we present a novel non-linear framework consisting of several components, such as data collection from various databases, time series decomposition techniques, feature construction methods and predictive modelling by means of random forests. Experimental results on global data sets indicate that with this framework it is possible to detect non-linear patterns that are much less visible with traditional Granger causality methods. In addition, we also discuss extensive experimental results that highlight 10 the importance of considering the non-linear aspect of climate–vegetation dynamics.
    [Show full text]
  • Sampling Distribution for Single-Regression Granger Causality Estimators
    Sampling distribution for single-regression Granger causality estimators A. J. Gutknecht∗†‡ and L. Barnett‡§ February 26, 2021 Abstract We show for the first time that, under the null hypothesis of vanishing Granger causality, the single- regression Granger-Geweke estimator converges to a generalised χ2 distribution, which may be well approximated by a Γ distribution. We show that this holds too for Geweke's spectral causality averaged over a given frequency band, and derive explicit expressions for the generalised χ2 and Γ-approximation parameters in both cases. We present an asymptotically valid Neyman-Pearson test based on the single- regression estimators, and discuss in detail how it may be usefully employed in realistic scenarios where autoregressive model order is unknown or infinite. We outline how our analysis may be extended to the conditional case, point-frequency spectral Granger causality, state-space Granger causality, and the Granger causality F -test statistic. Finally, we discuss approaches to approximating the distribution of the single-regression estimator under the alternative hypothesis. 1 Introduction Since its inception in the 1960s, Wiener-Granger causality (GC) has found many applications in a range of disciplines, from econometrics, neuroscience, climatology, ecology, and beyond. In the early 1980s Geweke introduced the standard vector-autoregressive (VAR) formalism, and the Granger-Geweke population log- likelihood-ratio (LR) statistic (Geweke, 1982, 1984). As well as furnishing a likelihood-ratio test for statis- tical significance, the statistic has been shown to have an intuitive information-theoretic interpretation as a quantitative measure of \information transfer" between stochastic processes (Barnett et al., 2009; Barnett and Bossomaier, 2013).
    [Show full text]
  • Analysis for a Subset of Systemically Important Banks
    10 Euronext 100 5 0 -5 -10 Does web anticipate stocks? Analysis for a subset of systemically important banks Michela Nardo and Erik van der Goot Michela2014 Nardo, Erik van der Goot Report EUR 27023 European Commission Joint Research Centre Institute for Protection and Security of the Citizen Contact information Michela Nardo Address: Joint Research Centre, via Fermi 2749 I-21027 Ispra (VA), Italy E-mail: [email protected] Tel.: +39 0332 78 5968 Erik van der Goot Address: Joint Research Centre, via Fermi 2749 I-21027 Ispra (VA), Italy E-mail: [email protected] Tel.: +39 0332 78 5900 JRC Science Hub https://ec.europa.eu/jrc Legal Notice This publication is a Technical Report by the Joint Research Centre, the European Commission’s in-house science service. It aims to provide evidence-based scientific support to the European policy-making process. The scientific output expressed does not imply a policy position of the European Commission. Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication. All images © European Union 2014 JRC 93383 EUR 27023 ISBN 978-92-79-44708-2 ISSN 1831-9424 doi: 10.2788/14130 Luxembourg: Publications Office of the European Union, 2014 © European Union, 2014 Reproduction is authorised provided the source is acknowledged. Abstract Is web buzz able to lead stock behavior for a set of systemically important banks? Are stock movements sensitive to the geo- tagging of the web buzz? Between Dec.
    [Show full text]
  • 07 Multivariate Models: Granger Causality, VAR and VECM Models
    07 Multivariate models: Granger causality, VAR and VECM models Andrius Buteikis, [email protected] http://web.vu.lt/mif/a.buteikis/ Introduction In the present chapter, we discuss methods which involve more than one equation. To motivate why multiple equation methods are important, we begin by discussing Granger causality. Later, we move to discussing the most popular class of multiple-equation models: so-called Vector Autoregressive (VAR) models. VARs can be used to investigate Granger causality, but are also useful for many other things in finance. Time series models for integrated series are usually based on applying VAR to first differences. However, differencing eliminates valuable information about the relationship among integrated series - this is where Vector Error Correction model (VECM) is applicable. Granger Causality In our discussion of regression, we were on a little firmer ground, since we attempted to use common sense in labeling one variable the dependent variable and the others the explanatory variables. In many cases, because the latter ‘explained’ the former it was reasonable to talk about X ‘causing’ Y. For instance, the price of the house can be said to be ‘caused’ by the characteristics of the house (e.g., number of bedrooms, number of bathrooms, etc.). However, one can run a regression of Y = stock prices in Country B on X = stock prices in Country A. It is possible that stock price movements in Country A cause stock markets to change in Country B (i.e., X causes Y ). For instance, if Country A is a big country with an important role in the world economy (e.g., the USA), then a stock market crash in Country A could also cause panic in Country B.
    [Show full text]
  • INVESTIGATION of HUMAN VISUAL SPATIAL ATTENTION with Fmri AND
    INVESTIGATION OF HUMAN VISUAL SPATIAL ATTENTION WITH fMRI AND GRANGER CAUSALITY ANALYSIS by Wei Tang A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, FL December 2011 Copyright by Wei Tang 2011 ii ACKNOWLEDGEMENTS I would like to thank all those who have helped to make this thesis reach its final com- pletion. My utmost gratitude goes to Dr. Steven Bressler, who has been advising me for five years. Without his knowledge and insights, and especially his enormous patience, I would have lost the battle against all those complicated problems. My appreciation goes to our collaborators at Washington University at St. Louis: Drs. Maurizio Corbetta, Gordon Shulman and Chad Sylvester, who designed and conducted the behavioral experiments and generously provided us the fMRI data, together with discussions on the results. I thank my friends from the laboratory, Tracy Romano, Marmaduke Woodman and Timothy Meehan, for giving valuable suggestions on the analysis methods. I also thank the Scientific Squir- rels Club of China for letting me present my work to general audience in my home country. At last, I would like to thank my parents and my sister, who have always been there with deep love to encourage and support me chasing my dream. iv ABSTRACT Author: Wei Tang Title: Investigation of Human Visual Spatial Attention with fMRI and Granger Causality Analysis Institution: Florida Atlantic University Dissertation Advisor: Dr. Steven L. Bressler Degree: Doctor of Philosophy Year: 2011 Contemporary understanding of human visual spatial attention rests on the hypothesis of a top-down control sending from cortical regions carrying higher-level functions to sen- sory regions.
    [Show full text]
  • On Granger Causality and the Effect of Interventions in Time Series
    Lifetime Data Anal DOI 10.1007/s10985-009-9143-3 On Granger causality and the effect of interventions in time series Michael Eichler · Vanessa Didelez Received: 8 December 2008 / Accepted: 6 November 2009 © Springer Science+Business Media, LLC 2009 Abstract We combine two approaches to causal reasoning. Granger causality, on the one hand, is popular in fields like econometrics, where randomised experiments are not very common. Instead information about the dynamic development of a sys- tem is explicitly modelled and used to define potentially causal relations. On the other hand, the notion of causality as effect of interventions is predominant in fields like medical statistics or computer science. In this paper, we consider the effect of exter- nal, possibly multiple and sequential, interventions in a system of multivariate time series, the Granger causal structure of which is taken to be known. We address the following questions: under what assumptions about the system and the interventions does Granger causality inform us about the effectiveness of interventions, and when does the possibly smaller system of observable times series allow us to estimate this effect? For the latter we derive criteria that can be checked graphically and are in the same spirit as Pearl’s back-door and front-door criteria (Pearl 1995). Keywords Causality · Causal inference · Decision theory · Graphical models · Time series M. Eichler Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD, Maastricht, The Netherlands V. Didelez (B) Department of Mathematics, University of Bristol, University Walk, BS8 1TW Bristol, UK e-mail: [email protected] 123 M.
    [Show full text]
  • A Non-Linear Granger-Causality Framework to Investigate Climate–Vegetation Dynamics
    Geosci. Model Dev., 10, 1945–1960, 2017 www.geosci-model-dev.net/10/1945/2017/ doi:10.5194/gmd-10-1945-2017 © Author(s) 2017. CC Attribution 3.0 License. A non-linear Granger-causality framework to investigate climate–vegetation dynamics Christina Papagiannopoulou1, Diego G. Miralles2,3, Stijn Decubber1, Matthias Demuzere2, Niko E. C. Verhoest2, Wouter A. Dorigo4, and Willem Waegeman1 1Depart. of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Ghent, Belgium 2Laboratory of Hydrology and Water Management, Ghent University, Ghent, Belgium 3Depart. of Earth Sciences, VU University Amsterdam, Amsterdam, the Netherlands 4Depart. of Geodesy and Geo-Information, Vienna University of Technology, Vienna, Austria Correspondence to: Christina Papagiannopoulou ([email protected]) Received: 11 October 2016 – Discussion started: 16 November 2016 Revised: 20 March 2017 – Accepted: 29 March 2017 – Published: 17 May 2017 Abstract. Satellite Earth observation has led to the creation in the global climate system. Plant life alters the characteris- of global climate data records of many important environ- tics of the atmosphere through the transfer of water vapour, mental and climatic variables. These come in the form of exchange of carbon dioxide, partition of surface net radia- multivariate time series with different spatial and temporal tion (e.g. albedo), and impacts on wind speed and direction resolutions. Data of this kind provide new means to fur- (Nemani et al., 2003; McPherson et al., 2007; Bonan, 2008; ther unravel the influence of climate on vegetation dynamics. Seddon et al., 2016; Papagiannopoulou et al., 2017). Because However, as advocated in this article, commonly used statis- of the strong two-way relationship between terrestrial vegeta- tical methods are often too simplistic to represent complex tion and climate variability, predictions of future climate can climate–vegetation relationships due to linearity assump- be improved through a better understanding of the vegetation tions.
    [Show full text]