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Characterizing spatio-temporal climate variability with complex network methods Reik V. Donner with Marc Wiedermann, Jonathan Donges, Alexander Radebach & others CITES 2017, Zvenigorod, 6 September 2017 How to infer information from climate

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Characterizing spatio-temporal climate variability with complex network methods

Reik V. Donner with Marc Wiedermann, Jonathan Donges, Alexander Radebach & others

CITES 2017, Zvenigorod, 6 September 2017

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Existing data cover various spatial and temporal scales:

  • Station-based observations of meteorological conditions
  • Remote sensing (satellite-based) recordings
  • Assimilation of observations into globally homogeneous data sets (reanalyses)
  • Climate model simulations
  • Spatially distributed paleoclimate archives (plus climate field reconstructions)

Problem: Integration of different spatial sites into joint analysis

  • Classical approach: (linear) multivariate statistics methods (principal component

analysis, canonical correlation analysis, and numerous variants thereof)

  • Last 15 years: data mining/machine learning techniques and other computer

science methods to also address nonlinearity in climate system (climate knowledge discovery, climate informatics)

  • Here: complex networks as a (statistical) physics-inspired third way

How to infer information from climate data?

Reik V. Donner, reik.donner@pik-potsdam.de

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Linear PCA: Diagonalization of lag-zero covariance matrix C of multivariate time series (matrix X wth standardized components)

  • Compute correlation matrix of all variables
  • Estimate eigenvalues and eigenvectors
  • Eigenvectors: additive decomposition into principal components (weighted

superpositions of original variables) with individual variances corresponding to associated eigenvalues ⇒ spatial EOF patterns + index/score time series describing magnitude and sign of individual EOF modes (characteristic for individual climate oscillations)

X X C

T

=

U U C

TΣ

= ) ,..., (

2 2 1 N

diag σ σ = Σ

with and

EOF analysis

Original motivation: extract dominating co-variability from spatio-temporal fields of climate observations records (dimensionality reduction)

Reik V. Donner, reik.donner@pik-potsdam.de

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EOF analysis

Example: leading EOF (EOF-1) of near-surface air pressure in Arctic => Dipole structure (Arctic Oscillation)

Reik V. Donner, reik.donner@pik-potsdam.de

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Limitations of EOF analysis

Purpose: extract dominating spatio-temporal (co-)variability modes from fields of climate observations

  • Linear decomposition/dimensionality reduction technique
  • Potential improvement: nonlinear extensions like kernel PCA, neural network PCA,

isometric feature mapping and other nonlinear dimensionality reduction methods

  • Intrinsic tendency to exhibit dipole (or multipole) structures enforced by
  • rthogonality constraint between modes
  • EOFs modes do not always coincide with specific climatic mechanisms
  • Relevance of EOF modes as dynamical modes (or even proper statistical modes)

questionable

  • Multiple superimposed patterns need to be considered
  • Spatial patterns = strength of co-variability, unclear relevance of associated

temporal patterns in other regions not highlighted by the same EOF

  • Integrated view on co-variability, pair-wise co-variability information is lost

Reik V. Donner, reik.donner@pik-potsdam.de

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Climate networks: The starting point…

(Bull. Amer. Meteor. Soc., 2006)

Reik V. Donner, reik.donner@pik-potsdam.de

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7 Reik V. Donner, reik.donner@pik-potsdam.de

Mathematical background

An unweighted network (graph) is described by

  • a set of nodes (vertices) V
  • a set of links (edges) E between pairs of vertices

Basic mathematical structure: adjacency matrix A Aij=1  nodes i and j are connected by a link Aij=0  nodes i and j are not connected by a direct link ⇒ binary matrix containing connectivity information of the graph ⇒ undirected graph: A symmetric Degree (centrality): number of neighbors of a vertex Local clustering coefficient: relative fraction of neighbors of a vertex that are mutual neighbors of each other

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Starting point: Spatially distributed climate time series ⇒ Spatial locations (individual series) = “nodes” of a network ⇒ Statistical similarity (e.g., correlation) between time series = “weights” of links ⇒ Remove all links with “weak” similarity = unweighted network ⇒ Analysis of structural properties of the resulting climate network (spatially coarse- grained representation of climate variability) Different options for threshold selection:

  • Global correlation threshold (% of strongest correlations)
  • Global edge density (implies global correlation threshold)
  • Global significance level of correlations (for removing artifacts due to different

serial correlation properties)

Climate networks

Reik V. Donner, reik.donner@pik-potsdam.de

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Climate networks: General workflow

(Donner et al., 2017)

Reik V. Donner, reik.donner@pik-potsdam.de

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Basic assumptions:

  • Relevant processes in the (continuous) climate system can be approximated by an

underlying spatial network structure (spatial coarse-graining is reasonable)

  • Statistical interdependences between climate variations at different locations

reveal corresponding network topology - “functional” network (statistics reflect dynamics) – also used in other fields (e.g., functional brain networks, economics) Different possible types of climate networks based on climatological variable and employed similarity measure (e.g. Pearson correlation, different types of mutual information, event synchronization)

Climate networks

Reik V. Donner, reik.donner@pik-potsdam.de

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Correlation climate networks vs. EOF analysis

Examples: Modulus of EOF-1 and degree field of correlation climate networks from monthly SLP (top) and SAT (bottom) anomalies (NCEP/NCAR reanalysis) (Donner et al., 2017)

Reik V. Donner, reik.donner@pik-potsdam.de

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Correlation climate networks vs. EOF analysis

Correlation climate networks and EOF analysis based on the same correlation matrix

  • EOF analysis: eigenvector decomposition – linear transformation
  • Climate network analysis: binarization by thresholding – nonlinear transformation

If EOF-1 dominates the data set (high fraction of explained variance): approximate relationship between degree field and modulus of EOF-1 (Donges et al., Climate Dynamics, 2015):

Reik V. Donner, reik.donner@pik-potsdam.de

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Correlation climate networks vs. EOF analysis

Added value of climate network analysis:

  • Commonly, more than just one EOF are statistically relevant – EOF-1 does not tell

the whole story

  • Network allows investigating spatial structure of links (e.g., where are strong

correlations with a given location/region located?)

  • EOF analysis just gives a single spatial pattern and time-dependent score per mode;

network analysis provides a multiplicity of characteristics that capture higher-order statistical properties of the spatial correlation structure

  • Aspects of spatio-temporal organization of climate variability hidden to EOF

analysis may be revealed

Reik V. Donner, reik.donner@pik-potsdam.de

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Spatial range of correlations

Surface air temperatures: correlations decay with spatial distance – implications for degree and edge length distributions

Reik V. Donner, reik.donner@pik-potsdam.de

Donges et al., EPL, 2009

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15 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

What can we learn from the temporal variation and spatial patterns of network properties?

Evolving global surface air temperature network

(Radebach et al., Phys. Rev. E, 2013)

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16 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

Climate network analysis for running windows in time: evolving climate networks Global network characteristics show distinct temporal variability profile strongly related to ENSO

Evolving global surface air temperature network

(Radebach et al., Phys. Rev. E, 2013)

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17 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

Climate network analysis for running windows in time: evolving climate networks Global network characteristics show distinct temporal variability profile strongly related to ENSO: El Nino and La Nina episodes can create hubs with long-range links (global impact)

Evolving global surface air temperature network

(Radebach et al., Phys. Rev. E, 2013)

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18 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

But: peaks in global network characteristics do not coincide 1:1 with timing of known El Nino and La Nina episodes Reason: peaks indicate the formation of “localized structures” of high connectivity

  • Can also arise after strong volcanic

eruptions (common regional cooling trend – increase of correlations)

  • Different types (“flavors”) of El Nino

and La Nina episodes: functional discrimination based on global impacts?

Evolving global surface air temperature network

(Radebach et al., Phys. Rev. E, 2013) Transitivity, NINO3.4, strat. opt. depth

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19 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

Strong eruptive volcanism: Mt Pinatubo as example ⇒ Injection of large amounts of aerosols into stratosphere ⇒ Large-scale regional cooling (with spatial shift and lag of 12-18 months) ⇒ Elevation of correlations in confined region ⇒ Introduction of spatially confined correlations in affected region (Kittel et al., in prep.)

Evolving global surface air temperature network

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20 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

New index for El Nino / La Nina flavor based on transitivity of weighted climate networks (weights = absolute correlations)

Evolving global surface air temperature network

(Wiedermann et al., Geophys. Res. Lett., 2016)

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21 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

  • Discrimination of El Nino and La Nina episodes into two types each
  • Transitivity characterizing the spatial localization of links in climate network and,

hence, provides a global impact-based perspective on ENSO patterns

  • High transitivity values: classical El Nino / La Nina pattern (strongly localized

structure)

  • Low transitivity values: El Nino / La Nina Modoki pattern (more spatially diffuse

teleconnections) Advantages:

  • Objective classification coinciding with the consensus among previous studies
  • Moderate computational efforts comparable with classical EOF analysis
  • Only two parameters to be chosen (link density and window width), results are

robust over climatologically reasonable range of both parameters

Evolving global surface air temperature network

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22 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

All times All “normal“ times Classical El Nino El Nino Modoki Classical La Nina La Nina Modoki (Wiedermann et al., Geophys. Res. Lett., 2016)

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23 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

Robustness of classification

(Wiedermann et al., Geophys. Res. Lett., 2016)

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24 Reik V. Donner, reik.donner@pik-potsdam.de

Coupled climate networks

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25 Reik V. Donner, reik.donner@pik-potsdam.de

Coupled climate networks

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26 Reik V. Donner, RD IV Transdisciplinary Concepts & Methods reik.donner@pik-potsdam.de

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27 Reik V. Donner, reik.donner@pik-potsdam.de

Atmosphere – ocean interactions

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28 Reik V. Donner, reik.donner@pik-potsdam.de

Hierarchical coupling structure

(Wiedermann et al., Int. J. Climatol., 2017)

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29 Reik V. Donner, reik.donner@pik-potsdam.de

Hierarchical coupling structure

(Wiedermann et al., Int. J. Climatol., 2017)

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30 Reik V. Donner, reik.donner@pik-potsdam.de

Scale-dependent climate networks

Climate variability patterns depend on temporal scale of dynamics – reflected in scale- dependent networks obtained for filtered climate data transitivity entropy

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31 Reik V. Donner, reik.donner@pik-potsdam.de

Cross-scale information transfer

Palus, PRL, 2014; Entropy, 2014: detection of statistical coupling between amplitudes/phases of oscillations at different time-scales based upon time-scale and amplitude-phase decomposition by complex continuous wavelet transform and (conditional) mutual information Next step: combine this information for multiple sites in cross-scale coupled climate network analysis

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32 Reik V. Donner, reik.donner@pik-potsdam.de

Interactions among different climate patterns

Idea: identification of key “modes” via dimensionality reduction/community detection in scale-dependent climate networks (and cross-scale coupled climate networks) Example: South American Monsoon System (new project)

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33 Reik V. Donner, reik.donner@pik-potsdam.de

Conclusions

  • (Correlation) climate network analysis as extension and complement of classical

EOF analysis

  • Extensions to other (nonlinear) similarity measures (e.g. event-related statistics)

and coupling structures among different subsystems

  • Global surface air temperature data: climate network transitivity as objective index

for classifying El Nino and La Nina episodes

  • Coupled climate network analysis allowing to resolve previously unknown

structural organization features among different climate subsystems Python package pyunicorn for climate network analysis at GitHub (Donges et al., Chaos, 2015)