Spatial extremes in climate analysis

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Spatial extremes in climate analysis

• Dag Johan Steinskog

Workshop, Summerschool St. Petersburg,
Russia 09.-13. September 2007
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Outline

• What are climate extremes?
• R and RCLIM
• Methodology of spatial extremes
• Factors controlling extremes
• Clustering of extremes
• Teleconnections of extremes

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What are climate extremes?
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Examples of wet and windy extremes
Hurricane
Convective severe storm
Extra-tropical cyclone
Polar low
Extra-tropical cyclone
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Examples of dry and hot extremes
Drought
Dust storm
Wild fire
Dust storm
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What do we mean by extreme?

• Large meteorological values
• Maximum value (i.e. a local extremum)
• Exceedance above a high threshold
• Record breaker (thresholdmax of past values)
• Rare event
• (e.g. less than 1 in 100 years p0.01)
• Large losses (severe or high-impact)
• (e.g. 200 billion if hurricane hits Miami)
• risk p(hazard) x vulnerability x exposure

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IPCC 2001 definitions

• Simple extremes
• individual local weather variables
• exceeding critical levels on a continuous
• scale
• Complex extremes
• severe weather associated with particular
• climatic phenomena, often requiring
• a critical combination of variables
• Extreme weather event
• An extreme weather event is an event
• that is rare within its statistical reference
• distribution at a particular place.
• Definitions of "rare" vary, but an extreme
• weather event would normally be as
• rare or rarer than the 10th or 90th percentile.
• Extreme climate event

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Origin of Extreme Events

• 1. Rapid growth due to instabilities
• Fast growth of weather systems caused by positive
  feedbacks
• e.g. convective instability, baroclinic growth,
  etc.
• 2. Displacement
• Survival of a weather system into a new spatial
  region or time period
• e.g. transition of a tropical cyclone into
  mid-latitudes.
• 3. Conjunction
• Simultaneous supposition of several non-rare
  events
• e.g. freak waves.
• 4. Intermittency
• Varying variance of a process in space or time
• e.g. precipitation.
• 5. Persistence or frequent recurrence
• Chronic weather conditions leading to a climate
  extreme
• e.g. drought, unusually stormy wet season,
  persistent blocking.

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3
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R and RCLIM
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What is R?

• R is an integrated suite of software facilities
  for data manipulation, calculation and graphical
  display. Among other things it has
• an effective data handling and storage facility,
• a suite of operators for calculations on arrays,
  in particular matrices,
• a large, coherent, integrated collection of
  intermediate tools for data analysis,
• graphical facilities for data analysis and
  display either directly at the computer or on
  hardcopy
• a well developed, simple and effective
  programming language (called S) which includes
  conditionals, loops, user defined recursive
  functions and input and output facilities.
  (Indeed most of the system supplied functions are
  themselves written in the S language.)

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Why R?

• It is free and based on S-PLUS!
• http//www.r-project.org
• Works on several platforms
• Windows
• Mac
• UNIX/Linux
• Several packages available
• http//cran.r-project.org
• A large and very active community working and
  updating the software

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What is RClim?

• Initiative by C. A. S. Coelho, C. A. T. Ferro, D.
  B. Stephenson and D. J. Steinskog.
• Goals
• to develop statistical methods for doing spatial
  extremes.
• read/write large gridded fields in netcdf format
• do nice geographical contour maps
• do general climate analysis at many grid points

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Development

• Spring 2005 Initiative started
• Spring 2006 Completed as it is today
  implemeted in KNMIs climate explorer
• August 2006 Paper submitted to Journal of
  Climate
• Future development Will be updated and expanded
  in the future methods for daily data.

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How to get started with RClim?

• Webpage http//www.secam.ex.ac.uk/index.php?nav6
  99
• Packages that must be installed
• rNetCDF
• evd
• ismev
• maps
• mapdata
• mapproj

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Installation

• Source the rclim.txt to install all functions.
• Can be found at http//www.nersc.no/dagjs/rcours
  e_nzu/DJS/Day4/rclim.txt
• Command
• source(rclim.txt)

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Complete course

• A complete course was given in Beijing, China in
  August 2007
• Webpage
• www.nersc.no/dagjs/rcourse_nzu
• Made by Hans Wackernagel and Dag Johan Steinskog

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Extreme value analysis
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Motivation

• Weather and climate time series on large
  grid-point arrays can be analysed in many ways
• Composites
• Correlation maps
• Principal component analysis
• Isolate leading patterns of climate variability
  (ENSO, NAO,)
• Why not use these methods when analysing extremes?

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• These methods are based on the whole distribution
  of a certain variable

And mask the extreme events in the tail of the
distribution
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How are tails
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related to the whole animal?
PDF Probability Density Function Or
Probable Dinosaur Function??
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Generalized Pareto Distribution
For sufficiently large thresholds, the
distribution of values above a sufficiently large
threshold u approximates the Generalized Pareto
Distribution (GPD)
Shape -0.4 upper cutoff Shape 0.0
exponential tail Shape 10 power law tail
Probability density function
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Example Central England Temperature

• n 3082 values
• Min -3.1C
• Max 19.7C
• 90th quantile 15.6C

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GPD fit to values above 15.6C

• Location parameter u15.6C
• Maximum likelihood estimates
• Scale parameter 1.38 /- 0.09C
• Shape parameter -0.30 /- 0.04C
• ? Upper limit estimate

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How good is the model fit?
upper limit 20.3C
u15.6C
? Good fit to 308 observed exceedances
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Return level plot
Deg C
(years)
? Fit can be used to make predictions of return
values
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Methodology of spatial extremes
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Need for extreme value theory methods

• New tools of extreme value theory (EVT)
  introduced (Coelho et al., 2007)
• Probability theory and statistical science that
  deals with the modelling and inference for
  extreme values
• Two main approaches
• Generalized extreme value (GEV)
• Maximum of blocks of data
• Generalized Pareto distribution (GPD)
• Values above a high threshold

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Dataset used in this example

• HadCRUT2v Monthly mean gridded surface
  temperature (Jones and Moberg, 2003)
• Available from http//www.cru.uea.ac.uk/cru/data/
  temperature/
• Time span January 1870 to December 2005
• Regular 5x5 global grid
• Long variation 136 years
• Grid points with more than 50 missing values and
  SH are omitted.

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Example

• European heat wave 2003
• Estimated mortality to be 35000-50000
• Map of temperature anomaly
• Beniston (2003) Normal summer late in this
  century

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Definition of extreme events

• Maximum value
• Not very reliable summary of the distribution of
  extreme events
• Non-resistant to outliers
• Excesses above a pre-defined threshold (t-u)
• Peak-over-threshold method (Coles 2001, chapter
  4)
• What about the block maxima approach?
• Annual blocks is not appropiate in this example,
  only 12 observations available each year
• Larger blocks? Decades? Reduce the sample size
  for estimation of GEV distribution parameters
• More appropriate for daily temperatures

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Choice of Threshold

• Climate data have a seasonal and trend component
• Without taking this into account in choice of
  threshold, a bias will occur
• Strategies for avoiding this
• Detrend the time series
• Time varying threshold

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Choice of threshold cont.

• Time varying threshold
• Approximately constant exceedance frequency
• Analysis is not biased towards the warmer climate
• Excesses are yielded relative to contemporary
  climate

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Choice of threshold cont.

• The threshold chosen in the course is
• 75th quantile time varying threshold
• Definition of threshold

- Long term trend component
- Mean annual cycle
- Constant increment to have a of the observed
values above the threshold
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GPD scale parameter estimate
? Large over extra-tropical land regions
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GPD shape parameter estimate
Generally negative ? finite upper temperature
limit
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Upper limit for excesses
? Largest over high-latitude land regions
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Return periods

• Using the GP distribution, it is possible to
  estimate the return period

• Return period is the frequency with which one
  would expect on average a given event to recur.

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Return periods for August 2003 event
? Central Europe return period of 133 years (c.f.
Schar et al 46000 years!)
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Note about return periods

• In Schär et al. (2004), a return period of the
  heat wave in Europe was estimated to be 46000
  years
• We estimated 133 years
• So who is right?
• Schär et al. (2004) used no EVD analysis!!
• Assumed the data had a Gaussian distribution

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Factors controlling extremes?
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Factors controlling extremes

• The relationship between extremes and factors
  (e.g. time and ENSO) can be examined by modelling
  the shape and scale parameters of the GP
  distribution as functions of these factors.
• For instance the following model can be used to
  analyse how the variability of summer
  temperature excesses is related to ENSO

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The role of large-scale modes
? ENSO effect on temperature extremes in NH
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Clustering of extremes
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Temporal clustering of extreme events

• Annual frequency of extreme events is a proxy for
  clustering of extremes
• Average number of summer exceedances is given to
  be

• The binary variable e1 if an extreme event is
  observed and e0 if an extreme event is not
  observed
• N is total number of summers with at leat one
  observed exceedance

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Average number of exceedances
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Teleconnections of extremes
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Teleconnections between extreme events

• xdependence - Compute extreme dependence measures
  between a given p x q x n three-dimensional array
  and a given time series of length n.
• Assume that we are interested e.g. in
  investigating how extreme temperature at one
  place are related to extreme temperature at
  another place

• The statistics provides a measure of extreme
  dependence for asymptotically dependent
  distributions.

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Teleconnections cont.

• However, X fails to provide information of
  discrimination for asymptotically independent
  distributions (Coles, 2001).
• Alternative method suggested

• Defined for the threshold on the range 0ltult1. The
  statistics ranges from -1 to 1.

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Teleconnections between extremes
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1-point association map for extreme events
? association with extremes in subtropical
Atlantic
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Methods in RCLIM
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Methods in RClim

• acs - Compute average cluster size for a given
  three-dimensional p x q x n array of excesses.
  First two dimensions p and q are space dimensions
  (e.g. longitude and latitude). Third dimension n
  is time.
• boundexcesses - Compute upper bound of excesses
  for a given 2 x p x q array of Generalized Pareto
  distribution parameters. First index of the first
  dimension of the array represents the scale
  parameter. Second index of the first dimension of
  the array represents the shape parameter.
• mygpd.fit - Same as gpd.fit function from ismev
  package but with standard error calculation
  disactivated.
• returnperiod - Compute return period for a given
  p x q matrix of excesses and a given 2 x p x q
  array of Generalized Pareto distribution
  parameters.
• tvt - Compute time-varying threshold for a given
  monthly time series.
• xdependence - Compute extreme dependence measures
  between a given p x q x n three-dimensional array
  and a given time series of length n.

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Methods in RClim cont.

• xdependence1 - Same as above, but also allows
  specification of fraction of non-missing values
  for the computation of the statistics. Grid
  points with larger fraction of missing values
  than specified are excluded.
• xexcess - Compute mean excess and variance of
  excess for a given n x p x q three-dimensional.
• xgev - Compute location, shape and scale
  parameters of a Generalized Extreme Value
  Distribution for block annual maxima or minima of
  a given p x q x n three-dimensional array.
• xindex - Compute the intervals estimator for the
  extremal index, an index for time clusters, for a
  given time series and threshold.
• xindexfield - Compute the intervals estimator for
  the extremal index at each grid point of a p x q
  x n three-dimensional array.
• xpareto - Compute shape and scale parameters of a
  Generalized Pareto Distribution for a given p x q
  x n three-dimensional array.

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Methods in RClim cont.

• xparetotvt - Fit Generalized Pareto distribution
  with time-varying threshold at each grid point
  for a given p x q x n three-dimensional array of
  montly data.
• xparetotvtcov - Fit Generalized Pareto
  distribution with time-varying threshold at each
  grid point for a given p x q x n
  three-dimensional array of montly data. Allows
  linear modelling of the paramters.

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Reference

• Coelho, C. A. S., C. A. T. Ferro, D. B.
  Stephenson and D. J. Steinskog Exploratory tools
  for the analysis of extreme weather and climate
  events in gridded datasets, Under revision
  Journal of Climate
• Contact info
• David Stephenson, d.b.stephenson_at_reading.ac.uk
• Dag Johan Steinskog, dag.johan.steinskog_at_nersc.no