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WMO course-Statistics and Climatology -
Lecture III
Dr. Bertrand Timbal Regional
Meteorological Training Centre, Tehran,
Iran December 2003
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Statistics of the Climate system---
Spatio-temporal linkages within the system
Statistics and Climatology Lecture III

• Overview
• Links within the system the example of ENSO
• Regression and correlation of variables
• Spatial structures reduction of the degree of
  freedom

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El Niño / La Niña a large scale feature
Schematic of summer La Niña conditions across the
Equatorial Pacific Ocean
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El Niño / La Niña a large scale feature
Schematic of summer EL Niño conditions across the
Equatorial Pacific Ocean
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El Niño a large scale feature

• Temperature, along an equatorial
• longitude-depth section
• Anomalies are relevant for interannual
  variability
• Observed with the TAO array of buoys in the
  Tropical Pacific
• Thermocline movements important for seasonal
  forecasting

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El Niño sub-surface ocean anomalies

• Anomalous warm water accumulated
• at depth in the West Pacific and travel across
  the basin along the thermocline
• The predictability comes from the slow moving
  ocean anomalies

97-98 El Niño formation
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Transition to the 98-99 La Niña
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El Niño air-sea interactions
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El Niño air-sea interactions
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El Niño Global Tele-connections
Courtesy of NOAA
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La Niña Global Tele-connections
Courtesy of NOAA
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El Niño impact on Australian rainfall
Stratification of the mean climate based on ENSO
phases
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La Niña impact on Australian rainfall
Stratification of the mean climate based on ENSO
phases
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El Niño global impact on rainfall
Probability of exceeding median rainfall for
Cold, Neutral and Warm conditions in the
Equatorial Pacific Ocean (Data for 1900-1997)
Stratification of the mean climate based on ENSO
phases.
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El Niño impact on Australian Wheat Yields
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• Links within the climate system exist
• El Niño is a planetary scale phenomenon
• Several variables exhibit coherent variations
  (correlation)
• Distant teleconnections are observed (lag
  correlation)
• Probabilities are shifted by ENSO phases
  (predictable)

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Statistics of the Climate system---
Spatio-temporal linkages within the system

• Overview
• Links within the system the example of ENSO
• Regression and correlation of variables
• Spatial structures reduction of the degree of
  freedom

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Simple model Least-Squares Regression
Regression
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Role of outliers

• Outlier detection method to find observations
  with large influence
• Problem often arises from either erroneous data
  or small sample
• Graphical visualisation is essential

r 0.457 r 0.336
In this example, out of 100 points, only one
data is different !
Courtesy of J. Stockburger
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Graphical visualisation of correlation
The relationship is not linear.
In all cases, the correlation is r0.816 but
Correlation is not robust and resistant
. Instead we can use the rank correlation
correlation based on ranked data
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An example of a non linear relation
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Correlation is not causation!
Is ENSO forced by Australian rainfall?
or Are Australian rainfall affected by ENSO?
Correlations between seasonal rain and SOI

• Correlation does not imply causation
• Simultaneous evolution
• Others techniques are needed
• Path analysis (Blalock, 1971)
• Temporal precedence

Courtesy of W. Drosdowsky
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Lag Correlation and auto-correlation
Lagged correlation between the SOI and cyclone
formation

• (Prior) Lag correlations exhibit the dependence
  between variables
• Predictability arises from lag correlation

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• Correlation in the climate system
• Correlation coefficientes express the part of
  the variation of two variables which are linked
  (no causality)
• Correlation assumes normality (!) and linear
  relation (!)
• A more robust coefficient is the rank
  correlation
• Lag correlation is useful for causality and
  predictability
• Auto-correlation of meteorological data has
  serious consequences for the use of statistics in
  climate

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Statistics of the Climate system---
Spatio-temporal linkages within the system

• Overview
• Links within the system the example of ENSO
• Regression and correlation of variables
• Spatial structures reduction of the degree of
  freedom

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Spatial structure in climate data

• Several motivations to identify large scale
  spatial features
• Data are not spatially independent spatial
  correlation
• Large scale structures are more coherent and
  predictable
• Extract the large scale climate signal
• Reduce the weather noise associated with small
  scales
• Smaller degree of freedom and reduced data set
• Identify useful relationships to exploit for
  climate forecasting

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Principal-Component (EOF) Analysis

• Objective
• To reduce the original data set to a new data
  set of (much) fewer variables
• To condense a large fraction of the variance of
  the original dataset
• To explore large multivariate data sets (spatial
  and temporal variation)

• Calculation
• PCA are done on anomalies
• Based on the covariance S or the
• correlation R matrix of a vector X XTX
• The principal components are the
• projection of X on the eigenvectors of S ei
• orthogonal one to an other new coordinate
  system
• maximise the variance measured by the
  eigenvalues (?i)

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Principal-Component (EOF) Analysis

• Eigenvectors (PCA) are orthogonal
• Strong constraint for small domain (Jolliffe,
  1989)
• Typically the 2nd PC is a dipole (not
  necessarily meaningful)
• The number of PCs to be consider is based on the
  eigenvalues

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EOFs of combined fields
Courtesy of M. Wheeler
200 hPa
850 hPa
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The phase-space representation of the MJO
M(t) RMM1(t),RMM2(t) Vector M traces -
large anti-clockwise circles about the origin
when the MJO is strong. - random jiggles around
the origin when the MJO is weak. For
compositing, we define the 8 equal-angle phases
as labeled, and described by the angle F
tan-1RMM2(t)/RMM1(t)
Courtesy of M. Wheeler
Southern Summer DJFMA
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MJO propagation based on vector M in the two
dimensional phase space OLR contour interval 4
Wm-2 blue negative 850 hPa wind Max vector
4.5 ms-1
Courtesy of M. Wheeler
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Rotated PCs

• Facilitate physical interpretation
• Review by Richman (1986) and by Jolliffe (1989,
  2002)
• New set of variable RPCs
• Varimax is a very classic rotation technique
  (many others)

First two rotated PCAs of Indian/Pacific SSTAs
using data from Jan 1949 to Dec 1991.
Courtesy of W. Drosdowsky
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Other multivariate analyses

• Extended EOFs and Complex (Hilbert) EOFs are two
  classical extensions of PCs
• Canonical Correlation Analysis extension of PCA
  to two multivariate data sets forecasting one
  variable with the other (book by Wilks, 1995).
• Principal Orthogonal Pattern (POP) and (PIP),
  SVD are other techniques used (book by von Storch
  and Navarra, 1995 and von Storch and Zwiers,
  1999)
• Discriminant analysis (e.g. the operational
  seasonal forecast of the BoM) the conditioning
  is on the predictand and in a sense the reverse
  conditional probabilities are estimated from the
  data, and Bayes theorem is used to invert these
  (article by Drosdowsky and Chambers, 2001)
• Analogue (lecture 7), clustering (book by Wilks,
  1995) and NHMM (next slide) are other techniques
  dealing with classification.
• All techniques can be use for forecasting and
  downscaling

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An other downscaling approach
Non-homogeneous Hidden Markov Model makes use of
non observed hidden weather states which are
related to observed rainfall structures
Courtesy of S. Charles
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Summary

• Many interactions in the system ? correlation
• Many issues with correlation robustness,
  causality
• Large scale structure exist ? multivariate
  analyses
• Useful for filtering, organizing and reducing
  the noise in data
• Forecasting uses many of these statistical tools