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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

Overview:

1. Links within the system: the example of ENSO

2. Regression and correlation of variables

3. Spatial structures: reduction of the degree of freedom

Review some classical statistical toolsstatistical tools

Statistics and Climatology: Lecture III

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Schematic of summer La Niña conditions across the Equatorial Pacific Ocean

El Niño / La Niña : a large scale feature

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Schematic of summer EL Niño conditions across the Equatorial Pacific Ocean

El Niño / La Niña: a large scale feature

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• 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

Thermocline: Layer of strongtemp gradient around 20C

El Niño: a large scale feature

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El Niño: sub-surface ocean anomalies

97-98 El Niño formation

• 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

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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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Probability of exceeding median rainfall for

Cold, Neutral and Warm conditions in the

Equatorial Pacific Ocean

(Data for 1900-1997)

El Niño: global impact on rainfall

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)

How to best express these relationships ?

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Statistics of the Climate system---

Spatio-temporal linkages within the system

Overview:

1. Links within the system: the example of ENSO

2. Regression and correlation of variables

3. Spatial structures: reduction of the degree of freedom

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Simple model: Least-Squares Regression

bxay ˆ

)(ˆ iii xyye

Regression: Correlation:

• Pearson ordinary correlation (r)

• a is the intercept for X=0

• b is the slope:

• r2 is the amount of variance explained

n

i

n

iii

n

iii

yxxy

yx

yx

SS

yxr

1 1

2'2'

1

''

)()(

),cov(

y

xxy S

Srb

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r = 0.457 r = 0.336

Courtesy of J. Stockburger

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

In this example, out of 100 points, only one data is different !

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False correlationbased on oneerroneous data

Perfect relationshipaffected byone data

In all cases, the correlation is r=0.816 but …

The relationshipis not linear.

Graphical visualisation of correlation

Correlation is not robust and resistant ….Instead we can use the rank correlation: correlation based on ranked data

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Annual SW WA Rainfall

350

550

750

950

1880 1900 1920 1940 1960 1980 2000

AswWArain 20 per. Mov. Avg. (AswWArain)

Annual SW WA Rainfall

350

550

750

950

1880 1900 1920 1940 1960 1980 2000

AswWArain 20 per. Mov. Avg. (AswWArain)

An example of a non linear relation

Perth Inflow v. sw WA Rainfall

0

250

500

750

1000

300 400 500 600 700 800 900 1000

Rainfall (mm/yr)

Inflow

(GL/y

)

Rainfall and river flow

Courtesy of S. Power

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Correlations between seasonal rain and SOI

Correlation is not causation!

• Correlation does not imply causation

• Simultaneous evolution

• Others techniques are needed:

• Path analysis (Blalock, 1971)• Temporal precedence

Is ENSO forced by Australian rainfall? orAre Australian rainfall affected by ENSO?

Courtesy of W. Drosdowsky

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Lag Correlation and auto-correlation

Lagged correlation between the SOI and cyclone formation

• Lag correlation of a series with itself is auto-correlation at lag-k:

• Meteorological variables are auto-correlated (persistence)

• Violate the independent data assumption effective sample size

• Hypothesis testing • Variance estimate

kxx

kk SS

xxr

),cov(

• (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:

1. Links within the system: the example of ENSO

2. Regression and correlation of variables

3. Spatial structures: reduction of the degree of freedom

III-26

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 (λi)

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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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200 hPa

850 hPa

EOFs of combined fields:

Courtesy of M. Wheeler

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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

Φ = tan-1[RMM2(t)/RMM1(t)]

Southern Summer = DJFMA Courtesy of M. Wheeler

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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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First two rotated PCAs of Indian/Pacific SSTAs using data from Jan 1949 to Dec 1991.

Rotated PCs

Courtesy of W. Drosdowsky

• 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)

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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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1016

1000

1012

1008

1016

1004

1012

1016

1016

1012

Typ

e 5

Typ

e 3

.2 .4 1.8.6

.2 .4 .6 1.8

H H

1020

1012

H

L

L

An other downscaling approachNon-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

Tool box to analyse our dynamic climate system …. and … basis for climate forecasting