Statistics as a means to construct knowledge in

climate and related sciences

-- a discourse --

Statistics as a means to construct knowledge in

climate and related sciences

-- a discourse --

Hans von StorchInstitute for Coastal ResearchGKSS, Germany

Hans von StorchInstitute for Coastal ResearchGKSS, Germany

9IMSC, Cape Town, 24-28 May 20049IMSC, Cape Town, 24-28 May 2004

The basic approach …

… is to combine systematically

empirical knowledge („data“)

with

dynamical knowledge („models“)

in order to determine

• characteristic parameters (“inference”)

• consistency of models and data (“testing”)

The knowledge represented by data and models are both uncertain.

This uncertainty makes us to resort to statistical concepts.

The resulting additional knowledge is

• best guesses of numbers (ideally together with confidence intervals)

• evaluation of the consistency of theoretical concepts with observational evidence.

These new knowledge claims are based on the amount of available data.

In general: If more data are available, the confidence in the numbers increases, but the consistency of the concepts decreases.

In general, the problem may be conceptualized by the state space formalism, with - a state space equation, e.g.,

Ψt+1 = F(Ψt, α, η) + ε (M)

with the state variable Ψt, external parameters η and internal parameters α. The term ε is a random component, which supposedly represents the uncertainty of the model M.

- an observation equation

xt = B(Ψt) + δ (B)

with the observable x, and the random component δ.

Examples:

1.Goodness of fit

2.Extreme value

3.PIPs and POPs

4.Downscaling

5.Detection and attribution

6.Determination of parameters

7.Analysis

(DJF) 3.04or

(JJA) 2.64fit with good

:M OWS

csextratropi in the windof casein

)x

( (x)

parameter scale shape;

on distributi Weibull

),(~ )(

)(1

x

W ef(B)

WuM

1. Goodness of fit

2. Extreme valuesLong memory?

Bunde et al., 2004: Return intervals of rare events in records with long-term persistence …

Distribution Pq(r) of return times between consecutive extreme values r. Rq is the expected value.

)].([

level. theexceeding of events obetween tw

time waitingoffunction density y probabilit the

0with 1

)()(

function) anceautocovari thebeing (

10with )()(

)/(

rPER

q

r

reR

rPB

kM

qq

Rr

qq

k

kq

Synthetic example with

k =0.4

722-1284 annual water levels of the Nile

Significance:

Extremes are not uniformly distributed in time, as described by a Poisson process, but appear in clusters.

Expected waiting time for next exceedance event conditional upon

length of previous waiting time r0.

Synthetic examples with k =0.4

722-1284 annual water levels of the Nile

Bunde et al., 2004: Return intervals of rare events in records with long-term persistence …

3. PIPS …

),,( (M) 1t tF

State space equation in low-dimensional subspace

Observational equation in high-dimensional space.

Parameters P, α determined such that

tt Px(B)

min),,(1 tt FPxE

… and POPsSpecial form

Ψ, λ complex numbers; (M) describes the damped rotation in a 2-dimensional space spanned by complex eigenvectors of E(xt+1xt

T) E(xtxt

T)-1. All eigenvectors form PT.

tt

t

Px

(B)

(M) 1t

Example: POP of MJO

Real and imaginary part of spatial pattern in equatorial velocity potential at 200 hPa

10-day forecast using state space equation in 2-d space

von Storch, H. and J.S. Xu, 1990: Principal Oscillation Pattern Analysis of the Tropical 30- to 60- day Oscillation: Part I: Definition of an Index and its Prediction. - Climate Dyn. 4, 175-190

4. DownscalingThe state space is simulated by ”reality” of by GCMs.

The observation equation relates large-scale variables, which are supposedly well observed (analysed) or simulated, to variables with relevant impact for clients.

Example: snow drops

Maak, K. and H. von Storch, 1997: Statistical downscaling of monthly mean air temperature to the beginning of the flowering of Galanthus nivalis L. in Northern Germany. - Intern. J. Biometeor. 41, 5-12

Large scale state: JFM mean temperature anomaly

Flowering date anomaly of snow drop (galanthus nivalis)

5. Detection and

attribution k

kkt gta )( (M)

adr

ktk gLA,

)( (B)

The state space dynamics is given by the assumption that the complete state of the atmosphere may be given by

The “patterns” gk represent the influence of a series of external influences, while ε represents the internal variability of the climate system. Ψ describes the full 3-d dynamics of the climate system.

The observation equation is formulated in a parameter space (A), and the state variable is projected on a space of observed variables (L[ψ] )

Here, L is the projector of the full space on the space of observed (and considered) variables, and gr,ad is the adjoint pattern of gk in the reduced space.

Detection means to test the null hypothesis

while attribution means the assessment that

Ak is consistent with ak.(i.e. Ak lies in a suitable small confidence “interval” of ak)

0:0 kAH

Detection and attribution (cont’d)Attribution diagram for observed 50-year trends in

JJA mean temperature.

The ellipsoids enclose non-rejection regions for testing the null hypothesis that the

2-dimensional vector of signal amplitudes estimated

from observations has the same distribution as the

corresponding signal amplitudes estimated from

the simulated 1946-95 trends in the greenhouse gas, greenhouse gas plus aerosol and solar forcing

experiments. Courtesy G. Hegerl.

Zwiers, F.W., 1999: The detection of climate change. In: H. von Storch and G. Flöser

(Eds.): Anthropogenic Climate Change. Springer Verlag, 163-209, ISBN 3-540-65033-4

6. Determination of parameters

In general, when many observations are available, optimal parameters α may be determined by finding those α which minimize the functional

),,(1 tt FBxE

Example: Determination of parameters – oceanic

dissipation

M2 tidal dissipation rates, estimated by combining Topex/Poseidon altimeter data with a hydrodynamical tide models. The solidlines encircle high dissipation areas in the deep ocean From Egbert and Ray [32]

Egbert GD, Ray RD (2000) Significant dissipation of tidal energy in the deepocean inferred from satellite altimeter data. Nature 45:775-778

7. AnalysisSkillful estimates of the unknown field Ψt are obtained by integrating the state-space equations and the observation equation forward in time:

)(

guessbest as and,

)(

),,(

1*

1*

11

*1

*1

*1

tttt

tt

tt

xxK

Bx

F

Example: spectralnudging in RCMsState space equation: RCM

Observable xt: large-scale features, provided by analyses or GCM output.

Correction step: nudging large-scales in spectral domain

Percentile-percentile diagram of local wind at an ocean location as recorded by a local buoy and as simulated in a RCM constrained by lateral control only, and constrained by spectral nudging

The purpose of statistics is …

• to specify pre-defined „models“ of reality by fitting characteristic numbers to observational evidence. developing and extending models and theories

• to analyze states and changes by interpreting empirical evidence in light of a pre-specified model. monitoring weather (and climate)

• to test theories and models as to whether they are valid in light of the empirical evidence. falsifying theories and models

Potential of „professional statisticians“

The specification of the models is usually not a statistical problem, but needs guidance by dynamical knowledge.

Therefore, when applying advanced method in climate science „professional“ statisticians often fail to achieve significant knowledge gains.

We need market places, where a) method-driven mathematical (and theoretical physics) statisticians meet problem-driven people from climate science b) other problem-driven scientists (e.g., geostatistians, econometricians) to allow the export of methods to climate science.

So what?