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Climate Change

Manfred Mudelsee

Climate Risk Analysis, Hannover, Germany

Alfred Wegener Institute, Bremerhaven, Germany

Potsdam, September 2010 Millennium Flood, July 1342 (Stadtarchiv Nürnberg)

Solomon et al. (Eds.) (2007) Climate Change 2007: The Physical Science Basis. Cambridge Univ. Press

Dresden, river Elbe

long-term

mean and variability

Brückner (1890) Geographische Abhandlungen 4:153

Hann (1901) Lehrbuch der Meteorologie, Tauchnitz

Köppen (1923) Die Klimate der Erde: Grundriss der Klimakunde, de Gruyter

Global Runoff Data Centre, Koblenz

Dresden, river Elbe

mean and variability risk

and extremes

Gardenier & Gardenier (1988) In: Encyclopedia of statistical sciences 8:141, Wiley

Mudelsee (2006) DKKV/ARL Workshop

Dresden, river Elbe

climate risk change?

Solomon et al. (Eds.) (2007) Climate Change 2007: The Physical Science Basis. Cambridge Univ. Press

{t(i), x(i)} sample (time series)

n sample size

q parameter

q estimator

CIq,12 = [ql; qu] confidence interval

12 confidence level

seq standard error

E[q ],biasq expectation, bias

n

i=1

^

^

^

^

^

^

^

“Estimates without error bars are useless.”

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Simple problem

derive CI analytically

(e.g., mean estimation, t distribution of q)

Advanced problem

resample data: bootstrap,

Bayesian

^

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

resample data shape

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

resample data shape

resample data blocks shape, persistence

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Data type Parametric model

Block maxima Generalized Extreme

Value distribution (GEV)

Peaks over Generalized Pareto

threshold (POT) distribution (GP)

p tail probability, risk

1/p return period (time units)

xp return level

u threshold

GEV distribution

GEV distribution, time-dependent

Coles (2001) An Introduction to Statistical Modeling of Extreme Values. Springer

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

GEV distribution

GEV distribution, time-dependent

“Recent achievements ...”

use of covariates

multivariate: difficult

Coles (2001) An Introduction to Statistical Modeling of Extreme Values. Springer

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

nonparametric:

kernel estimation

l occurrence rate, risk per time unit (Poisson process) h bandwidth

T time K kernel function

Tout extreme event date m number of extremes

Elbe, winter, class 2–3

m = 64, h = 35 yr

Diggle (1985) Applied Statistics 34:138

Elbe, winter, class 2–3

m = 64, h = 35 yr

Bootstrap resample

(with replacement, same size)

Elbe, winter, class 2–3

m = 64, h = 35 yr

Bootstrap resample

(with replacement, same size)

Elbe, winter, class 2–3

m = 64, h = 35 yr

Bootstrap resample

(with replacement, same size)

2nd

Bootstrap resample

Elbe, winter, class 2–3

m = 64, h = 35 yr

Bootstrap resample

(with replacement, same size)

2nd

Bootstrap resample

10 000

Bootstrap resamples

Elbe, winter, class 2–3

m = 64, h = 35 yr

Bootstrap resample

(with replacement, same size)

2nd

Bootstrap resample

10 000

Bootstrap resamples

90% confidence band

Elbe, winter, class 2–3

m = 64, h = 35 yr

Cowling et al. (1996) Journal of the American Statistical Association 91:1516

more maths

CI construction

bandwidth selection

boundary bias correction

90% confidence band

Elbe, winter, class 2–3

m = 64, h = 35 yr

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

“Recent achievements ...”

uncertain timescales (Tout*), paleoclimate

hybrid model

90% confidence band

Elbe, winter, class 2–3

m = 64, h = 35 yr

Smith (1989) Statistical Science 4:367

H0: “Constant occurrence rate, l(T).”

H1: “Increasing occurrence rate.”

U test statistic Under H0, statistic U

Tout extreme event date (“POT data”) is standard normally distributed.

m number of extremes

[T(1); T(n)] observation interval

n sample size

Cox & Lewis (1966) The Statistical Analysis of Series of Events. Methuen

H0: “Constant occurrence rate, l(T).”

H1: “Increasing occurrence rate.”

U test statistic Under H0, statistic U

Tout extreme event date (“POT data”) is standard normally distributed.

m number of extremes

[T(1); T(n)] observation interval

n sample size

Cox & Lewis (1966) The Statistical Analysis of Series of Events. Methuen

H0: “Constant occurrence rate, l(T).”

H1: “Increasing occurrence rate.”

U test statistic Under H0, statistic U

Tout extreme event date (“POT data”) is standard normally distributed.

m number of extremes

[T(1); T(n)] observation interval

n sample size

Cox & Lewis (1966) The Statistical Analysis of Series of Events. Methuen

Climate Risk Test: Mann–Kendall

H0: “Constant trend (mean).”

H1: “Increasing trend.”

Kendall (1938) Biometrika 30:81 Mann (1945) Econometrica 13:245

Climate Risk Test: Mann–Kendall

Zhang et al. (2004) Journal of Climate 17:1945

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Climate Risk Test: Monte Carlo

start

simulation

prescribed prescribed

l(T) PDFs

end

simulation

empirical power = #(H1) / nsim

generate time series

test H0 vs. H1

Climate Risk Test: Monte Carlo

nsim = 90000 n sample size

significance level = 0.10 mtrue prescribed number of extremes

sepower = 0.001 k scaling parameter l(T) ~ T1/k–1

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

“Recent achievements ...”

use Cox–Lewis, not Mann–Kendall

Zhang et al. (2004) Journal of Climate 17:1945

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Wildfires,

Canada

Girardin & Mudelsee (2008) Ecological Applications 18:391

Girardin et al. (2009) Global Change Biology 15:2751 Canadian Forest Service

Hurricanes,

Boston

Besonen et al. (2008) Geophysical Research Letters 35:L14705

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Climate change Climate risk change?

1. Estimates: give error bars

2. Tests: use Cox–Lewis,

not Mann–Kendall

3. Future achievements:

better estimators

systematic application,

data/models, small scale

• floods

• hurricanes

Climate change Climate risk change?

1. Estimates: give error bars

2. Tests: use Cox–Lewis,

not Mann–Kendall

3. Future achievements:

better estimators

systematic application,

data/models, small scale

• floods

• hurricanes

Thank you!

Climate Risk Test: Mann–Kendall

Appendix

H0: “Constant trend (mean).”

H1: “Increasing trend.”

x(3) x(1)

x(2) x(2)

x(4) x(3)

x(1) x(4)

U test statistic

U’ #(interchanges) Under H0, statistic U

n sample size is normally distributed.

Kendall (1938) Biometrika 30:81 Mann (1945) Econometrica 13:245

Climate Risk Test: Monte Carlo

Appendix

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Climate Risk Test: Monte Carlo

Appendix

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (1.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 90000, se = 0.001; significance level: 0.10.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (1.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 90000, se = 0.001; significance level: 0.10.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (1.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 47500, se = 0.001; significance level: 0.05.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (1.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 47500, se = 0.001; significance level: 0.05.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (3.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 90000, se = 0.001; significance level: 0.10.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (3.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 90000, se = 0.001; significance level: 0.10.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (3.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 47500, se = 0.001; significance level: 0.05.

Climate Risk Test: Monte Carlo

Appendix

n sample size

mtrue prescribed number of extremes, shifted (3.0) chi-squared in lognormal AR(1) noise

k scaling parameter for upwards trend, l(T) ~ T1/k–1

Empirical power: #(simulations) where H0: “no trend” is rejected against H1: “upwards

trend,” divided by nsim = 47500, se = 0.001; significance level: 0.05.

Appendix

Heatwaves

index variable = intensity x number of days

Kürbis et al. (2009) Theoretical and Applied Climatology 98:187

Book

http://www.manfredmudelsee.com/book

Climate Risk and Time Series Analysis

http://www.climate-risk-analysis.com

http://www.mudelsee.com

Mudelsee (2010) Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Springer

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