Presentation from the Kick-off Meeting "Seasonal to Decadal Forecast towards Climate Services: Joint Kickoff Meetings" for ECOMS, EUPORIAS, NACLIM and SPECS FP7 projects

1. Challenges in predicting
weather and climate extremes
David Stephenson
d.b.stephenson@exeter.ac.uk
Mathematics Research Institute
Seasonal to Decadal Forecast towards Climate Services: Joint
Kickoff Meetings, IC3 Barcelona, 6-9 November 2012. 1

2. Challenges
1. Which are the most appropriate methods to use to
define and estimate weather and climate extremes?
2. How are changes in extreme events related to changes
in the bulk of the distribution?
3. What can ensembles of imperfect models tell us about
real-world extreme events?
4. Should we recalibrate model simulated extremes, and if
so, how?
5. How predictable are weather and climate extremes?
2

3. What do we mean by “extreme”?
Large meteorological values
n Maximum value (i.e. a local extremum)
n Exceedance above a high threshold
n Record breaker (time-varying threshold
equal to max of previously observed
values)
Gare Montparnasse, 22 Oct 1895
Rare event in the tail of the distribution
(e.g. less than 1 in 100 years – p=0.01)
Large losses (severe or high-impact)
(e.g. $200 billion if hurricane hits Miami)
hazard, vulnerability, and exposure
Risk = ∑V (h( x, t ))e( x, t )
Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events
In Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp.
3

4. Methods used for weather and climate extremes
Various approaches are used such as:
n Sample statistics: “extreme” indices
n Changes in location & scale of
distribution
n Stochastic process models
n Basic extreme value modelling of tails
n GEV modelling of block maxima
n GPD modelling of excesses above high
threshold
n Point process model of exceedances
n More complex EVT models
n Inclusion of explanatory factors (e.g. trend,
ENSO, etc.)
n Spatial pooling
n Max stable processes
n Bayesian hierarchical models
n + many more
Katz, R.W. (2010) “Statistics of Extremes in Climate Change”, Climatic Change, 100, 71-76
4

5. Extreme indices are useful and easy but …
n They don’t always measure extreme values in the
tail of the distribution!
n They confound changes in frequency/rate and
intensity/magnitude
(e.g. EVT mean excess = σ /(1 − ξ ) )
n They strongly depend on threshold and so make
model comparison difficult
n They say nothing about extreme behaviour for
rarer extreme events at higher thresholds
n They don’t quantify sampling uncertainty - so
probability models are required to make
inference
We need instead to develop statistical
à
models of the process and then use
parameters from these models as indices.
5

6. … and the indices are not METRICS!
One should avoid the word “metric” unless
the statistic has distance properties! Index,
statistic, or measure is a more sensible name!
Oxford English Dictionary:
Metric - A binary function of a topological
space which gives, for any two points of the
space, a value equal to the distance between
them, or a value treated as analogous to
distance for analysis.
Properties of a metric:
d(x, y) ≥ 0
d(x, y) = 0 if and only if x = y
d(x, y) = d(y, x)
d(x, z) ≤ d(x, y) + d(y, z) 6

7. How might extreme events change?
Changes in location, scale,
and shape all lead to
big changes in the tail of the
distribution.
Some physical arguments
exist for changes in location
and scale.
E.g. Clausius-Clapeyron for
change in scale parameter
for precipitation.
7

8. How are the tails of the distribution …
8

9. related to the whole animal?
Change in scale Change in shape
PDF = Probability Density Function
Or … Probable Dinosaur Function??
9

10. Attributing changes in quantiles
Describe the changes in quantiles in terms of changes in the
location, the scale, and the shape of the parent distribution:
ΔIQR
ΔX α = ΔX 0.5 + ( X α − X 0.5 )
IQR
+ shape changes
The quantile shift is the sum of:
• a location effect (shift in median)
• a scale effect (change in IQR)
• a shape effect
Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for
exploring changing probability distributions of weather, J. Climate, 18, 4344 4354.
Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under
changing climatic conditions, Global and Planetary Change, 44, pp 1-9 10

11. Example: Regional Model Simulations of daily Tmax
T90 ΔT90 (2071-2100 minus 1971-2000)
ΔT90- ΔT90-Δm-(T90-m)
Δm Δs/s
à Changes in location, scale and shape all important 11

12. Estimating changes in extremes
How to infer distribution of future observations O’ from
distributions of present day observations O, model output
G and future model output G’?
1. No calibration
Assume O’ and G’ have identical distributions (i.e. no model biases!)
i.e. Fo’ = FG’
2. Bias correction
Assume O’=B(G’) where B(.)=Fo-1 (FG(.)) G G’
3. Change factor O’=B(G’)
Assume O’=C(O) where C(.)=FG’-1 (FG(.))
4. Other O
O’=C(O) O’
e.g. EVT fits to tail and then adjust EVT parameters
=?
Ho CK, Stephenson DB, Collins M, Ferro CAT, Brown SJ. (2012)
Calibration strategies: a source of additional uncertainty in climate change projections,
Bulletin of the American Meteorological Society, volume 93, pages 21-26. 12

13. Location, Scale and Shape adjustment
Adjust quantiles for changes in Location (L), Location and Scale (LS), and
Location, Scale and Shape (LSS) of the distributions.
Bias correction Change factor
O' = o + (G '− g ) O' = g ' + (O − g )
so sg '
O' = o + (G '− g ) O' = g ' + (O − g )
sg sg
Shape changes corrected by using
Box-Cox transformations
to map the data to normal before
doing LS transformation
λO
~ O −1
O=
λO
13

14. Change in 10-summer level 2040-69 from 1970-99
No calibration Bias correction Change factor
Tg’ - To
à Substantial differences between different estimates! 14

15. Universal process for extremes
N=number of points For a large number n of
with Z>z independent and identically
distributed values and a
sufficiently high threshold z:
N ~ Poisson(Λ)
Λn e − Λ
Pr(N = n) =
t=t1 t=t2 n!
−1/ ξ
⎡ ⎛ z − µ ⎞ ⎤
Λ = (t2 − t1 ) ⎢1 + ξ ⎜ ⎟ ⎥
⎣ ⎝ σ ⎠ ⎦ +
NB: “extreme” is a property
Stochastic point process model leads to of a process ... not a binary
n Generalized Extreme Value models for maxima
n Generalized Extreme Value models for r-largest values
quality of events!
n Generalised Pareto Distribution for excesses above a threshold 15

16. Probabiility models for maxima and excesses
lim n, z → ∞
−1/ ξ
Λn e − Λ ⎡ ⎛ z − µ ⎞ ⎤
Pr(N = n) = Λ = (t2 − t1 ) ⎢1 + ξ ⎜ ⎟ ⎥
n! ⎣ ⎝ σ ⎠ ⎦ +
⇒ Pr {max( Z ) ≤ z} = Pr {N ( z ) = 0} = e −Λ
Generalized Extreme Value (GEV) distribution
⇒ Pr {Z > z | Z > u} = Λ ( z ) / Λ (u )
−1/ ξ
⎡ ⎛ z − u ⎞ ⎤
= ⎢1 + ξ ⎜ %
σ = σ + ξ (u − µ )
⎣ % ⎟ ⎥ +
⎝ σ ⎠ ⎦
Generalized Pareto Distribution (GPD)
Note: extremal properties are
characterised by only three parameters
(for ANY underlying distribution!) 16

17. Example: GPD modelling with covariates
Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J.
(2008): Methods for exploring spatial and temporal variability of
extreme events in climate data, Journal of Climate, 21, pp 2072-2092
Observed surface temperatures 1870-2005
Monthly mean gridded surface temperature (HadCRUT2v)
n 5 degree resolution
n Summer months only: June July August
n Grid points with >50% missing values and SH are omitted.
Maximum monthly temperatures
Maximum temperature
80
60
40
20
0
-150 -100 -50 0 50 100 150
0 5 10 15 20 25 30 35 40
Celsius 17

18. GPD scale and shape estimates
−1
⎡ ⎛ z − u ⎞ ⎤ ξ
Pr( Z > z | Z > u ) = ⎢1 + ξ ⎜
⎣ ⎝ σ ⎟ ⎥ +
⎠ ⎦
log σ = α 0 + α1 x ξ = ξ0
Scale parameter is large over high-
latitude land areas AND shows
some dependence on ENSO.
Shape parameter is mainly
negative suggesting finite upper
temperature.
Spatial pooling used to get more
reliable shape estimates
18

19. Teleconnections of extremes
Bivariate measure of extremal dependency:
2 log Pr(Y > u )
χ= −1
log Pr(( X > u ) & (Y > u ))
Coles et al., Chi bar (75th quantile) Central Europe
b)
Extremes, (1999)
80
60
40
20
0
-150 -100 -50 0 50 100 150
-0.4 -0.1 0.1 0.4 0.7 1
à association with extremes in subtropical Atlantic 19

20. Worse things than extreme climate …
Thanks for your attention
d.b.stephenson@exeter.ac.uk
20

21. References
Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In
Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp.
Definitions of what we mean by extreme, rare, severe and high-impact events
Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing
probability distributions of weather, J. Climate, 18, 4344 4354.
Attribution of changes in extremes to changes in bulk distribution
Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic
conditions, Global and Planetary Change, 44, pp 1-9
Time-varying attribution of changes in heat wave extremes to changes in bulk distribution
Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and
temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092
GPD fits to gridded data including changing thresholds and covariates. Spatial pooling and teleconnection methods.
Antoniadou, A., Besse, P., Fougeres, A.-L., Le Gall, C. and Stephenson, D.B. (2001): L Oscillation Atlantique Nord
NAO: et son influence sur le climat europeen, Revue de Statistique Applique , XLIX (3), pp 39-60
One of the earliest papers to use climate covariates in EVT fits – extreme value distribution of central England
temperature depends on NAO.
Stuart Coles, An Introduction to Statistical Modeling of Extreme Values, Springer.
Excellent overview of extreme value theory.
21

22. Non-stationarity due to seasonality and long term trends
Example: Grid point in Central Europe (12.5ºE, 47.5ºN)
2003 exceedance
Excess
(Ty,m – uy,m)
75th quantile (uy,m = 16.2ºC) a)
20
15
10
Temperature
Long term trend in mean
(Celsius)
5
0
-5
2001 2002 2003 2004 2005 2006
year 22

23. Spatial pooling
Pool over local grid points but allow for
spatial variation by including local spatial
covariates to reduce bias (bias-variance
tradeoff).
For each grid point, estimate 5 GPD
parameters by maximising the following
likelihood over the 8 neighbouring grid
points:
1
Lij = ∏ f (y i + Δi , j + Δj ; σ i + Δi , j + Δj , ξ i + Δi , j + Δj )
Δj = −1
Δi = −1
log σ i + Δi , j + Δj = σ i0, j + σ ix, j ( xi + Δi − xi ) + σ iy, j ( y j + Δj − y j )
ξ i + Δi , j + Δj = ξ 0
i, j
No spatial pooling: 2 parameters from n data values
Local pooling: 5 parameters from 9n data values
23

24. How significant is ENSO on extremes?
à Null hypothesis of no effect can only be rejected with
confidence over tropical Pacific and Northern Continents
24

25. Return periods for August 2003 event
Return period for the excess
Excess ExcessesAugust 2003
a) August 2003:
for above 75% threshold for August 2003
b) August 2003: Return period
80
80
60
60
40
40
20
20
0
0
-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150
0 1 2 3 4 1 5 10 50 150 500
Celsius years
à Europe return period of 133 years (<< 46000 years from Normal!)
25