why you need an umbrella on hot days · 2015. 7. 9. · low sample size does not underestimate high...
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Why you need an umbrella on hot days
http://7-themes.com/7024707-child-with-umbrella.html
application of extreme value statistics to precipitation
Berry Boessenkool, uni-potsdam.de, June 2015
github.com/brry
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
60 years of hourly records at 14 stations across Germany (figs: Potsdam)P
reci
pita
tion
[mm
/h]
(logs
cale
)
Precipitation [mm/h] (logscale)
2
5
10
20
Daily average temperature [°C]
0 10 20 30
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
precipitation intensities follow CC-scaling of air moisture with temperatureP
reci
pita
tion
[mm
/h]
(logs
cale
)
Precipitation [mm/h] (logscale)
2
5
10
20
Daily average temperature [°C]
0 10 20 30
99 % QuantileVPsat [hPa]
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
VPsat: Clausius-Clapeyron governed saturated Vapor Pressure
High precipitation quantiles drop at high temperaturesP
reci
pita
tion
[mm
/h]
(logs
cale
)
Precipitation [mm/h] (logscale)
2
5
10
20
Daily average temperature [°C]
0 10 20 30
99.99 %Q99.99990VPsat [hPa]CC−rate
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
High precipitation quantiles drop at high temperaturesP
reci
pita
tion
[mm
/h]
(logs
cale
)
Precipitation [mm/h] (logscale)
2
5
10
20
Daily average temperature [°C]
0 10 20 30
99.99 %Q99.99990VPsat [hPa]CC−rate
Dependency on sample size?
3 66 215 579 1139164117651171 627 307 112 33 7 1
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
Fit distribution functions (available in lmomco)extremeStat::distLfit
Precipitation [mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.5 1 2 5 10 20 50
(Empirical) Probability Density Function (PDF)
wakweikappe3ln3gnogumgev
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
sorted by goodness of fit (RMSE)
Distribution quantiles differ a lotextremeStat::distLquantile
Precipitation [mm/h]
0.00
0.02
0.04
0.06
0.08
5 10 20 50 100
wakweikappe3ln3gnogumgev
(Empirical) Probability Density Function (PDF)
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
lines:Q99.9%
Censored quantiles (of the highest values of a sample) are robustextremeStat::distLquantile(..., truncate=0.9)
Precipitation [mm/h] truncated to top 10%
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
5 10 20 50
gpakapwakweipe3ln3gnoexp
(Empirical) Probability Density Function (PDF)
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
Low sample size underestimates high quantilesEmpirical quantile: order based quantile function in R::stats
0 200 400 600 800sample size n
random sample 99.9% quantile [mm/h]
2
5
10
20
50
100 empirical quantile
smoothing bandwidth = 15distribution quantile: 22.9 mm/hmedian of 200 simulationssimulation mean (not smoothed)
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
Low sample size does not underestimate high quantiles if you useParametric quantile: quantile from distribution fitted to sample
0 200 400 600 800sample size n
random sample 99.9% quantile [mm/h]
2
5
10
20
50
100 empirical quantile
smoothing bandwidth = 15distribution quantile: 22.9 mm/hmedian of 200 simulationssimulation mean (not smoothed)
0 200 400 600 800sample size n
parametric quantileparametric
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
In a synthetic, continuously rising PT-relationship,simulated empirical quantiles show observed drop
5 10 15 20 25
Precipitation [mm/h]
Temperature [°C]
5
10
20
50CC−scalingDistribution QEmpirical Q
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
In a synthetic, continuously rising PT-relationship,parametric quantiles do not drop at high temperatures
5 10 15 20 25
Precipitation [mm/h]
Temperature [°C]
5
10
20
50CC−scalingDistribution QEmpirical QParametric Q
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
parametric truncated quantiles applied to original dataset:very high quantiles might actually continue to rise with temperature
2
5
1090 %Q
366
215579
11391641
17651171
627307
112
33
71
5
10
20 99 %Q
WeibullWakebyKappaEmpirical
5 10 15 20 25 30
5
10
20
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100
200 99.9 %Q
5 10 15 20 25 30
5
10
20
50
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50099.99 %Q
Pre
cipi
tatio
n [m
m/h
]
Daily average temperature [°C]
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
That’s why you need an umbrella on hot days
precipitation intensity drop at high temperaturesmay be an effect of sample size
and not an actual meteorological boundary
use parametric quantiles for extreme rainfall intensity estimation
github.com/brry/prectemp
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?
devtools::install github("brry/extremeStat")
distLquantile(x, probs=0.999, truncate=0.8, plot=TRUE)
Distribution Q 99.9%wak 23.903843kap 23.857294wei 25.809802gpa 21.305499pe3 27.480982ln3 32.388869gno 32.388869gev 35.685466gum 23.770059glo 45.456163exp 36.798357ray 17.076555
gam 17.612948lap 19.765904rice 13.889985nor 13.746608
revgum 9.940204quantileMean 23.424439
Density distributions of xtrunc
Pro
babi
lity
Den
sity
Fun
ctio
n (P
DF
)
0.00
0.05
0.10
0.15
0.20
10 20 50
wakkapweigpape3
Boessenkool, useR!2015: precipitation intensity drop at high temperatures - a statistical artefact?