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Example: precipitation
What information can we use?
f(x) = 1/µ exp{-x/µ}
µ is the wet-day mean
How to estimate parameters for precipitation.
Number of data points → expectation values (central limit theorem)
Parameters as variables
Parameters as variables
Simple model for wet-day amount
Simple model for wet-day amount
“Heavy precipitation”
A test
Different quantities
x = fx = fww µµ
Mean = (wet-frequency) • (wet-mean)
How sensitive are the parameters?
Probabilities for all days
Pr(X>x) = fPr(X>x) = fw w ee
-x/-x/µµ
How often does it rain? ffww (fraction) (fraction)
How often does it rain?
The 24-hr precipitation amounts
Pr(X>x) = fPr(X>x) = fw w ee
-x/-x/µµ
How much does it rain when it rains?
µ µ (mm/day)(mm/day)
How much does it rain when it rains?
The question: more heavy rain events?
• Observations: annual mean µ- Downscaled from 107 CMIP5 GCMs (RCP4.5)
PRELIMINARY RESULTS
Temperature
library(esd)data(ferder)y=anomaly(ferder)hist(coredata(y),breaks=seq(-21,15,by=0.5),col='pink',freq=FALSE,xlab='Temperature anomaly (deg C)',main='Ferder lighthouse')lines(x,dnorm(x,mean(y,na.rm=TRUE),sd=sd(y,na.rm=TRUE)),lwd=4)text(-20,0.15,expression(f(x)==1/(sqrt(2*pi) * sigma) * e^-((x - mu)^2/(2 *sigma^2))),pos=4)
Temperature
Two parameters: µ and σ
Temperature
Dependency on the large-scale conditions?
From parameters to weather
How to use the probabilities
Pr(X>x) = fPr(X>x) = fw w ee
-x/-x/µµ
How to use the probabilities
Pr(X>x) = fPr(X>x) = fw w ee
-x/-x/µµ
Pr(X>x) = Pr(X>x) = 1/(21/(2σσ ) e) e-(x--(x-µµ )/)/σσ
How to predict number of events
p=Pr(X>x) = fp=Pr(X>x) = fw w ee
-x/-x/µµ
Pr(X=k) = Pr(X=k) = nnCCkk p pkk
(1-p)(1-p)n-kn-k
The number of heavy rain events
Predicting number of heavy rain events
Disaggregation: synthesizing daily series from parameters
Weather generators
Different kinds
The specific question
Input from climate parameters– Temporal structure?
• ρ1, LTP, f
w, n
cdd, n
cwd
– Spatial structure?
Validation
Different types:
Traditional:• Cross-validation
– Correlation, RMSE.
• GCM: common EOFs
Non-traditional:• Probabilities: binomial
• Non-stationarity test.
