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Recently, I received comments (here and on Twitter) about my previous graphs on the

temperature in Paris. I mentioned in a comment (there) that studying

extremas (and more generally quantiles

or interquantile evolution) is not the same as studying the variance.

Since I am not a big fan of the variance, let us talk a little bit

about extrema behaviour.

In order to study the average temperature it is natural to look at the

linear (assuming that it is linear, but I proved that it could

reasonably be assumed as linear in the paper) regression, i.e. least

square regression, which gives the expected value. But if we care about

extremes, or almost extremes, it is natural to look at quantile

regression.

For instance, below, the green line

is the least square regression, the red one is 97.5%

quantile,

and the blue

on the 2.5% quantile regression.

It looks like the slope is the same, i.e. extremas are increasing as

fast as the average…

tmaxparis=read.table("temperature/TG_SOUID100845.txt",

skip=20,sep=",",header=TRUE)

head(tmaxparis)

Dparis=as.Date(as.character(tmaxparis$DATE),"%Y%m%d")

Tparis=as.numeric(tmaxparis$TG)/10

Tparis[Tparis==-999.9]=NA

I=sample(1:length(Tparis),size=5000,replace=FALSE)

plot(Dparis[I],Tparis[I],col="grey")

abline(lm(Tparis~Dparis),col="green")

library(quantreg)

abline(rq(Tparis~Dparis,tau=.025),col="blue")

abline(rq(Tparis~Dparis,tau=.975),col="red")

(here I plot randomly some points to avoid a too heavy figure, since I

have too many observations, but I keep all the observations in the

regression !).

Now, if we look at the slope for different quantile level (Fig 6 in the

paper, here,

but on minimum daily temperature, here I look at average daily

temperature), the interpretation is different.

s=0

COEF=SD=rep(NA,199)

for(i in seq(.005,.995,by=.005)){

s=s+1

REG=rq(Tparis~Dparis,tau=i)

COEF[s]=REG$coefficients[2]

SD[s]=summary(REG)$coefficients[2,2]

}

with the following graph below,

s=0

plot(seq(.005,.995,by=.005),COEF,type="l",ylim=c(0.00002,.00008))

for(i in seq(.005,.995,by=.005)){

s=s+1

segments(i,COEF[s]-2*SD[s],i,COEF[s]+2*SD[s],col="grey")

}

REG=lm(Tparis~Dparis)

COEFlm=REG$coefficients[2]

SDlm=summary(REG)$coefficients[2,2]

abline(h=COEFlm,col="red")

abline(h=COEFlm-2*SDlm,lty=2,lw=.6,col="red")

abline(h=COEFlm+2*SDlm,lty=2,lw=.6,col="red")

Here, for minimas (quantiles associated to *low* probabilities, on the left), the

trend has a higher slope than the average, so in some sense, warming of

minimas is stronger than average temperature, and on other hand, for

maximas (*high* probabilities on

the right), the slope is smaller – but positive – so summer are warmer,

but not as much as winters.

Note also that the story is different for minimal temperature

(mentioned in the paper) compared with that study, made here on average

daily temperature (see comments)… This is not a major breakthrough in climate research, but this is all I got…

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