(This article was first published on

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.**Freakonometrics - Tag - R-english**, and kindly contributed to R-bloggers)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")

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]

}

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")

*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...

To

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