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…

here is the slope of quantiles regressions on daily

maximasand daily

minimas