In the paper on the heat wave in Paris (mentioned here) I discussed changes in the distribution of temperature (and autocorrelation of the time series).

During the workshop on *Statistical Methods for Meteorology and Climate Change* today (here) I observed that it was still an important question: is climate change affecting only averages, or does it have an impact on extremes ? And since I’ve seen nice slides to illustrate that question, I decided to play again with my dataset to see what could be said about temperature in Paris.

Recall that data can be downloaded here (daily temperature of the XXth century).

tmaxparis=read.table("/temperature/TX_SOUID100124.txt", skip=20,sep=",",header=TRUE) Dmaxparis=as.Date(as.character(tmaxparis$DATE),"%Y%m%d") Tmaxparis=as.numeric(tmaxparis$TX)/10 tminparis=read.table("/temperature/TN_SOUID100123.txt", skip=20,sep=",",header=TRUE) Dminparis=as.Date(as.character(tminparis$DATE),"%Y%m%d") Tminparis=as.numeric(tminparis$TN)/10 Tminparis[Tminparis==-999.9]=NA Tmaxparis[Tmaxparis==-999.9]=NA annee=trunc(tminparis$DATE/10000) MIN=tapply(Tminparis,annee,min) plot(unique(annee),MIN,col="blue",ylim=c(-15,40),xlim=c(1900,2000)) abline(lm(MIN~unique(annee)),col="blue") abline(lm(Tminparis~unique(Dminparis)),col="blue",lty=2) annee=trunc(tmaxparis$DATE/10000) MAX=tapply(Tmaxparis,annee,max) points(unique(annee),MAX,col="red") abline(lm(MAX~unique(annee)),col="red") abline(lm(Tmaxparis~unique(Dmaxparis)),col="red",lty=2)

On the plot below, the dots in red are the annual maximum temperatures, while the dots in blue are the annual minimum temperature. The plain line is the regression line (based on the annual max/min), and the dotted lines represent the average maximum/minimum daily temperature (to illustrate the global tendency),

It is also possible to look at annual boxplot, and to focus either on minimas, or on maximas.

annee=trunc(tminparis$DATE/10000) boxplot(Tminparis~as.factor(annee),ylim=c(-15,10), xlab="Year",ylab="Temperature",col="blue") x=boxplot(Tminparis~as.factor(annee),plot=FALSE) xx=1:length(unique(annee)) points(xx,x$stats[1,],pch=19,col="blue") abline(lm(x$stats[1,]~xx),col="blue") annee=trunc(tmaxparis$DATE/10000) boxplot(Tmaxparis~as.factor(annee),ylim=c(15,40), xlab="Year",ylab="Temperature",col="red") x=boxplot(Tmaxparis~as.factor(annee),plot=FALSE) xx=1:length(unique(annee)) points(xx,x$stats[5,],pch=19,col="red") abline(lm(x$stats[5,]~xx),col="red")

Plain dots are average temperature below the 5% quantile for minima, or over the 95% quantile for maxima (again with the regression line),

We can observe an increasing trend on the minimas, but not on the maximas !

Finally, an alternative is to remember that we focus on annual maximas and minimas. Thus, Fisher and Tippett theory (mentioned here) can be used. Here, we fit a GEV distribution on a blog of 10 consecutive years. Recall that the GEV distribution is

install.packages("evir") library(evir) Pmin=Dmin=Pmax=Dmax=matrix(NA,10,3) for(s in 1:10){ X=MIN[1:10+(s-1)*10] FIT=gev(-X) Pmin[s,]=FIT$par.ests Dmin[s,]=FIT$par.ses X=MAX[1:10+(s-1)*10] FIT=gev(X) Pmax[s,]=FIT$par.ests Dmax[s,]=FIT$par.ses }

The location parameter is the following, with on the left the minimas and on the right the maximas,

while the scale parameter is

and finally the shape parameter is

On those graphs, it is very difficult to say anything regarding changes in temperature extremes… And I guess this is a reason why there is still active research on that area…

One as to remember that variance is a common tool to study dispersion, but so is the interquantile distance. And both can vary in different direction…. Consider the following data, with 2.5%, 5%, 95% and 97.5% quantiles. The interquantile distance is increasing as we go on the right,

But the variance is decreasing,

(here I considered Student distribution with a scale parameter decreasing with the number of degrees of freedom). My point is that studying maximas/minimas (i.e. quantile related since there are 365 days per year) is not the same as studying the variance.

Here I studied maximas and minimas because a lot of researchers are working on that. Nevertheless, since the temperature series is highly nonstationnary (increasing trend and seasonality), I do not really like the idea of considering the series to study its distribution. And if we do remove the trend and the cycle, we get what was mentioned in the paper. But maybe I should post a more detailed note on that….

Yet, if the average temperature is increasing , which seems very well documented elsewhere , unless the distribution changes the minimum and maximum would increase as well. These data imply that the variance decreased, no? I’d say that is a very unexpected result

ANSWER: actually (1) the increase of the average is small, extremely small, but significant, (2) it is possible to find distribution with extremes getting closer, and variance increasing… I did not calculate the variance, but if you want, I can….