# Forecasting Natural Catastrophes (is rather difficult)

Following my previous post, I wanted to spend more time, on the time series with “global weather-related disaster losses as a proportion of global GDP” over the time period 1990-2016 that Roger Pilke sent me last night.

```db=data.frame(year=1990:2016, ratio=c(.23,.27,.32,.37,.22,.26,.29,.15,.40,.28,.14,.09,.24,.18,.29,.51,.13,.17,.25,.13,.21,.29,.25,.2,.15,.12,.12))```

In my previous post, I spend some time explaining that we should provide some sort of ‘confidence interval’ when we try to predict a pattern. That was what we call ‘model uncertainty’. But there are two (important) issues that I did not mention. (1) it is a time series, so why not use techniques dedicated to time series objects ? (2) we do not really care actually about ‘model uncertainty’ (unless we want to assess if a decreasing trend is significant, or not), and we care more about real prediction uncertainty: in the next ten years, what could be the range for the this ratio, with some given probability (say 95%)? Could we say that with 95% chance the global weather-related disaster losses as a proportion of global GDP should be (each year) within 0 and 0.35 or 0 and 0.7?

A first idea might be to use exponential smoothing techniques (without a seasonal component here).

```ratio=ts(db\$ratio,start=1990,frequency=1) plot(ratio,xlim=c(1990,2030)) hw=HoltWinters(ratio,gamma=FALSE) phw=predict(hw,n.ahead=15,prediction.interval = TRUE) plot(hw,phw,xlim=c(1990,2030)) polygon(c(2017:2031,rev(2017:2031)), c(phw[,2],rev(phw[,3])),border=NA,col=rgb(0,0,1,.2))```

The decreasing trend is coming from the fact that exponential smoothing is here a linear regression, with weight exponentially decaying with time (the older, the smaller the weight). But we cannot use that prediction, since the ratio cannot (obviously) be negative. So why not consider, here, the logarithm of the ratio

```plot(db\$year,db\$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio") hw=HoltWinters(log(ratio),gamma=FALSE) phw=predict(hw,n.ahead=15,prediction.interval = TRUE) abline(v=2016,lty=2,col="grey") lines(2017:2031,exp(phw[,2]),col="blue") lines(2017:2031,exp(phw[,3]),col="blue") lines(c(1992:2016,2017:2031),c(exp(hw\$fitted[,1]),exp(phw[,1])),col="red") polygon(c(2017:2031,rev(2017:2031)),exp(c(phw[,2],rev(phw[,3]))),border=NA,col=rgb(0,0,1,.2)) abline(h=0,lty=2)```

The confidence band is huge, here. What if we consider some ARIMA model here?

```fit=auto.arima(ratio) farma=forecast(fit,15) farma=cbind(as.numeric(farma\$fitted)[1:15],as.numeric(farma\$lower[,1]),as.numeric(farma\$upper[,1]),as.numeric(farma\$lower[,2]),as.numeric(farma\$upper[,2])) plot(db\$year,db\$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio") abline(v=2016,lty=2,col="grey") lines(2017:2031,farma[,4],col="blue") lines(2017:2031,farma[,5],col="blue") lines(2017:2031,farma[,1],col="red") polygon(c(2017:2031,rev(2017:2031)),c(farma[,4],rev(farma[,5])),border=NA,col=rgb(0,0,1,.2)) abline(h=0,lty=2)```

Here, there is an intercept, but no dynamics for the time series (which is considered, here, as a pure white noise). We get exactly the same if we consider the average value of the series

```fit=lm(ratio~1,data=db) s=summary(fit)\$sigma plot(db\$year,db\$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio") abline(v=2016,lty=2,col="grey") ndb=data.frame(year=2017:2031) pf=predict(fit,newdata=ndb) farma=cbind(pf,pf-1.96*s,pf+1.96*s) lines(2017:2031,farma[,2],col="blue") lines(2017:2031,farma[,3],col="blue") lines(1990:2031,c(predict(fit),farma[,1]),col="red") polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2)) abline(h=0,lty=2)```

Here, we get back to my previous post, if we want to consider a possible trend (and not only an intercept)

```fit=lm(ratio~year,data=db) s=summary(fit)\$sigma plot(db\$year,db\$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio") abline(v=2016,lty=2,col="grey") ndb=data.frame(year=2017:2031) pf=predict(fit,newdata=ndb) farma=cbind(pf,pf-1.96*s,pf+1.96*s) lines(2017:2031,farma[,2],col="blue") lines(2017:2031,farma[,3],col="blue") lines(1990:2031,c(predict(fit),farma[,1]),col="red") polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2)) abline(h=0,lty=2)```

Again, the confidence region is not based