This is the "Hockey Stick Graph"

It was first presented in 1999 by Mann, Bradley & Hughes as evidence for man-made global warming in the ongoing climate debate.

Despite the controversy surrounding reconstruction studies, we are not here to take any side in the argument. What we'll be doing today is to demostrate using this chart Segmented Linear Regression.

So let us get started!

The "hockey stick" is something that can be modelled using a

Spline a piece-wise defined polynomial function satisfying continuity at all breakpoints

which in its simplest form is just two linear segments joined at one single knot point

as represented by this equation: \[ Y_i = \beta_0 + \beta_1 X_i + \gamma_1 (X_i - \xi)_+ + \epsilon_i \] where \(\xi\) is a known knot point

Here's an implementation in R:

# This is the code to reproduce the MBH99 hockey stick diagram shown earlier
sourceURL <- paste("http://www.meteo.psu.edu/holocene/public_html",
                   "/shared/research/ONLINE-PREPRINTS/Millennium",
                   "/DATA/RECONS/nhem-recon.dat")
dat <- read.table(sourceURL, col.names=c("year","temp"))
dat <- transform(dat, splineTerm = (year - 1900) * (year > 1900))
fit <- lm(temp~year+splineTerm, data=dat)
with(dat,plot(year,temp))
lines(dat$year, fit$fitted.values, col="red", lwd=3)

So far so good?

?? But what if we don't know

where's the knot point ??

No worries. Because an App has been created just to make things a little easier for you!