[R] stationary "terminology" time series question

Prof Brian Ripley ripley at stats.ox.ac.uk
Fri Jun 27 08:49:10 CEST 2008

Stationarity is a statement about a stochastic process, not about a single 
realization. It is a statement about what might have happened, not what 
did happen.

A sine wave with a random (uniform) wave is stationary, and indeed the 
superposition of such waves is the spectral decomposition. A sine wave 
with a fixed phase is not.

So stationarity is a modelling assumption.  This comes up often in the 
geosciences when you have just one realization.  For example look at earth 
temperature series -- whether they are stationary is a modelling 
assumption, and may in part depend on the timescale involved.  But for 
example James Lovelock's Gaia hypothesis implies stationarity.

On Thu, 26 Jun 2008, markleeds at verizon.net wrote:

> This is not exactly an R question but the R code below may make my question 
> more understandable.
> If one plots sin(x) where x runs from -pi to pi , then the curve hovers 
> around zero obviously. so , in  a"stationary in the mean" sense,
> the series is stationary. But, clearly if one plots the acf, the 
> autocorrelations at lower lags are quite high and, in  the "box jenkins"
> sense, this series is clearly not stationary in terms of its acf. so, i'm 
> confused in terms of what ithe statistical definition of stationary is
> as box jenkins define it ?

You are crediting Box and Jenkins (sic) with something that was long 
established before them.  Using the ACF needs only second-order 
stationarity, without which it is not defined.

> I don't have their text in front of me but I don't remember them having an 
> example such as below when they talk about needing to difference series
> to achieve stationarity.  thanks for any insights or a text that talks about 
> this.

Almost all good texts do.  Perhaps Box and Jenkins have confused you by 
majoring on ARIMA models, which can be made stationary by differencing -- 
not a general attribute but useful for the sales forecasting series that 
(I am told) was their primary motivation.

> x <- seq(pi,-pi,by=-pi/4)
> y <- sin(x)
> plot(x,y)
> acf(y)
> P.S: this question arose because a colleague asked me to look at the plot of 
> his series and the associated acf and he claims it's a stationary series and
> I'm trying to explain to him that it is not and to try to use the acf to 
> build a model for it is not reasonable.

Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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