R help - Jun 2008 - stationary "terminology" time series question

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 ?

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.

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.

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