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, [EMAIL PROTECTED] 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, [EMAIL PROTECTED]
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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