Dear List
I only received one reply to my question about Box-Tiao intervention analysis
(thanks Laimonis). At the end of the day I decided on a suitable model using
GAM, mainly because of a range of minor "atypical events" present in the
series (but at unknown intervals) in addition to the known interruptions. I
did get a couple of requests to post any information I might find, so:
Original Question:
> I am attempting to fit a Box-Tiao intervention model to an interrupted
time> series. I am unsure how to set up the indicator variables to represent
the> various types of intervention. I assume that a point intervention would
take
> the form:
> 0 0 0 0 0 1 0 0 0 0 0
> for the length of the time series, where 1 indicates a temporary point>
interruption, and:
> 0 0 0 0 0 1 1 1 1 1 1
> would indicate a permanent point interruption.
>
> How, though, do I construct variables to indicate gradual permanent and>
gradual temporary intervention? I can't think how the indicator > variable(s)
should look, or indeed how to estimate the "gradual".
>
> Also, has anyone worked up code (for a teaching exercise, for example) for>
Exercise 13.5 in Venables and Ripley's MASS (3rd Ed, p.431). It might be>
useful for me to see how someone else analyses this type of data.
>
> I am using S-Plus 2000 (Rel 3) for Windows. Please respond directly and I>
will summarise for the list.
From: "Laimonis Kavalieris" <lkavalieris@maths.otago.ac.nz>
One way to introduce gradual changes is to use models such as
y(t) = c +| a y(t-1) + bx(t)+ e(t)
where x(t) is the indicator variable. Suppose the intervention occurs
at time T. Prior to the intervention the mean of the process is
c/(1-a). At time T+j , i.e. j observations after the intervention add
(a^j)*b to the mean, so that the post intervention mean converges
geometrically to (c+b)/(1-a). The easy way of fitting this model is to set the
regression variables (xreg) in the arima function to be the x(t) and
lagged y(t-1), i.e. simply regress y(t) on x(t) and y(t-1). This
assumes that the residuals are uncorrelated, but upon examining the acf
and pacf of the residuals you can often identify an appropriate model
for the noise term that can be incorporated into the arima model. Of
course there is the problem of identifying the nature of the
intervention model in the first place - if you expect intervention
effects to converge geometrically to a new level then the above model
may be ok.
Laimonis Kavalieris
Department of Mathematics and Statistics
University of Otago
Dunedin NZ
John McKinlay
Research Scientist
Western Australia Marine Research Laboratories
Email: jmckinlay@fish.wa.gov.au
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