Re: How should I do this?
- From: "Damien Mather" <dmather@xxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 23 Sep 2005 01:46:34 +1200
Bonjour François.
This is nowhere near a complete answer but maybe of some help in getting
started. I have successfully modeled rainfall as a mixture of distributions:
Distribution 1 is the distribution of rainy days in a year, i.e. the
(intermediate) dependent variable is "did it rain or not?" 0/1. I used a
generalized linear mixed model with a binomial link function, first and
second order harmonic dummy variables over each year (4 seasons with
sharpish transitions, I guess!), lag1 day of the dependent (did it rain
yesterday? 0/1) and an AR(1) covariance structure specification.
Distribution 2 is conditional on the first event (a rainy day) and is (you
guessed it already, right?) intensity (how much did it rain, given that it
did (rain)) with a log-gamma link function, first and second order harmonic
dummy variables over each year (4 seasons for both occurrence and intensity,
I guess!), lag n of the dependent (how much did it rain on the last rainy
day? +ve real number) and an AR(1) covariance structure specification, but
strictly between rainy days.
I would be really interested in what closed-form distribution results from
the mixture of these two generalized linear mixed models. Is it still in the
exponential family? I guess it would be even 'rougher' than a negative
binomial distribution. All I know is that the models work, and neither are
greatly over- or under- dispersed.
These models together can predict rainfall on any specified day of the year
given, for your site:
....
the day of year
average annual rainfall
average number of rainy days per year
the parameters of the log-gamma distribution
the AR(1) covariance parameters for both models
did it rain yesterday?
how much did it rain last rainy day?
....
with a known uncertainty.
In order to support my own design decisions I use the rather slow and clumsy
method of simulating each day's rainfall and seeding the next day's
prediction, for whole years, and iterating that with random start points for
new year's day. OK, I've embarrassed myself now in front of all these
excellent statisticians! I hope you find this helpful after all!
I used the SAS GLIMMIX macro for this, but I'm fairly sure WinBUGS would be
more directly helpful, especially for producing the final mixed
distribution.
You would need many years of daily rainfall data for various sites around
your region of interest to re-fit your local mixed model parameters. I used
30 years of daily data for at least one site in each Australian state.
Damien Mather
Department of Marketing
University of Otago
Dunedin
New Zealand
"François Grondin" <francoisdotgrondin@xxxxxxxxxxx> wrote in message
news:pakYe.181275$wr.122630@xxxxxxxxxxx
> Hello everyone.
>
> A colleague submitted me this problem. There was a rain that fell on a
> given
> city with a river that goes through. The sewer network in the city is
> designed to handle a rain that could occur once every 10 years. This means
> that the sewer won't overflow except in the river if necessary.
>
> The rain that fell that day was categorized to occur once every 5 years,
> so
> the sewer network can handle it. The river reached a level that could be
> reached once every 5 years.
>
> The problem is that the rain never stopped. It lowered in intensity and
> became a rain that can occur 6 times per year, so it's a usual rain event.
> But the level of the river continued to rise and reached a level that
> could
> occur once every 20 years. The result : because the level of the river was
> too high, the sewer network was never able to overflow in the river as it
> should do, but in the streets, which is, as you could imagine, a very bad
> situation.
>
> My question : knowing the frequencies of the rain and of the level of the
> river (for example once every 5 years), is there any way to determine the
> frequency of such rain event? Is this combination of rain events
> equivalent
> to a rain that occur once every 100 years? I believe that these two rains
> can't be considered independant, so I can't multiply the frequencies. Can
> I
> find at least a frequency bound? Where should I start? I have no idea of
> the
> rainfall intensity nor the level of the river, only their frequencies.
>
> Thanks for your help.
>
> François
>
>
.
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