Breakpoints and non linear regression

Hello,
I already tried and looked at the bfast package (very nice package by the
way!) as I am working on VI time series as well.
Good! :-)
However, my model is definitely not linear,
Not even after taking logs or some other transformation?

In principle, the breakpoint ideas can of course also be applied to 
non-linear models but so far in my applications I could always find 
transformations that lead rather naturally to roughly piecewise linear 
relationships.
so in worst case scenario my idea was to use the bfast package to find 
the breakpoints (with the harmonic fit) and then to fit the seasonal 
part in each segment with my model (so basically almost what you are 
suggesting - using harmonic to find breakpoints).
Yes, but for the log-transformed data...
But the breakpoints will not be dependent on my model, so this may be an
issue, isn't it ?
Yes.
The asymmetric gaussian fit has been recognized as being one of the best 
fit for VI time series, and I used this method for periodic fit (so far 
it was used only as a smoothing function of the time series, not as a 
fit for the seasonal component).?

The point would be to combine this method with an iterative breakpoint 
method such as bfast to detect abrupt changes, but to do that I need to 
find breakpoints in the seasonal trend with a non linear model (that is 
the tricky part :) ).
In principle, you can set up the same type of procedure that bfast uses 
with a non-linear model - as long as the objective function is additive in 
the observations. But I wouldn't know of a (fast enough) fitting function 
for such a segmented model in R.

hth,
Z

Thanks !

On Fri, Nov 9, 2012 at 2:00 PM, Achim Zeileis <Achim.Zeileis at uibk.ac.at>
wrote:
      On Fri, 9 Nov 2012, thomas88 wrote:

            Hello,

            I have done some research about breakpoints (I am
            not a statistician) and I
            found out about the breakpoint, strucchange and
            segmented packages in R
            allowing to find breakpoints assuming linear model.

            However, I would like to fit a periodic time series
            with a non linear
            (periodic) model, and I was wondering how I could
            find breakpoints for this
            model in R. Is it even possible ?

            My model is an asymmetric gaussian fitting (cf
http://www.nateko.lu.se/personal/Lars.Eklundh/Institutionssida/IEEE_TGRS_ti
            mesat.pdf)
            with a linear-time-dependant amplitude (I am fitting
            this model over the
            whole time series).

            *My ideas
            *

            1) I guess I am more interested in the breakpoints
            of the "amplitude" of my
            periodic function, so that I could assume a model of
            the form:

            Y ~ (a + b*t)*f(t), with |f(t)| <= 1, where f is a
            periodic function to be
            fitted to a non linear model, but where no
            breakpoints should occur.

            So basically, the breakpoints would only happen in
            the (a,b) pair of
            coefficients, which would be a linear regression.
            However, as f is unknown,
            this makes things harder and I don't have a lot of
            extremas (min/max) to
            detect breakpoints robustly...

            2) To detect breakpoint with an harmonic model and
            then to apply my non
            linear regression on each segment.

      I would probably first try whether the following leads to
      reasonable fits

      Y(t) = A * exp(b * t) * H(t)

      i.e., a multiplicative model with an exponential trend and some
      harmonic trend. By taking logs you then get

      log Y(t) = log(A) + b * t + log(H(t))
      ->
      log(Y(t)) = a + b * t + h(t)

      so that you can fit a model with a linear trend plus harmonic
      season to the log-series. And, of course, the harmonic trend can
      then be built up up sin/cos at different frequencies and you
      could fit the whole thing as a linear model to the log-series.

      It's not quite the same model that you propose above but might
      be an approach worth exploring. You could also look at the
      "bfast" package which has a function bfastpp() for setting up
      trend and harmonic season for a time series. And it also allows
      for iterative fitting of separate trend and season breakpoints
      in the time series.

      hth,
      Z

            These two ideas could potentially work, however
            these are workarounds.

            Thank you for your advices !

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Breakpoints and non linear regression

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