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Breakpoints and non linear regression

4 messages · thomas88, Thomas Coquet, Achim Zeileis

thomas88
# 
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_timesat.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.

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

Thank you for your advices !



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Achim Zeileis
#
On Fri, 9 Nov 2012, thomas88 wrote:

            

      
      
      
    

        
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

            

      
      
      
    

        

      
      
    

  
  


  


      

    

    

      
      

        
      

    

    

      
      

        


  

        

      Achim Zeileis
    

    

      
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On Fri, 9 Nov 2012, Thomas Coquet wrote:

        

            

      
      
      
    

        
Good! :-)

            

      
      
      
    

        
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.

            

      
      
      
    

        
Yes, but for the log-transformed data...

            

      
      
      
    

        
Yes.

            

      
      
      
    

        
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