[R] fit a "threshold" function with nls

[Bert Gunter]

Véronique:

I've cc'ed this to a true expert (Ravi Varadhan) who is one of those
who can give you a definitive response, but I **believe** the problem
is that threshhold type function fits have  objective functions whose
derivatives are discontinuous,and hence gradient -based methods can
run into the sorts of problems that you see.

**If** this is so, then you might do better using an explicit
non-gradient optimizer = rss minimizer via one of the optim() suite of
functions (maybe even the default Nelder-Mead); but this is definitely
where the counsel of an expert would be valuable. Also check out the
CRAN Optimization Task View for advice on optimization options.

Cheers,
Bert

On Mon, Oct 15, 2012 at 9:43 AM, Véronique Boucher Lalonde
<veronique.boucher.lalonde at gmail.com> wrote:
> I am trying to model a dependent variable as a threshold function of
> my independent variable.
> What I mean is that I want to fit different intercepts to y following 2
> breakpoints, but with fixed slopes.
> I am trying to do this with using ifelse statements in the nls function.
> Perhaps, this is not an appropriate approach.
>
> I have created a very simple example to illustrate what I am trying to do.
>
> #Creating my independent variable
> x <- seq(0, 1000)
> #Creating my dependent variable, where all values below threshold #1 are 0,
> all values above threshold #2 are 0 and all values within the two
> thresholds are 1
> y <- ifelse(x < 300, 0, ifelse(x>700, 0, 1))
> #Adding noise to my dependent variable
> y <- y + rnorm(length(y), 0, 0.01)
> #I am having trouble clearly explaining the model I want to fit but perhaps
> the plot is self explanatory
> plot(x, y)
> #Now I am trying to adjust a nonlinear model to fit the two breakpoints,
> with known slopes between the breakpoints (here, respectively 0, 1 and 0)
> threshold.model <- nls(y~ifelse(x<b1, 0, ifelse(x>b2, 0, 1)),
> start=list(b1=300, b2=700), trace=T)
>
> I have played around with this function quite a bit and systematically get
> an error saying: singular gradient matrix at initial parameter estimates
> I admit that I don't fully understand what a singular gradient matrix
> implies.
> But it seems to me that both my model and starting values are sensible
> given the data, and that the break points are in fact estimable.
> Could someone point to what I am doing wrong?
>
> More generally, I am interested in knowing
> (1) whether I can use such ifelse statements in the function nls
> (1) whether I can even use nls for this model
> (3) whether I can model this with a function that would allow me to assume
> that the errors are binomial, rather than normally distributed
> (ultimately I want to use such a model on binomial data)
>
> I am using R version 2.15.1 on 64-bit Windows 7
>
> Any guidance would be greatly appreciated.
> Veronique
>
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>
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-- 

Bert Gunter
Genentech Nonclinical Biostatistics

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