Jari,

Yes, you right, I am sorry I didn't say that I was talking about Euclidean
measure of distance in that passage, I know that other distance are
different, and finally I understood the non-linear regression stuff, now is
much more clear!

If you have some good book titles I would appreciate a lot... :)))

Thank you so much,

G.








> ... I supposed, that If we use as many dimensions as there are variables,
> > then we can perfectly reproduce the observed distance matrix. Isn't it?
> Gian, Not quite so. I think it would be useful to consult a good book, but
> here some explanation.
>
> The NMDS is not a simple "reproduction" method, but it is a non-linear
> regression problem. For n points and k dimensions we fit a nonlinear
> regression with n*k parameters fitted to n*(n-1)/2 observations. It doesn't
> require much intuition to see that this is not well defined for k
> approaching n, and then the non-linear regression fails. For details, the
> non-linear regression function is isoreg() in R, and the model fitting
> happens with optim() using method = "BFGS" (Broyden, Fletcher, Goldfarb &
> Shanno). All this is not very obvious because it is done within a C
> function
> in the MASS package. The NMDS is nonlinear just in order to be able to
> produce a good mapping with low values of k: so stick with low values of k.
>
> If you want to have complete mapping of dissimilarities, you should use
> metric scaling. Then you typically ignore the latter axes. However, even
> here the situation is not as clear as you write. If you use Euclidean
> distances, then the number of variables give the number of dimensions of
> metric scaling. With Euclidean distances, the complete solution also
> exactly
> reproduces the observed distances. However, with non-Euclidean
> dissimilarities (like Bray-Curtis in your case) the situation is more
> complicated. Metric scaling and complete mapping is Euclidean, and if your
> dissimilarities are non-Euclidean, you have a problem (that you usually
> ignore). Firstly, the number of above zero eigenvalues and corresponding
> real eigenvalues is not directly defined by the number of variables.
> Secondly, you cannot reproduce the observed dissimilarities from real
> eigenvectors because that reproduction is Euclidean and your measure was
> non-Euclidean. For exact reproduction, you should subtract the distances in
> imaginary space (negative eigenvalues) from distances in the real space
> (positive eigenvalues). We actually do it exactly like this in the
> betadisper() function in vegan, and for this reason the wcmdscale()
> function
> of vegan also returns information on complex eigenvectors and negative
> eigenvalues.
>
> For your other post that came when I wrote this: stress 11.6 is really
> fine.
> I think that if you get stress down to 5% (0.05) or less, then there is
> something fishy in your data or in your model specification, like
> overfitting.
>
> Cheers, Jari Oksanen
>
> > But,
> > of course, our goal is to reduce the observed complexity of nature, that
> is,
> > to explain the distance matrix in terms of fewer underlying dimensions...
> > So what is best at the end??
> > And also wich is the function for plotting the stress values versus the
> > number of dimnsions and how to read the plot?
> > I hope I was clear, thank you so much!
> > Yours,
> >
> > G.
>

	[[alternative HTML version deleted]]

