Well, in your terms (below) I would NOT claim to be "someone who 'knows'"
(and in this notoriously tricky territory to suggest otherwise would be
vainglorious megalomania), but I think some useful things can be said, even
if I have to guess some of the story.
The gradient matrix, X say, is the n x p matrix of partial derivatives of
the model function with respect to the parameters, where the rows correspond
to the observations and the the columns to the parameters. In linear
regression it is the ordinary model matrix and is constant with respect to
the parameters. In non-linear regression calculations it depends on the
current approximation to the parameter estimates and hence changes as the
iterations proceed. Locally it plays a very similar role to the model
matrix in linear regression, in fact. "Singular gradient matrix" really
means "the gradient matrix has less than full column rank" (it's a
rectangular matrix, so 'singular' or 'non-singular' are not really terms
that apply). In turn this means that the algorithm has got itself to a
point in the parameter space where there appears to be a rank loss in the
model surface, and so the parameters are (computationally at least) locally
not identified and their least squares estimates not uniquely defined
locally. In other words the iterative process does not have a unique
direction in which to look and rather than choose any one arbitrarily it
gives up.
In brief the initial values you gave it has led the algorithm to a point on
the model surface where the coordinate system provided by the parameters
appears to break down and it has no clear idea where to go next. The only
solution you have available is to start from somewhere else and hope for
better luck next time.
The thing you have to keep in mind (and I can't say this often enough,
really) is that although it looks simple enough to describe, non-linear
regression is extremely tricky stuff because it is so open-ended: the model
function is *any* function of parameters and covariates with no assumptions
other than mild differentiability. It is not surprising that nls() and ms()
often need kid glove treatment (and there may well be better algorithms for
some classes of model), but the truly amazing thing to me is that so often
they work as well as they do.
Bill Venables. (someone who is definitely still learning...)
-----Original Message-----
From: ROS Mathieu [mailto:mim78@cict.fr]
Sent: Friday, July 23, 1999 6:59 PM
To: E. Conrad Lamon III
Cc: S-news@wubios.wustl.edu
Subject: Re: [S] nls( ) singular gradient matrix
On Thu, 22 Jul 1999, E. Conrad Lamon III wrote:
>
> While trying to fit a nonlinear model using nls( ), I get an error message
> that says I have a singular gradient matrix. I am not sure what causes
> this problem. Should I try different initial values of the parameters, or
> is the problem something else?
yes, you have to put different initial values, but I'd like someone who
'knows' to tell me why this message appears when the initial values
are not close enough to the 'real' ones ?
thanks,
Mathieu
-------------------------------------------------------
Mathieu Ros
mathieuros@bigfoot.com
Universite Paul Sabatier, Toulouse.
-------------------------------------------------------
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