Re: Fitting: unbalanced data point density and weights
- From: Richard Ulrich <Rich.Ulrich@xxxxxxxxxxx>
- Date: Wed, 14 May 2008 17:37:15 -0400
On Wed, 14 May 2008 08:28:27 -0400, Paul Rubin <rubin@xxxxxxx> wrote:
Kristof Lebecki wrote:
Hello everybody,
I have a set of points {x,y}. When I plot them in a log-log scale they
apparently agree with a power dependence function F(x) = a*x^p. So, I am
fitting these points with this function.
(Details: Mathematica package, function NonlinearRegress, its description
for the help file: finds a least-squares fit to a list of data for a model
that is a not linear combination of the given basis functions; the
coefficients of the linear combination are the parameters of the fit).
The problem is that my data are evidently bad balanced: I have much more
points on the "left" side (small x value), their errors are also much
smaller (I use the errors to define weights, different for every point). On
the other side, the x-density of the "right" points is much smaller. The
points form a slight bow in the log-log scale - as the result, the fitted
function matches "perfectly" left points while the other points are badly
matched.
The question is: what to do to keep the data points appropriately balanced?
A friend of mine saw once an article about this subject, but you know: "gone
with the wind". ;)
Any advice? Should I present you a sketch of the problem?
(I hope this group fits to that subject).
If you regress log y on log x (linear regression), do you still see the
same phenomenon?
You could perhaps use weighted least squares, assigning greater weight
to the points with larger x values. Another kludge would be to
replicate the points with larger x (just put extra copies of them in the
sample), until the sample was more balanced.
The underlying reason here is that the analysis is
one that "minimizes the residuals" -- in some sense.
In Least Squares, the residuals are squared and summed,
and the effect *can* be much the same in Maximum Likelihood.
Or, the proper ML model may provide "weights" that are
more appropriate than the squares of the actual residuals.
Paul gives a few approaches to re-emphasizing certain
residuals -- "weighting" is common; extra copies of cases
is (as he says) a "kludge" to approximate a weighting function.
How important is it that you keep your original metric?
In my biostatistical applications, I would be able to assume
that I can take the logs of the variables -- or some other
transformation -- and achieve my proper weighting. And
analyze the new variables.
I don't think I would advocate either of those approaches, though, at
least not at first. If the log-log linear regression again shows a good
fit for small x and a poor fit for large x, I would take that to mean
that the regression function is not really a power function, and I would
try something else -- perhaps a sum of two power functions with
different exponents?
One of the important guides to a model is "what makes
sense for the variables". That depends especially on the
variables, and somewhat on the conventions of the field
that uses the variables. If you mention what you have,
someone here might be familiar with it.
--
Rich Ulrich
http://www.pitt.edu/~wpilib/index.html
.
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