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Re: LSF method vs. averaging



Quoting Savinainen Antti <antti.savinainen@KUOPIO.FI>:

why is the LSF method considered to pro=
vide a better estimate in this particular case and also in general th=
an averaging?

0) I have a slight personal preference for the term "curve fitting"
(preferred over "least-squares fitting"). So please allow me to
use that term. In simple cases it makes sense to do curve fitting
using least squares, but the curve fitting notion is more general.

1) Suppose the curve we use for our curve-fitting is a horizontal
line, i.e. the simplest case. In that case curve fitting (using
least squares) is provably identical to averaging. You can prove
this easily using calculus, or prove it almost as easily without
calculus (hint: complete the square).

2) Now suppose there are multiple quantities to be considered,
e.g. a slope *and* an intercept. The concept of "averaging"
does not directly or easily apply ... so we need to find a
more general concept ... and the concept of curve fitting fills
the bill quite nicely. You can think of curve fitting as the
multi-dimentional generalization of averaging.

============

3) If you allow me to pursue this topic a ways, going well
beyond the question that was asked: An even more general
view of this sort of data analysis is _model building_. We
construct a theoretical model having a few free parameters
and we see which parameter-settings best agree with observations.

Simple averaging corresponds to a very simple model, namely
that the data is well described by a constant with some
symmetrically-distributed additive noise. If you have a more
sophisticated model, more involved methods will be needed to
ascertain the best-fit parameters.