From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Subject: Re: Least Square Errors
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota
Date: Thu, 10 Sep 1992 03:29:11 GMT

rizzo@cbnewsf.cb.att.com (anthony.r.rizzo) writes:

>This is probably something that's a piece of cake for all you
>stat-pros out there, but for a mere engineer the question requires
>some thought and, possibly, some help.

Such problems always require thought, even for stat "pros".

>I have experimental data, a calibration curve of sorts, for the
>thermal output of a strain gauge.  I've fitted a 4th degree polynomial
>to the data, by the method of least square errors, and I'm using
>the polynomial to correct strain gauge readings taken at various
>temperatures.  My quandary is that the polynomial does not pass
>through the one point of which I'm dead certain, (20,0).  The instrument
>with which the data were collected was zeroed at 20 C.  So, the
>curve, ideally, should pass through (20,0). 

To answer a question like this you must really think about why you
think there are errors in the data to begin with.  You have already
done some of this since you tell us that you are "dead certain" that
(20,0) is on the curve.  But least squares (or constrained least
squares) implicitly assumes that all of the other errors have equal
variance.  Given your description of the problem I suspect that it
might be more reasonable to assume that points that are close to the
"zeroing" point would have smaller errors than points that are far
away.

If this is the case, then you should consider weighting observations
taken at temperatures close to 20 C more heavily than observations
elsewhere in whatever statistical procedure you employ.  (However,
going overboard in this respect exposes you to excessive dependence on
just a few of the observations.)

On the other hand, you should ask yourself why you are dead certain
that the true curve goes through (20,0).  You seem to be implicitly
assuming that there were no measurement errors present either in the
strain gauge reading or in the calibration instrument when you did the
zeroing.

>Two options are available to me.  First, I can simply change the
>value of the constant term in my polynomial, so as to shift
>the curve up or down by the required amount.  But this will give
>me a new curve that DOES NOT minimize the squares of the errors.
>Second, I can re-derive the equations such that the fitted curve
>is CONSTRAINED to pass through (20,0).  (This would not be unlike
>the application of boundary conditions by the theoretical method
>in finite element problems.)  Doing so should insure that
>the curve pass through (20,0), while still giving me coefficients
>that minimize the square of the errors.  Now my questions:  

Certainly there are other possibilities.  Option (1) is equivalent to
assuming that (a) there is no measurement error at (20,0) and (b)
there is a constant bias in the errors for all of the other
observations.  Option (2) implicitly assumes that (a) there is no
measurement error at (20,0) and (c) all of the other observations have
error that is unbiased and of equal variance.  It is not clear to me
that any of the assumptions (a), (b) or (c) is reasonable.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209
Newsgroups: sci.math.stat