From: rizzo@cbnewsf.cb.att.com (anthony.r.rizzo)
Subject: Re: Least Square Errors
Organization: AT&T
Date: Thu, 10 Sep 1992 15:17:22 GMT

In article <thompson.716095751@daphne.socsci.umn.edu> thompson@atlas.socsci.umn.edu writes:
>rizzo@cbnewsf.cb.att.com (anthony.r.rizzo) writes:
>
>>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.

The points that are closer to the zeroing point have smaller
errors in an absolute sense.  As a percentage of the readings
the errors aren't necessarily smaller.

>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.)

I wasn't aware of "constrained" least squares.  And I hadn't
considered "weighted" least squares.  The latter sounds like
a rather painless way to go.  It also seems to make more sense
in my case, since I'm more interested in accuracy near 20C.

>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.

The bridge circuit was balanced at 20C.  This is comparable to
saying that the thermal output of the gauge-sample system is
zero by definition at 20C.  The actual strain measurements,
after the thermal output curve is generated, are then taken
at temperatures other than 20C, with the sample subjected to
structural loads.  But the bridge circuit is balanced again
at 20C before the structure is loaded.  Of course, there is
some variation here.  But that is much smaller than what exists
during the test.

>>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.  

You've very eloquently stated my reasons for not using option (1).
I have no reason to believe that there is any bias in the
erros for all the other points.

>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.

Assumption (a) comes close to being true.  The measurement error
at the zeroing point is much smaller than the erros in the other
observations.  Before zeroing the instrument, I had the luxury of
letting everything come to thermal equilibrium.  I could only
come close to thermal equilibrium during the actual observations.
This alone is reason for the assumption that the measurement error
at (20,0) is very small in comparison to the other observations.

Assumption (c) comes close.  I have no
reason to suspect a bias in the errors of the other observations.
If there was some systematic error, I'm not aware of it.
I can make no statement about the variance, other than to say
that the points appear to lie on a curve.

>--
>T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
>Department of Economics        phone:  (612) 625-0119
>University of Minnesota        fax:    (612) 624-0209

Thanks!

Tony

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