🚀 go-pugleaf

In article <1992Sep9.150541.15735@cbfsb.cb.att.com>, rizzo@cbnewsf.cb.att.com (anthony.r.rizzo) writes:

> My quandary is that the polynomial does not pass
> through the one point of which I'm dead certain, (20,0).  

Let me summarize the posting, just to be sure we're working on the same
problem.  You have some data to which you would like to fit a 4-th degree
polynomial, but the curve doesn't go where you expect it to.

There are only two possibilities (well, 3, because I'm going to split #2):

1.  The data must lie within sampling variability of the curve, or

2.  You've got model failure.  If it's model failure, there are two
     possiblities
        
     2.i.  You've specified the wrong model, i.e., it's not a 4-th degree
            polynomial, or
     2.ii. Something went wrong in the data collection process, i.e., you've
            managed to generate some invalid data.

What you propose to do is reasonable, and amounts to constrained least
squares.  There is a fair body of literature on constrained least
squares.  Your proposed method is also a degenerate case of a weighted
least squares problem, in which not all dependent values are measured
with the same lack of precision, and so that a poorer fit is tolerated at
those more imprecisely measured points.

[Don't apologize for being an engineer; I am one also -- who has heavily
delved into the statistical arena]

CN Haas
Environmental Engineering
Drexel University
Philadelphia

In article <1992Sep9.142202.352@nutmeg.hnrc.tufts.edu>, I wrote:

<some stuff . . .>

As a post script, let me add that if the data were within sampling variability
of the fitted curve, that is, I were confident that this was not a case of
model failure, I would seriously consider fitting the constrained model.
The consideration would revolve around whether the response should be *exactly*
0 at X=20.  I don't know enough about your problem to make that judgement by
myself.

In article <1992Sep9.150541.15735@cbfsb.cb.att.com>, rizzo@cbnewsf.cb.att.com (anthony.r.rizzo)
asks about the propriety of constraining a fitted quartic to pass through
a given point:
> 
This sort of thing is often done in linear regression, usually to make the
line pass through the origin. There is no objection to doing this, but it
makes the fit slightly worse elsewhere.

I have greater reservations about using a quartic. The higher degree
the polynomial, the more unstable it is. I hope you fitted it by using
orthogonal polynomials which helps with numerical stability.

I don't say it is wrong to use high degree polynomials, just be careful;
draw and examine residual plots etc (you should do this anyway).

Anthony,

Your problem is of wide interest to many people, not
sure I can help, but I have been working on a related
problem...below is the context of this problem, something
I have been trying to sort out wrt. to my own work.  Perhaps
others could contribute...perhaps I whistling something 
other that Dixie?

Problem:  INTERPOLATION

Approach: Non-stochastic vs. Stochastic

  Non-Stochastic techniques
     polynomial fitting
     splines

  Stochastic techniques
     moving average
     kriging

Using polynomial fitting one does not preserve the original
data points --- therefore it is very useful as a global or
smoothing technique. This is also true of moving average
techniques.

Splines do preserve the original data points and spline 
fitting is a huge area (literature wise).  Generally splines
are fit locally so that changing or removing points does not
affect the entire data set.

IMHO the stochastic methods are most interesting, the class
of techniques called BLUE (best linear unbiased estimators) 
particularly those called kriging (or Gandin optimal methods
in meteorology) allow you to derive error variance estimates
so that you can where your curve is likely to be less 
reliable.


I am not sure if it is important in your case, but remember
that if you are fitting polynomials you are implicitly imposing
a certain structure on your data (true also in the case of
sin and cos functions).  

I have found the following to be extremely helpful in all
areas of interpolation, but particularly the two dimensional
cases:

Davis, J.C. 1986. Statistics and Data Analysis in Geology,
2nd Ed., John Wiley and Sons, New York.
-it also comes with software and sample data sets

Cheers,
Chris

In article <1992Sep9.192431.13546@cbis.ece.drexel.edu> chaas@coe.drexel.edu (Chuck Haas) writes:
>What you propose to do is reasonable, and amounts to constrained least
>squares.  There is a fair body of literature on constrained least
>squares.  Your proposed method is also a degenerate case of a weighted
>least squares problem, in which not all dependent values are measured
>with the same lack of precision, and so that a poorer fit is tolerated at
>those more imprecisely measured points.

Chuck Haas is certainly right about there being a fair body of
literature on the subject of constrained least squares.  I am
appending a dump from my personal bibliography.  Since I am a
numerical analyst, it largely concerns computational methods and
perturbation theory.  There is more in the statistics journals.

Pete Stewart

\begin{thebibliography}{10}

\bibitem{barl:88}
J.~L. Barlow.
\newblock Error analysis and implementation aspects of deferred correction for
  equality constrained least squares problems.
\newblock {\em SIAM Journal on Numerical Analysis}, 25:1340--1358, 1988.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{baha:88}
J.~L. Barlow and S.~L. Handy.
\newblock The direct solution of weighted and equality constrained
  least-squares problems.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  9:704--716, 1988.

\bibitem{banp:88}
J.~L. Barlow, N.~K. Nichols, and R.~J. Plemmons.
\newblock Iterative methods for equality-constrained least squares problems.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  9:892--906, 1988.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{bjor:84}
{\AA}.~Bj{\"o}rck.
\newblock A general updating algorithm for constrained linear least squares
  problems.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  5:394--402., 1984.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{bjor:88b}
{\AA}.~Bj{\"o}rck.
\newblock A bidiagonalization algorithm for solving ill-posed system of linear
  equations.
\newblock {\em BIT}, 28:659--670, 1988.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{clos:88}
D.~I. Clark and M.~R. Osborne.
\newblock On linear restricted and interval least-squares problems.
\newblock {\em IMA Journal on Numerical Analysis}, 8:23--36, 1988.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{deap:78}
P.~Deuflhard and V.~Apostolescu.
\newblock An underrelaxed {Gauss-Newton} method for equality constrained
  nonlinear least squares.
\newblock In J.~Stoer, editor, {\em In Proceedings 8th IFIP Conference on
  Optimization Techniques}, volume~7, pages 22--32. Springer Verlag, 1978.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{eila:75}
E.~L. Eichhorn and C.~L. Lawson.
\newblock An {ALGOL} procedure for solution of constrained least squares
  problems.
\newblock Computing Memorandum No. 374, JPL, Pasadena, California, 1975.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{elde:77a}
L.~Eld{\`e}n.
\newblock {\em Numerical Analysis of Regularization and Constrained Least
  Square Methods}.
\newblock PhD thesis, Linkoping Studies in Science and Technology, Linkoping,
  Sweden, 1977.
\newblock Cited in \cite{govl:89}.

\bibitem{elde:80}
L.~Eld\'en.
\newblock Perturbation theory for the least squares problem with linear
  equality constraints.
\newblock {\em SIAM Journal on Numerical Analysis}, 17:338--350, 1980.

\bibitem{elde:83}
L.~Eld\'en.
\newblock A weighted pseudoinverse, generalized singular values, and
  constrained least squares problems.
\newblock {\em BIT}, 22:487--502, 1983.

\bibitem{flet:81}
R.~Fletcher.
\newblock {\em Practical Methods of Optimization, Vol. 2, Constrained
  Optimization}.
\newblock John Wiley, New York, 1981.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{gand:81}
W.~Gander.
\newblock Least squares with a quadratic constraint.
\newblock {\em Numerische Mathematik}, 36:291--307, 1981.

\bibitem{gay:84}
D.~M. Gay.
\newblock A trust-region approach to linearly constrained optimization.
\newblock In D.F. Griffiths, editor, {\em In Proceedings of the 1983 Dundee
  Conference on Numerical Analysis}, pages 72--105, New York, 1984. Springer
  Verlag.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{gimu:76}
P.E. Gill and W.~Murray.
\newblock Nonlinear least squares and nonlinearly constrained optimization.
\newblock In {\em In Proceedings Dundee Conference on Numerical Analysis 1975},
  New York, 1976. Springer Verlag.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{govl:89}
G.~H. Golub and C.~F. Van~Loan.
\newblock {\em Matrix Computations}.
\newblock Johns Hopkins University Press, Baltimore, Maryland, 2nd edition,
  1989.

\bibitem{govm:91}
G.~H. Golub and U.~von Matt.
\newblock Quadratically constrained least squares and quadratic problems.
\newblock {\em Numerische Mathematik}, 59:561--580, 1991.

\bibitem{hans:86}
R.~J. Hanson.
\newblock Linear least squares with bounds and linear constraints.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  7:826--834, 1986.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{haha:81}
K.~H. Haskell and R.~J. Hanson.
\newblock An algorithm for linear least squares problems with equality and
  nonnegativity constraints.
\newblock {\em Mathematical Programming}, 21:98--118, 1981.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{hofl:79}
J.~N. Holt and R.~Fletcher.
\newblock An algorithm for constrained non-linear least-squares.
\newblock {\em J. Inst. Maths. Applics.}, 23:449--463, 1979.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{kape:78}
L.~Kaufman and V.~Pereyra.
\newblock A method for separable nonlinear least squares problems with
  separable nonlinear equality constraints.
\newblock {\em SIAM Journal on Numerical Analysis}, 15:12--20, 1978.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{kopa:81}
S.~Kourouklis and C.~C. Paige.
\newblock A constrained approach to the general {Gauss-Markov} linear model.
\newblock {\em Journal of the American Statistical Association}, 76:620--625,
  1981.

\bibitem{lewe:70}
{\"O.}~Leringe and P.-\AA. Wedin.
\newblock A comparison between different methods to compute a vector $x$ which
  minimizes {$\|Ax-b\|_2$} when {$Gx=h$}.
\newblock Technical report, Department of Computer Science, Lund University,
  1970.

\bibitem{lots:83}
P.~L{\"o}tstedt.
\newblock Perturbation bounds for the linear least squares problem subject to
  linear inequality constraints.
\newblock {\em BIT}, 23:500--519, 1983.

\bibitem{lots:84}
P.~L{\"o}tstedt.
\newblock Solving the minimal least squares problem subject to bounds on the
  variables.
\newblock {\em BIT}, 24:206--224, 1984.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{mahd:81}
N.~Mahdavi-Amiri.
\newblock {\em Generally Constrained Nonlinear Least Squares and Generating
  Test Problems: Algorithmic Approach}.
\newblock PhD thesis, The John Hopkins University,, 1981.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{mova:91}
M.~Moonen and J.~Vandewalle.
\newblock A square root covariance algorithm for constrained recursive least
  squares estimation.
\newblock {\em Journal of VLSI Signal Processing}, 3:163--172, 1991.

\bibitem{olru:86}
D.~P. O'Leary and B.~W. Rust.
\newblock Confidence intervals for inequality-constrained least squares
  problems, with applications to ill-posed problems.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  7:473--489, 1986.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{schi:83}
K.~Schittkowski.
\newblock The numerical solution of constrained linear least-squares problems.
\newblock {\em IMA J. Numer. Anal.}, 3:11--36, 1983.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{schi:85}
K.~Schittkowski.
\newblock Solving constrained nonlinear least squares problems by a general
  purpose {SOP}-method.
\newblock Technical report, Institute f{\"u}r Informatik, Universit{\"a}t
  Stuttgart, 1985.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{scst:79}
K.~Schittkowski and J.~Stoer.
\newblock A factorization method for the solution of constrained linear least
  squares problems allowing subsequent data changes.
\newblock {\em Numerische Mathematik}, 31:431--463, 1979.

\bibitem{shmw:87}
T.~J. Shepherd and J.~G. McWhirter.
\newblock A pipelined array for linearly constrained least squares
  optimization.
\newblock In T.~S. Durrani, J.~B. Abbiss, J.~E. Hudson, R.~W. Madan, J.~G.
  McWhirter, and T.~A. Moore, editors, {\em Mathematics in Signal Processing},
  pages 607--635, Oxford, 1987. Clarendon Press.
\newblock Cited in \cite{mova:91}.

\bibitem{stoe:71}
J.~Stoer.
\newblock On the numerical solution of constrained least-squares problems.
\newblock {\em SIAM Journal on Numerical Analysis}, 8:382--411, 1971.

\bibitem{vloa:85}
C.~F. Van~Loan.
\newblock On the method of weighting for equality constrained least squares.
\newblock {\em SIAM Journal on Numerical Analysis}, 22:851--864, 1985.

\bibitem{wedi:79}
P.-{\AA}. Wedin.
\newblock Notes on the constrained linear least squares problem. a new approach
  based on generalized inverses.
\newblock Technical Report UMINF--75.79, Institute of Information Processing,
  University of Ume\aa, 1979.

\bibitem{wedi:85}
P.-{\AA}. Wedin.
\newblock Perturbation theory and condition numbers for generalized and
  constrained linear least squares problems.
\newblock Technical Report UMINF--125.85, Institute of Information Processing,
  University of Ume\aa, 1985.

\bibitem{wedi:87}
P.-{\AA}. Wedin.
\newblock Perturbation theory and condition numbers for generalized and
  constrained least squares problems.
\newblock Technical Report S-901-87, Institute of Information Processing,
  University of Ume\aa, 1987.

\bibitem{wrho:85}
S.~J. Wright and J.~N. Holt.
\newblock Algorithms for nonlinear least squares with linear inequality
  constraints.
\newblock {\em SIAM Journal on Scientific and Statistical Computing},
  6:1033--1048, 1985.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\end{thebibliography}

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

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

In article <1992Sep9.150541.15735@cbfsb.cb.att.com> rizzo@cbnewsf.cb.att.com (anthony.r.rizzo) writes:

(description of polynomial regression problem greatly abstracted by current
poster)

>with which the data were collected was zeroed at 20 C.  So, the
>curve, ideally, should pass through (20,0).  

>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.

Right you are. Don't do this, you don't have to.

>Second, I can re-derive the equations such that the fitted curve
>is CONSTRAINED to pass through (20,0). . . .

You can easily do this by recoding the predictors and running the regression
without an intercept term. Instead of X, X**2, X**3, X**4, create a new
predictor Q = X - 20, and run your regression on Q, Q**2, Q**3, and Q**4.

Run it with the intercept term first. Check the partial t-statistic for the
intercept term. If it's not statistically significant, it should be okay to
run the regression again, this time without the intercept. You'll get the
answer you want.

You can expand the polynomial in Q to get a polynomial in X without
too much trouble; it's only fourth degree.

> Now my questions:  

>1) Is this legitimate, or am I just whistling Dixie?  

It is completely legitimate to recode the predictors in any way you like.
You must be careful when you recode the response. You only need to recode
the predictors in this case. (What's wrong with "Dixie"? Don't you like
Dan Emmett?) Be sure to test the significance of the intercept term, though.
If it is significant, it indicates either (a) lack of fit, or (b) the
system drifted during the process, rendering the zeroing at 20 degrees
invalid. (If you made subsequent measurements at 20 degrees throughout the
process you can get an idea of how likely (b) is.)

>2) Is this a common thing to do?  If so, any references?

Sure, it's common. It's so common, I do it without even thinking too much
about it, so I don't have a reference for you. Sorry!

==Doctor FORTRAN