Least-squares fitting with errors in the response and predictor
Least squares regression is commonly used in metrology for calibration and estimation. In regression relating a response y to a predictor x, the predictor x is often measured with error that is ignored in analysis. Practitioners wondering how to proceed when x has non-n...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
EDP Sciences
2012-01-01
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| Series: | International Journal of Metrology and Quality Engineering |
| Subjects: | |
| Online Access: | https://www.metrology-journal.org/articles/ijmqe/pdf/2012/02/ijmqe120010.pdf |
| _version_ | 1818594703187116032 |
|---|---|
| author | Burr T. Croft S. Reed B.C. |
| author_facet | Burr T. Croft S. Reed B.C. |
| author_sort | Burr T. |
| collection | DOAJ |
| description | Least squares regression is commonly used in metrology for calibration and estimation. In
regression relating a response y to a predictor x, the
predictor x is often measured with error that is ignored in analysis.
Practitioners wondering how to proceed when x has non-negligible error
face a daunting literature, with a wide range of notation, assumptions, and approaches.
For the model ytrue = β0 + β1 xtrue,
we provide simple expressions for errors in predictors (EIP) estimators
\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}
β̂0, EIP
for β0 and
\hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}
β̂1, EIP
for β1 and for an
approximation to covariance
(
\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}
β̂0, EIP
,
\hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}
β̂1, EIP
). It is assumed that there are measured data
x = xtrue + ex,
and
y = ytrue + ey
with errors ex in x and
ey in y and the
variances of the errors ex and
ey are allowed to depend on
xtrue and ytrue, respectively.
This paper also investigates the accuracy of the estimated cov(
\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}
β̂0, EIP
,
\hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}
β̂1, EIP
) and provides a numerical Bayesian alternative using
Markov Chain Monte Carlo, which is recommended particularly for small sample sizes where
the approximate expression is shown to have lower accuracy than desired. |
| first_indexed | 2024-12-16T11:04:20Z |
| format | Article |
| id | doaj.art-a7c9573aa2494677ae691f50a163ded5 |
| institution | Directory Open Access Journal |
| issn | 2107-6839 2107-6847 |
| language | English |
| last_indexed | 2024-12-16T11:04:20Z |
| publishDate | 2012-01-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | International Journal of Metrology and Quality Engineering |
| spelling | doaj.art-a7c9573aa2494677ae691f50a163ded52022-12-21T22:33:55ZengEDP SciencesInternational Journal of Metrology and Quality Engineering2107-68392107-68472012-01-013211712310.1051/ijmqe/2012010ijmqe120010Least-squares fitting with errors in the response and predictorBurr T.0Croft S.1Reed B.C.2Statistical Sciences, Los Alamos National LaboratorySafeguards Science and Technology, Los Alamos National LaboratoryDepartment of Physics, Alma CollegeLeast squares regression is commonly used in metrology for calibration and estimation. In regression relating a response y to a predictor x, the predictor x is often measured with error that is ignored in analysis. Practitioners wondering how to proceed when x has non-negligible error face a daunting literature, with a wide range of notation, assumptions, and approaches. For the model ytrue = β0 + β1 xtrue, we provide simple expressions for errors in predictors (EIP) estimators \hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $} β̂0, EIP for β0 and \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $} β̂1, EIP for β1 and for an approximation to covariance ( \hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $} β̂0, EIP , \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $} β̂1, EIP ). It is assumed that there are measured data x = xtrue + ex, and y = ytrue + ey with errors ex in x and ey in y and the variances of the errors ex and ey are allowed to depend on xtrue and ytrue, respectively. This paper also investigates the accuracy of the estimated cov( \hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $} β̂0, EIP , \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $} β̂1, EIP ) and provides a numerical Bayesian alternative using Markov Chain Monte Carlo, which is recommended particularly for small sample sizes where the approximate expression is shown to have lower accuracy than desired.https://www.metrology-journal.org/articles/ijmqe/pdf/2012/02/ijmqe120010.pdfleast squareregressionbayesian estimationerrors |
| spellingShingle | Burr T. Croft S. Reed B.C. Least-squares fitting with errors in the response and predictor International Journal of Metrology and Quality Engineering least square regression bayesian estimation errors |
| title | Least-squares fitting with errors in the response and
predictor |
| title_full | Least-squares fitting with errors in the response and
predictor |
| title_fullStr | Least-squares fitting with errors in the response and
predictor |
| title_full_unstemmed | Least-squares fitting with errors in the response and
predictor |
| title_short | Least-squares fitting with errors in the response and
predictor |
| title_sort | least squares fitting with errors in the response and predictor |
| topic | least square regression bayesian estimation errors |
| url | https://www.metrology-journal.org/articles/ijmqe/pdf/2012/02/ijmqe120010.pdf |
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