### Abstract

It is assumed that stochastically independent measurements are made of the distance from a reference centre to n points spaced at equi-angular intervals along an arc of length θ on the circular contour of a part. The least squares estimates of the radius of the circular contour and the coordinates of the true centre relative to the reference centre are derived assuming that the distance between the true and reference centres is small, relative to the radius of the part. These estimates are shown to be expressible as linear combinations of the measured distances with coefficients that are explicit functions of n and θ. The estimates are shown to be systematically in error (biased) unless the true and reference centres coincide. Formulas are given for the calculation of the bias error in any situation. The random error of the estimated quantities is expressed in terms of the random error of an individual measurement and the values of n and θ. The correlation coefficient between the estimated quantities is also found. The results enable the determination, in advance of making measurements, of whether the likely random and systematic errors will be tolerable for a given choice of n and θ and a given centring accuracy.

Original language | English (US) |
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Pages (from-to) | 215-220 |

Number of pages | 6 |

Journal | Precision Engineering |

Volume | 1 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1979 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)

### Cite this

*Precision Engineering*,

*1*(4), 215-220. https://doi.org/10.1016/0141-6359(79)90102-8

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*Precision Engineering*, vol. 1, no. 4, pp. 215-220. https://doi.org/10.1016/0141-6359(79)90102-8

**Systematic and random errors in least squares estimation for circular contours.** / McCool, J. I.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Systematic and random errors in least squares estimation for circular contours

AU - McCool, J. I.

PY - 1979/10

Y1 - 1979/10

N2 - It is assumed that stochastically independent measurements are made of the distance from a reference centre to n points spaced at equi-angular intervals along an arc of length θ on the circular contour of a part. The least squares estimates of the radius of the circular contour and the coordinates of the true centre relative to the reference centre are derived assuming that the distance between the true and reference centres is small, relative to the radius of the part. These estimates are shown to be expressible as linear combinations of the measured distances with coefficients that are explicit functions of n and θ. The estimates are shown to be systematically in error (biased) unless the true and reference centres coincide. Formulas are given for the calculation of the bias error in any situation. The random error of the estimated quantities is expressed in terms of the random error of an individual measurement and the values of n and θ. The correlation coefficient between the estimated quantities is also found. The results enable the determination, in advance of making measurements, of whether the likely random and systematic errors will be tolerable for a given choice of n and θ and a given centring accuracy.

AB - It is assumed that stochastically independent measurements are made of the distance from a reference centre to n points spaced at equi-angular intervals along an arc of length θ on the circular contour of a part. The least squares estimates of the radius of the circular contour and the coordinates of the true centre relative to the reference centre are derived assuming that the distance between the true and reference centres is small, relative to the radius of the part. These estimates are shown to be expressible as linear combinations of the measured distances with coefficients that are explicit functions of n and θ. The estimates are shown to be systematically in error (biased) unless the true and reference centres coincide. Formulas are given for the calculation of the bias error in any situation. The random error of the estimated quantities is expressed in terms of the random error of an individual measurement and the values of n and θ. The correlation coefficient between the estimated quantities is also found. The results enable the determination, in advance of making measurements, of whether the likely random and systematic errors will be tolerable for a given choice of n and θ and a given centring accuracy.

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U2 - 10.1016/0141-6359(79)90102-8

DO - 10.1016/0141-6359(79)90102-8

M3 - Article

AN - SCOPUS:0018530360

VL - 1

SP - 215

EP - 220

JO - Precision Engineering

JF - Precision Engineering

SN - 0141-6359

IS - 4

ER -