The paper studies the problem of estimating the center and radius of a circle based on observations near its circumference. Least squares estimators have been used. Asymptotic properties of the estimators such as asymptotic biases, variances and covariances are studied. The study confines to observations that are independently and identically distributed (i.i.d.) and is based on two error models, the Cartesian and radial models.
Details: Denote the observations near the circumference of a circle with center \((\xi,\eta)\) and radius \(\rho\) by \((X_iY_i)\), \(1\leq i\leq n\). The general Cartesian model is:
\[ X_i = \xi + \rho \cos \theta_i+\omega_i \cos \Phi_i \] and
\[ Y_i=\eta +\rho \sin \theta_i+\omega_i \sin \phi_i, \]
\(1\leq i\leq n\), where \(\omega_i\)’s and \(\theta_i\)’s are assumed to be mutually independent sets of i.i.d. random variables, independent of \(\theta_i\)’s, with \(\phi_i's\simeq Uniform(0,2\pi]\), \(\omega_i\geq 0\) a.e. When \(\omega^2_i/\sigma^2\simeq \chi^2(2 df)\), the Gaussian model results. The paper restricts to the Cartesian structural model which assumes that the \(\theta_i\)’s are i.i.d. random variables, in turn making \((X_iY_i)\), \(1\leq i\leq n\), i.i.d.
The general radial model is:
\[ X_i=\xi +R_i \cos \tau_i, \quad Y_i=\eta +R_i \sin \tau_i, \quad 1\leq i\leq n, \]
with \(R_i = \sqrt{\left[(X_i-\xi)^2+(Y_i-\eta)^2\right]}\), i.i.d. random variables, independent of \(\tau_i's\), \(1\leq i\leq n\), and \(E[R_i]=\rho\). The paper discusses the radial structural model obtained by assuming independence of the \(\tau_i's\). This also implies that the \((X_iY_i)\)’s are i.i.d. random variables.
Some results: Assume both \(\omega_i\) and \(\theta_i\) have absolutely continuous distributions and \(\omega_i\) has positive support at \(\rho\). Then \(R_i\) and \(\tau_i\) are statistically independent iff \(\theta_i\simeq Uniform(0,2\pi].\)
Asymptotic properties: Denote by S(u,v,r) the sum of squares of distances measured by a metric, from data points to the circumference of the circle with center (u,v) and radius r. Let \((\xi^*,\eta^*,\rho^*)\) be the unique value of (u,v,r) for which E[S(u,v,r)] is least. Then the least squares estimator \(\left({\hat \xi},{\hat \eta},{\hat \rho}\right)\) converges in probability to \(\left(\xi^*,\eta^*,\rho^*\right)\) as \(n \to \infty\). When \(\mu(s,t) = | s^m-t^m|\) is the metric, \((\xi^*,\eta^*)\) minimises \(\text{Var}\left[R(u,v)\equiv \sum R(u,v)^m\right]\) and \(\rho^*=E[R(\xi^*,\eta^*)^m]^{1/m}\). Assume that \(X\) and \(Y\) are not (almost surely) linearly related. The least squares estimator \(\left({\hat \xi}_2,{\hat \eta}_2\right)\) (with \(m=2\)) of \((\xi,\eta)\) converges to \((\xi^*,\eta^*)\). A necessary and sufficient condition for the consistency of \(\left({\hat \xi}_2,{\hat \eta}_2\right)\) is: \(\text{Cov}\left[R^2, R \cos \tau\right] = \text{Cov}\left[R^2, R \sin \tau\right] =0\) \(\Leftrightarrow\) (in the Cartesian case) \(E[\cos\theta] = E[\sin\theta] = 0\) and (in the Radial case) \(E[\cos\tau] E[\sin \tau] = 0\). \({\hat\rho}_2\) converges to \(\rho^*\). Formulas for \(\xi^*\), \(\eta^*\), and \(\rho^*\) are given.
Variances and covariances for consistent estimators: If \(R\) and \(\tau\) are independent, \(\sqrt{\left({\hat \xi}-\xi,{\hat \eta}-\eta,{\hat \rho}-E[R]\right)}\) is asymptotically normally distributed with mean \(\underline0\) and cov matrix \(\lambda^{-1}\Sigma \lambda\) (formulas for \(\Sigma\) and \(\lambda\) are provided). If \(\tau\simeq \text{Uniform}(0,2\pi)\), then \(\sqrt{n\left({\hat \xi}_2-\xi,{\hat \eta}_2-\eta\right)}\) is asymptotically jointly normally distributed with mean \(\underline0\), zero covariance and common variance \((\mu_2+\mu \mu_2)/(2\mu^2)\) where \(\mu =E[R^2]\), \(\mu_k=kth\) moment of \(R^2\). Some other results on asymptotic relative efficiencies and small simulation studies are discussed. There is an appendix deriving some results for the Cartesian model.