Circular and Linear Regression —— Fitting Circles and Lines by Least Squares

----- 循环回归和线性回归

ISBN: 9781439835906 出版年：2010 页码：285 Chernov, Nikolai CRC Press

Introduction and Historic Overview Classical regression Errors-in-variables (EIV) model Geometric fit Solving a general EIV problem Nonlinear nature of the "linear" EIV Statistical properties of the orthogonal fit Relation to total least squares (TLS) Nonlinear models: general overview Nonlinear models: EIV versus orthogonal fit Fitting Lines Parametrization Existence and uniqueness Matrix solution Error analysis: exact results Asymptotic models: large n versus small sigma Asymptotic properties of estimators Approximative analysis Finite-size efficiency Asymptotic efficiency Fitting Circles: Theory Introduction Parametrization (Non)existence Multivariate interpretation of circle fit (Non)uniqueness Local minima Plateaus and valleys Proof of two valley theorem Singular case Geometric Circle Fits Classical minimization schemes Gauss-Newton method Levenberg-Marquardt correction Trust region Levenberg-Marquardt for circles: full version Levenberg-Marquardt for circles: reduced version A modification of Levenberg-Marquardt circle fit Spath algorithm for circles Landau algorithm for circles Divergence and how to avoid it Invariance under translations and rotations The case of known angular differences Algebraic Circle Fits Simple algebraic fit (Kasa method) Advantages of the Kasa method Drawbacks of the Kasa method Chernov-Ososkov modification Pratt circle fit Implementation of the Pratt fit Advantages of the Pratt algorithm Experimental test Taubin circle fit Implementation of the Taubin fit General algebraic circle fits A real data example Initialization of iterative schemes Statistical Analysis of Curve Fits Statistical models Comparative analysis of statistical models Maximum likelihood estimators (MLEs) Distribution and moments of the MLE General algebraic fits Error analysis: a general scheme Small noise and "moderate sample size" Variance and essential bias of the MLE Kanatani-Cramer-Rao lower bound Bias and inconsistency in the large sample limit Consistent fit and adjusted least squares Statistical Analysis of Circle Fits Error analysis of geometric circle fit Cramer-Rao lower bound for the circle fit Error analysis of algebraic circle fits Variance and bias of algebraic circle fits Comparison of algebraic circle fits Algebraic circle fits in natural parameters Inconsistency of circular fits Bias reduction and consistent fits via Huber Asymptotically unbiased and consistent circle fits Kukush-Markovsky-van Huffel method Renormalization method of Kanatani: 1st order Renormalization method of Kanatani: 2nd order Various "Exotic" Circle Fits Riemann sphere Simple Riemann fits Riemann fit: the SWFL version Properties of the Riemann fit Inversion-based fits The RTKD inversion-based fit The iterative RTKD fit Karimaki fit Analysis of Karimaki fit Numerical tests and conclusions Bibliography Index