Method of Least Squares

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Method of Least Squares

• Advanced Topic of Lecture on Astrometry

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Data Analysis by Model Fitting

• Examples
• Linear Motion ... Star Position
• Keplerian Ellipse ... Binary Orbit
• Keplerian Parabola ... Cometary orbit
• Constant Offset ... Correction to Models
• Model Constants/Parameters ... GM
• Initial Conditions ... Dynamical Models

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Method of Least Squares

• Gauss (1801) Orbit Determination of Ceres
• Typical Minimization Problem
• Parameter, Objective FunctionF(?)
• Minimization ? Zero-Derivative ? Linear Eqs ?
  Solution
• Linear Eqs Normal Eqs

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Zero Derivatives

• Mini-/Maxi-mization Zero Derivatives
• Taylor Expansion
• Solution by Newton Method Linear Eqs
• Normal Eqs.

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Normal Equations

• Positive Definite Symmetric Matrix H
• Solution (Modified) Cholesky Method, ...
• Warning! Rank Deficiency/Degeneration
• Cures
• General Inverse Matrix
• Orthogonal Basis Expansion
• Examination of Parameter Correlation
• Good Initial Guess

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Variation of LSQ

• Weighted LSQ
• Chi Square Fitting
• Non-Linear LSQ
• Gaussian Approximation
• LSQ Associated with Dynamical Systems
• Variational Equation

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Uncertainty Estimation

• Covariance Matrix
• Index of Correlation among Parameters
• Practical Uncertainty