Method of Least Squares

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# Method of Least Squares - PowerPoint PPT Presentation

Method of Least Squares. Advanced Topic of Lecture on Astrometry. 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

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### Method of Least Squares

Advanced Topic of Lecture on Astrometry

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
Method of Least Squares
• Gauss (1801): Orbit Determination of Ceres
• Typical Minimization Problem
• Parameter, Objective Function:Φ(λ)
• Minimization → Zero-Derivative → Linear Eqs → Solution
• Linear Eqs = Normal Eqs
Zero Derivatives
• Mini-/Maxi-mization = Zero Derivatives
• Taylor Expansion
• Solution by Newton Method: Linear Eqs
• Normal Eqs.
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
Variation of LSQ
• Weighted LSQ
• Chi Square Fitting
• Non-Linear LSQ
• Gaussian Approximation
• LSQ Associated with Dynamical Systems
• Variational Equation
Uncertainty Estimation
• Covariance Matrix
• Index of Correlation among Parameters
• Practical Uncertainty