Method of Least Squares

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

2. 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

3. 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

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

5. 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

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

7. Uncertainty Estimation • Covariance Matrix • Index of Correlation among Parameters • Practical Uncertainty