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