Method of Least Method of Least Squares Squares

Method of Least Method of Least

Squares Squares

Advanced Topic of Lecture on Advanced Topic of Lecture on

Astrometry

Astrometry

Data Analysis by Model Data Analysis by Model

Fitting Fitting

 Examples Examples

 Linear Motion ... Star Position Linear Motion ... Star Position

 Keplerian Ellipse ... Binary Orbit Keplerian Ellipse ... Binary Orbit

 Keplerian Parabola ... Cometary orbit Keplerian Parabola ... Cometary orbit

 Constant Offset ... Correction to Models Constant Offset ... Correction to Models

 Model Constants/Parameters ... GM Model Constants/Parameters ... GM

 Initial Conditions ... Dynamical Models Initial Conditions ... Dynamical Models

Method of Least Squares Method of Least Squares

 Gauss (1801): Orbit Determination of Ceres Gauss (1801): Orbit Determination of Ceres

 Typical Minimization Problem Typical Minimization Problem

 Parameter, Objective Function: Parameter, Objective Function: Φ Φ ( ( λ λ ) )

 Minimization → Zero-Derivative → Linear Eq Minimization → Zero-Derivative → Linear Eq s → Solution

s → Solution

 Linear Eqs = Normal Eqs Linear Eqs = Normal Eqs

  

^, 

j

f t g

 ^  ^

     

Zero Derivatives Zero Derivatives

 Mini-/Maxi-mization = Zero Derivatives Mini-/Maxi-mization = Zero Derivatives

 Taylor Expansion Taylor Expansion

 Solution by Newton Method: Linear Eqs Solution by Newton Method: Linear Eqs

 Normal Eqs. Normal Eqs. H      b 

 

0 0

j

i i j i j

 

   

 

       

     

         

       ^

0



 



Normal Equations Normal Equations

 Positive Definite & Symmetric Matrix: H Positive Definite & Symmetric Matrix: H

 Solution: (Modified) Cholesky Method, ... Solution: (Modified) Cholesky Method, ...

 Warning! Warning! Rank Deficiency/Degeneration Rank Deficiency/Degeneration

 Cures: Cures:

 General Inverse Matrix General Inverse Matrix

 Orthogonal Basis Expansion Orthogonal Basis Expansion

 Examination of Parameter Correlation Examination of Parameter Correlation

 Good Initial Guess Good Initial Guess

Variation of LSQ Variation of LSQ

 Weighted LSQ Weighted LSQ

 Chi Square Fitting Chi Square Fitting

 Non-Linear LSQ Non-Linear LSQ

 Gaussian Approximation Gaussian Approximation

 LSQ Associated with Dynamical Systems LSQ Associated with Dynamical Systems

 Variational Equation Variational Equation

Uncertainty Estimation Uncertainty Estimation

 Covariance Matrix Covariance Matrix

 Index of Correlation among Parameters Index of Correlation among Parameters

 Practical Uncertainty Practical Uncertainty