Statistical Methods in Astronomy-Lecture 1: Numerical Techniques

1. Examples of Techniques you might use:
   • Fitting
   • Interpolating / Extrapolating
   • Smoothing
   • Derivatives / Integrals
   • Solutions to equations
   • Books:
     • Bevington & Robinson: Data Reduction and Error Analysis for Physical Sciences (McGraw-Hill)
     • Press et al, Numerical Receipes in Fortran and C (Cambridge)
     • Wolfram, Mathematica (Addison-Wesley)
2. Differentiation / Integration
   • Differentiation
   • Differential Equations
   • Integration (Trapezoidal, Simpsons Rule)
3. Roots to Non-Linear Equations
   • Trial and Error on an Interval
   • Newton-Raphson
4. Exact data fitting
   • Polynomial
   • Cubic Spline
5. Least Squares Fitting
   • Straight Line
   • Transform to Straight Line
   • Numerical recipes: fit and fitexy
   • Robust estimation (removing outliers)
   • Fitting a Polynomial (routine lfit)
6. Non-Linear Functions
   • An artform!
   • Consider choice of starting parematers, method of solution, step size
   • Be aware of multiple solutions
   • Convergence conditions
   • Grid Search techniques
   • Gradient Search techniques
   • Numerical receipes routine mrqmin applies "Levenberg-Marquardt" method.
7. Mathematica Example
   • Fits 2 Black Bodies to broad band photometry. Example can readily be modified to other functions and data sets.
   • Mathematica Macro
   • Example run (gzipped postscript)

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Michael Burton M.Burton@unsw.edu.au 1998 March 12