1. 9. Data Fitting Fitting Experimental Spectrum
Fitting Exponential Decay
Theory: Probability & Statistics
Least-Square Fitting
Calling LAPACK from C

2. 9.1. Fitting Experimental Spectrum
Two types of techniques for fitting data :
Interpolation: polynomials passing through all data points.
Least-squares: model with adjustable parameters.
Model :

3. Lagrange Interpolation
nth order polynomial passing through n + 1 data points { ( xi , fi ) ; i = 0,1,...,n } :
where
i  j   k  i ,   x 1 2 4
f(x)
12
24
60
E.g.

4. Assuming f describe the 1st n derivatives exactly, then the remainder theorem gives :
Ex Do §

5. Cubic Splines
Ref: D.Kincaid, W.Cheney, "Numerical Analysis", § 6.4, Brook (1991)
Given a set of n + 1 data points { ( xi , yi ) ; i = 0,1,..., n },
the cubic spline S(x) interpolates each of the n intervals with a 3rd degree polynomial such that
S passes through all n + 1 data points.
f  & f  are continuous in all n  1 interior points.
# of parameters = P = 4 n.
# of constraints = C = 2n + 2(n 1) = 4n 2
Natural spline : S 0 = S n = 0

6. For Si on interval [ xi , xi+1 ] :
Since Si is a 3rd degree polynomial, Si is linear in x. 
where   

7. With z0 = zn = 0,
is a tri-diagonal linear system for the unknowns { zi ; i = 1,..., n1 }
Once it’s solved, say, by Gaussian elimination ( Thomas algorithm ), one has
Ex Do a spline fit for the cross section data ( see § 9.1.5).

8. 9.2. Fitting Exponential Decay
Theory :
stochastic events: 
Task:
‘Fit’ theory to experimental data by choosing the ‘best’ values of N0 and .

9. 9.3. Theory: Probability & Statistics
Basics ( discrete events ) :
Probability P(x) of event x (occurring) :
0  P(x)  1.
1  P(x)  Probability of event x not occurring.
In the sampling of a population,
P(x) of a particular sample having value x is :
Mean value of a function f(x) is :

10. Binomial Distribution
Let the probability of success in each trial be p.
After N trials, the probability of having succeeded x times is

11. Gaussian & Poisson Distribution
See F.Reif, “Fundamentals of Statistical & Thermal Physics, §1.6 & Prob 1.9.
As N   while   Np remains finite, PB becomes the Gaussian distribution
As N   while p  0, PB becomes the Poisson distribution

12. 9.4. Least-Square Fitting
Given ND data points { xi , yi ± i } , i = 1,..., ND ,
& a model function g(x) = g(x; {m} ) of Np parameters m, m = 1,..., Np,
find the values of {m} by minimizing
Thus,
m = 1,..., Np
Ex Do Exercise in §

13. Linear Regression   

14. To minimize subtractive cancellation error, let  
where

15. Nonlinear Fit of Breit-Wigner to Cross Section
Breit-Wigner resonance formula :

16.  

17. 9.5. Calling LAPACK from C