Ordinary least-squares fit

Objective

Fit a linear combination of given functions, ∑ c_kf_k(x)|_k=1..m, to a (measured) data set {x_i, y_i, δy_i}_i=1..n, where δy_i are the uncertainties of the measurement (error-bars).

Tasks

1. (6 points) Ordinary least-squares fit by QR-decomposition
   • You can use at your choice procedurial, object-oriented, or functional programming stile.
   • Make sure that your QR-decomposition routines work for tall matrices.
   • Implement a function that makes a least-squares fit—using your QR-decomposition routines—of a given data-set, {x_i, y_i, δy_i}_i=1...n , with a linear combination F(x)≐∑ _k=1..m c_k f_k(x) of given functions f_k(x)|_k=1..m .
   • The parameters to the function should be the data to fit, {x_i, y_i, δy_i}, and the set of functions, {f_k}, the linear combination of which should fit the data. The function must calculate the vector of the coefficients, {c_k}.
   • In 1902 [Rutherford and Soddy] measured the radioactivity of the (then not well explored) element ThX and obtained the following results,

     Time t (days)                     : 1   2   3  4  6  9    10   13   15
     Activity y of ThX (relative units): 117 100 88 72 53 29.5 25.2 15.2 11.1
     

     

     From this data they correctly deduced that radioactive decay follows exponentil law, y(t)=ae^-λt (equation (1) in the article).

     Now, assume that the incertainty δy of the measurement was, say, about 5%,

     δy=y/20
     

     and fit the data with exponential function in the usual logarithmic way, ln(y)=ln(a)-λt. The uncertainty of the logarithm should be probably taken as δln(y)=δy/y.

     Plot the experimental data (with error-bars) and your best fit.

     From your fit find out the half-life time of ThX.

     ThX is today known as ²²⁴Ra – compare your result with the modern value.

2. (3 points) Uncertainties of the fitting coefficients

   • Modify you least-squares fitting function such that it also calculates the covariance matrix and the uncertainties of the fitting coefficients.
   • Calculate the uncertainty of the half-life value for ThX from the given data.
   • Does it agree with the modern value within the estimated uncertainty?

3. (1 points) Evaluation of the quality of the uncertainties on the fit coefficients

   • Plot your best fit,
     ∑ _k=1..m c_k f_k(x),
     together with the fits where you change the fit coefficients by the estimated δc, that is,
     ∑ _k=1..m (c_k±δc_k) f_k(x).