780.20: 1094 Session 10

Handouts: Pang excerpt, pendulum power spectrum graphs, printouts of multifit_test.cpp and multimin_test.cpp.

In this session, we'll take a look at nonlinear least-squares fitting with GSL.

Your goals for today:

• Take a quick look at some pendulum power spectra.
• Try out nonlinear minimization on a demonstration problem.
• Try out nonlinear least-squares fitting on a demonstration problem.
• Think about applying nonlinear minimization to a real physics problem (from the Pang excerpt).

Please work in pairs (more or less). The instructors will bounce around 1094 and answer questions.

Pendulum Power Spectra [Session 8 Revisited]

Look at the handout with the three rows of plots showing the time dependence of a pendulum on the left and the corresponding power spectrum on the right plotted in terms of frequency = 1/period.

1. How can you most accurately determine the single frequency in the first row from the graph on the left?
   
   
   
   
2. Identify two of the frequencies in the second row.
   
   
   
   
3. What is the characteristic of the power spectrum in the third row that is consistent with a chaotic signal?
   
   
   
   

Multidimensional Minimization with GSL Routines

The basic problem is to find a minimum of the scalar function f(xvec) [scalar means that the result of evaluating the function is just a number] where xvec = (x_0, x_1, ... , x_N) is an N-dimensional vector. This is a much harder problem for N > 1 than for N=1. The gsl library offers several algorithms; the best one to use depends on the problem to be solved. Therefore, it is useful to be able to switch between different choices to compare results. Gsl makes this easy. Here we'll try a sample problem. For more details, see the online gsl manual under "Multidimensional Minimization".

The routines used here find local minima only, and only one at a time. They have no way of determining whether a minima is a global one or not (we'll return to this point later).

1. Look at the test code "multimin_test.cpp" and compare it to the online gsl documentation under "Multidimensional Minimization". Identify the steps in the minimization process.
2. Create a multimin_test project, add multimin_test.cpp, compile and run it. Your results may differ from what is in the comments of the code. Adjust the program so that your answer is good to 10^-6 accuracy. What is it that is found to that accuracy? (E.g., is it the positions of the minimum or something else?)
   
   
   
   
3. Modify the function minimized so it is a function of three variables (e.g., x, y, z), namely a three-dimensional paraboloid with your choice of minimum. Change the code so that the dimension of xvec is a parameter. (Be sure to change ALL of the relevant functions.) Verify that the minimum is still found.
4. What algorithm is used initially to do the minimization? Modify the code to try a couple of the other algorithms. Which is "best" for this problem?
   
   
   
   
5. In Session 14, we'll return to try to reproduce the results illustrated in Fig. 4.2 of the Pang handout (stable structures of Na₃Cl₂⁺ and Na₃Cl₃). Take a look at that now to see if you have any immediate questions.

Nonlinear Least-Squares Fitting with GSL Routines

Using the nonlinear least-squares fitting routines from gsl is similar to using the gsl minimizer (and involves a special case of minimization). Your main job is to figure out from the online documentation and this example from gsl (only slightly modified from what you would find in the documentation) how to use the routines. Here we briefly explore the sample problem, which also illustrates how to create a "noisy" (i.e., realistic) test case.

The model is that of a weighted exponential with a constant background:
y = A e^{-lambda t} + b
You'll generate pseudo-data at a series of time steps t_i (in practice, t_i is taken to be 0, 1, 2, ...). Noise is added to each y_i in the form a random number distributed like a gaussian with a given standard deviation sigma_i. We'll revisit the gsl functions that generate this random distribution in the next session; for now just accept that it works.

1. Take a look at the multifit_test.cpp code (there is a printout), make a multifit_test project and run it, and figure out the flow of the program. What is the fitting function to be minimized? What role does sigma_i play?
   
   
   
   
2. What is the "Jacobian" for? (Remember to check the online documentation!)
   
   
   
   
3. Modify the code so that you can plot both the initial data and the fit curve using gnuplot. Look up how to add error bars for the data in gnuplot. [Hint: Try "help set style", "help errors", and "help plot errorbars" for some clues.]
4. The covariance matrix of the best-fit parameters can be used to extract information about how good the fit is. How is it used in the program to estimate the uncertainties in the fit parameters? How do these uncertainties scale with the number of time-steps used in the fit? How do they scale with the magnitude of the noise? (I.e., if you change the "amplitude" of the gaussian noise, how do the errors in the fit parameters change?)
   
   
   
   
5. Modify the function by adding a fourth parameter and try to fit (and plot!) again. For example, you could change "b" to "b sin(omega t)" with omega another parameter, or "A" to "A cos(omega t)". Your choice!