to Dave's research index.
Simplex V Powell's Minimization of Pandel Timing Likelihood
David Steele & Annie Malkus
23 August 2001
to Dave's research index.
This page provides details of a comparison of the performance of the Simplex and Powell's method minimizers on low level AMANDA-II data. In summary, the Simplex minimizer, when used with an optimized first "step" size, results in better angular resolution than Powell's method for raw AMANDA-II nusim MC. The usual basic distributions (i.e. zenith distribution) resulting from the two methods are comparable. Simplex minimization is faster than Powell's method. Simplex appears to be a better choice of minimizer for use in low level processing.
We performed identical reconstructions of low level AMANDA-II MC and 1999 data with only the minimizer (and minimizer parameters) changed between runs.
Prior to comparing results of simplex minimization with powell minimization, we optimized the "step" size of the simplex minimizer on the pandel timing likelihood function. Step sizes are specified with a recoos command line flag such as:
-x x:step=40,y:step=40,z:step=40,zenith:step=0.1,azimuth:step=0.2
where the spatial step sizes are in meters and the angular step sizes in degrees. These "step sizes" are not step sizes in the usual sense. As simplex runs, it determines its own step sizes, eventually converging to the minimum with smaller and smaller step sizes. The parameters specified in the recoos command line are used to set the initial verticies of the simplex.
Convential wisdom is that the initial simplex size should be on the same scale or smaller than the uncertainty or resolution in the reconstruction parameters. In the case of AMANDA, this a few meters in space and a degree in angle. We found that, contrary to intuition, a bigger initial spatial size works better than the conventional choice.
Shown here is the angular resolution of AMANDA-II signal Monte Carlo using the Pandel timing likelihood function reconstructed with the simplex minimizer with two different spacial step sizes. It is clear the the larger step size results in a better angular resolution.
We have tried many step sizes in both the spatial and angular parameters and found that the optimum step sizes for minimizing the pandel timing likelihood are 40 m for the spatial parameters, 0.1 degrees for zenith, and 0.2 degrees for azimuth. A possible explanation for this counterintuitive result is that using the larger initial simplex helps the minimizer find the true global minimum by intially spreading itself out over a larger portion of the likelihood space.
In the rest of the analysis, all simplex minimizations are done using the optimized initial size.
We reconstructed 16,800 nusim AMANDA-II Monte Carlo events using the two minimizers on the Pandel timing likelihood function. These results are for muon reconstruction with NO gridsearch interations. Below are angular resolution, likelihood, and Ndir C distributions, the solid line showing results with the Powell minimizer, the dotted with simplex. The top median quoted is for the Powell minimizer and the bottom for simplex.
The simplex minimizer seems to give a better angular resolution on the lowest level data.
The likelihood distributions of the two methods disagree slightly in shape. The distribution made with Powell minimization has a slight bump at a likelihood of about 12, whereas simplex minimization gives a smooth distribution.
Ndir C distributions agree fairly well, with the Simplex method resulting in slightly higher values of Ndir on average.
Zenith, likelihood, and Ndir C distributions for 1999 minimum bias data reconstructed with the two different minimizers agree well.
We have found that the performance of the simplex minimizer on low level data is comparable, if not marginally better than the performance of the Powell's method minimizer. Use of the faster simplex minimizer is recommended for low level data filtering to reduce processing time.
to the research index.