MCMC reconstruction of the 2 HE cascade events

Dmitry Chirkin, UW Madison

Markov Chain Monte Carlo1. Start with some initial values,

e.g., x,y,z=COG; ,=0,0; E=1.e5, t=0or even x,y,z=0,0,0 (as in the example on first slide).

2. Simulate EM cascade with these parameters (point 1).

3. Compute the likelihood L1 quantifying the difference between this simulation and data (here: same as in SPICE fits, or can be of your choice, e.g., 2).

4. Sample the next point 2 from a proposal distribution, e.g., gaussian centered on the cascade parameters at point 1.

5. Calculate the likelihood L2 at new point 2. If L2>L1 jump to the new point 2, otherwise stay at 1.

6. Repeat from step 2 until chain converges to stationary state.

Configuration choices

• Simulation: ppc with SPICE Lea

• Likelihood: as in SPICE fitdepends on number of data events nd and simulated events ns

combine 25 ns. time bins with bayesian blocks algorithm

• Start with COG; calculate best time offset t and scale energy to maximize likelihood after each simulation (this reduces the number of parameters that are varied in MCMC). The energy scaling is achieved by fitting for ns.

Uncertainties and systematics

The result of MCMC is a set of points that are distributed near the best fit set of parameters. The spread of MCMC points characterizes the parameter uncertainties.

This is how to include systematic uncertainties, e.g., ice model:before running the simulation at points 1 and 2 first pick the model with pre-determined probabilities, e.g., 60% Spice Lea and 40% WHAM or sample directly from the error ellipse in a and e.

Only statistical uncertainties are presented here.

Results: xy

Event 118545 Event 119316

Blue: 1…500 Red: 501…1000

Results:

Event 118545 Event 119316

Blue: 1…500 Red: 501…1000

Results: E

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Results: x

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Results: y

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Results: z

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Results:

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Results:

Event 118545 Event 119316

Blue: 1…1000 Red: 501…1000

Summary

Event 118545 Event 119316

x = -202.7 +- 3.4y = 95.8 +-3.4z = 121.1 +- 3.9th = 75.1+-14.3ph = -46.0 +- 15.0E = 1093956 +- 2.5%

x = -75.2 +- 5.3y = 261.4 +- 4.6z = 24.4 +- 5.4th = 31.3 +- 12.5ph = -57.7 +- 40.2E = 1342765 +- 9.4%

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Simulation at best point vs. data

Event 118545 Event 119316

Concluding remarks

These results are preliminary.

Will run for other ice models.

The likelihood value at minimum can be used to rank ice models.

More plots at http://icecube.wisc.edu/~dima/work/IceCube-ftp/mcmc/.