A new approach for the gamma tracking clustering by the deterministic annealing method François Didierjean IPHC, Strasbourg

1. Principle of the deterministic annealing method in (a few !) equations  2. Description of the algorithm  3. 2D example 4. Simulated events with AGATA detector

* set of input data (N points) goal : to assign each data point to a cluster. * set of n points, called prototypes : centers of the cluster A few definitions : * the global distortion association probability of v i with c j squared Euclidean distance

The goal of clustering is to minimize the global distortion D with the minimum of prototypes. In analogy with the statistical physics : FREE ENERGY S : Shannon entropy T : Lagrange multiplier, called now temperature The minimization of the free energy, F, consists in maximizing the entropy, S, and minimizing the distortion, D.

Deterministic annealing method : minimize F while lowering T 1. minimizing F fixing the prototype position : 2. minimizing F fixing the association probability :

As the temperature is lowered, the system undergoes a sequence of phase transitions, which consists of a natural split of the cluster. phase transition [K. Rose el al. Phys. Rev. Lett 65 (1990) 945] : A cluster centered at prototype c j undergoes a splitting phase transition when the temperature reaches the critical value T crit : 2 max. max is the largest eigenvalue of the covariance matrix :

D.A. ALGORITHM Set of 5 points (1D) V = {x i } = {0.6, 1.2, 1.5, 2.8, 3.2} initializing step : T  (all data points belong to the same cluster)  1 prototype located in the center of the data distribution

search for prototype position at constant T value convergence test : step (k)   [k] 1 -  [k-1] 1 <  T = 4  1 = 1.86

check the condition for the phase transition 2 max = 1.93 < T = 4 (critical temperature not reached) cooling step T  a T (a < 1) …… T = 1.85  1 = max = 1.93 > T = 1.85 (critical temperature reached)  splitting of the prototype

adding a new prototype position F T = 2 T = 1.86    2 [0] =  1 +   1 [0]   1 - 

1/T position Parameters of the D.A. method : * cooling factor * minimal temperature to stop the algorithm

p(c 1 /v 1 )=1. p(c 2 /v 1 )= p(c 3 /v 1 )= p(c 4 /v 1 )= p(c 1 /v 2 )= p(c 2 /v 2 )=1. p(c 3 /v 2 )= p(c 4 /v 2 )= p(c 1 /v 3 )= p(c 2 /v 3 )= p(c 3 /v 3 )=1. p(c 4 /v 3 )= p(c 1 /v 4 )= p(c 2 /v 4 )= p(c 3 /v 4 )= p(c 4 /v 4 )=1. cluster 1 cluster 2 cluster 3 cluster 4 v1v1 v2v2 v3v3 v4v4

Simulated events from AGATA : GEANT 3 code x y crystal n°19 crystal n°10 cristal n°88 crystal n°90 crystal n°148 crystal n°127 crystal n°138 simple event :  multiplicity : M  = 5 gammatypecrystalsectordeposed E 1Compton Compton Compton Photo abs Compton Compton Compton Compton Photo abs Compton Compton Compton Compton Compton Photo abs Compton Compton Compton Compton Photo abs

CONCLUSION Starting with one unique cluster, the phase transition mechanism of the deterministic annealing allows to create new cluster progressively while the temperature decreases. And, at the end of the process, the best partition is found. The first test on the deterministic annealing algorithm with 1 MeV  -ray and multiplicity 5 allows to show that the clustering method is promising.

TO DO LIST 1.Check the parameters of deterministic annealing method with different mumtiplicities using GEANT 3 : - cooling factor - minimal temperature to stop the process 2. Comparison between the 2 clustering methods with data given by the collaboration (GEANT 4) : - forward tracking / relative angle distance - deterministic annealing