MODELOS ESPACIALES Y USO DE LOS SISTEMAS DE INFORMACIÓN

Fig. 4.5 shows differences of the image parameters between the gamma-ray events and hadron + muon events. Red dots and points show the gamma-ray MC events while blue ones show events from the observed data, which are completely dominated by hadrons and muons. The top left panel of the figure shows LENGTH as a function of log

10

(SIZE). At log 10

(SIZE) < 2, no difference is visible, while at log10(SIZE) 2:5, a population of events with large LENGTH is visible. This population can be explained by muon events. For arc-like images of muon events, LENGTH becomes very large for a given SIZE and most of them have a log

10

(SIZE) of 2:5. In addition to this muon branch, a further discrepancy is visible at log10

(SIZE)> 2:8. This is due to the intrinsic difference between hadron events and gamma- ray events. Therefore,LENGTHis a good parameter for gamma-ray/(hadron + muon) discrim- ination at log

10

(SIZE) > 2. The same behavior is seen inWIDTH vs log 10

(SIZE)plot in the top right panel of the figure. WIDTH is also a good parameter for the discrimination at log

10

(SIZE)>2. It should be noted that at log 10

(SIZE)>2:8, where the intrinsic differences between gamma-ray and hadron images are shown, theLENGTH=WIDTH ratio is larger for

114 4. Analysis Method of MAGIC Data

gamma-rays than for hadrons. This is reflecting the difference in the transverse spread of the air showers, as discussed in Sect. 3.1.8. The middle left panel shows distribution ofCONC. Gamma-rays are relatively more concentrated. This is attributed to the differences in the trans- verse spread and the disorder of the air showers, as discussed in 3.1.8. CONC is also useful for the discrimination. The middle right panel shows Time Gradientas a function ofDIST. Since gamma-rays come only from the source, DIST is correlated with the impact parameter (see Fig. 4.10). On the other hand,TimeGradientis also correlated with the impact parameter (see [23]). Therefore, the correlation betweenTimeGradientandDIST can be seen, which is not the case for hadrons and muons. The combination ofTime Gradient andDIST helps to discriminate between gamma-ray events and hadron + muon events. The bottom left panel shows TimeR MS as a function of log

10

(SIZE). At log 10

(SIZE) < 2and at log 10

(SIZE) >2:8, data show larger Time R MS than for gamma-ray MC. This can mainly be explained by the hadron component, for which a largerTimeR MSis expected due to the disorder of the hadron- induced air showers (see Sect. 3.1.8). On the other hand, at log10

(SIZE) 2:5, data show a smaller Time R MS than for gamma-ray MC. This can be attributed to the muon component, for which a smallerTimeR MSis anticipated because all the detected Cherenkov photons orig- inate from a single particle. The ratio between the hadron component and the muon component is different at different SIZEs, as can be seen in, e.g., theLENGTH vs log

10

(SIZE) rela- tion. TimeR MS is helpful for the discrimination mainly at log

10

(SIZE) > 2:8. The bottom right panel showsALPHAas a function of log

10

(SIZE). A clear difference is seen, as can be understood by the definition ofALPHA.

Atlog 10

(SIZE)<2:5, since the difference in each parameter (exceptALPHA) is not very large, a simple cut in any one of the parameters does not significantly increase the signal-noise ratio. A combination of many parameters is necessary to improve the telescope sensitivity for such small SIZEs. Even for larger SIZEs, the best background rejection would be achieved by making use of differences in many parameters. Therefore, in the standard MAGIC analysis, the multi-dimensional (hadron+muon)/gamma-ray separation is applied for allSIZEs, based on the Random Forest method with many image parameters (see the next sub-section).

4.4 Hadron/Gamma-ray Separation 115

Conc

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Normalized Number of Events

0 0.005 0.01 0.015 0.02 0.025 CONC MC gamma data (hadrons) CONC

Figure 4.5: Differences of the image parameters between gamma-rays MC (red dots) and hadron + muons (blue dots). Black points show the mean values of the vertical axis parameters for each horizontal axis bins. Top left: LENGTH vs log10

(SIZE). Top right: WIDTH vs log 10

(SIZE). Middle left: Distribution

of CONC. Middle right: Time Gradient vs DIST. Bottom left: Time RMS vs log10

(SIZE)Bottom right:

Alpha RMS vs log10

116 4. Analysis Method of MAGIC Data

4.4.2

Random Forest Method

The details of the method and the application to the MAGIC analysis is well described in [48], [46] and [97]. Here I will describe the basic procedure. In this section, “hadron events” include the muon events as well.

First Step: Random Forest Generation

Firstly, the hadron-gamma classification trees are generated. Osteriais used for this. The gener- ation procedure of a tree is schematically shown in the left panel of Fig. 4.6.

1. Gamma-ray and hadron event samples are prepared (S gand

S

h). Since the observation data themselves are dominated by hadron events, a part of them are used as hadron samples, while gamma-ray samples are produced by MC.

2. Several parameters to be used for the gamma/hadron separation, such asWIDTH,LENGTH, CONCandTimeGradientare chosen.

3. One parameter is randomly chosen from the separation parameters. S g ( S h) is split into S L g andS R g (S L h andS R h

) by the chosen parameter at the value which minimizes the Gini indexQ Ginidefined as Q Gini =2 #S L g #S L h #S L g +#S L h + #S R g #S R h #S R g +#S R h ! (4.6)

where #S is the number of events in a sample S. Each term of Eq. 4.6 indicates the impurity of the event class (hadron or gamma-ray) for each divided sample (S

L andS

R ) 4. The next parameter is chosen randomly from the separation parameters.Leftbranch (S L g andS L h) and R ight branch (S R g and S R

h) are further split by it in the same way as process 3. This “branching” continues until the number of samples at the end of the branch is very small (normally 3) or occupied by only one class (gamma-rays or hadrons). In this way, one classification tree is produced.

5. Each end of the branches is assigned a value~

h, which is#S end h =(#S end h +#S end g ), where S end

is the sample at the end of the branch. ~

hindicates the purity of hadrons at the end of the branch.

6. Many (normally 100) of the classification trees are produced by repeating the above pro- cess from 2 to 5. Since branching parameters are always randomly chosen, they are not identical. The name of the method “Random Forest” comes from the fact that there are many trees produced with randomly chosen branching (separation) parameters.

4.4 Hadron/Gamma-ray Separation 117 Second Step: Application to the Data

Now, the produced trees are applied to each observed event. This is done byMelibea. Application of one tree to one event is illustrated in the right panel of Fig. 4.6.

1. One tree is applied to one event. According to the image parameters, the event reaches one end of a tree and obtains the value~

h.

2. The process 1 is done with all the trees andHadronnessis calculated as

Hadronness= N X i=1 ~ h i =N (4.7)

whereN is the number of trees and ~ h

i is ~

hfromith tree. As can be deduced from the definition of ~

h, Hadronness is the measure of the likelihood that the event is a hadron, ranging from 0 to 1. Hadronnesscan also be calculated for gamma ray events generated by MC. The distributions ofHadronessfor gamma-ray events and data (dom- inate by hadrons) are compared in Fig 4.7. SIZE,LENGTH,WIDTH,CONC andDIST were used as the separation parameter for this figure. One can clearly see that for log10

(SIZE) >2.5 (SIZE > 300), the Random Forest method recognizes well gamma-ray events as such. Selection of events withHadronness, for example below 0.1, can reject 95% of the hadron background events keeping 85% of the gamma-ray signal events. SIZE above 300 corre- sponds roughly to primary gamma-ray energies above 200 GeV, which is too high for the Crab pulsar. SIZE range below 100 is essential for it but separation is almost powerless. There- fore, in order to avoid additional systematic errors,Hadronessis not used for the Crab pulsar analysis, whereas it is used for the nebula analysis.

118 4. Analysis Method of MAGIC Data Sh SgLSh L _ShSg SgLL Sh LL SgRR ShRR SgRL Sg LR Sh LR SRLh ~ h = ~h = ~h = ~ h = g S R R = 10 = 0 = 11 0.0 0.5 > 0.1 < 0.1 WIDTH CONC > 0.3 < 0.3 > 0.2 < 0.2 = 1 = 1 = 0 = 2 = 1 0.33 1.0 # # # # # # # # ~ h = 0.0 ~_{h = 0.5} ~ h = 1.0 > 0.1 < 0.1 WIDTH CONC > 0.3 < 0.3 > 0.2 < 0.2 = ~ h 0.33

Event (WIDTH = 0.08, CONC = 0.23)

Figure 4.6: Left: A simplified example of the classification tree. WIDTH and CONC are used for the separation. For the first branching, WIDTH is used. The best separation value that minimizes the Gini index defined by Eq. 4.6 is 0.1. For the second branching, CONC is used. The best separation value is 0.3 and 0.2 forS

L

andS R

, respectively. The branching stops here because the samples either are dominated by the same class or contain too few events. Then,~

his calculated for each end. Right: Application of the

tree to the data. Since the event has WIDTH = 0.08 and CONC = 0.23, the tree assigns~

h =0:33for it.

Many (normally 100) different trees are applied to the event and Hadronness for the event is the average of multiple~

h’s.

Figure 4.7: Hadronness distribution as a function of log10

(SIZE). Red dots and blue dots indicate MC

gamma-ray events and hadron + muon events from the observed data, respectively. The separation is very good at log10

(SIZE)>2:5(SIZE>300) but it gets worse for smaller SIZE. This can be understood