# Entropic Approach to Multiscale Clustering Analysis

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## Abstract

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## 1. Introduction

## 2. Model Selection with the Kullback–Leibler Divergence

## 3. Extreme Value Statistics

## 4. The Multiscale Autocorrelation Function

**Figure 2.**MAF: average chance probability (solid line), with 68% region around the mean value, estimated from isotropic and anisotropic skies generated as explained in the text. The dashed line indicates the value of the dispersion adopted to generate the corresponding mock map: (

**a**) 5, (

**b**) 10 and (

**c**) 20 degrees; (

**d**) isotropic map. (Adapted from [9], reprinted by permission from IOP Publishing, Figure 4).

**Figure 3.**MAF: test power as a function of the percentage of isotropic contamination in a sky of 100 events and 10 randomly distributed sources. We fix the angular scale of clustering in simulations, namely $\rho ={3}^{\circ}$ (left panel) and $\rho ={10}^{\circ}$ (right panel), and we show the test power corresponding to different values of the test significance, ranging from $\alpha =0.1$ to $\alpha ={10}^{-7}$, at the corresponding angular scales.

**Figure 4.**As in Figure 3, but varying the angular scale of clustering in simulations from $\rho ={3}^{\circ}$ to $\rho ={20}^{\circ}$, and fixing the test significance, namely $\alpha =0.1$ (left panel) and $\alpha ={10}^{-5}$ (right panel).

**Figure 5.**MAF. (

**a**) Distribution of $\tilde{p}\left({\Theta}^{\u2605}\right)$ for $n=40,60,80,100$ and 500 events; (

**b**) Distribution of $max\left\{s\right(\Theta \left)\right\}$ for $n=40,60,80,100$ and 500 events. Solid line correspond to the least-square fit of the Gumbel density with parameters $\mu =1.743\pm 0.002$ and $\sigma =0.470\pm 0.002$ (${\chi}^{2}/\text{ndf}=1.1\times {10}^{-5}$). (Adapted from [9], reprinted by permission from IOP Publishing, Figure 6).

## 5. Application to the Physics of UHECRs

**Figure 6.**Expected clustering signal, as a function of the angular scale, from a sky of protons with $E\ge 100$ EeV (in the field of view of Pierre Auger Observatory) and for values of the Hubble parameter ranging from 50 to 100 km/s/Mpc. Sources of 44% of events are AGN within 200 Mpc in the SWIFT-BAT 58-months catalog, whereas the remaining 56% of events are isotropically distributed. The intrinsic luminosity of AGN is taken into account. The clustering for $N=100$ (left panel) and $N=50$ (right panel) is considered. The signal at each angular scale is obtained by averaging over ${10}^{4}$ Monte Carlo realizations.

**Figure 7.**Chance probability (not penalized for the scan over Θ) for clustering as a function of the angular scale, estimated from the set of 27 UHECR events detected with the Pierre Auger Observatory. Results obtained from multiscale autocorrelation function (MAF) and two-points angular correlation function (ACF) are compared.

## 6. Conclusions

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De Domenico, M.; Insolia, A. Entropic Approach to Multiscale Clustering Analysis. *Entropy* **2012**, *14*, 865-879.
https://doi.org/10.3390/e14050865

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De Domenico M, Insolia A. Entropic Approach to Multiscale Clustering Analysis. *Entropy*. 2012; 14(5):865-879.
https://doi.org/10.3390/e14050865

**Chicago/Turabian Style**

De Domenico, Manlio, and Antonio Insolia. 2012. "Entropic Approach to Multiscale Clustering Analysis" *Entropy* 14, no. 5: 865-879.
https://doi.org/10.3390/e14050865