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Multiscale Autocorrelation Function: a new approach to anisotropy studies Manlio De Domenico 12 , H. Lyberis 34 1 Laboratory for Complex Systems, Scuola Superiore di Catania, Catania, Italy 2 Istituto Nazionale di Fisica Nucleare, Sez. di Catania,
1Laboratory for Complex Systems, Scuola Superiore di Catania, Catania, Italy 2Istituto Nazionale di Fisica Nucleare, Sez. di Catania, Catania (Italy) 3CNRS/IN2P3 - IPN Orsay, Paris (France) 4Dipartimento di Fisica, Universit´
a di Torino, Torino (Italy)
1 It depends on one parameter, i.e. the clustering scale Θ 2 It is based on information theory and extreme value theory 3 It is unbiased against the null hypothesis of isotropy 4 It provides high discrimination power against the alternative
5 It is semi-analytical 6 It requires few minutes to analyze (and penalize) data sets up
Main Ref.: M.D.D, A. Insolia, H. Lyberis, M. Scuderi arXiv:1001.1666 Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 2 / 22
Divide the observed sky into equal-area boxes. The number of boxes defines the angular scale Θ of the analysis. Procedure: ψi: density of events falling in the box Bi ψi: expected density of events falling in the box Bi from an isotropic distribution
i ψi(Θ) log ψi(Θ) ψi(Θ) Kullback-Leibler Divergence (1951)
We define the Multiscale Autocorrelation Function (MAF) s(Θ) = |Adata(Θ) − Aiso(Θ) | σAiso(Θ)
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 3 / 22
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 4 / 22
Fixed grid may cut existing clusters, causing loss of information or reducing the signal To avoid this: at any scale Θ each point is extended into 8 exposure-weighted points whose distance from the original one is Θ/2 (dynamical binning) MAF uses the density of extended points
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 5 / 22
Our numerical studies show that such a dynamical binning (by means of extended points) approach recovers the correct information on the amount of clustering in the data Θ⋆, where the significance is minimum, is the significative clustering scale: the scale at which occurs a greater number of points respect to that one occuring by chance, with no regard for a particular configuration of points, e.g. doublets or triplets. 3 skies of 60 events: 20% of events from a single source with 3
ρ = 4◦, 10◦, 25◦; 80% are isotropic.
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 6 / 22
Numerical experiments under the Null Hypothesis In general, the scale Θ⋆, where minimum chance probability occurs, is
min{p(Θ)} = arg min
Θ {Pr (siso(Θ′) ≥ sdata(Θ)|H0, ∀Θ′ ∈ P)}
corresponding to isotropic skies, should be equally likely. The distribution of min{p(Θ)} is flat, as expected, with no regards for the data set size = ⇒ unbiased against H0
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 7 / 22
For each Θ ∈ P, we investigate the density of s(Θ) = |Adata(Θ)−Aiso(Θ)|
σAiso (Θ)
The distribution of s(Θ) is half-normal: G1/2[s(Θ)] = 2 √ 2π e− s2(Θ)
2
for all Θ ∈ P
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 8 / 22
We investigate the density of max{s(Θ)}, used for penalizing p−values Density is the generalized Gumbel distribution: (Fisher-Tippet Type I from Extreme Value Theory) g(z) = 1 ˜ σ exp [−z − ez] , z = max{s(Θ)} − ˜ µ ˜ σ NOT dependent on data set size!
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 9 / 22 ˜ µ = 1.737 ± 0.001 ˜ σ = 0.464 ± 0.001 χ2/ndf ≈ 10−4
Summary
1
MAF is unbiased against the null hypothesis
2
Penalization procedure can be analytically performed: p (max{s(Θ)}) = 1 − exp
max{s(Θ)} − ˜ µ ˜ σ
We find excellent agreement between estimation through Montecarlo realizations and through analytical estimation Analytical computation is ≈ 15 times faster
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 10 / 22
Ultra High-Energy Cosmic Rays Detecting anisotropy of UHECR is important for understanding creation and propagation mechanisms, for indirectly investigating extra-galactic magnetic fields, ... We test MAF against anisotropic mock maps of UHECR according to some physical constraints:
1
Reference catalog of candidate sources: Active Galactic Nuclei (AGN) with known redshift z < 0.047 (≈ 200 Mpc) from Palermo SWIFT-BAT hard X-ray catalogue
[Cusumano, G. et al, Astron. Astrop. 510 (2010)] 2
# of events proportional to source flux Φ and to z−2
3
Magnetic deflections
4
Isotropic background contamination
5
Distribution of events weighted by exposure of world-wide surface detectors
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 11 / 22
AGN are candidate sources
G.R.Farrar and P.L.Biermann, PRL 81 (1998) 3579 P.G.Tinyakov and I.I.Tkachev, JETP Lett. 74 (2001) 445 V.Berezinsky et al, astro-ph/0210095 (2002) D.F.Torres et al, ApJ 595 (2003) P.Auger Coll., Science 318 (2007) 938 P.Auger Coll., Astrop. Phys., 29 (2008) 188 I.Zaw, G.R.Farrar, J.E.Greene ApJ (2009) 696 P.Auger Coll., In Press (2010), arXiv:1009.1855
SWIFT-BAT It provides the most complete and uniform all-sky hard X-ray survey up to date
[P.Auger Coll., In Press (2010), arXiv:1009.1855]
top: VCV; bottom: SWIFT-BAT
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 12 / 22
SD detect Extended Air Shower of particles produced by UHECR, by mean of a large array of individual stations
Full-time operating and fully efficient SD do not
depends on the relative exposure ω(δ) ∝ cos φ0 cos δ sin αm + αm sin φ0 sin δ, (1) where φ0 is the detector latitude and αm = ξ > 1 π ξ < −1 cos−1 ξ
(2) with ξ ≡ cos θmax − sin φ0 sin δ cos φ0 cos δ .
[P. Sommers, Astrop. Phys. 14, 271 (2001)] Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 13 / 22
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 14 / 22
Experiment φ0 θmax
λ #Ev. Volcano R. 35.15◦N 70◦ 0.2 × 1016 1.000⋆ 6 Yakutsk 61.60◦N 60◦ 1.8 × 1016 0.625† 20
53.97◦N 74◦ − 1.000⋆ 7 AGASA 35.78◦N 45◦ 4.0 × 1016 0.750† 29 SUGAR 30.43◦S 70◦ 5.3 × 1016 0.500⋆ 13
35.20◦S 60◦ 28.4 × 1016 1.200† 27
⋆[M. Kachelrieß and D. Semikoz, Astrop. Phys. 26, 10 (2006)] †[V. Berezinsky, Nucl. Phys. B - Proc. Supp. 188, 227 (2009)]
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 15 / 22
UHECR 102 UHECR with rescaled energy E ′ ≥ 4.0 × 1019 eV from world-wide SD
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 16 / 22
Catalog+UHECR Nearby AGN within 200 Mpc (z < 0.047) and UHECR Catalog + UHECR With flux-weighted catalog
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 17 / 22
Test accepts H0 Test rejects H0 H0 is true OK: 1 − α CL α: Type I Error H1 is true β: Type II Error OK: 1 − β Power
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 18 / 22
Simulation Setup The anisotropic mock map of 103 skies is generated according to:
1
Events gaussianly distributed (on the sphere) within ̺ = 3◦ around the sources
2
30% of events are isotropically distributed
3
70% of events are distributed according to our constraints
4
Power is estimated for different values of α
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Mock Map: 103 skies of 102 events Constraints
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 20 / 22
Power 1 − β: probability to (correctly) reject H0 when it is, in fact, false (or prob. to accept H1 when it is, in fact, true)
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 21 / 22
Multiscale Autocorrelation Function New technique not depending on particular configurations of points (e.g. doublet, triplet); physical info: significative clustering scale Physical and astrophysical constraints can be easily taken into account (by weighting densities...) Summary
1
Unbiased method against H0 / High discrimination power on anisotropy signal
2
Semi-analitical: drastically reduce CPU time We thank P. Auger Collaboration for fruitful discussion and O. Deligny for his invaluable help.
Main Ref.: M.D.D, A. Insolia, H. Lyberis, M. Scuderi arXiv:1001.1666 Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 22 / 22
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 23 / 22
CPU Time required by MAF analysis
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 24 / 22
Most significative clustering scale Θ in power analysis (α ≤ 1%)
Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 25 / 22
Let P and Q be two probability distributions, with densities p(x) and q(x), respectively. The Kullback-Leibler (KL) divergence is a measure quantifying the error in approximating the density p(x) by means of q(x), and it is defined as DKL(p||q) = Z p(x) log p(x) q(x) dx (3) Let ˜ P the empirical distribution of random outcomes xi (i = 1, 2, ..., n) of the true distribution P, putting the probability 1
n on each outcome as
˜ p(x) = 1 n
n
X
i=1
δ(x − xi ) (4) and let QΘ be the statistical model for the data, depending on the unknown parameter Θ. It follows DKL(˜ p||qΘ) = −H(˜ p) − Z ˜ p(x) log q(x|Θ)dx (5) where H(˜ p) is the information entropy of ˜ p, not depending on Θ, whereas ˜ p and qΘ = q(x|Θ) are the corresponding densities of ˜ P and QΘ, respectively. Putting Eq. (4) in the right-hand side of Eq. (5): DKL(˜ p||qΘ) = −H(˜ p) − 1 n
n
X
i=1
log q(xi |Θ) = −H(˜ p) − 1 n Lq(Θ|x) (6) where Lq(Θ|x) is the log-likelihood of the statistical model. It directly follows that arg min
Θ DKL(˜
p||qΘ) = 1 n arg max
Θ Lq(Θ|x)
(7) where the function arg min(arg max)f (Θ) retrieves the minimum (maximum) of the function f (Θ). Hence, another way to obtain the maximum likelihood estimation it to minimize the KL divergence. Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 26 / 22
Extreme value theory is the research area dealing with the static analysis of the extremal values of a stochastic
probability to obtain an outcome greater or equal than Mn is: Pr(Mn ≤ x) = Pr(x1 ≤ x, x2 ≤ x, ..., xn ≤ x) = F n(x) It can be shown that the limiting distribution F n(x) is degenerate and should be normalized. However, if there exists sequences of real constants an > 0 and bn such that Pr „ Mn − bn an ≤ x « = F n(anx + bn) then lim
n− →∞ F n(anx + bn) = G(x)
(8) The function G(x) is the generalized extreme value (GEV) or Fisher-Tippett distribution G(z) = 8 > > < > > : exp “ −e−z ” ξ = 0 exp " − (1 − ξz)
1 ξ
# ξ = 0 , z = x − µ σ (9) defined for 1 − ξz > 0 if ξ = 0 and for z ∈
R if ξ = 0, where µ, σ and ξ are the location, scale and shapeparameters, respectively. The Gumbel distribution is related to the distribution of maxima and it is retrieved for ξ = 0. The corresponding probability density g(x) is easily obtained from G as g(x) = 1 σ exp » − x − µ σ − exp „ x − µ σ «– (10) Relation to mean ˜ µ and to standard deviation ˜ σ of the distribution (γ = 0.577215... is the Euler constant): ˜ µ = µ + γσ, ˜ σ2 = π2 6 σ2 (11) Manlio De Domenico, H. Lyberis Multiscale Autocorrelation Function: a new approach to anisotropy studies 27 / 22