A mul&scale autocorrela&on func&on for anisotropy studies

Transcription

1 A mul&scale autocorrela&on func&on for anisotropy studies Mario Scuderi 1, M. De Domenico, H Lyberis and A. Insolia 1 Department of Physics and Astronomy & INFN Catania University ITALY DAA2011 Erice, 17 Apr

2 Outline: MAF Mul&scale Anisotropy Func&on Descrip&on of the method A first test Improvement (dynamical coun&ng procedure) MAF: Features and interpreta&on Quan&&es involved in detec&ng anisotropy Recovering the clustering scale Sta&s&cal features Under the null hypothesis Under the alterna&ve hypotesis 2

3 Take Home Message: MAF Features Method for anisotropy signal detec&on (small dataset) One parameter dependence (clustering scale) High discrimina&on power MAF Sta&s&cs Unbiased against the null hypothesis Penaliza&on analy&cally performed 3

4 MAF Mul&scale Autocorrela&on Func&on: defini&on Divide the observed sky Σ into N equal area boxes B i. Each box covers the solid angle Ω Σ The number of boxes defines the angular scale Θ by: 4

5 MAF Mul&scale Autocorrela&on Func&on: defini&on ψ i (Θ) ψ i (Θ) : the density of events falling into the box B i. : expected density of events falling in the box B i from an isotropic distribu&on of n points quan&fies the error in selec&ng the density ψ i (Θ) to approximate the density ψ i (Θ) 5

6 MAF Mul&scale Autocorrela&on Func&on: defini&on NOTE: s(θ) depends on the angular scale Θ Chance probability of s(θ): If H 0 denotes the null hypothesis of an underlying isotropic distribu&on for the data, the chance probability at the angular scale Θ, properly penalized because of the scan on Θ, is the probability 6

7 MAF: a first test 5000 isotropic realiza&on 5000 realiza&on :70% isotropic 30% normally distributed from 10 sources (spreading 5 0 ) 7

8 Dynamical coun&ng procedure New way to count events inside each box (dynamical coun&ng). Each point is extended into 8 weighted points whose distance from the original one is Θ/2 The weight of a single point is spread on the neighbours cells An Example: sta&c coun&ng is not able to recover the differences between the two configura&ons Monte Carlo skies producing the configura&on in Figure (a) are not frequently expected s(θ) should be greater than that one es&mated from the sta&c method lower chance probability in the case (a) 8

9 Detec&ng Anisotropy of small dataset Some quan&&es involved in detec&ng anisotropies: Anisotropic sky 40% of events are isotropic 60% of events are normally distributed, with dispersion ρ, around 10 random sources ρ = 5 0 ρ = 10 0 ρ = events 100 events A"ribute required High Power (1 β) Low significance (α) Clustering scale Power (1 β): the probability to reject H 0 when it is false Significance (α): probability to reject H 0 when it is true (type I error) H 0 = isotropy 9

10 Interpreta&on of MAF Clustering scale The angular scale Θ where the significance (α) is minimum => clustering scale It is the scale at which occurs a greater accumula&on of points respect to that one occurring by chance 5000 simula&ons of 100 events each (30% anisotropic, 10 sources ) p ~ (Θ ) = argmin Θ ρ = 5 0 ρ = 10 0 ρ = 20 0 isotropic p(θ) 10

11 Sta&s&cal features under H 0 (isotropy) Under H 0, all p values p ~ (Θ ) = argmin p(θ) Θ corresponding to isotropic skies, should be equally likely. The distribu&on of p ~ (Θ ) regards for the data set size. Unbiased against H 0 is flat, as expected, with no We inves&gate the density of max {s (Θ)}, used for penalizing p values The cumula&ve density of maxima is the generalized Gumbel distribu&on and the corresponding probability density g(x) is the probability to obtain a maximum value of s(θ) (at any Θ), greater or equal than a given value max {s(θ)} is Distribu&on of max {s(θ)} for n = 40, 60, 80, 100 and 500 events. (parameters µ = ± and σ = ± 0.002) Penaliza&on procedure can be analy&cally performed 11

12 Sta&s&cal features under H 1 (anisotropy) Mock maps of anisotropic sky are generated according to expected highest energy Cosmic Rays arrival direc&on. Ref. arxiv: A sky is labeled as anisotropic if, for a fixed value of the significance α, the penalized chance probability is lesser or equal than α, p ~ (Θ ) = argmin p(θ) α Θ for some angular scale Θ. For n>60 events a power of 90% with significance α=1% 12

13 Summary and Conclusion MAF Features A new technique to detect anisotropy of a small dataset, namely Mul&scale Anisotropy Func&on is introduced Capable to assign a clustering scale to the distribu&on under inves&ga&on (not depending on the par&cular model used to generate the anisotropic sky) Not depending on par&cular configura&ons of points (e.g. doublets and/or triplets) MAF Sta&s&cs Unbiased against the null hypothesis Penaliza&on procedure can be analy&cally performed High discrimina&on power against the alterna+ve hypothesis of anisotropy (even in presence of small datasets) 13

14 BAKUP 14

15 Hypothesis Tes&ng: Sta&s&cal Errors Vede isotropia quando c è Vede anisotropia quando non c è Significanza Vede isotropia quando non c è Vede anisotropia quando c è 15

16 Differen&al? Expected 10 Detected 10 => Isotropy at 6 0 Expected 5 Detected 9 => Anisotropy at 3 0 Expected 5 Detected 1 => Anisotropy at

17 17