Maximum Likelihood Blocks for Time PDFs

1. Maximum Likelihood BlocksforTime PDFs Developments in flare detection inspired in the Scargle’s methods of Maximum Likelihood Blocks AgustínSánchezLosa 06-May-2011

2. Flare Analysis • Find coincidences of gamma flares with neutrino events in ANTARES • Correlation proportional to the intensity of source’s flare light • Flare periods and intensities have to be identified by denoising the light curves. • Use of light curves measured by satellites (FERMI, SWIFT,...) 06-May-2011 AgustínSánchezLosa

3. Flare Analysis • Different methods described and used in the literature: • “Studies in astronomical time series analysis V” • Scargle J D 1998ApJ601, 151 • “IceCube: Multiwavelength search for neutrinos from transient point sources” • Resconi E 2007 J. Phys.: Conf. Ser. 60 223 • “On the classification of flaring states of blazars” • Resconi E 2009 arXiv:0904.1371v1 • “Studies in astronomical time series analysis VI” • Scargle’s Draft(2006) at his web page: “http://astrophysics.arc.nasa.gov/~jeffrey/” • Et cetera. 06-May-2011 AgustínSánchezLosa

4. Maximum Likelihood Blocks • Scargle’sDraft describe a general method for different data types and some clues in the way to stop the algorithm: • Finally considered method is the one for binned data perturbed by a known Gaussian error • General idea: • Data available (light curves): {xn,σn,tn} → {flux,error,time} • Divide the data in significant blocks of more or less constant rate in light emission, chosen and guided by a proper likelihood function • Flare duration irrelevant: not sensitive to the time value or gaps but the consecutive order of the flux values and their error 06-May-2011 AgustínSánchezLosa

5. Maximum Likelihood Blocks • Algorithm: • Divide in each iteration the data interval from only one block until N blocks (where N is the number of total data points) • Do it in an order that try to represent in each new step qualitatively the different rate periods as best as possible (chosen likelihood) • Decide when to stop 06-May-2011 AgustínSánchezLosa

6. Maximum Likelihood Blocks • The likelihood for each block k, assuming that each given data n in that block comes from a constant rate λ perturbed by a Gaussian error, is: • The constant rate λ that maximize that likelihood is: • The total logarithmic likelihood, dropping the constant terms that are going to contribute always the same amount, is: 06-May-2011 AgustínSánchezLosa

7. Maximum Likelihood Blocks • The remaining free parameters are the beginning of each block, or CPs (change points), and the total number of blocks, M, i.e. the total amount of CPs • The likelihood maximize with so many blocks as the total amount of data, i.e. a block or a CP for each data point, M = N • Once this is done remains the choice of the number of blocks to use, M (further explained) 06-May-2011 AgustínSánchezLosa

8. Algorithm • Brute force: try all possible CPs in the data, and chose the ones that maximize the likelihood. Too much expensive in computing time. • One CP per iteration: in every step is maintained all the previously found CPs as the most optimum ones and only a new CP is added, the one who maximize the likelihood. The most optimum in time computing possible, and not a bad approximation at all, but some flares are more difficult to be found. This is the developed method used and tested until now. • Two CPs per iteration: in every step “the 2 most optimum CPsinside each block” are compared and chosen the pair which maximize the likelihood. Shows better capacity to detect evident flares in less number of blocks, M. Still in debugging phase. 06-May-2011 AgustínSánchezLosa

9. Testing... Likelihood Sampleforthe FERMI source 3C454.3 06-May-2011 AgustínSánchezLosa

10. Testing... Likelihood “found-a-flare-gap” Sampleforthe FERMI sourcePMNJ2345-15555 06-May-2011 AgustínSánchezLosa

11. Stop criteria • The chosen algorithm determinethe order in which the CPs are added • The choice of when stop have been studied for the “One CP per iteration” algorithm: • Likelihood threshold or similar: Stop when the likelihood value is a given percentage of the maximum likelihood. Not useful criterion due to the different evolution of the likelihoods and not predictable presence of “found-a-flare-gaps”. • Scargle’s Prior Study: Made a study of the most optimum γfor this prior: to add to the logarithm likelihood in order to create a maximum. Pretty complicated and does not work with “flat-flared” light curves. • Fixed Scargle’s Prior: In the Scargle’s Draft is mentioned that γ should yields γ≈N, and that has been observed in the previous stop criterion. With this fixed value more missed flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves. 06-May-2011 AgustínSánchezLosa

12. Stop criteria • The chosen algorithm determinethe order in which the CPs are added • The choice of when stop have been studied for the “One CP per iteration” algorithm: • Likelihood threshold or similar: Stop when the likelihood value is a given percentage of the maximum likelihood. Not useful criterion due to the different evolution of the likelihoods and not predictable presence of “found-a-flare-gaps”. • Scargle’s Prior Study: Made a study of the most optimum γfor this prior: to add to the logarithm likelihood in order to create a maximum. Pretty complicated and does not work with “flat-flared” light curves. • Fixed Scargle’s Prior: In the Scargle’s Draft is mentioned that γ should yields γ≈N, and that has been observed in the previous stop criterion. With this fixed value more missed flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves. PMNJ2345-15555 3C454.3 06-May-2011 AgustínSánchezLosa

13. Stop criteria Simulation of the light curve • The chosen algorithm determinethe order in which the CPs are added • The choice of when stop have been studied for the “One CP per iteration” algorithm: • Likelihood threshold or similar: Stop when the likelihood value is a given percentage of the maximum likelihood. Not useful criterion due to the different evolution of the likelihoods and not predictable presence of “found-a-flare-gaps”. • Scargle’s Prior Study: Made a study of the most optimum γfor this prior: to add to the logarithm likelihood in order to create a maximum. Pretty complicated and does not work with “flat-flared” light curves. • Fixed Scargle’s Prior: In the Scargle’s Draft is mentioned that γ should yields γ≈N, and that has been observed in the previous stop criterion. With this fixed value more missed flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves. Application of thealgorithm Withdifferentγvaluesdifferentoptimum blocks forthesamples and different error between real light curve and obtained blocks Withdifferentγvaluesdifferentoptimum blocks forthesamples and different error between real light curve and obtained blocks 06-May-2011 AgustínSánchezLosa

14. Stop criteria • The chosen algorithm determinethe order in which the CPs are added • The choice of when stop have been studied for the “One CP per iteration” algorithm: • Likelihood threshold or similar: Stop when the likelihood value is a given percentage of the maximum likelihood. Not useful criterion due to the different evolution of the likelihoods and not predictable presence of “found-a-flare-gaps”. • Scargle’s Prior Study: Made a study of the most optimum γfor this prior: to add to the logarithm likelihood in order to create a maximum. Pretty complicated and does not work with “flat-flared” light curves. • Fixed Scargle’s Prior: In the Scargle’s Draft is mentioned that γ should yields γ≈N, and that has been observed in the previous stop criterion. With this fixed value more missed flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves. 06-May-2011 AgustínSánchezLosa

15. To Do List • Finish to debug the “Two CPs per iteration” algorithm and test it • Decide the final algorithm and stop criterion • Develop a base line estimator in order to define the flare period: bellow a threshold given by a few sigmas from the base line the time PDF is set to zero. • Start real analyzing with those time PDFs 06-May-2011 AgustínSánchezLosa

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