Maximum likelihood block method for denoising gamma-ray light curve

Collaboration MeetingMoscow, 6-10 Jun 2011

Agustín Sánchez LosaIFIC (CSIC – Universitat de València)

Outline

• Flare Analysis• Maximum Likelihood Blocks

– Likelihood– Algorithms

• Brute force• 1 Change Point• 2 Change Point

– Stop criteria• Likelihood threshold• Prior study• Fixed Prior

• To-Do List

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 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,...)

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 3

Flare Analysis• Different methods already described and used in the literature:– “Studies in astronomical time series analysis V”• Scargle J D 1998 ApJ 601, 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/”

– Etcetera.

• Scargle’s Draft describe a general method for different data types and some clues in the way to stop the algorithm

• Finally studied method is the one for binned data perturbed by a known Gaussian error

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 4

Maximum Likelihood Blocks• General idea:– Data available (light curves): {xn,σn,tn} → {flux,error,time}– Divide the data in significant blocks of approximately 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

• Algorithm:– Divide 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, every time the number of blocks is increased,

try to represent qualitatively the different rate periods as best as possible (chosen likelihood)

– Decide when to stop

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 5

Maximum Likelihood BlocksEach cell represent a {xn,σn,tn} data

A group of cells conform a block

The first cell of a block define a Change Point (CP)

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 6

Another blockAnother CP

Every time the number of blocks is increased look, with a likelihood, for the cells in which isthe most probable changes in the flux rate, considered constant

inside the block, i.e. look for the most optimum CPs

Maximum Likelihood Blocks• The likelihood for each block k, assuming that 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:

n

x

n

k n

kn

eL

2

21

21

k

nn

nnn

k

k

k

kkx

L

2

log

nn

nnn

n n

n n

n

kx

x

2

2

1

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 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

k

nn

nnn

k

k

k

kkx

L

2

log

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 8

Algorithm• Brute force: try all possible CPs in the data, and chose the ones that

maximize the likelihood.Too expensive in computing time.

• One CP per iteration: in every step is maintained all the previously found CPs as the optimum ones and only a new CP is added, the one who maximize the likelihood.The best in time computing possible, and not a bad approximation at all, but some flares are more difficult to be found.

• Two CPs per iteration: in every step “the 2 best CPs inside 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.

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 9

1CP

2CP

1CP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 10

One CP testing...

Sample for the FERMI source 3C454.3

Likelihood

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

1CP

130days

1CP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 11

One CP testing... Likelihood

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

1CP

Sample for the FERMI source PMNJ2345-1555

“found-a-flare-gap”

50days

Two CP testing...

2CP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 12

Sample for the FERMI source 3C454.3

Likelihood

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

2CP

130days

Two CP testing...

2CP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 13

Likelihood

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

2CP

Sample for the FERMI source PMNJ2345-1555

50days

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 14

Two CP testing...

Small sample for the FERMI source 3C454.3

2CP

300days

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 15

Two CP testing...

Small sample for the FERMI source 3C454.3

2CP

300days

Stop criteria• The chosen algorithm determine the order in which the CPs are added• The choice of when stop have been studied for the “Two CP per

iteration” algorithm:– Likelihood threshold or similar: Stop when the likelihood value is a given

percentage of the maximum likelihood.Not really useful criterion due to the different evolution of the likelihoods.

– Scargle’s Prior Study: Made a study of the most optimum γ for 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 all flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves.

loglog NP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 16

• The chosen algorithm determine the order in which the CPs are added• The choice of when stop have been studied for the “Two CP per

iteration” algorithm:– Likelihood threshold or similar: Stop when the likelihood value is a given

percentage of the maximum likelihood.Not really useful criterion due to the different evolution of the likelihoods.

– Scargle’s Prior Study: Made a study of the most optimum γ for 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 all flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves.

loglog NP

Stop criteria0208-512

3C454.3

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 17

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

Number of blocks

1-Lo

g(L)

/Log

(Lm

ax)

• The chosen algorithm determine the order in which the CPs are added• The choice of when stop have been studied for the “Two CP per

iteration” algorithm:– Likelihood threshold or similar: Stop when the likelihood value is a given

percentage of the maximum likelihood.Not really useful criterion due to the different evolution of the likelihoods.

– Scargle’s Prior Study: Made a study of the most optimum γ for 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 all flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves.

loglog NP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 18

Stop criteriaSimulation of the

light curve Application of the algorithm

With different γ values different

optimum blocks for the samples and different error

between real light curve and obtained

blocks

loglog NP With different γ values different optimum blocks for the samples

and different error between real light

curve and obtained blocks

• The chosen algorithm determine the order in which the CPs are added• The choice of when stop have been studied for the “Two CP per

iteration” algorithm:– Likelihood threshold or similar: Stop when the likelihood value is a given

percentage of the maximum likelihood.Not really useful criterion due to the different evolution of the likelihoods.

– Scargle’s Prior Study: Made a study of the most optimum γ for 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 all flares are now detected but implies a bigger number of unnecessary blocks for the “easy-flares” light curves.

Stop criteria

loglog NP

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 19

40days

70days

100days

1000days

To Do List

• Develop a base line estimator in order to define the flare period and build time PDFs

• Start real data analyzing with those time PDFs

6-10 Jun 2011 Agustín Sánchez Losa – Moscow Collaboration Meeting 20

Thank youfor your attention