# Presentation: 878281

## Novel Methods for Analysis Of Event/Point Data

-  Introduction; Events and Points
-  New Concepts (to me)
-  Photonic astronomy
-  Fractal-Based Point Processes
-  Problems
-  Power spectra from time-tagged data (worked)
-  Measuring delays in time series (worked)
-  Deconvolution of known point-spread function (in progress)
-  Large Scale Structure (Viewpoints); Causal Set QG
- AISRP: May, 2008		Jeff Scargle	NASA Ames Research Center

## Handbook of Statistics for Event Data

-  Astronomical Introduction: photonic astronomy
-  Data spaces
-  Points
-  Events
-  Data modes
-  To bin or not to bin, that is the question ...
-  Observational errors
-  Statistical models
-  Confidence measures, upper limits
-  Computational methods
-  0 dimension: repeated measurements
-  1 dimension: time series, spectra, etc.
-  2 dimensions: images, maps, etc.
-  3+ dimensions: redshift surveys, data mining, anomaly detection, etc.
-  Appendices: Case studies; Probability; Theory; Algorithms; Tutorials

## Events and Points

- 	Event = Point + Other information
- 	Discrete
- 	Point Density
- 	Density Gradients
- 	Adjacency
- 	Segmentation

## Photonic Astronomy

-  Most knowledge of the Universe arrives via photons.
-  Both individual photons and photon streams
- 	are more complex and convey more information
- 	than has generally been appreciated.
-  Quantum optics may open new fundamental information channels.

## Information content of light. I

*Information content of light. I*

**D.Dravins, ESO Messenger 78, 9 (1994)**

## Instruments measuring first-order spatial coherence

- <date/time>

- Galileo’s telescopes (1609)

**Instruments measuring first-order spatial coherence**

- Hubble Space Telescope (1990)

## Instruments measuring first-order temporal coherence

- <date/time>

*HARPS*  (2003)

- Fraunhofer’s spectroscope (1814)

**Instruments measuring first-order temporal coherence**

## Information content of light. II

- <date/time>

*Information content of light. II*

**D.Dravins, ESO Messenger 78, 9 (1994)**

## <date/time>

**R.Hanbury Brown, J.Davis, L.R.Allen, MNRAS 137, 375 (1967)**

## Slide 10

## The Hanbury Brown-Twiss Effect

- First intensity correlation experiment + observation.
- Measured angular diameters of 32 bright stars.
- Controversial at first.
- Solid theory and laboratory experiments.
- Basis of a huge amount of experimental nuclear and particle physics.

## Roy Glauber

- <date/time>

*Roy Glauber*
- Nobel prize in physics
- Stockholm, December 2005

## PHOTON  STATISTICS

- R. Loudon *The Quantum Theory of Light* (2000)

**PHOTON  STATISTICS**

## Slide 14

## Slide 15

## Slide 16

* **** ***
***  ******SECONDS & MILLISECONDS***
- Lunar & planetary-ring occultations
- Rotation of cometary nuclei
- Pulsations from X-ray pulsars
- Cataclysmic variable stars
- Pulsating white dwarfs
- Optical variability around black holes
- Flickering of high-luminosity stars
- X-ray binaries
- Optical pulsars
- Gamma-ray bursts

##   MILLI-, MICRO- & NANOSECONDS

* **** ******MILLI-, MICRO- & NANOSECONDS***
- Millisecond pulsars ?
- Variability near black holes ?
- Surface convection on white dwarfs ?
- Non-radial oscillations in neutron stars ?
- Surface structures on neutron-stars ?
- Photon bubbles in accretion flows ?
- Free-electron lasers around magnetars ?
- Astrophysical laser-line emission ?
- Spectral resolutions reaching R = 100 million ?
- Quantum statistics of photon arrival times ?

## Things to be Investigated

-  Off-line, Multi-Detector Intensity Interferometers (Ofir et al.)
-  Single event photon HBT Interferometry (Shanxi U. QO Lab)
-  Quantum theory of emission ... synchrotron radiation, etc.
-  Quantum theory of detection process
-  Inter-photon interval distributions.
-  Other QO effects: orbital angular momentum (Harwit), etc.
-  Can GLAST image GRBs?

## Bibliography

- Roy Glauber: *Quantum Theory of Optical Coherence*: Selected Papers and Lectures, 2007
- 2005 Nobel Prize Lecture “One Hundred Years of Light Quanta” http://nobelprize.org/nobel_prizes/physics/laureates/2005/glauber-lecture.html
- Hanbury Brown: The Intensity Interferometer (1974), Taylor and Francis/Wiley
- Dainis Dravins: Photonic Astronomy and Quantum Optics (2007), astro-ph/0701220
- Labeyrie et al.: An Introduction to Optical Stellar Interferometry (2006), Cambridge U. Press
- Mandel and Wolf: Optical coherence and quantum optics (1995), Cambridge U. Press
- The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions, Gordon Baym nucl-th/9804026
- Intensity interferometry in subatomic physics, David Boal, Rev. Mod. Phys, 62 (1990), 553-602

## Fractal-Based Point Processes

*Steven Lowen and MalTeich*
*Wiley Series in Probability and Statistics, 2005*

## Fourier vs. Wavelet Power Distributions

| Transform | Fourier | Wavelet |
| --- | --- | --- |
| Variance function | Power Spectrum | Scalegram |
| Track non-stationarities with | Time-frequency distribution | Time-scale distribution (scalogram) |
| Target signal | Periodic | Random |
| Dependent variable | Mean-square Fourier amplitude | Mean-square wavelet coefficient |
| Independent variable | Frequency | Log(scale) |
| Poisson noise correction? | Subtract constant from spectrum | Subtract constant from scalegram |

## Slide 23

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## Slide 26

## How best to measure Energy-Dependent Lags?

- The data:  time and energy tagged  --   ti   Ei   i = 1, 2, ... , N
- _Usual approach_: 	Bin the data in both time and energy
- 			Find peak in cross-correlation function (across E bands)
- _Entropy approach_
- 		define transformation of time:  t'i = f( ti ) = ti + L(, Ei )
-  		(lag L is a function of a parameter )
- If  is other than the correct value, the light curve for the transformed
- times will be smeared out.  Hence the entropy of the light curve will
- be minimum for the correct value:
-   optimum =  argmin[ Entropy ( histogram(  ti + L(, Ei ) )  ]
- 				lag estimate is then just  L( optimum, E )

## How best to measure Energy-Dependent Lags?

- Entropy method:
- 1. Make histogram  nk = number of transformed times t'i in bin k
- 2. Convert to a probability distribution:   pk  =  nk /  nk
- 3. Compute entropy:  S() =  pk log( pk )
- 4. Minimize S() with respect to 
- Examples:
- 	t'i = f( ti ) = ti +  Ei 		(first term in Taylor expansion)
-  	t'i = f( ti ) = ti +  / Ei2 	(finite photon mass dispersion rel.)

## Matlab code

- pp = hist( times, bins );
- iu = find( pp > 0 );
- % Shannon information
- cost = - sum( pp(iu) .* log( pp(iu) ) ) * dd_xx;
- % Renyi information: Cover and Thomas, (16.83)
- cost = log( sum( pp(iu) .^ alpha  ) * dd_xx ) / ( 1 - alpha );
- % total variation
- cost = sum( abs( diff( pp ) ) );
- % Fisher information; eq. (1.24) of Frieden
- cost = 4 * sum( ( diff( sqrt( pp ) ) ) .^ 2  );
- % Interval Distribution Curvature:
- hh = hist( diff( sort( times) ), bins );
- pp = polyfit( bins, hh, 2 );
- cost = - pp(3) - pp(2);

##  Cost function vs   (ms/GeV) for binned data from a synthetic GRB (imposed dispersion of 20 ms/GeV ) as will be seen by GLAST.

-  Cost function vs   (ms/GeV) for binned data from a synthetic GRB (imposed dispersion of 20 ms/GeV ) as will be seen by GLAST.

## Distribution of estimated dispersion (ms/GeV) for 100 synthetic GRBs with true value  = 20.  Means (variances) are indicated for each cost function.

- Distribution of estimated dispersion (ms/GeV) for 100 synthetic GRBs with true value  = 20.  Means (variances) are indicated for each cost function.

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