John Rice

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Detecting Periodicity in Point Processes

• John Rice
• University of California, Berkeley
• All animals act as if they can make decisions.
  (I. J. Good)
• Joint work with Peter Bickel and Bas Kleijn.
  Thanks to Seth Digel, Patrick Nolan and Tom
  Loredo

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Outline
Introduction and motivation A family of score
tests Assessing significance in a blind
search The need for empirical comparisons
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Motivation
Many gamma-ray sources are unidentified and may
be pulsars, but establishing that these sources
are periodic is difficult. Might only collect
1500 photons during a 10 day period.
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Difficulties

• Frequency unknown
• Spins down
• Large search space
• Glitches
• Celestial foreground

Computational demands for a blind search are very
substantial. A heroic search did not find any
previously unknown gamma-ray pulsars in EGRET
data. (Chandler et al, 2001).
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Detection problem
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Unpleasant fact There is no optimal test.
A detection algorithm optimal for one function
will not be optimal for another function. No
matter how clever you are, no matter how rich the
dictionary from which you adaptively compose a
detection statistic, no matter how multilayered
your hierarchical prior, your procedure will not
be globally optimal. The pulse profile ?(t) is
an infinite dimensional object. Any test can
achieve high asymptotic power against local
alternatives for at most a finite number of
directions. In other words, associated with any
particular test is a finite dimensional
collection of targets and it is only for such
targets that it is highly sensitive. Consequence
You have to be a closet Bayesian and choose
directions a priori.
Lehman Romano. Testing Statistical Hypotheses.
Chapt 14
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Specifying a target
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Likelihood function and score test

• Let the point spread function be w(ze). The
  likelihood given times (t), energies (e), and
  locations (z) of photons

where wj w(zj ej).. A score test (Rao test)
is formed by differentiating the log likelihood
with respect to ? and evaluating the derivative
at ? 0
Neglible if period ltlt T
Unlike a generalized ratio test, a Rao test does
not require fitting parameters under the
alternative, but only under the null hypothesis.
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Phase invariant statistic

• Square and integrate out phase. Neglecting the
  second term
• Apart from psf-weighting was proposed by Beran as
  locally most powerful invariant test in the
  direction ?( ) at frequency f. Truncating at n1
  gives Rayleigh test. Truncating at nM gives
  ZM2. Particular choice of coefficients gives
  Watsons test (invariant form of Cramer-von
  Mises).

Mardia (1972). Statistics of Directional Data
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Relationship to tests based on density estimation

• In the unweighted case the test statistic can be
  expressed as
• A continuous version of a chi-square goodness of
  fit test using kernel density estimate rather
  than binning. But note that a kernel for density
  estimation is usually taken to be sharply peaked
  and thus have substantial high frequency content!
  Such a choice of ?( ) will not match low
  frequency targets.

Kernel density estimate
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Power

• Let
• And suppose the signal is
• and
• Then

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Tradeoffs

• The n-th harmonic will only contribute to the
  power if ?n is substantial and if n? is small.
  That is, inclusion of harmonics is only helpful
  if the signal contains substantial power in those
  harmonics and if sampling is fine, n??lt 1
  compared to the spacing of the Fourier
  frequencies, 1/T otherwise the cost in variance
  of including higher harmonics may more than
  offset potential gains. Viewed from this
  perspective, tests based on density estimation
  with a small bandwidth are not attractive unless
  the light curve has substantial high frequency
  components and the target frequency is very close
  to the actual frequency.

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Integration versus discretization

• Rather than fine discretization of frequency,
  consider integrating the test statistic over a
  frequency band using a symmetric probability
  density g(f).

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• Requires a number of operations quadratic in the
  number of photons. However the quadratic form
  can be diagonalized in an eigenfunction
  expansion, resulting in a number of operations
  linear in the number of photons.
• (In the case that g() is uniform, the
  eigenfunctions are the prolate spheroidal wave
  functions.) Then
• Power is still lost in high frequencies unless
  the support of g is small.
• This procedure can be extended to integrate over
  tiles in the plane when

MultiTaper
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Example Vela
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Assessing significance

• At a single frequency, significance can be
  assessed easily through simulation. In a
  broadband blind search this is not feasible and
  furthermore one may feel nervous in using the
  traditional chi-square approximations in the
  extreme tail (it can be shown that the limiting
  null distribution of the integrated test
  statistic is that of a weighted sum of chi-square
  random variables). We are thus investigating the
  use of classical extreme value theory in
  conjunction with affordable simulation.

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Gumbel Approximation
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Example
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Tail Approximations
According to this approximation, in order for a
Bonferonni corrected p-value to be less than
0.01, a test statistic of about 11 standard
deviations or more would be required.
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log- log F(t) versus t
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Need for theoretical and empirical comparisons

• Since no procedure is a priori optimal,
  comparisons are needed.
• Suppose we are considering a testing procedure
  such as that we have described and two Bayesian
  procedures
• A Gregory-Loredo procedure based on a step
  function model for phased light curve
• A prior on Fourier coefficients, eg independent
  mean zero Gaussian with decreasing variance
• Also note that within each of these two, the
  particular prior is important. Even in
  traditional low-dimensional models, the Bayes
  factor is sensitive to the prior on model
  parameters, in contrast to its small effect in
  estimation.
• Kass Raftery (1995). JASA. p. 773-

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Example

• We run the procedure discussed earlier with
• We also use a Bayesian procedure
• The signal has
• How do the detection procedures compare?

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• To compare a suite of frequentist and Bayesian
  procedures, we would like to like to understand
  the behavior of the Bayes factors if there is no
  signal and if there is a signal. (Box suggested
  that statistical models should be Bayesian but
  should be tested using sampling theory). Theory
  for the Bayesian models above?
• It might be possible to convert p-values to Bayes
  factors.
• It might be possible to evaluate posterior
  probabilities of all the competing models and
  perform composite inference (would involve
  massive computing).
• Inference is constrained by computation.
• Touchstone blind comparisons on test signals.
  We understand that GLAST will be making such
  comparisons.

Box (1981). In Bayesian Statistics Valencia
I Good (1992) JASA
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Conclusion
Problems are daunting, but with imagination, some
theory, and a lot of computing power, there is
hope for progress.
NERSC's IBM SP, Seaborg, has 6,080 CPUs .