Lecture 5: source detection.
• Test the null
hypothesis (NH).
Cutoff at 5% level.
– The NH says: let’s
suppose there is no
source there – ie,
model is just
background.
– Calculate χ2 for this
null model.
– Calculate the
probability of χ2
exceeding the value
obtained. (Sometimes
called a P-value.)
NASSP Masters 5003F - Computational Astronomy - 2009
Source detection.
– If this probability (the P-value) is smaller than
a previously chosen cutoff, call this a positive
detection.
• BUT! Note that there is no certainty.
– Sometimes the null model will by chance give
a large χ2 => ‘false positives.’ For given data,
background and cutoff, there will be a fixed
number of false positives expected in the
source list.
• => ‘reliability’. More on this later.
– Sometimes a real source will give a small nullhypothesis χ2 => ‘false negatives’, real
sources which are missed.
• => ‘completeness’. More on this later.
NASSP Masters 5003F - Computational Astronomy - 2009
Problems with the NH approach:
• We don’t have exact knowledge of the
background.
– Have to estimate it either from
• separate data – in which case we need separate
data!
• or from the same data… but this may be
dominated by the source...
– Or our background model may be wrong.
• Same issues as other model fitting. In
particular:
– χ2 has to be used with care when the noise is
Poisson.
NASSP Masters 5003F - Computational Astronomy - 2009
But where are the sources?
• A low probability for the null hypothesis tells us,
at best, that there is a source somewhere.
• Finding the source(s) consists rather of looking
for peaks in a random signal.
• The simplest example is when the noise is
uncorrelated and the source peaks have width=0.
NASSP Masters 5003F - Computational Astronomy - 2009
A generic source-detection algorithm
• We shall assume that:
– The data is ‘binned’ (eg CCD data).
– We have a good independent estimate of the
background.
– The sources are sparsely distributed – such
that we can deal with them one at a time.
– The shape of the source profile is known.
– The source position is unknown.
– The source amplitude is unknown (but >0).
NASSP Masters 5003F - Computational Astronomy - 2009
Generic source-detection algorithm:
The algorithm has 3 steps:
1:
Calculate a sliding-window map.
2:
Find the peaks in this map.
3:
Rejects
For each peak, calculate the
probability that it could arise by
chance from the background
(the null hypothesis P-value).
No
P < Pcutoff?
Yes
Choose
a Pcutoff
Sources
NASSP Masters 5003F - Computational Astronomy - 2009
1: The sliding window.
y
U
y
U
y
U
NASSP Masters 5003F - Computational Astronomy - 2009
1: The sliding window.
Same thing.
• For each position of the sliding window, a
single number U is calculated from the
values falling within the window.
• The output is a map of the U values.
• The intent is to:
– Raise the signal-to-noise
– Improve sensitivity
– Amplify the sources at the expense of the
noise.
• Sliding-window processing only has value
when the source has a width > 1 pixel.
• Edges need special treatment.
NASSP Masters 5003F - Computational Astronomy - 2009
1: Window functions
• A weighted sum (= a convolution).
– Simplest with all weights = 1: “sliding box”.
– Optimum weights – a “matched filter”:
• For uniform Gaussian noise, wopt = s.
• Trickier to optimize for Poisson noise.
• Per-window null-hypothesis χ2.
– With either an independent value of bkg (in
which case degrees of freedom = number of
pixels Nw in the window), or…
– …one fitted from the data (deg free = Nw-1).
• Likelihood (same bkg provisions as χ2).
NASSP Masters 5003F - Computational Astronomy - 2009
1: Window functions
Parent function
Data
NASSP Masters 5003F - Computational Astronomy - 2009
1: Window functions
Parent function
Chi squared, size=100
Matched filter, size=10
Log-likelihood, size=100
NASSP Masters 5003F - Computational Astronomy - 2009
2: Peak finding
Gaussian noise, convolved with a gaussian filter.
…don’t get the gaussians mixed up!
NASSP Masters 5003F - Computational Astronomy - 2009
2: Peak finding
• No single neat prescription.
• Naive prescription:
– Pixel i is a peak pixel if yi > any other y within
a patch of pixels from i-j to i+j.
• But what value to choose for j?
• Things to avoid are:
– j too small – results in more than 1 peak per
source;
– j too large – misses a close adjacent source.
NASSP Masters 5003F - Computational Astronomy - 2009
2: Peak finding
Box too small:
Box too large:
NASSP Masters 5003F - Computational Astronomy - 2009
3: Decision time – is it a source or not?
• To calculate a P-value we need the
probability distribution of peaks in the postwindow map of U values (given the null
hypothesis).
• This is not the same as the probability
distribution of the original data values…
• …nor is it even the same as the probability
distribution of U values.
• In fact, little work seems to have been
done on ppeaks. (Though there is quite a lot
on the distribution of extrema – not quite
the same thing.)
NASSP Masters 5003F - Computational Astronomy - 2009
3: The decision
‘Map’ vs ‘peak’ distributions for Gaussian noise.
Black: all pixels
Red: peaks
NASSP Masters 5003F - Computational Astronomy - 2009
3: Cash to the rescue
• First of all, remember that our model m
has p parameters θ = [θ1, θ2,… θp].
• Cash theory – form a ratio between 2
likelihoods:
– The numerator is calculated with all p
parameters fixed at their ‘null hypothesis’
values.
– For the denominator, a subset, q in number, of
the parameters are adjusted to give the
highest likelihood value.
• -2log(this ratio) behaves like χ2 with q
degrees of freedom.
NASSP Masters 5003F - Computational Astronomy - 2009
3: Cash to the rescue
• A practical recipe for applying Cash to
source detection goes as follows:
– Choose a window area surrounding each
peak.
– Within this window, calculate Lnull with model
mi = bi (the background map values).
– Calculate Lbest by fitting a model
mi = bi + θ1 s(ri – θr)
• Degrees of freedom ν = 1 (the amplitude) + d (the
dimensions of the spatial fit).
– The Cash statistic 2(Lbest-Lnull) behaves like χ2
with 1+d deg. free.
NASSP Masters 5003F - Computational Astronomy - 2009
3: Cash to the rescue
• The only difficult point
(which is a problem
for every method) is
to calculate the
fraction of pixels
which are peaks.
– Monte Carlo
– Possibly a Fourier
technique?
• Also, don’t want to
use the fit for final
parameter values. A
Mighell fit is better.
NASSP Masters 5003F - Computational Astronomy - 2009
•
•
•
•
•
•
•
Useful references:
W Press et al, “Numerical Recipes in
Fortran”
P Bevington, “Data reduction and error
analysis for the physical sciences”
W Cash, Ap J 228, 939 (1979)
K J Mighell, Ap J 518, 380 (1999)
I M Stewart, A&A 454, 997 (2006)
I M Stewart, A&A, in print (2009)
Wikipedia
NASSP Masters 5003F - Computational Astronomy - 2009