Transcript ppt

Lecture 8
• Source detection
NASSP Masters 5003S - Computational Astronomy - 2009
Different sorts of model
All models
Background + signal
Background + many
similar signals
NASSP Masters 5003S - Computational Astronomy - 2009
b + s vs b + Σsi
mr   br, θ b   Asrr0 
mr   br, θb   i Ai srr0,i 
s may often be assumed to be:
-slowly varying with r;
-with compact support
NASSP Masters 5003S - Computational Astronomy - 2009
Source detection
• The basic idea is related to Null
Hypothesis testing…
– But if the sources can be assumed to be
localized, we can cut the data up and test
each source-sized bit at a time.
Survival function
• sliding window.
Some missed jargon:
the probability at the intercept
is called the P-value (you can
google it)
NASSP Masters 5003S - Computational Astronomy - 2009
Testing the NH:
• Not all tests are equally good at finding
signals!
• Eg Cash statistic is better than χ2 (in
circumstances where the Cash test is
appropriate – eg bkg is a subset of the
signal model).
• Cash stat makes use of knowledge about
the signal shape – in general any stat which
does similar (eg a matched filter) will also
perform well.
• There is an infinite variety of ‘statistics’ to
choose from.
NASSP Masters 5003S - Computational Astronomy - 2009
Source detection.
– If the SF probability in each patch (the Pvalue) is smaller than a previously chosen
cutoff, we can 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 the
separate data! (Don’t always have the luxury.)
• 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?
• Applying some sort of NH test in a sliding window
will return a new random signal – now correlated..
• Finding the sources consists rather of looking for
peaks in this random signal.
• The simplest example is when the noise is
uncorrelated and the source peaks have width=0.
NASSP Masters 5003F - Computational Astronomy - 2009
Looking for sources 1 channel at a time:
• In each channel, we test the NH with N=1.
– Since there are no fitted parameters, υ=1
also.
– If the source occupies a single channel, this
procedure is optimal.
– If, however, the source is spread over several
channels (as is usual), this procedure is not
efficient.
– We want a statistic which uses the maximum
amount of information about the source
shape.
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
• How best to do it?
• There’s 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.
• This probably looks familiar to you.
• 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
• 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.
From my 2009 Cash paper.
NASSP Masters 5003F - Computational Astronomy - 2009
What is the best detection method?
From my 2009 Cash paper.
NASSP Masters 5003F - Computational Astronomy - 2009