NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 3 First, a bit more python. Then some...

NASSP Masters 5003F - Computational Astronomy - 2009

Lecture 3

• First, a bit more python.

• Then some noise statistics.


Python ins and outs• We’re going to mostly read our data from

FITS files, using a module called pyfits.– http://www.stsci.edu/resources/

software_hardware/pyfits

• We’ll crunch up the data using a module called numpy.– http://numpy.scipy.org/

• For graphical output we’ll use module ppgplot, (2 ps) which is a (fairly crude) wrapper to a package called PGPLOT.– http://efault.net/npat/hacks/ppgplot/– http://www.astro.caltech.edu/~tjp/pgplot/BREAKING NEWS! pylab is better for graphical output.


Manuals and example code

• You won’t need to download manuals for pyfits, numpy, scipy or pylab. I’ll do so and make them available from my home page. It’ll be much quicker for you to copy (or simply access) them from there.

• Some of these manuals are huge – I recommend you neither print or read them in entirety, but rather:– Read the bits you need using acroread.– Look at the example code I’ll provide you.


Random variables – probability density

x

p(x)Average μ:

Variance σ2:

Units: probabilityper unit x.

xpxdxx



x

p(x)Estimate of μ:

Estimate of σ2:

N

iixN 1

22 ˆ1

1ˆ



x

p(x)



x

p(x)

x0


p(x)

Noise• 2 most important distributions:

– Gaussian• …but not necessarily ‘white’

2

2

2exp

2

1,

xxp


Noise– Poisson

(= i)

…but note Central Limit theorem.


Gauss and Poisson


Combinations of random variables

Weighted average:

Uncertainty propagation: if y = f(a,b),

Uncertainty in this:

Best SNR when:

where:


Filtering and correlation

Uncorrelated ie ‘white’ noise (with a Gaussian probability distribution.)



Fourier transformed – looks the same – same power at all f – hence ‘white’.



Autocorrelation function.

Power spectrum:

where F is the Fourier transform of signal f, eg:

The autocorrelation function is:



Filtering operation correlates the noise. It is still Gaussian but no longer white.



FT is dominated in this case by low frequencies.



Autocorrelation function is broadened.


Signals and noise

Natural signal

+ background

Instrumentalfiltering/correlation

General:

+ uncorrelatednoise

+ uncorrelatednoise

Natural signal

+ background


XMM-Newton:

+ uncorrelatednoise

+ uncorrelatednoise

Natural signal

+ background


Interferometry:

+ uncorrelatednoise

+ uncorrelatednoise


Signal detection

Much harder.An obvious source.


Signal detection• Parent function; data; model.

• Probability that the data results from a certain model:

• Reduced chi2: divide by deg free.

• Recall that Psignal = 1-Pno signal.

• Many types of signal, but only 1 no-signal.

• Hence test model with no source.

• Called the ‘null hypothesis’.

N

i i

ii my

12

22


Signal detection

• Null hypothesis requires:– Perfect knowledge of background B.– A good estimate of σ.

• Sparse sources among gaussian: ok.

• But…– what about poisson bins with zero counts?

• Answer: Maximum Likelihood.

– what about crowded fields?• Answer: Bayes..?

– can we trust the distribution?• Answer: Monte Carlo.

NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 3 First, a bit more python. Then some...

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