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LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Object detection for LOFAR

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Outline

1

LOFAR Object Detection Bayesian Framework Nested Sampling LOFAR peculiarities... The Likelihood The Prior Ratio

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Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The detection of discrete objects in a diffuse background is a generic problem in astrophysics. Problems arise when the background emission varies on similar scale lengths to the objects themselves ...and when the level of instrumental noise is larger than or comparable to the amplitude of the object.
Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Bayesian Framework

Outline

1

LOFAR Object Detection Bayesian Framework Nested Sampling LOFAR peculiarities... The Likelihood The Prior Ratio

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3

4

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Bayesian Framework

Bayes' Formula: Pr(|D , H ) = Pr(D |, H )Pr(|H ) Pr(D |H ) L() () Z

P () =

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Bayesian Framework

Bayes' Formula: Pr(|D , H ) = Pr(D |, H )Pr(|H ) Pr(D |H ) L() () Z

P () =

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Bayesian Framework

Historically, MCMC sampling algorithms such as Metropolis-Hastings have been used to calculate the posterior distribution. Obtaining the evidence has been done as a complicated piece of extra work using that posterior combined with the prior. This meant that evidence calculation was a long and arduous process Getting an error on your evidence required multiple runs of the process

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Bayesian Framework

The Evidence: Z= The model selection ratio: R= Pr(H1 |D ) Pr(D |H1 )Pr(H1 ) Z1 Pr(H1 ) = = Pr(H2 |D ) Pr(D |H1 )Pr(H1 ) Z2 Pr(H2 ) L() ()d D

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Nested Sampling
.. Nested sampling reparameterizes the problem of computing the evidence. Z = LdX , where dX = ()d Much faster than conventional sampling. Computes the evidence first, then the posterior as a by-product.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

Nested Sampling
..

Star t with N points drawn from the prior space Find the lowest likelihood value, record it and then replace it with a new point, i , drawn with the constraint Li > Li -1 ...and repeat

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Outline

1

LOFAR Object Detection Bayesian Framework Nested Sampling LOFAR peculiarities... The Likelihood The Prior Ratio

2

3

4

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

LOFAR Likelihood

Most object detection problems have relatively simple likelihoods: either Gaussian or Poisson. In the LOFAR RM data cubes we are dealing with polarized intensity. P = Q2 + U 2 D=
2 2 (A + N (0, 1 ))2 + (B + N (0, 2 ))2

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

If we make the assumptions that 1 = 2 = , and that A = A0 sin(), B = A0 cos() then we find that the probability density function becomes: P(A0 |D ) =
D I 2 0 DA0 2

e

-

(D 2 +A2 ) 0 2 2

= RICIAN distribution

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

If we make the assumptions that 1 = 2 = , and that A = A0 sin(), B = A0 cos() then we find that the probability density function becomes: P(A0 |D ) =
D I 2 0 DA0 2

e

-

(D 2 +A2 ) 0 2 2

= RICIAN distribution

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

As A0 / grows large the Rician distribution tends to a Gaussian distribution. We need not compute the posterior over the whole data cube. We may cut our model off at a given distance and set it to zero.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

As A0 / grows large the Rician distribution tends to a Gaussian distribution. We need not compute the posterior over the whole data cube. We may cut our model off at a given distance and set it to zero.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

As A0 / grows large the Rician distribution tends to a Gaussian distribution. We need not compute the posterior over the whole data cube. We may cut our model off at a given distance and set it to zero.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

As A0 / grows large the Rician distribution tends to a Gaussian distribution. We need not compute the posterior over the whole data cube. We may cut our model off at a given distance and set it to zero.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

As A0 / grows large the Rician distribution tends to a Gaussian distribution. We need not compute the posterior over the whole data cube. We may cut our model off at a given distance and set it to zero.

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Pre-filtering

Assuming some model for our sources, say A t (x - X ; R ). We can substitute this into the log-likelihood (assuming a constant prior) and differentiate: ln P -1 t - At T N -1 t A = DN This gives us an analytic estimate for amplitude: T N -1 ^ A = DT N -1 tt t The correponding log-likelihood is: ^ ^ ln L(X , A, R ) = c + ln[I0 [A2 t T N
-1

^ t ]] - At T N

-1

t

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Outline

1

LOFAR Object Detection Bayesian Framework Nested Sampling LOFAR peculiarities... The Likelihood The Prior Ratio

2

3

4

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Object detection quantification
In most model comparison the prior ratio can be assumed to equal one. For object detection this is not the case. For each possible object in our field we are comparing two hypotheses which have different a priori probabilities. For Instance: Let's say our objects are randomly distributed in space. The probability of there then being N objects is Poisson distributed: -µ µN Pr(N |µ) = e N !

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

For Instance (2)... Our choices of model: H1 = `no object is present' H2 = `at least one object is present' Using our Poisson probability:
H1 H2

= exp µ - 1 µ for µ

1

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Now we can calculate our prior ratio we can calculate the model selection ratio, R . This gives us the relative probabilities of our detection being a true object detection as opposed to a false one. We can then define a threshold probability, pi , such that candidate objects whose probability lies below that threshold are rejected. This probability is defined as pi =
Ri 1+Ri

This probability then allows us to quantify the number of false positives detected in our data, as: < nFP >=
N i =1,pi pth

1 - pi

Object detection for LOFAR


LOFAR Object Detection Nested Sampling LOFAR peculiarities...

The Likelihood The Prior Ratio

Conclusions

Large data cubes will need a fast object detection algorithm Nested sampling provides an efficient method for doing this Using the evidence, Z, also provides a way of determining the probability of each object being a true detection.

Object detection for LOFAR