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Statistics for Astronomy VII:
B. Nikolic b.nikolic@mrao.cam.ac.uk
Astrophysics Group, Cavendish Laboratory, University of Cambridge http://www.mrao.cam.ac.uk/~bn204/lecture/astrostats.html

12 November 2008


Goals for this Lecture

Measuring Pressure-Broadened Lines

P(D) analysis


Outline

Measuring Pressure-Broadened Lines

P(D) analysis


Problem Recap
Pressure variation
250 250 200 200

Temperature variation

150 Tb (K) Tb (K) 100

150

100

50

50

0 175 177.5 180 182.5 (GHz) 185 187.5 190

0 175 177.5 180 182.5 (GHz) 185 187.5 190

Amount of Water
250
250

Filters
200

200

150 Tb (K)
Tb (K)

150

100

100

50

50

0 175 177.5 180 182.5 (GHz) 185 187.5 190

0 175 177.5 180 182.5 (GHz) 185 187.5 190


Sources of errors
Devices to make absolute measurements of brightness in the radio par t of the spectrum are called radiometers.


The intrinsic noise of radio detectors is given by: Tsys Trec + T T = = Bt Bt
src

(1)



for wide-band devices this can be ver y low (e.g., mK range) The measurement described here is however primarily limited by calibration errors, arising from:




Uncer tainty in coupling between the receiver and the sky ( length-scales in optical system so diffraction) Standing waves Errors in the internal calibration system

These are unlikely to be Normally distributed!


Likelihood calculation



Calibration errors not normally distributed but unknown until device design is finalised (and possibly not thereafter either) It is unlikely, however, the errors will have fat tails normal distribution not a poor first approximation
4



log P (x | ) = - where 1 K

i =1

[xi - Gi ( )]2 2 2

(2)


Example of a MCMC Chain
Calculated using the Metropolis-Hastings algorithm

n 1.00014321 1.00474235 1.0037564 ··· 0.99196826 0.99196355 0.99675269

T 270.1529 270.2211 270.2000 ··· 259.6134 259.6944 259.6910

7431 8522 7779 8244 2168 0638

P 599.8585 599.8214 599.6376 ··· 564.3115 564.2889 564.3103

2731 1295 5512 5122 6232 8744


Example of a MCMC Chain
Calculated using the Metropolis-Hastings algorithm

Illustration of a chain that star ts far from the high-posterior region:
550 545 275 540 P (mbar) 535 530 265 525 520 0.75 260 0.75 T (K) 1.25 c (mm) 1.5 1.75 270 280

1

2

1

1.25 c (mm)

1.5

1.75

2


Posterior distribution with no prior information
n
340

T
8 · 10
4

P
800 8 · 10
4

0.02
320 6 · 10 750
4

0.015
T (K) 300 4 · 10 280
4

0.01

P (mbar)

700

6 · 10

4

650

4 · 10

4

f

0.005
260

2 · 10

4

600

2 · 10

4

n

0 0.975

0

550 0 1 1 1.025 1.05 c (mm) 800 1.075 1.1

0 0 1

1

1.025

1.05 c (mm)

1.075

1.1

1.125

1

1.025

1.05 c (mm)

1.075

1.1

0.025
750

1 · 10

5

0.02
P (mbar) 700 7.5 · 10
4

0.015 f

650

0.01
600

5 · 10

4

0.005
550

2.5 · 10

4

T

0 240 260 280 300 T (K) 320 340 360

0 260 280 300 T (K) 320 340 0 1

0.04

0.03

0.02

f

0.01

P

0 500 600 700 P (mbar) 800 900


Posterior with priors

Priors: P (n) = P (T ) = P (p ) = c1 0 c2 0 c3 0 if 0 n 5 mm otherwise if 260 T 280 K otherwise if 570 p 610 mB otherwise (3) (4) (5)


Posterior with priors
n
0.04
277.5 3 · 10 275
4

T
605 600 595
2 · 104 272.5

P
2 · 10
4

0.03
T (K)

1.5 · 10

4

P (mbar)

270 267.5

590 585 1 · 10
4

0.02

f

0.01

265 262.5

1 · 104

580 575

5 · 10

3

n

0 0.97

0

0
0 1

0.98

0.99

1 c (mm)

1.01

1.02

1.03

0.98

0.99

1 c (mm)

1.01

1.02

0.98

0.99

1 c (mm)

1.01

1.02

0

1

0.025

605 600

3 · 10

4

0.02
595 P (mbar)

2.5 · 10 2 · 10

4

4

0.015 f

590 1.5 · 10 585 580 1 · 10 5 · 10

4

0.01

4

0.005
575

3

T

0 260 265 270 T (K)
0.015 0.0125 0.01 0.0075 0.005 0.0025 0 570 580 590 P (mbar) 600 f

0

275

280

265

270 T (K)

275

0

1

P

610


Comparison: Prior Vs. No Prior
No Prior
800 8 · 10
4

Prior
605 600 6 · 10
4

2 · 104

750

P (mbar)

P (mbar)

700

595 590 585 580 575

1.5 · 104

650

4 · 10

4

1 · 104

600

2 · 10

4

5 · 103

550 1 1.025 1.05 c (mm) 1.075 1.1

0 0 1 0.98 0.99 1 c (mm) 1.01 1.02

0 0 1



Complex posterior distribution High correlation between pressure and water a mo u n t



Much smaller range on the axes Pressure range controlled by the prior Low correlation between pressure and water a mo u n t








Comparison: Flat Vs. Log-Flat Prior
n
0.04
275 272.5

T
8000

P
605 600
6000

6000

0.03
270 T (K)

595 P (mbar) 590 585
265

4000

0.02

267.5

f

4000

0.01
262.5

2000

580 575

2000

n

0 0.97

0

0
0 1

0.98

0.99 c (mm)

1

1.01

1.02

0.98

0.99

1 c (mm)

1.01

0.98

0.99

1 c (mm)

1.01

0

1

0.025

605 600

1 · 10 8 · 10 6 · 10 4 · 10 2 · 10

4

3

0.02
595 P (mbar)

3

0.015 f

590 585 580

0.01

3

0.005
575

3

T

0 260 265 270 T (K) 0.025 275 280
265 T (K) 270 275

0 0 1

0.02

0.015 f 0.01 0.005

P

0 570 580 590 P (mbar) 600 610


Comparison: Flat Vs. Log-Flat Prior

Flat
0.04 0.03

Log-Flat
0.04 0.03

0.01

f 0.98 0.99 1 c (mm) 1.01 1.02 1.03

0.02

f

0.02

0.01

0 0.97

0 0.97

0.98

0.99 c (mm)

1

1.01

1.02


Marginalisation
605 600 595 P (mbar) 590 585 580 575 0 0.98 0.99 1 c (mm) 1.01 1.02 0 1 1 · 10
4

2 · 10

4



1.5 · 10

4

A large par t of the parameter space is allowed However marginalisation to produce just P (n), shows amount of water is ver y well constrained, with almost normal-like errors


3

5 · 10

0.04

0.03

0.02

f

0.01

0 0.97

0.98

0.99

1 c (mm)

1.01

1.02

1.03


Marginalisation
605 600 595 P (mbar) 590 585 580 575 0 0.98 0.99 1 c (mm) 1.01 1.02 0 1 1 · 10
4

2 · 10

4



1.5 · 10

4

A large par t of the parameter space is allowed However marginalisation to produce just P (n), shows amount of water is ver y well constrained, with almost normal-like errors P (p ) poorly constrained but that is not significant for this problem


3

5 · 10

0.015 0.0125 0.01 0.0075 0.005 0.0025 0 570 580 590 P (mbar) 600 610 f




Outline

Measuring Pressure-Broadened Lines

P(D) analysis


P(D) analysis
See P. A. G. Sheuer (1957, 1974) and J. J Condon (1974)




Consider a sur vey of an area of the sky: some sources are detected What does the par t of the area in which no sources are detected tell you?


P (D ) probability of deflection (strip-char t terminology?)

Obviously nothing about individual sources But, may give information about the statistics of undetected sources



Key idea: the observed fluctuations are due to a combination of system noise and random number of weak sources combining within the `beam' ( point-source response) of the telescope


If the weak sources dominate, the observation is `confusion limited'


Source statistics
Euclidean universe: hypothesise homogeneous universe with sources per unit volume, all of same strength P . How many sources do we see [N (S )] up to a flux limit S ? N = V 4 V = R 3 3 P S= 4 R 2 P 3/2 N (> S ) 3/2 S P 3/2 N (S )dS 5/2 S (3) (4) (5) (6) (7)


Confusion noise illustration
Surface statistics beam 20px

10
1000 1.5 · 10
-3

1000
1000 0.006 1.25 · 10
-3

800

800

0.005

600 signal y

1 · 10

-3

0.004 600 signal 400 200 0.001 0 0 1 0 200 400 x 600 800 1000 y 0.003

7.5 · 10

-4

400 5 · 10 200 2.5 · 10
-4

0.002

-4

0 0 200 400 x 600 800 1000

0

0 0 1

10
1000 800

4
1000 0.02 800 0.015

10

5
0.12

0.1

600 signal y y

600 signal 400

0.08

0.01 400

0.06

200

0.005

200

0.04

0 0 200 400 x 600 800 1000 0 1

0 0 200 400 x 600 800 1000

0.02 0 1


Confusion noise illustration II
Euclidean statistics, beam 20px

10
1000 1000 0.04 800 800

1000
1.5 1.25

600 signal y

0.03 y

1 600 signal 400 0.5 200 0.25 0 0 1 0 200 400 x 600 800 1000 0 1 0.75

400

0.02

200

0.01

0 0 200 400 x 600 800 1000

0

10
1000 800

4
2 1000 1.75 800 1.5

10

5

4

1.25 600 signal 1 y y 600 signal 400

3

400

0.75

2

0.5 200 0.25 200 1

0 0 200 400 x 600 800 1000 0 1

0 0 200 400 x 600 800 1000 0 1


P(D) histograms

20px beam
4 · 104 4 · 104 3 · 104 3 · 104

40px beam

2 · 104

2 · 104

N

1 · 104

N

1 · 104

0 -1000

0

1000 Flux

2000

3000

0 -5000

-2500

0 Flux

2500

5000


Approximate analysis I


X is uniformly distributed over interval [0, ] = P (x )dx = const



What is the distribution of Y = X -1/ ? Use conservation of probability under change of variables: |P (y )dy | = |P (x )dx | P (y ) = This implies P (y )dy y and, P (> y ) y -
- -1

(8) (9)

dx P (x ) dy



Can turn uniform numbers into numbers distributed according to a power law


Approximate analysis II


The above allows integration of `typical' sources within a beam Consider 1, 2, · · · , N : they are uniformly distributed in the range 0 to N = 1-1/ , 2-1/ , · · · , N -1/ are distributed according to population which has a cumulative distribution propor tional to P (S ) S - The expected flux in a beam is then ver y approximately:
-1/





S

x
1

dx

(10)

[ Note, if > 1 and the upper limit were really infinity this would diverge ­ Olber's paradox]


Approximate analysis III


What is the variance of S , i.e., the width of histograms? Typical sources in a beam form sequence: 1-1/ , 2-1/ , 3-1/ · · · , N -1/ Any par ticular beam could have one, zero or two of each... Think of the final flux as a random walk of steps: 1-1/ , 2-1/ , 3-1/ · · · , N -1/ The variance is the sum of squares of the steps: Var F 1
-2/





+2
-2

-2/

+3

-2/

+ ··· + N

-2/

(11) (12)

x
1

This integral converges to 3 for = 3/2 and to a number of order unity for most feasible


= Width of the histogram is of order of the strength of the strongest source that a typical beam has


Sketch of the exact analysis
Characteristic functions... If all sources were same strength (Poisson Statistics), then characteristic function of P(D) would be: d ( ) = exp e Recall if Y =
k i

-1

(13)

ak Xk then: Y (t ) =
k

Xk (ak t )

(14)

ak source strength. Xk all same except for which governed by the N(S) distribution. Come to result: d ( ) = exp [n ( ) - n (0)] Derive n from, for example, N S transform to get P(D).
-

(15)

, then inverse Fourier


Radio counts at 1.4 GHz

[Figure by J. J. Condon]