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Astronomical Data Analysis > Definitions and Explanations > Flashcards

Definitions and Explanations Flashcards

1
Q

poisson statistics

A

the number of photons arriving at our detector from a given source will fluctuate

treat the arrival rate of photons statistically

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2
Q

poisson statistics assumptions

A
  1. Photons arrive independently in time
  2. Average photon arrival rate is constant
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3
Q

as Rτ increases

A

the shape of the Poisson distribution becomes more symmetrical

tends to a normal or gaussian

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4
Q

variance

A

is a measure of the spread in the Poisson distribution

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5
Q

Laplace’s basis for plausible reasoning

A

Probability measures our degree of belief that something is true

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6
Q

probability density function

A

when measuring continuous variables which can take on infinitely many possible values

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7
Q

sketch of a poisson PDF

A

see notes

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8
Q

sketch of a uniform PDF

A

see notes

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9
Q

sketch of a central/normal or gaussian pdf

A

see notes

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10
Q

sketch of a cumulative distribution function (CDF)

A

see notes

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11
Q

the 1st moment is called

A

the mean or expectation value

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12
Q

the 2nd moment is called

A

the mean square

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13
Q

the median divides

A

the CDF into two equal halves

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14
Q

the mode is

A

the value of x for which the pdf is a maximum

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15
Q

central limit theorem

A

explains the importance of a normal pdf in statistics

but still based on the asymptotic behaviour of an infinite ensemble of samples that we didn’t actually observe

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16
Q

Isoprobability contours for the bivariate normal pdf

A

p > 0 : positive correlation y tends to increase as x increases

p < 0 : negative correlation y tends to decrease as x increases

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17
Q

as |p| -> 1

A

contours become narrower and steeper

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18
Q

the principle of maximum likelihood

A

is a method to estimate the parameters of a distribution which fit the observed data

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19
Q

if we obtain a very small P-value

A

we can interpret this as providing little support for the null hypothesis,

which we may then choose to reject

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20
Q

Monte-carlo methods

A

method for generating random variables

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21
Q

can test psuedo-random numbers for randomness in several ways

A

a) histogram of sampled values
b) correlations between neighbouring pseudo-random numbers
c) autocorrelation
d) chi squared

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22
Q

Markov Chain Monte Carlo

A

method for sampling from PDFs

  1. start off at some randomly chosen value (a(1),b(1))
  2. compute L(a(1),b(1)) and gradient
  3. Move in direction of steepest +ve gradient
  4. repeat from step 2 until (a(n),b(n)) converges on maximum likelihood
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23
Q

MCMC provides

A

a simple metropolis algorithm for generating random samples of points from L(a,b)

24
Q

MCMC

A
  1. sample random initial point P(1) = (a(1),b(1))
  2. Centre a new pdf, Q, called the proposal density, on P(1)
  3. Sample tentative new point P’=(a’,b’) from Q
  4. Compute R = L(a’,b’)/L(a(1),b(1))

see notes for diagram

if R > 1 : P’ is uphill we accept P’
if R < 1 : P’ is downhill we may reject P’

25
Q

whether we accept R < 1

A

generate a random number x ~ U[0,1]
if x < R then accept P’
if x > R then reject P’

26
Q

correlation theorem

A

the FT of the first time domain function, multiplied by the complex conjugate of the FT of the second time domain function is equal to the FT of their correlation

27
Q

the broader the gaussian in the time domain

A

the narrower the gaussian in the frequency domain

28
Q

sketch the probability density function of the CDF

A

should be a normal distribution with two probability density functions

29
Q

how a CDF of a random variable can be used to generate a random sample

A

to generate a sample for p(x), sample y from U[0,1], a unitary number in range 0 -> 1

compute x = P^-1(y) or y = P(x)

Then x~p(x)

see notes for graph

30
Q

how is a maximum likelihood constructed

A
  1. first consider the best model which fits the data
  2. a visual inspection can be used to see if its a uniform, normal…
  3. then calculate the likelihood for the chosen distribution
  4. the individual likelihoods are then multiplied and a minimisation is used to ‘maximise’ the likelihood.
31
Q

quantisation noise likelihood function

A

uniform distirubtion

32
Q

(x(i) - µ)

A

is the residual used in the least squares problem

33
Q

histogram of sampled values

A

will show that all values in the interval are equally likely to occur and the histogram should be flat

34
Q

correlations between neighbouring pseudo-random numbers

A

plotting x(i) versus x(i+1) the data should be randomly scattered and show no pattern

35
Q

autocorrelation

A

the autocorrelation should be unity for zero lag, and zero for all other values

36
Q

chi-squared test

A

a confidence limit of p>0.05 will show whether the hypothesis is believable

37
Q

statistics that describe noisy data

A

noise comes from random distributions i.e. uniform or normal distribution

38
Q

the upper frequency is given by

A

the Nyquist-Shannon sampling theorem

39
Q

the lower frequency is given by

A

the lower bound is set by the total data length

40
Q

low pass filter

A

from Nyquist-Shannon sampling theorem a low pass filter will generate a sinc function in the time domain

when converting to the frequency domain, thus will just become a top-hat function

this is the representation of our ideal filter.

41
Q

low pass filter sketch

A

see notes

42
Q

why is a low pass filter important

A

to reduce the noise
to remove the problem of aliasing

43
Q

sketch of no correlation

A

see notes

44
Q

sketch of positive correlation

A

see notes

45
Q

sketch of negative correlation

A

see notes

46
Q

central limit theorem

A

for any pdf with finite variance σ^2 , as M -> ∞

µ(hat) follows a normal pdf with mean µ and variance σ^2 / M

47
Q

probability density function sketch

A

see notes

48
Q

means for poisson

A

number of photons/second counted by a CCD

number of galaxies/degree^2

49
Q

if correlation coefficient = 0

A

then x and y are independent

50
Q

the residuals are equally likely

A

to be positive or negative and all have equal variance

51
Q

weighted least squares

A

makes good use of small data sets

52
Q

ordinary least and weighted lest squares plot

A

see notes

53
Q

chi 2 used when

A

we know there are definite outcomes

no errors on measurement

54
Q

reduced chi 2 used when

A

we know there is uncertainty or variance in a measured quantity

errors on measurement

55
Q

reduced chi 2 degrees of freedom

A

are the number of data points

Astronomical Data Analysis (12 decks)

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