Definitions and Explanations Flashcards
poisson statistics
the number of photons arriving at our detector from a given source will fluctuate
treat the arrival rate of photons statistically
poisson statistics assumptions
- Photons arrive independently in time
- Average photon arrival rate is constant
as Rτ increases
the shape of the Poisson distribution becomes more symmetrical
tends to a normal or gaussian
variance
is a measure of the spread in the Poisson distribution
Laplace’s basis for plausible reasoning
Probability measures our degree of belief that something is true
probability density function
when measuring continuous variables which can take on infinitely many possible values
sketch of a poisson PDF
see notes
sketch of a uniform PDF
see notes
sketch of a central/normal or gaussian pdf
see notes
sketch of a cumulative distribution function (CDF)
see notes
the 1st moment is called
the mean or expectation value
the 2nd moment is called
the mean square
the median divides
the CDF into two equal halves
the mode is
the value of x for which the pdf is a maximum
central limit theorem
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
Isoprobability contours for the bivariate normal pdf
p > 0 : positive correlation y tends to increase as x increases
p < 0 : negative correlation y tends to decrease as x increases
as |p| -> 1
contours become narrower and steeper
the principle of maximum likelihood
is a method to estimate the parameters of a distribution which fit the observed data
if we obtain a very small P-value
we can interpret this as providing little support for the null hypothesis,
which we may then choose to reject
Monte-carlo methods
method for generating random variables
can test psuedo-random numbers for randomness in several ways
a) histogram of sampled values
b) correlations between neighbouring pseudo-random numbers
c) autocorrelation
d) chi squared
Markov Chain Monte Carlo
method for sampling from PDFs
- start off at some randomly chosen value (a(1),b(1))
- compute L(a(1),b(1)) and gradient
- Move in direction of steepest +ve gradient
- repeat from step 2 until (a(n),b(n)) converges on maximum likelihood
MCMC provides
a simple metropolis algorithm for generating random samples of points from L(a,b)
MCMC
- sample random initial point P(1) = (a(1),b(1))
- Centre a new pdf, Q, called the proposal density, on P(1)
- Sample tentative new point P’=(a’,b’) from Q
- 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’
whether we accept R < 1
generate a random number x ~ U[0,1]
if x < R then accept P’
if x > R then reject P’
correlation theorem
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
the broader the gaussian in the time domain
the narrower the gaussian in the frequency domain
sketch the probability density function of the CDF
should be a normal distribution with two probability density functions
how a CDF of a random variable can be used to generate a random sample
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
how is a maximum likelihood constructed
- first consider the best model which fits the data
- a visual inspection can be used to see if its a uniform, normal…
- then calculate the likelihood for the chosen distribution
- the individual likelihoods are then multiplied and a minimisation is used to ‘maximise’ the likelihood.
quantisation noise likelihood function
uniform distirubtion
(x(i) - µ)
is the residual used in the least squares problem
histogram of sampled values
will show that all values in the interval are equally likely to occur and the histogram should be flat
correlations between neighbouring pseudo-random numbers
plotting x(i) versus x(i+1) the data should be randomly scattered and show no pattern
autocorrelation
the autocorrelation should be unity for zero lag, and zero for all other values
chi-squared test
a confidence limit of p>0.05 will show whether the hypothesis is believable
statistics that describe noisy data
noise comes from random distributions i.e. uniform or normal distribution
the upper frequency is given by
the Nyquist-Shannon sampling theorem
the lower frequency is given by
the lower bound is set by the total data length
low pass filter
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.
low pass filter sketch
see notes
why is a low pass filter important
to reduce the noise
to remove the problem of aliasing
sketch of no correlation
see notes
sketch of positive correlation
see notes
sketch of negative correlation
see notes
central limit theorem
for any pdf with finite variance σ^2 , as M -> ∞
µ(hat) follows a normal pdf with mean µ and variance σ^2 / M
probability density function sketch
see notes
means for poisson
number of photons/second counted by a CCD
number of galaxies/degree^2
if correlation coefficient = 0
then x and y are independent
the residuals are equally likely
to be positive or negative and all have equal variance
weighted least squares
makes good use of small data sets
ordinary least and weighted lest squares plot
see notes
chi 2 used when
we know there are definite outcomes
no errors on measurement
reduced chi 2 used when
we know there is uncertainty or variance in a measured quantity
errors on measurement
reduced chi 2 degrees of freedom
are the number of data points
