Pulsars and neutron stars/Statistical and analysis methods for pulsar research

IntroductionEdit

Pulsar searching and timing requires the analysis of time series of data. In this section we present commonly used equations, algorithms, numerical methods, methodologies and routines.

Basic time series analysisEdit

We assume that we have a time series of   samples. Each sample,  , has a time   and its value  . The mean of the values (note that we are starting the element counter from zero):

 

The standard deviation represents the amount of variation in a data set.

 

This can also be calculated using:

 

DistributionsEdit

 -distributionEdit

The  -distribution is defined by the number of degrees of freedom,  . The mean of the distribution is   and the variance  . For a power-spectrum estimate the distribution of each point is given by a  -distribution with 2 degrees-of-freedom (corresponding to an exponential distribution with the rate parameter  ):

 


The mean of this is 2 and the variance is 4. It is common to normalise the distribution so that the mean=1. The normalised chisquare(2) has   which has a mean=1 and variance=1. The 95% confidence limits are 0.025 and 3.67.

Fourier transforms and power spectraEdit

The Discrete Fourier Transform (DFT)Edit

For a regularly sampled time series of values   of N data points, the discrete Fourier transform (DFT) is:

 

(Note that this is the definition that is used in the forward transform for the fftw libraries). Note that the   values are complex:

 

Note that for pulsar searching it is common to normalise all the Fourier coefficients,   by the factor (see Ransom et al. 2012)

 

Least-squaresEdit

Kolmogorov-Smirnov testEdit

Bayesian and frequentist methodologyEdit