Pulsars and neutron stars/Statistical and analysis methods for pulsar research
Introduction[edit | edit source]
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 analysis[edit | edit source]
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:
Distributions[edit | edit source]
-distribution[edit | edit source]
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 spectra[edit | edit source]
The Discrete Fourier Transform (DFT)[edit | edit source]
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)