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

## Introduction[edit]

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]

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]

### -distribution[edit]

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]

### 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)