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

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

$$\bar{y_j} = \frac{1}{N}\sum_{j=0}^{N-1} y_j$$

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

$$\sigma = \sqrt{\frac{1}{N}\sum_{j=0}^{N-1}(y_i - \bar{y})^2}$$

This can also be calculated using:

$$\sigma = \sqrt{\left(\frac{1}{N}\sum_{j=0}^{N-1} x_i^2\right) - \left(\frac{1}{N}\sum_{j=0}^{N-1}y_i\right)^2}$$

$$\chi^2$$-distribution
The $$\chi^2$$-distribution is defined by the number of degrees of freedom, $$k$$. The mean of the distribution is $$k$$ and the variance $$2k$$. For a power-spectrum estimate the distribution of each point is given by a $$\chi^2$$-distribution with 2 degrees-of-freedom (corresponding to an exponential distribution with the rate parameter $$\lambda = 0.5$$):

$$p(x) = \frac{1}{2}e^{-x/2}.$$

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 $$p(x) = e^{-x}$$ which has a mean=1 and variance=1. The 95% confidence limits are 0.025 and 3.67.

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

$$ F_k = \sum_{j=0}^{N-1} y_j \exp\left(-2\pi \sqrt{-1} jk/N\right) $$

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

$$F_k = R_k + i I_k.$$

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

$$\gamma = (N\bar{y^2_j})^{1/2}$$