Editing Colors of noise (section)

=== Lag(1) autocorrelation method (non-overlapped) ===
The first method for doing noise identification is based on a paper by W.J Riley and C.A Greenhall.<ref>{{Cite book |last1=Riley |first1=W.J. |last2=Greenhal |first2=C.A. |title=18th European Frequency and Time Forum (EFTF 2004) |chapter=Power law noise identification using the lag 1 autocorrelation |date=2004 |url=https://digital-library.theiet.org/content/conferences/10.1049/cp_20040932 |language=en |publisher=IEE |pages=576–580 |doi=10.1049/cp:20040932 |isbn=978-0-86341-384-1}}</ref> First the lag(1) autocorrelation function is computed and checked to see if it is less than one third (which is the threshold for a stationary process):

<math>R_1 = \frac{\frac{1}{N}\sum_{t=1}^{N-1}(z_t - \bar z)*(z_{t+1} - \bar z)}
{\frac{1}{N}\sum_{t=1}^{N}{(z_t - \bar z)}^{2}}</math>

where <math>N </math> is the number of data points in the time series, <math>z_t </math> are the phase or frequency values, and <math>\bar z </math> is the average value of the time series. If used for clock stability analysis, the <math>z_t </math> values are the non-overlapped (or binned) averages of the original frequency or phase array for some averaging time and factor. Now discrete-time fractionally integrated noises have power spectral densities of the form <math>(2sin(\pi f))^{-2\delta} </math> which are stationary for <math>\delta < .25 </math>. The value of <math>\delta </math> is calculated using <math>R_1 </math>:

<math>\delta = \frac{R_1}{1+R_1} </math>

where <math>R_1 </math> is the lag(1) autocorrelation function defined above. If <math>\delta > .25 </math> then the first differences of the adjacent time series data are taken <math>d </math> times until <math>\delta < .25 </math>. The power law for the stationary noise process is calculated from the calculated <math>\delta </math> and the number of times the data has been differenced to achieve <math>\delta < .25 </math> as follows:

<math>p = -2(\delta + d)  </math>

where <math>p </math> is the power of the frequency noise which can be rounded to identify the dominant noise type (for frequency data <math>p </math> is the power of the frequency noise but for phase data the power of the frequency noise is <math>p+2 </math>).