Editing Colors of noise (section)

== Identification of power law frequency noise ==
Identifying the dominant noise type in a time series has many applications including clock stability analysis and market forecasting. There are two algorithms based on autocorrelation functions that can identify the dominant noise type in a data set provided the noise type has a power law spectral density. 

=== 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>).

=== Lag(m) autocorrelation method (overlapped) ===
This method improves on the accuracy of the previous method and was introduced by Z. Chunlei, Z. Qi, Y. Shuhuana. Instead of using the lag(1) autocorrelation function the lag(m) correlation function is computed instead:<ref>{{Cite book |last1=Zhou Chunlei |last2=Zhang Qi |last3=Yan Shuhua |chapter=Power law noise identification using the LAG 1 autocorrelation by overlapping samples |date=August 2011 |pages=110–113 |title=IEEE 2011 10th International Conference on Electronic Measurement & Instruments |chapter-url=http://dx.doi.org/10.1109/icemi.2011.6037776 |publisher=IEEE |doi=10.1109/icemi.2011.6037776|isbn=978-1-4244-8158-3 }}</ref>

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

where <math>m</math> is the "lag" or shift between the time series and the delayed version of itself. A major difference is that <math>z_t </math> are now the averaged values of the original time series computed with a moving window average and averaging factor also equal to <math>m</math>. The value of <math>\delta </math> is computed the same way as in the previous method and <math>\delta < .25 </math> is again the criteria for a stationary process. The other major difference between this and the previous method is that the differencing used to make the time series stationary (<math>\delta < .25 </math>) is done between values that are spaced a distance <math>m</math> apart:

<math>z_1 = z_{1+m}-z_1, z_2 = z_{2+m} - z_2..., z_{N-m} = z_N - z_{N-m} </math>

The value of the power is calculated the same as the previous method as well.