Editing Circular error probable (section)

==Conversion==
While 50% is a very common definition for CEP, the circle dimension can be defined for percentages. [[Percentile]]s can be determined by recognizing that the horizontal position error is defined by a 2D vector which components are two orthogonal [[normal distribution|Gaussian]] [[random variable]]s (one for each axis), assumed [[uncorrelated]], each having a standard deviation <math>\sigma</math>. The ''distance error'' is the magnitude of that vector; it is a property of [[multivariate normal distribution|2D Gaussian vectors]] that the magnitude follows the [[Rayleigh distribution]], with scale factor <math>\sigma</math>. The ''distance [[root mean square]]'' (DRMS), is <math>\sigma_d=\sqrt{2}\sigma</math> and doubles as a sort of standard deviation, since errors within this value make up 63% of the sample represented by the bivariate circular distribution. In turn, the properties of the Rayleigh distribution are that its percentile at level <math>F \in [0\%, 100\%]</math> is given by the following formula:

:<math>Q(F, \sigma) = \sigma \sqrt{-2\ln(1 - F/100\%)}</math>

or, expressed in terms of the DRMS:

:<math>Q(F, \sigma_d) = \sigma_d \frac{\sqrt{-2\ln(1 - F/100\%)}}{\sqrt{2}}</math>

The relation between <math>Q</math> and <math>F</math> are given by the following table, where the <math>F</math> values for DRMS and 2DRMS (twice the distance root mean square) are specific to the Rayleigh distribution and are found numerically, while the CEP, R95 (95% radius) and R99.7 (99.7% radius) values are defined based on the [[68–95–99.7 rule]]

{|class="wikitable"
|-
! Measure of <math>Q</math>
! Probability <math>F \, (\%)</math>
|-
| DRMS
| 63.213...
|-
| CEP
| 50
|-
| 2DRMS
| 98.169...
|-
| R95
| 95
|-
| R99.7
| 99.7
|}

We can then derive a conversion table to convert values expressed for one percentile level, to another.<ref name=gps>Frank van Diggelen, "[http://gpsworld.com/gps-accuracy-lies-damn-lies-and-statistics/ GPS Accuracy: Lies, Damn Lies, and Statistics]", ''GPS World'', Vol 9 No. 1, January 1998</ref><ref name="gnss">Frank van Diggelen, "GNSS Accuracy – Lies, Damn Lies and Statistics", ''GPS World'', Vol 18 No. 1, January 2007. Sequel to previous article with similar title [http://www.gpsworld.com/gpsgnss-accuracy-lies-damn-lies-and-statistics-1134] [http://www.frankvandiggelen.com/wp-content/uploads/2009/03/2007-gps-world-accuracy-article-0107-van-diggelen-1.pdf]</ref> Said conversion table, giving the coefficients <math>\alpha</math> to convert <math>X</math> into <math>Y=\alpha.X</math>, is given by:

{|class="wikitable"
|-
! From <math>X \downarrow</math> to <math>Y \rightarrow</math>
! RMS (<math>\sigma</math>)
! CEP
! DRMS
! R95
! 2DRMS
! R99.7
|-
! RMS (<math>\sigma</math>)
| 1.00
| 1.18
| 1.41
| 2.45
| 2.83
| 3.41
|-
! CEP
| 0.849
| 1.00
| 1.20
| 2.08
| 2.40
| 2.90
|-
! DRMS
| 0.707
| 0.833
| 1.00
| 1.73
| 2.00
| 2.41
|-
! R95
| 0.409
| 0.481
| 0.578
| 1.00
| 1.16
| 1.39
|-
! 2DRMS
| 0.354
| 0.416
| 0.500
| 0.865
| 1.00
| 1.21
|-
! R99.7
| 0.293
| 0.345
| 0.415
| 0.718
| 0.830
| 1.00
|}

For example, a GPS receiver having a 1.25&nbsp;m DRMS will have a 1.25&nbsp;m × 1.73 = 2.16&nbsp;m 95% radius.