PROBABILITY CONTENT OF ERROR ELLIPSE AND ERROR CONTOUR (navell-08.mcd)

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1 PROBABILITY CONTENT OF ERROR ELLIPSE AND ERROR CONTOUR (navell-8.mcd) 6.. Conditions of Use, Disclaimer This document contains scientific work. I cannot exclude, that the algorithm or the calculations contain errors. I expressly exclude any form of liability for dammage which might arrise in practical applications by the use of this algorithm. The content of this document may only be used for private, non-commercial purposes. Ingo Harre, Bremen, January /December 6 Ingo.Harre@mar-it.de NAVELL This Mathcad sheet belongs to the NavGen suite of analysis tools for the determination of the accuracy of position determination. It provides results of numerical integration of the two-dimensional probability density function of position fixes over certain error contours. Input data are the probability density functions in the two orthogonal axes with assumed Gaussian shape, characterised by the parameters σ x and σ y ( left figure above). The pair of orthogonal error distributions determines the 3D CDF, shown in the right figure. In this document the D PDF is integrated over specific error contours, i.e. error ellipses ( left figure) and error circles, yielding the 'confidence content' or 'error probability' related to the error contour under examination. For each error contour to be examined, five different ratios of the underlying standard deviations σ y /σ x are applied to show the variation of confidence content with the ellipticity of the D error distribution. Input Standard Deviations Standard deviations of σ x : k :.5 Ratio of SDs position coordinates σ yi : k σ i x.75..5

2 Genuine Error Ellipse Semi major axis σ x : Semi minor axis σ y..5. related drms error drms : σ i x + σ yi drms ( ) a. Integration Contour: Single Standard Deviations Error Ellipse (The integration contour is the same as the genuine error ellipse) Semi axes a : σ i x b : σ i yi Case displayed i : 3 Excentricity: k 3 Definition of plot variables j :.. j 5 d : j, 5 ( ) d : σ j, yi Error ellipse ( σ x ) d. Red 5, 5, d. Red,, ( ) d : σ j, yi Integration contour ( σ x ) (error ellipse) d. Red 5, 5, d., Red, ( ) ( ). ( ) ( )..5.5 (Error ellipse and integration contour coincide)

3 3 j, Probability integral Integration boundaries a : i σ x b : i σ yi a i a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y σ ( yi ) dy dx drms i P i Result: the confidence content (or error probability) is.39. This means the probability that the position lies within the error contour is 39%. This value is independent of the ratio of the underlying standard deviations. The confidence content of the error ellipse can be calculated more easily by the equivalent Rayleigh cumulative probability function (CDF). with σ : σ x + ( σ y ) and R : σ σ. P( R) : exp R σ P( R).393

4 b. Integration Contour: Double Standard Deviations Error Ellipse Semi axes a : i σ x b : i σ yi Case displayed i : 3 Excentricity: k 3 Definition of plot variables j :.. j 5 d : j, 5 d : σ j, y ( ) i ( ) ( σ x ) Error ellipse d. 5, d., ( ) ( ). Red 5, Red, ( ) i d : σ j, y ( ) ( σ x ) Integration contour (error ellipse) d. 5, d., ( ) ( ). Red 5, Red,.5.5 solid red line: genuine error ellipse dashed blue line: integration contour Probability integral Integration boundaries a : i σ x b : i σ yi a i b i

5 5 a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y σ ( yi ) dy dx P i Result: the confidence content (or error probability) is.865. This means the probability that the position lies within the error contour is 86.5%. This value is independent of the ratio of the underlying standard deviations. The confidence content of the error ellipse can be calculated more easily by the equivalent Rayleigh cumulative probability function (CDF). with P( R) : exp σ : σ x R σ + ( σ y ) and P( R).865 R: σ c. Integration Contour: 95% - Error Ellipse Probability integral Integration boundaries a : i b : i.9 σ x.9 σ yi Iteratively produced Sigma-multiplier for p,95 integration boundaries.

6 6 a i a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y ( σ yi ) dy dx P i Result: the confidence content (or error probability) chosen is.95. This means the probability that the position lies within the error contour is 95%. This value is independent of the ratio of the underlying standard deviations. The respective multiplier for both standard deviations is.9 This value is used in NavGen to produce the 95% error ellipse shown in the diagram of the individual position fixes. The multiplier for the standard deviations yielding an error ellipse with a confidence content of 95 % can be calculated more easily by resolving the Raleigh CDF for the radius R: P( R) :.95 σ : R: ln ( PR ( )) σ R.8

7 7 a. Integration Contour: Circle drms Distance Root Mean Square (drms) is a commonly used error measure for position fixes, as it is quite easy to calculate. It replaces two-dimensional statistics by one-dimensional, and in fact defines an error circle, no matter what the ratio of the underlying standard deviations is. The calculation below shows that the confidence content of the related error circle is dependent on the ratio of the standard deviations. Hence, caution must prevail if the error distribution is not circular, i.e. not having identical error distributions. σ x : σ yi : k σ i x k..5. Radius of error circle drms : σ i x + ( σ yi ) drms Case displayed i : axis ratio: k i j :.. σ x. σ yi drms.8 i Definition of plot variables j 5 d : j, 5 maxd ( ). d : σ j, yi ( ) ( σ x ) Error Ellipse d. 5, d., ( ) ( ). Red 5, Red, d : drms j, i ( ) ( ) Integration contour (error circle) d. 5, ( ).8 Red 5, d. 8, ( ). Red 8,

8 solid red line: genuine error ellipse dashed blue line: integration contour Probability integral Integration boundaries a : i drms i b : i drms i a i a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) e x y + σ x σ yi ( ) dy dx drms i P i

9 9 Result: the confidence content (or error probability) varies with the ratio of the underlying standard deviations betwen.63 and.683. The drms error measure does not possess a fixed probability. For equal standard deviations a probability of 63. % applies. As the probability content of the drms error circle varies with the ratio of the underlying SDs, the Rayleigh distribution cannot be used to determine the probability content. The author has developed the following approximation for the probability content as a function of the SD ratio k: ( ) ( 5.876) P ai : ( 5.876) ( ) drms P P a b. Integration Contour: Circle drms Case displayed i : Excentricity: k i Definition of plot variables j :.. 6 j 3 d : j, maxd ( ) 3. d : σ j, yi ( ) ( σ x ) Error Ellipse d. 3, d., ( ) ( ). Red 3, Red, d : ( drms) j, i ( ) Integration contour (error circle) d. 3, ( ).36 Red 3, d , ( ). Red 537,

10 3 3 3 Probability integral Integration boundaries a i : ( drms) i : ( drms) i a i a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y σ ( yi ) dy dx ( drms) i P i

11 Result: the confidence content (or error probability) varies with the ratio of the underlying standard deviations betwen.955 and.98 The drms error measure does not possess a fixed probability. For equal standard deviations a probability of 89. % applies. As the probability content of the drms error circle varies with the ratio of the underlying SDs, the Rayleigh distribution cannot be used to determine the probability content. The author has developed the following approximation for the probability content as a function of the SD ratio k: P ai : k k.6968 i i ( ) ( ) 3..6 drms.36 P.97 P a a. Integration Contour: Circle CEP5 h*drms CEP5 is produced iteratively by varying h. As the probability varies with the shape of the error ellipse, an error probability of 5% is produced for all k. Probability integral Integration boundaries h : (factors varied iteratively to produce a probability of 5%)

12 a : i b : i ( h drms i i ) ( ) h drms i i a i a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y σ ( yi ) dy dx drms i h drms i i P i h CEP5 Result: There is no constant relationship between CEP5 and drms. CEP5 can be determined from drms for a given ratio of the standard deviations k by multiplying drms with the applicable factor h. For the case of a circular error distribution (k ) - and just only for this case - we can check the result above by applying the Rayleigh CDF. P( R) :.5 σ : R: ln ( PR ( )) σ R.77 The multiplier for drms is then R.833 drms q.e.d

13 3 3b. Integration Contour: Circle CEP95 v*cep5 CEP95 is produced iteratively by varying h. As the probability varies with the shape of the error ellipse, an error probability of 95 % is produced for all k. Probability integral Integration boundaries h : (factors varied iteratively to produce CEP5 from drms) v :.339 (factors varied iteratively to produce CEP95 from CEP5).8.79 a : i b : i ( h v drms i i i ) ( ) h v drms i i i a i

14 a i P : i π σ x σ yi a i ( ) ( ) x ( a i ) x ( a i ) x + σ x e y ( σ yi ) dy dx drms i h v i i P i v Result: There is no constant relationship between CEP95 and CEP5. CEP95 can be determined from CEP5 (or from drms) for a given ratio of the standard deviations by multiplying CEP5 (drms) with the applicable factor v (h*v). For the case of a circular error distribution (k ) - and just only for this case - we can check the result above by applying the Rayleigh CDF. P( R) :.95 σ : R: ln ( PR ( )) σ R.8 The multiplier for drms is then R.73 q.e.d drms ###

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