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CHAPTER 4 STOCHASTIC PROCESSES
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graph, the 50% and 95% ellipses are indicated as are the transformations of the v frame axes The previous analysis of this section has discussed the computation of 2 2 the 50% and 95% uncertainty ellipses When the variances 1 = 2 = 2 are equal then RCEP = 11774 and R95 = 24477
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Computation of the radius of the circle expected to contain a given percentage of the points is not straightforward Various approximate formulae exist A few examples follow Let = max( n , e ) and = min( n , e ) De ne = / The rst formula is RCEP = 0589( n + e ) which is accurate to 3% for [02, 10] The second formula is RCEP = 0615 + 0562 which is much more accurate than the previous formula for [03, 10] For [00, 01], RCEP 0675 In addition to the statistics previously discussed, two additional measures are sometimes of interest: drms and 2drms The distance root-meansquare (drms) is the root-mean-square (RMS) value of the norm of the horizontal errors The formula for the drms measure is drms =
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The 2drms error measure is twice the drms value: 2drms = 2(drms) The reason for mentioning the 2drms is that the name is subject to misinterpretation Note in particular that 2drms is not the two-dimensional RMS position error If the horizontal error distribution is circular (ie, n = e = ), then 2 drms = 2drms = 2 2 4913 Scalar Analysis
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Let x represent one component of the position error, the component-wise error density is: 1 x2 p(x) = exp 2 (4136) 2 2 2
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49 DETAILED EXAMPLES Statistic Radius Probability RMS 393 drms 2 632 2drms 2 2 982 CEP 2 ln(2) 500 R95 2 ln(20) 950
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Table 41: Summary of various horizontal error statistics in w coordinates and their relationships to the error standard deviation assuming a circular error distribution (ie, = I) Given a set of samples {xi }N of the random variable x, the RMS error is i=1 rmsx = x = 1 N
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For large values of N , the RMS value is expected to converge to From distribution tables for Gaussian random variables (see p 48 in [107]), P rob{|x| < 0674 } = 500% P rob{|x| < 2 } = 955% P rob{|x| < } = 683% P rob{|x| < 3 } = 997%
Therefore, if the altitude h is estimated to be 100m with = 5m, then the probability that h [95, 105]m is 68%, that h [90, 110]m is 95%, etc 4914 Summary Comparison
Table 41 summarizes the various horizontal error statistics de ned above in relation to the one dimensional error standard deviation The table is useful for converting between error statistics Example 425 If an analyst is interested in nding the R95 error statistic that is equivalent to a stated drms error statistic of 100 m, then from Table 41: 10 drms = 10 = 2 R95 = 10 2ln(20) 2
Therefore, the equivalent R95 statistic is 173 m Repeating the process of Example 425 for each of the other possible combinations of accuracy statistics in two dimensions results in Table 42
154 drms 2drms CEP R95 10 07 04 08 04
CHAPTER 4 STOCHASTIC PROCESSES drms 14 10 05 12 06 2drms 28 20 10 24 12 CEP 12 08 04 10 05 R95 24 17 09 21 10
Table 42: Summary of conversion factors from the statistic listed in the in the leftmost column to the statistic indicated in the top row, assuming a circular error distribution (ie, = I)
To use this table, nd the row corresponding to the given statistic Multiply this row by the numeric value of that statistic to obtain all the other statistics as indicated in the (top) header row For example, if the R95 statistic is 10 m, them the other statistics are = 04m, drms = 06m, 2drms = 12m, and RCEP = 05m
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