# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Bias and cocked hats**

**From:**Bill Noyce

**Date:**2007 Apr 19, 08:47 -0400

Thanks for running these simulations. The description is rather terse, so let me see if I understand it. You're trying to answer the question, "What is the probability that the cocked hat surrounds the true position, when there's some systematic error in addition to random scatter in the observations?" You've chosen a "best case" of three observations equally-spaced in azimuth, and the error of each observation consists of a fixed bias (common to all three) plus a normally-distributed random error. Summarizing the results, we see that when the bias is zero, the chance of the cocked hat surrounding the true position is 25% -- confirming what we already knew. When the bias is more than 3 sd, the cocked hat surrounds the true position in over 99% of the cases. When the bias is 0.8 sd, the cocked hat surrounds the true position in 50% of the cases. It doesn't matter whether the bias is positive or negative (I assume differences of 1% are the expected fluctuation from the Monte Carlo process). And the actual magnitudes of the errors don't matter (for determining this percentage) -- only their relative sizes. It would be interesting to see similar results when the observations are spaced only 60 degrees in azimuth, so the common error tends to push the cocked hat away from the true position. -- Bill N. On 4/18/07, d waldenwrote: > > For three equally spaced observations with constant bias from -1 min to 1 min and independent normally distributed random errors with standard deviation labeled sd results from the previously listed FORTRAN Monte-Carlo simulation using 100,000 runs are as shown. > > Results can be nondimensionalized by sd. > > > > > sd=.5 0.25 1 > -1 93% 100% 60% > -0.9 90% 100% 55% > -0.8 85% 100% 50% > -0.7 78% 99% 45% > -0.6 69% 98% 40% > -0.5 60% 93% 36% > -0.4 50% 85% 32% > -0.3 40% 69% 29% > -0.2 32% 50% 27% > -0.1 27% 32% 26% > 0 25% 25% 25% > 0.1 27% 33% 26% > 0.2 33% 50% 27% > 0.3 41% 70% 29% > 0.4 50% 84% 33% > 0.5 60% 93% 36% > 0.6 70% 98% 41% > 0.7 78% 99% 45% > 0.8 84% 100% 50% > 0.9 90% 100% 55% > 1 93% 100% 60% > > > ________________________________ Ahhh...imagining that irresistible "new car" smell? > Check out new cars at Yahoo! Autos. > > > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---