NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Error Estimation
From: Hanno Ix
Date: 2013 Jan 9, 20:05 -0800
Again, when measuring the absolute error of distances there will no negative distance.
To actually measure this distribution I imagine having a package of cylindrical cookie-cutters.
With these cookie cutters I now cut out of that bell shaped hill rings concentric with the peak but with increasing radius'.
Result:
2. Similarly, since sigmaR decreases with the number N of observations: How many measurements should we take
From: Hanno Ix
Date: 2013 Jan 9, 20:05 -0800
Your postings make for interesting reading. You also triggered some thoughts here.
I apologize if this has been discussed in the group before or if it goes too far off the track.
I am not quite sure what exactly the error is that you a trying to evaluate statistically.
I assume you are talking about the absolute error viz. the deviation of the logged location from the true location.
It is always zero or positive, i.e. never negative.
In the CelNav case, location describes a position on the surface of the Earth which is understood
as 2-dimensional. 2-D statistics however seem to have some surprising characteristics.
For the sake of simplicity let's say our location is exactly at
Lat = 0 and long = 0.
Now, given 100 CelNav measurements at that location we find that these measurements
can be pictured by a bell-shaped hill of sand, if you will, where each sand corn occupies
the individually measured location. Ideally, the "hill's peak" will be at the true location and the "hill" will be
rationally symmetrical. Does the peak somehow represent the average absolute error?
Does it represent the most likely absolute error?
Again, when measuring the absolute error of distances there will no negative distance.
Strictly, just for that reason alone the distribution of the absolute error cannot be Normal/Gaussian.
So, we expect some other distribution.
To actually measure this distribution I imagine having a package of cylindrical cookie-cutters.
They are all of the same height, however the radius of their circular cross sections changes from, say,
0.1 sea mile, to 0.2 sm, to 0.3 sm and so forth in linear fashion. Crazy perhaps but a good mental tool.
With these cookie cutters I now cut out of that bell shaped hill rings concentric with the peak but with increasing radius'.
Note: The width of the rings stays the same, here 0.1sm, only the height changes with the radius of the rings.
What's left then is counting the number y of sand corns in each ring and plotting this number along the y-axis
and the respective ring radius along the x-axis.
What ever we do, we make rarely no error and rarely very, very big errors. Similarly, we expect the number
of sand corns in the rings captured by the narrowest cookie cutters to be very small. The same for the widest cookie cutters.
The plot absolute error vs. distance/radius will therefore look like this: It starts at y = 0 sand corns at ring radius x = 0,
climbs up to a peak at a certain distance/radius and then falls back asymptotically to y = 0 with further increasing radius.
This plot, i.e. distribution of the absolute error, will be quite different from looking bell shaped . It will asymmetrical and
look more like the rock of Gibraltar! It is actually known and carries the name Rayleigh distribution (RD).
http://en.wikipedia.org/wiki/Rayleigh_distribution
For a given RD with N measurements we have:
sigmaR = sqrt ( SUM(distances^2)/ ( 2*N) ); different from the normal distribution but related!
meanR = sigmaR*1.253... ; most likely absolute error of distance
meanR = sigmaR*1.253... ; most likely absolute error of distance
maxR = 0,606..../sigmaR
Result:
1. We can now answer a rather interesting question: Given our skills, means and circumstances as expressed by sigmaR ,
whenever we have N measurements, what is the probability that we exceed an absolute error which we consider still save?
2. Similarly, since sigmaR decreases with the number N of observations: How many measurements should we take
in one session in order for the vast majority of measurements, say 95% or more, to be of less than the
tolerable max absolute error? What is the risk if we take less measurements?
One nagging question, though, is still open for me: Is my sigmaR small enough? I mean, is it likely that the number of
CelNav measurements on my trip will statistically "never" reach the likely number between two dangerously wrong measurements?
Fortunately, one can find some answers to these questions just by simulations like the above on
spreadsheets, MATLAB, Mathematica, etc.
h