This is a summary of the "averaging" thread that took place
from about July 1999 to November 1999 in
www.sci.geo.satellite-nav. All information here is readily
verifiable by reading through the full text in www.deja.com.
I have summarized this and made it available to avoid
repitition.
1. Someone asked whether averaging improved accuracy.
2. Sam Wormley said that it did, so long as the readings were
statistically independent. People like Joe Mehaffey were to
soon jump on this bandwagon. It is true in statistics that
some formula that can be used to predict accuracy only work
if the points are statistically independent. They incorrectly
generalised this to say that NO improvement whatsoever was
achievable with points that weren't totally statistically
independent. At the time, Sam (and others, including myself)
believed that the points only became statistically independent
after 15 minutes, which would mean that an average for less
than 15 minutes would produce NO reduction in the effects of
S/A.
3. I disagreed with Sam's assertion that no improvement was
possible with statistically dependent points.
4. I gave an intuitive example as to why it was better to
take an average than either of two unknown statistically
dependent points.
5. They still didn't understand why they were wrong, and
thought that my example was just one example, it didn't
work in general.
6. I created a mathematical proof that showed why it was
also true in general.
7. They still didn't understand the maths, and Joe resorted
to abuse as a substitute for alternative maths, or finding
some error in my own. At one point Sam tried quoting the
Mean Central Tendency theorem, still missing the point
completely.
8. Andrew Kalinowski created his own mathematical proof, based
on an incorrect assumption, that "proved" that averaging made
the accuracy WORSE. Both Sam and Joe somehow managed to
immediately see the relevance of this maths and supported such
a ludicrous proposition.
9. The matter was finally resolved, not by them understanding
the mathematical flaws, but by David Wilson posting empirical
data that showed that I was right, averaging stastically
dependent data improved accuracy.
10. In attempt to save face, Sam and Joe immediately started
to bluff that what they meant all along was that short term
averaging produced no worthwhile/significant/etc benefits.
This wasn't helped by some comments by David Wilson, who
despite doing no cost analysis or usage analysis had decided
that short-term averaging wasn't worthwhile (the figures tell
a very different story, depending on waypoint usage and cost
function).
The only contributions to this thread that were actually
accurate and helpful came from:
1. Paul Edwards, who initially realised that averaging
statistically dependent points would offer some improvement
(although he was unable to quantify how much). Also (later)
for analysis of data to produce a complete set of 16 graphs.
2. Dave Martindale who pointed out the geometric advantages
afforded by averaging.
3. James Giles for realising that one of the big benefits was
not so much the average reduction, but the effect on the
maximum error, and doing data analysis of same.
4. John Galvin, for providing empirical data.
5. David Wilson, for both data and analysis.
6. Sam Wormley, for making some raw data available for others
to analyse.
Questions still unanswered:
1. What is the correct mathematical analysis (theorems or
fomulas) that could have seen this issue resolved right at
the start?
2. Some empirical data to show how much time is saved from
real searches of real tourist waypoints.