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Fred Hebard wrote-
"All of this discussion could be informed immensely by some data and 
associated analyses.  Data talk."

And in a later posting, objected to the use of simulated data, rather than 
real data.

In my view, both have their place. With simulated data, it's possible to 
know the exact value of the original quantity, before it's subjected to 
deliberate perturbations, so you can discover how well an analysis 
procedure recovers that original quantity from the perturbed data.

But in general, I agree with Fred. I can't think of any more appropriate 
set of data to investigate than that set of 9 observations which has been 
proffered by Peter Fogg on many occasions over the last 3 years, most 
recently on 13 Dec 2010, attached as "102278.example slope.jpg" under 
threadname "[NavList] A 'real-life' example of slope", and attached here 
under the same label. That plots and also tabulates the 9 observations, 
together with values for latitude and azimuth, which if correct result in a 
calculated slope of 32 arc-minutes in the 5 minite period of observation.

I have replotted those points as an Excel chart, attached as "slope fit 
3.xls", and for those who don't have Excel, as a simple picture, "slope fit 
3.gif", which shows exactly the same thing. 5 hours should be added to the 
time in minutes on the bottom scale to correspond to time of day, and 66º 
to the altitudes shown in arc-minutes.

What I have done is to plot all 9 points of that series, discarding none. 
Error-bars have been estimated, based on the observed scatter, of +/- 4.45 
arc minutes. They have then been analysed in a number of different ways.

First, they have been treated as most navigators would do, by simple 
averaging. This boils them down to a single mean point we will call P, at 
which the averaged time is 5h 28m 10s, and the averaged altitude is 66º 
25.9'. The standard deviation of that mean is reduced, compared with that 
of each individual observation, by a factor of 3 (= root 9) to +/- 1.48 
arc-minutes, as shown by its error-bar. This single point is then chosen to 
represent the altitude in subsequent calculations.

Second, a line, constrained to have a slope of 32' in 5 min of time, has 
been fitted to those points as well as possible, minimising the squared 
deviations from it, and plotted as a dotted line. That slope was chosen  to 
accord with Peter Fogg's own estimate. It's on the basis of those 
deviations from that line, that the standard deviation of the data-points 
has been assessed. Whatever its slope, every such line has to pass through 
the point P, as Lars Bergman pointed out in a posting on 9th December. It 
will be clear to most navigators that the observed points appear perfectly 
compatible with that line, in the way they scatter around it. The largest 
departure is that of point 1, which differs by 1.85 standard deviations. 
Just according to regular Gaussian statistics, we would expect 
one-fifteenth of the points (on average) to differ by that much or more, so 
it should cause no great surprise to find one such, in a sample of 9. It 
provides no grounds whatever for rejecting that point as any sort of 
deviant outlier.

Third, we have to consider how Peter Fogg has analysed this observation, 
which presents some problems. He has discarded not just observation 1, but 
also no. 3. His grounds for doing so have nowhere been stated clearly, 
despite numerous requests. Statements about his procedures have used the 
word "intuition", more than once. Through the remaining 7 points he has 
attempted to fit a straight line, as shown here by dashes. Unfortunately, 
though he has specified that line to have a slope of 32' over 5 minutes of 
time, his own plot (and therefore mine as well) has actually been drawn 
with a slope of 34. (This is the second example in which he has drawn an 
erroneous calculated slope.) Whether that error has contributed to his 
rejection of points 1 and 3, only he can tell us. After all this, his 
slope-fit passes about 1.4' away from point P. Which result is most 
true-to-life is impossible to say.

Fourthly, we can ask for a best-fit straight-line to the data, allowing the 
best-slope to be freely chosen instead of being constrained to 32. Just a 
glance at the data is enough to indicate that the chosen slope of 32' per 5 
min does not accord particularly well with the observed data, and a reduced 
slope would fit it better. When we do so, the resulting continuous line, 
showing a significantly better fit to the 9 observations, has a slope of 
only 24. That's by no means conclusive; no more than suggestive, that the 
calculated slope of 32 may be somewhat suspect. It would be worth checking 
it out once again, to be sure. The quoted latitude of 34º corresponds with 
the (South) lat. of his home port of Sydney, so is unlikely to be wrong, 
but what about the calculated azimuth of 149º? We haven't been given 
sufficient information to check that for ourselves; perhaps we can be 
provided with the missing details.

================

I hope this has provided Fred, and maybe others, with enough data to be 
able to assess whether Peter Fogg's data-rejection is justified, whether 
his procedure (whatever it may be) offers any improvement over standard 
statistics, and whether all the prolonged resulting hoo-hah has been 
worthwhile.

George.

contact George Huxtable, at george{at}hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message ----- 
From: "Peter Fogg" 
To: 
Sent: Monday, December 13, 2010 7:49 AM
Subject: [NavList] A 'real-life' example of slope


| The attached file, an example of slope in action, comes from a post I 
made
| on 10 March 2007 [NavList 2278].  It may seem like a poor round of sights
| compared with Antoine's, but remember the crucial difference often 
ignored
| by our armchair navigators: the relative stability of the platform used.
| Unless the sea conditions are abnormally calm, the near-perfection of
| Antoine's sights is in practice unachievable from the deck of a smallish
| sailing boat, in my experience.
|
| The analysis below comes from [NavList 2455] of 22 March 2007:
|
| Sights 1 and 3 have been discarded, as they cannot be matched to the
| slope. This slope, a fact, is then best matched to the pattern of the
| other sights that exhibit random error.
|
| What is the alternative to this technique? In this example, taking
| just the one sight could have been equivalent to choosing any one of
| these sights at random. What were the odds of obtaining as good an
| observation as the slope will produce with just the one sight?
|
| Of a poor sight (#1&3):                      2 out of 9;         22%
| Of a mediocre sight (#2,4,6,7,8,9):          6 out of 9;         67%
| Of poor or mediocre:                         8 out of 9;         89%
| Of an excellent sight (#5):                  1 out of 9;         11%
|
| So at the cost of a little extra calculation and the drawing up of a
| simple graph this 11% chance has been converted to a 100% chance of a
| similar result to what appears to be the best sight of the bunch,
| together with all the other advantages of KNOWING a lot more about
| this round of sights, and being able to derive extra information (eg;
| standard deviation) at will.
|
| Is this a typical example? No. Typically there are fewer sights in the
| 5 minutes, and NONE of the individual sights is as good as the derived
| slope; confirmed by comparing the resulting position lines to a known
| position.
| 

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