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Re: The Darn Old Cocked Hat - the sequel 1
From: Hanno Ix
Date: 2013 Mar 15, 15:42 -0700
Hi all:

Maybe we are getting close!

Please understand: my memos are concerned with absolute
distances (i.e. sm) from the TL, not on positions (i.e. cordinates) - as weird as it sounds.
Actually: on the probabilities thereof.

David,  the integral you cite:

"integrate r * p(r) * 2Pir dr from 0 to infinity"

turns out to be a constant which the Rayleigh distribution is not.
The Rayleigh distribution is defined as, in your notation

p(r) =  r *  (  1/Pi *exp( - (r^2)) )

It is zero at TL.

http://mathworld.wolfram.com/RayleighDistribution.html

I still claim:

'The Rayleigh distribution expresses the probability  of an error of size r
where r is always positive and measured in units of lengths "
r implies just 1 coordinate, distance.

This error is occasionally called *RMS error.*  .The term occurs frequently
elsewhere but I have not seen it in CelNav texts. Perhaps it is a helpful term.
( May I please urge you to make in Excel two cols of  Gaussian random numers,
mean = 0, σ = 1 and calculate in the third col the square root of the sum
of squares of the first two. Tha'ts the meaning of the abbreviation RMS)

Rayleigh includes this fact: the  probability of the RMS error being zero is zero.

Bowing to Karl: I cannot know any of this when the TL is unknown.
However, Rayleigh is my best guess in situations where TL is unknown
having measured it when TL was known.

I believe my Fig 3 caused confusion. It tried to express  that the RMS error
is independent of directions. Or: Rayleigh is applicable no matter the azimuth.
And Yes, the term distribution in the caption is a misnomer.

Fig 3 is meant as a 3D histogram of the RMS error, one with an huge number of small bins
spread around TL and showing the probability of the RMS error around TL.
And since that one is zero at zero we have here a hole at zero .
Hilarious! And true!  You have not missed much if you forget Fig 3.

In contrast, Karl's picture shows the probability-density of position around TL
which is not the probability itself. I picture  probability-density as a measure of consistency.
Whatever it is, this one doesn't have any holes.

For those who care: I think of probability as relative frequency.
Also, of course a point is not a distribution. But I don't think I said that.
I hate statics's terminology; it uses similar terms for very different things.
The hole thing is so abstract. As I have noticed before:
Communication about statistics is probably more complicated than statistics themselves. :)

Regards,

h

From: David Fleming <d.l.fleming.1---.com>
To: hannoix---.net
Sent: Friday, March 15, 2013 6:54 AM
Subject: [NavList] Re: The Darn Old Cocked Hat - the sequel 1

DOCH Again
Hanno Ix has revived this topic by his recent post:
He claimed:
a) true location (TL) is not a normal distribution
b) probability of fix hitting TL is exceedingly small
Statement a) is unclear.
First a point is not a distribution. But the probability of the TL occuring at points in space is a distribution.
We can use rectangular coordinate system and we will find:

p(x,y) = 1/Pi * exp( - (x^2 +y^2))

a Gaussian distribution
For the Probability Density distribution in xy plane with coordinates measured in lengths of σ .
In order to find probability multiply by area, dx dy.

Or we can use polar coordinate system in which case we have:

p(r) = 1/Pi *exp( - (r^2))

a Rayleigh distribution
For the Probability Density distribution in polar coordinates, again unit r is σ .

As noted by John Karl and Geoffrey Kolbe, this distribution has no hole, ala claim b) .
To find probability in annular region, multiply by area 2π r dr. (there is the hole for r =0).
and we should expect probability of fix being an exact point to be 0.
to compute average distance from fix to TL
integrate r * p(r) * 2Pir dr from 0 to infinity
you find r =.886 for average r ( fix to TL)

in units of σ

John Karl's 2010 analysis is fine as far as it goes, and that is certainly further than is required by the practical navigator. Others have commented on how to think about the fact that we may not be inside the DOCH.
Not surprisingly the analysis can be made more complicted without improving the utility of the analysis. Not accounted for is the DR. That, like a sight, represents imperfect knowledge about our position and could be viewed as a different gaussian distribution along the LOP seperate from the distribution affecting the intercept.
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