Re: Another little puzzle

From	Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups	comp.programming
Subject	Re: Another little puzzle
Date	2022-12-31 00:24 +0000
Organization	A noiseless patient Spider
Message-ID	<878rioifnh.fsf@bsb.me.uk> (permalink)
References	(1 earlier) <algorithm-20221221130021@ram.dialup.fu-berlin.de> <average-20221228125821@ram.dialup.fu-berlin.de> <87tu1diu2s.fsf@bsb.me.uk> <864jtdtkt5.fsf@linuxsc.com> <87o7rlhtsv.fsf@bsb.me.uk>

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Ben Bacarisse <ben.usenet@bsb.me.uk> writes:

> Tim Rentsch <tr.17687@z991.linuxsc.com> writes:

A couple of further thoughts only one of which is directed at you
Tim (at the end)...

Given the general conception of a mean (rather than any other kind of
summary statistic) as minimising the sum of squares of some "distance"
metric:

  Sum_{i=1,n} difference(A, t(i))^2

we can characterise the two contenders by what distance is being
minimised.  For the less well discussed "conventional average" (to use
your terminology) we are minimising the sum of squares of the shorter
arc lengths between A and the t(i).

For the "vector average", we convert the t(i) to unit vectors u(i) and
we calculate the mean if the u(i) to get a vector m.  The "average", A,
is just the direction of this vector -- another point on the unit
circle.  In this case we are minimising the sum of squares of the
/chord/ lengths between A and the t(i).

The mean vector itself (which may not lie on the unit circle) minimises
the sum of the squares of the length of the vector differences m-u(i):

  Sum_{i=1,n} |m - u(i)|^2

and any other vector along the same line as m (i.e. c*m for real c)
minimises

  Sum_{i=1,n} |c*m - u(i)|^2

This includes, of course, m projected out to the unit circle.

This distinction between arc lengths and chord lengths helps to
visualise where these averages differ, and why the conventional average
may seem more intuitive.

Incidentally, I found another book on statistics on spheres, and that
gets the average over in a few paragraphs.  It states, without
considering any alternatives, that the average is the vector average.  I
can't find anyone using or citing the arc-length minimising average,
despite it being natural on a sphere to find the mid-point of, say, a
set of points on the Earth's surface.

Tim, my best shot at calculating this other average sorts the points to
find the widest gap.  I suspect your algorithm is similar since you say
it is O(n log n).

-- 
Ben.

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Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-30 01:00 +0000
  Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-29 23:25 -0800
    Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-30 14:04 +0000
      Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-31 00:24 +0000
        Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-31 06:42 -0800
          Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2022-12-31 17:04 +0100
            Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-01 01:24 +0000
            Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-08 07:45 -0800
              Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-08 17:17 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 20:41 +0000
                Re: Another little puzzle Richard Heathfield <rjh@cpax.org.uk> - 2023-01-08 21:14 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-08 22:31 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-09 03:25 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-09 11:22 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-09 20:37 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-10 09:06 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-11 02:41 +0000
                Re: Another little puzzle "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de> - 2023-01-11 10:01 +0100
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-12 01:00 +0000
                Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-10 23:36 -0800
                Re: Another little puzzle Y A <ya00000100000@yahoo.com> - 2023-01-11 02:39 -0800
          Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-01 01:10 +0000
            Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-08 07:17 -0800
              Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 19:43 +0000
                Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-01-08 19:59 +0000
                Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2023-01-10 23:25 -0800
      Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-31 06:20 -0800
        Re: Another little puzzle Augǝl <angel0000000001000000000000@mail.ee> - 2022-12-31 10:23 -0800

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