From: news.dead.person.stones@darjeeling.plus.com
On 24/12/2022 22:30, Ben Bacarisse wrote:
> Mike Terry writes:
>
>> On 24/12/2022 14:21, Tim Rentsch wrote:
>>> Ben Bacarisse writes:
>>>
>>>> Tim Rentsch writes:
>>>>
>>>>> ram@zedat.fu-berlin.de (Stefan Ram) writes:
>>>>>
>>>>>> ram@zedat.fu-berlin.de (Stefan Ram) writes:
>>>>>>
>>>>>>> Given n times of the 24-hour day, print their average.
>>>>>>> For example, the average of "eight o'clock" and
>>>>>>> "ten o'clock" (n=2) would be "nine o'clock".
>>>>>>> (You can choose any representation, for example "HH:MM"
>>>>>>> or "seconds since midnight".)
>>>>>>
>>>>>> Thanks for all replies!
>>>>>>
>>>>>> I waited a few days before answering to allow
>>>>>> sufficient time to think about the problem.
>>>>>>
>>>>>> There were not enough tests written and run. As a result,
>>>>>> the puzzle has not yet been solved (unless I have overlooked
>>>>>> a contribution or misworded expectations).
>>>>>>
>>>>>> So, here are two possible test cases.
>>>>>>
>>>>>> average( 23.5, 1.5 )== 0.5
>>>>>> average( 11.5, 13.5 )== 12.5
>>>>>>
>>>>>> (I use hours as units, so "0.5" means, "half past midnight".)
>>>>>>
>>>>>> I hope that these test cases encode sensible expectations
>>>>>> for an average of two times on a 24-hour clock in the spirit
>>>>>> of the example given in the OP, which was, "the average of
>>>>>> eight o'clock and ten o'clock would be nine o'clock", since
>>>>>> these test cases just have rotated that example by 3.5 and
>>>>>> 15.5 hours.
>>>>>>
>>>>>> I believe that I have not seen an algorithm so far in this
>>>>>> thread that would pass these tests.
>>>>>
>>>>> As before, the problem is underspecified.
>>>>
>>>> Some remarks not specifically in reply to you, Tim...
>>> I didn't really say very much. Your comments are a lot more
>>> interesting.
>>>
>>>> The input is a collection, t(n), of n > 1 numbers in [0, 24). The
>>>> average should be a number, A, in [0, 24) that minimises
>>>>
>>>> Sum_{i=1,n} distance(A, t(i))
>>>>
>>>> (or Sum_{i=1,n} difference(A, t(i))^2 if you prefer to think in terms
>>>> of variance). So far, this is just what an average is. The key point
>>>> is what is the distance (or difference) whose sum (or sum of squares)
>>>> we want to minimise? For times, I would say it is the length of the
>>>> shorter arc round an imaginary 24-hour clock face.
>>> Minimizing the square of the distance (aka length of the shorter arc)
>>> is one metric, and I think a reasonable one. (Minimizing the absolute
>>> value of the distance gives a result more like the median than the
>>> mean, IIANM.)
>>
>> Yes - perhaps Ben was thinking of aking
>>
>> Sum_{i=1,n} signed_distance(A, t(i)) = 0
>>
>> rather than minimising the (absolute) distance. That would match up
>> with minimising the sum of squares of the distance (mean). Or he just
>> meant the median as you say, which is different from the mean. Both
>> have plausible "average" properties,
>
> Well, I was trying to be deliberately vague (hence my talking about "an
> average" rather than mean, median or whatever) but I was no vague enough
> with the formula.
>
> I wasn't thinking of a zero net distance because some metrics (though
> probably not applicable in this situation) don't always allow for a zero
> sum. I should have referred to minimising some kind distance sum, be it
> the absolute value of a signed sum or, as in the solution I was
> proposing, the sum of directional differences (i.e. full vectors).
>
>> and meet my "minimal" criteria
>> for what I think anyone would require of an "average" applying to a
>> modular arithmetic:
>>
>> - if all x_i are within 5 minutes of each other [ALLOWING FOR WRAPPING
>> AROUND THE END OF THE MODULO LOOP] then the "average" will be within
>> those same 5 minutes. NOT e.g. 12 hours or 4 hours or whatever away.
>
> And when the x_i are widely spaced the average might be considered
> nearly meaningless. If you have {0, 8, 16} hours then what is the
> "average"? Are there cases where every set of circular values has a
> meaningful "average"? If not, you really need more than a single number
> answer.
>
> (These are not questions to you, specifically, I'm interested to know of
> interpretations that go beyond the way I see things.)
>
>> - if all x_i are translated by a fixed offset like 1 hour, the average
>> will translate by the same offset, i.e. it maintains the same "modulo"
>> relation with the translated x_i.
>>
>> hmm, thinking now I'll add the similar:
>>
>> - if all x_i are "reflected" about a chosen time, then the new average
>> will be the reflected time of the original average.
>>
>> (those aren't intended to be a definition of a "modular average", just
>> to exclude suggestions that blatently lack the right symmetry for such
>> an average.)
>>
>>>> The problem has a natural interpretation in terms of angles. Whatever
>>>> the circular quantity is, convert the values to unit vectors round a
>>>> circle. For times of day, just scale [0, 24) to [0, 2*pi). The
>>>> average is then just the direction of the average vector, converted
>>>> back to a time of day.
>>> Averaging the "time unit vectors" is another plausible metric.
>>
>> Yes I think this one would work best in most situations, given that
>> the OP didn't have a precise mathematical requirement for his solution
>> - probably he wants something that just works reasonably as expected.
>> (It's my favourite answer.)
>>
>> Another way of thinking of this approach is "balancing" the 24-hour
>> modular circle, where we've put unit weights at each of the times x_i.
>> E.g. we look for a balancing line through the centre of the circle.
>
> Is that the same? For {0, 8, 16} any of the three lines through 0, 8 or
> 16 balances, but the vector sum interpretation is undefined. And for
> {0, 6, 12, 18} there are two balance lines, neither of which seems to be
> a natural answer for the average.
Yes, it's logically the same, in so far as WHEN the x_i aren't already
balanced [aka they /have/ a
meaningful average] we get the same average both ways. (Adding vectors is
effectively a way of
calculating moments of the weights in the balancing view of things.)
With modular arithmetic there is a scenario where everything is already
balanced, and so it isn't
clear what an average would represent, if anything. Both {0, 8, 16} and {0,
6, 12, 18} are already
perfectly balanced - ANY line through the origin would balance for either
example. That is just a
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