From: news.dead.person.stones@darjeeling.plus.com   
      
   On 21/12/2022 21:54, Dmitry A. Kazakov wrote:   
   > On 2022-12-21 20:55, Mike Terry wrote:   
   >   
   >> The OP was probably deliberately rather vague on this point!  The easiest   
   definition for "average"   
   >> literally doesn't make sense in a modulo context:  we can ADD the modular   
   values, but in general   
   >> dividing in such a mathematical setting doesn't make much sense, so   
   >>   
   >>      Average ([i=1,2,..n] x_i)  =[def]=  Sum ([i=1,2,..n] x_i) /n   
   >>   
   >> would be inappropriate due to the /n operation.   
   >   
   > Unless you calculate everything as reals or integers and then take the   
   remainder of 24. Which is   
   > kind of most natural definition, at least to me.   
      
   That is unambiguous, but to my mind not at all what is wanted - an artificial   
   cut-point has been   
   introduced to the continuous mod-24 "loop", converting all the points into   
   non-modular lengths (say   
   real numbers), and then those are averaged.  That produces "silly" conclusions   
   as people have   
   listed.  I get that people want to read the original question in a literal   
   fashion, because at least   
   that would make it clear - but to me that is very much not what was intended   
   or actually said.   
      
   (Well, I agree very little was actually /said/ but near the end of the post I   
   give a couple of [what   
   I consider] minimal requirements for what a modulo average should achieve, and   
   I bet the OP would   
   agree with those!)   
      
   >   
   >> HOWEVER, there's another characterisation for the average of a set, which   
   looks more promising in   
   >> a modular (or other more general) setting:  the average is the value of V   
   which MINIMISES THE   
   >> "ERROR" calculated by   
   >>   
   >>      error = Sum ([i=1,2,..n] (V - x_i)^2)   
   >>   
   >> that is, minimises the sum of squares of differences from V.   
   >   
   > This has exactly same problem as above, because subtraction and   
   multiplication (power of two) have   
   > different semantics in modular arithmetic.   
      
   Hmm, I don't agree - we've moved from a "calculate this formula for averaging   
   numbers in R"   
   perspective, which DOESN'T WORK IN THE MODULO SYSTEM [due to the division] to   
   a "choose V which   
   minimises such-and-such error function" which DOES work, provided we have a   
   way of making the error   
   function clear.   
      
   Note that in the error =  formula above, the "(V - x_i)^2" terms are the   
   squares of the distance   
   between V and X_i.  I agree V-x_i would be ambiguous in the modulo system, but   
   that formula is the   
   formula for REAL number averages (the "mean" as you point out below).  In the   
   modulo system we need   
   a replacement for "distance" that is well defined and appropriate.   
      
   So provided we can propose a sensible metric on the modular system, the error   
   function approach   
   makes perfect sense.  That's not at all like the "convert modulo number to   
   real number and then take   
   arithmetic mean, and then reduce that modulo whatever" definition, which will   
   give different results   
   depending on where you make the required "cut" in the modulo loop.   
      
   So to be clearer perhaps I should have written:   
      
          error = Sum ([i=1,2,..n] Distance(V,x_i)^2)   
      
   ALSO NOTE:  this error calculation ISN'T A MODULO CALCULATION!  It's a   
   calculation in the real   
   numbers R.  So Distance is a real number, and squaring is being done in the   
   real numbers, and the   
   error function is a function mapping to real numbers that is being minimised   
   as a value in R (by a   
   choice of V in the modulo system).  [That all relates to your remark about   
   different semantics in   
   modular arithmetic.]   
      
   >   
   >> [Incidentally, if we minimise the sum of absolute differeces from V, that   
   characterises the mode   
   >> (aka "midpoint") of the sample, but I think the median is more what is   
   wanted here...]   
   >   
   > Actually it is the same beast. Median is the least C-norm defined as |x-y|.   
   >   
   > Mean is the least Euclidean norm (squares), the thing you proposed.   
   >   
   > Mean and average are same for real numbers.   
      
   Yeah, I agree - Thanks, I'd totally muddled the definitions for those terms!!    
   Well, I'll say it's   
   been a while :)  I should have said "mean".  A quick check shows "mode" is   
   actually "the most common   
   value" which is not of much interest here.   
      
   >   
   > You are right that median (and C-norm) can be unambiguously defined in   
   modular numbers. But nobody   
   > would ever qualify this as a puzzle! (:-))   
   >   
   > [...]   
   >   
   >> Now, as to how to CALCULATE the V above???   
   >   
   > As I said, in real numbers V is the mean, i.e.   
   >   
   >     V = Sum (Xi) =def= mean({Xi | i=1..n})   
   >        i=1..n   
      
   But that doesn't work sensibly for a modulo system where arithmetic wraps.    
   That's the whole point   
   of this thread, it seems to me.   
      
   Hmmm, ok clearly for a modulo system, if we translate all the x_i by a fixed   
   amount, THE "AVERAGE"   
   SHOULD ALSO TRANSLATE BY THE SAME FIXED AMOUNT.  Otherwise it is simply not a   
   sensible definition of   
   average with that modulo system.  Your definition does not have the   
   translation property while my   
   one does - that's why I propose it for this problem.   
      
   I'd agree that there's plenty of room for discussing what is the right   
   "distance" function or   
   precisely how that feeds into the "error" function (squares vs. abs values or   
   whatever).  But also I   
   think my suggestion for those is good enough...   
      
   >   
   > Any numeric method is in the end mapping Xi to some normal number,   
   calculating a classic mean,   
   > mapping back. The problem is with the forward mapping being ambiguous. One   
   method to disambiguate is   
   > staying within the same day. Another could be taking a median and   
   considering 12 hours before and 12   
   > hours after it. And there are many more.   
      
   Maybe so - I'd just say here that you are discussing an /algorithm/ to compute   
   a result, rather than   
   the definition of what is to be considered the correct answer, which was the   
   focus in my post.  I   
   agree any algorithm could well incorporate calculating the mean of certain   
   real numbers, but   
   /correctness/ is judged by whether the algorithm produces the V with minimum   
   error function.   
   [Otherwise people just make an arbitrary coding decision, and say by   
   definition their code is   
   correct because it correctly codes what they want it to :) ]   
      
   Going a bit further than my last post, I'd say that the average as I've   
   defined it would probably   
   turn out to be the real-number average, if only we could make the cut in the   
   modulo loop 12 hours   
   away from the point V.  Of course the point V is the value to be determined,   
   which isn't known up   
   front, so /defining/ V like this would be circular!  It does seem that the   
   effect of moving V in all   
   the calculations would be more easily evaluated, if the times x_i were SORTED,   
   so we could quickly   
   find how many x_i are in different intervals.  Just thinking...   
      
   >   
   > P.S. There are differences between the average and mean. OP referred the   
   average, which may mean   
   > something completely counterintuitive (no pun intended (:-))   
   >   
      
      
   [continued in next message]   
      
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