How to resolve the algorithm Diversity prediction theorem step by step in the REXX programming language

Published on 12 May 2024 09:40 PM
#Rexx

How to resolve the algorithm Diversity prediction theorem step by step in the REXX programming language

Table of Contents

Problem Statement

The   wisdom of the crowd   is the collective opinion of a group of individuals rather than that of a single expert. Wisdom-of-the-crowds research routinely attributes the superiority of crowd averages over individual judgments to the elimination of individual noise,   an explanation that assumes independence of the individual judgments from each other. Thus the crowd tends to make its best decisions if it is made up of diverse opinions and ideologies.

Scott E. Page introduced the diversity prediction theorem:

Therefore,   when the diversity in a group is large,   the error of the crowd is small.

For a given   true   value and a number of number of estimates (from a crowd),   show   (here on this page):

Use   (at least)   these two examples:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Diversity prediction theorem step by step in the REXX programming language

Source code in the rexx programming language

/* REXX */
Numeric Digits 20
Call diversityTheorem 49,'48 47 51'
Say '--------------------------------------'
Call diversityTheorem 49,'48 47 51 42'
Exit

diversityTheorem:
  Parse Arg truth,list
  average=average(list)
  Say 'average-error='averageSquareDiff(truth,list)
  Say 'crowd-error='||(truth-average)**2
  Say 'diversity='averageSquareDiff(average,list)
  Return

average: Procedure
  Parse Arg list
  res=0
  Do i=1 To words(list)
    res=res+word(list,i)  /* accumulate list elements */
    End
  Return res/words(list)  /* return the average */

averageSquareDiff: Procedure
  Parse Arg a,list
  res=0
  Do i=1 To words(list)
    x=word(list,i)
    res=res+(x-a)**2      /* accumulate square of differences */
    End
  Return res/words(list)  /* return the average */


/*REXX program calculates the  average error,  crowd error,  and  prediction diversity. */
                               numeric digits 50 /*use precision of fifty decimal digits*/
call diversity 49,   48  47  51                  /*true value and the crowd predictions.*/
call diversity 49,   48  47  51  42              /*  "    "    "   "    "        "      */
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
avg:   $= 0;           do j=1  for #;  $= $ +  word(x, j)        ;   end;     return $ / #
avgSD: $= 0;  arg y;   do j=1  for #;  $= $ + (word(x, j) - y)**2;   end;     return $ / #
/*──────────────────────────────────────────────────────────────────────────────────────*/
diversity: parse arg true, x;    #= words(x);     a= avg()   /*get args; count #est; avg*/
           say '   the  true   value: '   true  copies("═", 20)  "crowd estimates: "   x
           say '   the average error: '   format( avgSD(true) , , 6) / 1
           say '   the  crowd  error: '   format( (true-a) **2, , 6) / 1
           say 'prediction diversity: '   format( avgSD(a)    , , 6) / 1;        say;  say
           return                                            /*   └─── show 6 dec. digs.*/

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