Diversity prediction theorem

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: "The squared error of the collective prediction equals the average squared error minus the predictive diversity". Therefore, when the diversity in a group is large, the error of the crowd is small.

- Average Individual Error: Average of the individual squared errors

- Collective Error: Squared error of the collective prediction

- Prediction Diversity: Average squared distance from the individual predictions to the collective prediction

So, The Diversity Prediction Theorem: Given a crowd of predictive models

Collective Error = Average Individual Error - Prediction Diversity

wikipedia paper

C

Accepts inputs from command line, prints out usage on incorrect invocation. <lang C> /*Abhishek Ghosh, 25th October 2017*/

include<string.h>
include<stdlib.h>
include<stdio.h>

float mean(float* arr,int size){ int i = 0; float sum = 0;

while(i != size) sum += arr[i++];

return sum/size; }

float variance(float reference,float* arr, int size){ int i=0; float* newArr = (float*)malloc(size*sizeof(float));

for(;i<size;i++) newArr[i] = (reference - arr[i])*(reference - arr[i]);

return mean(newArr,size); }

float* extractData(char* str, int *len){ float* arr; int i=0,count = 1; char* token;

while(str[i]!=00){ if(str[i++]==',') count++; }

arr = (float*)malloc(count*sizeof(float)); *len = count;

token = strtok(str,",");

i = 0;

while(token!=NULL){ arr[i++] = atof(token); token = strtok(NULL,","); }

return arr; }

int main(int argC,char* argV[]) { float* arr,reference,meanVal; int len; if(argC!=3) printf("Usage : %s <reference value> <observations separated by commas>"); else{ arr = extractData(argV[2],&len);

reference = atof(argV[1]);

meanVal = mean(arr,len);

printf("Average Error : %.9f\n",variance(reference,arr,len)); printf("Crowd Error : %.9f\n",(reference - meanVal)*(reference - meanVal)); printf("Diversity : %.9f",variance(meanVal,arr,len)); }

return 0; } </lang> Invocation and Output :

C:\rosettaCode>diversityTheorem.exe 49 48,47,51
Average Error : 3.000000000
Crowd Error : 0.111110263
Diversity : 2.888888597
C:\rosettaCode>diversityTheorem.exe 49 48,47,51,42
Average Error : 14.500000000
Crowd Error : 4.000000000
Diversity : 10.500000000

C++

include <iostream>
include <vector>
include <numeric>

float sum(const std::vector<float> &array) {

   return std::accumulate(array.begin(), array.end(), 0.0);

}

float square(float x) {

   return x * x;

}

float mean(const std::vector<float> &array) {

   return sum(array) / array.size();

}

float averageSquareDiff(float a, const std::vector<float> &predictions) {

   std::vector<float> results;
   for (float x : predictions)
       results.push_back(square(x - a));
   return mean(results);

}

void diversityTheorem(float truth, const std::vector<float> &predictions) {

   float average = mean(predictions);
   std::cout
       << "average-error: " << averageSquareDiff(truth, predictions) << "\n"
       << "crowd-error: " << square(truth - average) << "\n"
       << "diversity: " << averageSquareDiff(average, predictions) << std::endl;

}

int main() {

   diversityTheorem(49, {48,47,51});
   diversityTheorem(49, {48,47,51,42});
   return 0;

} </lang>

Output:

average-error: 3
crowd-error: 0.11111
diversity: 2.88889
average-error: 14.5
crowd-error: 4
diversity: 10.5

C#

<lang csharp> using System; using System.Linq; using System.Collections.Generic;

public class MainClass {

   static double Square(double x) => x * x;

   static double AverageSquareDiff(double a, IEnumerable<double> predictions)
       => predictions.Select(x => Square(x - a)).Average();

   static void DiversityTheorem(double truth, IEnumerable<double> predictions)
   {
       var average = predictions.Average();
       Console.WriteLine($@"average-error: {AverageSquareDiff(truth, predictions)}

crowd-error: {Square(truth - average)} diversity: {AverageSquareDiff(average, predictions)}");

   public static void Main() {

DiversityTheorem(49, new []{48d,47,51});

   	DiversityTheorem(49, new []{48d,47,51,42});
   }

}</lang>

Output:

average-error: 3
crowd-error: 0.11111
diversity: 2.88889
average-error: 14.5
crowd-error: 4
diversity: 10.5

Clojure

John Lawrence Aspden's code posted on Diversity Prediction Theorem. <lang Clojure> (defn diversity-theorem [truth predictions]

 (let [square (fn[x] (* x x))
       mean (/ (reduce + predictions) (count predictions))
       avg-sq-diff (fn[a] (/ (reduce + (for [x predictions] (square (- x a)))) (count predictions)))]
   {:average-error (avg-sq-diff truth)
    :crowd-error (square (- truth mean))
    :diversity (avg-sq-diff mean)}))

(println (diversity-theorem 49 '(48 47 51))) (println (diversity-theorem 49 '(48 47 51 42))) </lang>

Output:

{:average-error 3, :crowd-error 1/9, :diversity 26/9}
{:average-error 29/2, :crowd-error 4, :diversity 21/2}

JavaScript

ES5

<lang JavaScript>'use strict';

function sum(array) {

   return array.reduce(function (a, b) {
       return a + b;
   });

}

function square(x) {

   return x * x;

}

function mean(array) {

   return sum(array) / array.length;

}

function averageSquareDiff(a, predictions) {

   return mean(predictions.map(function (x) {
       return square(x - a);
   }));

}

function diversityTheorem(truth, predictions) {

   var average = mean(predictions);
   return {
       'average-error': averageSquareDiff(truth, predictions),
       'crowd-error': square(truth - average),
       'diversity': averageSquareDiff(average, predictions)
   };

}

console.log(diversityTheorem(49, [48,47,51])) console.log(diversityTheorem(49, [48,47,51,42])) </lang>

Output:

{ 'average-error': 3,
  'crowd-error': 0.11111111111111269,
  diversity: 2.888888888888889 }
{ 'average-error': 14.5, 'crowd-error': 4, diversity: 10.5 }

ES6

<lang JavaScript>(() => {

   'use strict';

   // mean :: Num a => [a] -> b
   const mean = xs => {
       const lng = xs.length;

       return lng > 0 ? (
           xs.reduce((a, b) => a + b, 0) / lng
       ) : undefined;
   }

   // meanErrorSquared :: Num a => a -> [a] -> b
   const meanErrorSquared = (observed, predictions) =>
       mean(predictions.map(x => Math.pow(x - observed, 2)));

   // diversityValues :: Num a => a -> [a] ->
   //     {mean-Error :: b, crowd-error :: b, diversity :: b}
   const diversityValues = (observed, predictions) => {
       const predictionMean = mean(predictions);

       return {
           'mean-error': meanErrorSquared(observed, predictions),
           'crowd-error': Math.pow(observed - predictionMean, 2),
           'diversity': meanErrorSquared(predictionMean, predictions)
       };
   }

   // TEST

   // show :: a -> String
   const show = x => JSON.stringify(x, null, 2);

   return show([{
       observed: 49,
       predictions: [48, 47, 51]
   }, {
       observed: 49,
       predictions: [48, 47, 51, 42]
   }].map(x => {
       const dctData = diversityValues(x.observed, x.predictions),
           dct = {};

       return (
           Object.keys(dctData)
           .forEach(k => dct[k] = dctData[k].toPrecision(3)),
           dct
       );
   }));

})();</lang>

Output:

[
  {
    "mean-error": "3.00",
    "crowd-error": "0.111",
    "diversity": "2.89"
  },
  {
    "mean-error": "14.5",
    "crowd-error": "4.00",
    "diversity": "10.5"
  }
]

Julia

Works with: Julia version 0.6

<lang julia>function diversitytheorem(truth::T, pred::Vector{T}) where T<:Number

   avg = mean(pred)
   avgerr = mean((pred .- truth) .^ 2)
   crderr = (avg - truth) ^ 2
   divers = mean((pred .- avg) .^ 2)
   return avgerr, crderr, divers

end

for (t, s) in [(49, [48, 47, 51]),

              (49, [48, 47, 51, 42])]
   avgerr, crderr, divers = diversitytheorem(t, s)
   println("""
   average-error : $avgerr
   crowd-error   : $crderr
   diversity     : $divers
   """)

end</lang>

Output:

average-error : 3.0
crowd-error   : 0.11111111111111269
diversity     : 2.888888888888889

average-error : 14.5
crowd-error   : 4.0
diversity     : 10.5

Kotlin

Translation of: TypeScript

<lang scala>// version 1.1.4-3

fun square(d: Double) = d * d

fun averageSquareDiff(d: Double, predictions: DoubleArray) =

   predictions.map { square(it - d) }.average()

fun diversityTheorem(truth: Double, predictions: DoubleArray): String {

   val average = predictions.average()
   val f = "%6.3f"
   return "average-error : ${f.format(averageSquareDiff(truth, predictions))}\n" +
          "crowd-error   : ${f.format(square(truth - average))}\n" +
          "diversity     : ${f.format(averageSquareDiff(average, predictions))}\n"

}

fun main(args: Array<String>) {

   println(diversityTheorem(49.0, doubleArrayOf(48.0, 47.0, 51.0)))
   println(diversityTheorem(49.0, doubleArrayOf(48.0, 47.0, 51.0, 42.0)))

}</lang>

Output:

average-error :  3.000
crowd-error   :  0.111
diversity     :  2.889

average-error : 14.500
crowd-error   :  4.000
diversity     : 10.500

Perl 6

<lang perl6>sub diversity-calc($truth, @pred) { my $ae = avg-error($truth, @pred); # average individual error my $cp = ([+] @pred)/+@pred; # collective prediction my $ce = ($cp - $truth)**2; # collective error my $pd = avg-error($cp, @pred); # prediction diversity return $ae, $ce, $pd; }

sub avg-error ($m, @v) { ([+] (@v X- $m) X**2) / +@v }

sub diversity-format (@stats) { gather {

   for <average-error crowd-error diversity> Z @stats -> ($label,$value) {
       take $label.fmt("%13s") ~ ':' ~ $value.fmt("%7.3f");
   }

} }

.say for diversity-format diversity-calc(49, <48 47 51>); .say for diversity-format diversity-calc(49, <48 47 51 42>);</lang>

Output:

average-error:  3.000
  crowd-error:  0.111
    diversity:  2.889
average-error: 14.500
  crowd-error:  4.000
    diversity: 10.500

REXX

version 1

<lang rexx>/* 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 */</lang>

Output:

average-error=3
crowd-error=0.11111111111111111089
diversity=2.8888888888888888889
--------------------------------------
average-error=14.5
crowd-error=4
diversity=10.5

version 2

<lang rexx>/*REXX program calculates: average error, crowd error, and prediction diversity.*/ numeric digits 50 /*set precision at fifty decimal digits*/ call diversity 49, 48 47 51 /*true value, and crowd predictions. */ call diversity 49, 48 47 51 42 /* " " " " " */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ avg: $=0; do k=1 for #; $=$ + word(ests,k) ; end; return $/# avgSD: $=0; do j=1 for #; $=$ + (word(ests,j) - arg(1))**2; end; return $/# /*──────────────────────────────────────────────────────────────────────────────────────*/ diversity: parse arg true, ests; #=words(ests) /*get args; count number of estimates.*/

          say '   the  true   value: '  true  copies('═', 20)  'crowd estimates: '   ests
          avg=avg()                             /* [↓]  avgSD=avg of squared difference*/
          say '   the average error: '      format( avgSD(true)   , , 6) / 1
          say '   the  crowd  error: '      format( (true-avg)**2 , , 6) / 1
          say 'prediction diversity: '      format( avgSD(avg)    , , 6) / 1;  say;   say
          return                                /*only show up to 6───┘  decimal digits*/</lang>

output when using the default inputs:

   the  true   value:  49 ════════════════════ crowd estimates:  48 47 51
   the average error:  3
   the  crowd  error:  0.111111
prediction diversity:  2.888889


   the  true   value:  49 ════════════════════ crowd estimates:  48 47 51 42
   the average error:  14.5
   the  crowd  error:  4
prediction diversity:  10.5

Sidef

Translation of: Perl 6

<lang ruby>func avg_error(m, v) {

   v.map { (_ - m)**2 }.sum / v.len

}

func diversity_calc(truth, pred) {

   var ae = avg_error(truth, pred)
   var cp = pred.sum/pred.len
   var ce = (cp - truth)**2
   var pd = avg_error(cp, pred)
   return [ae, ce, pd]

}

func diversity_format(stats) {

   gather {
       for t,v in (%w(average-error crowd-error diversity) ~Z stats) {
           take(("%13s" % t) + ':' + ('%7.3f' % v))
       }
   }

}

diversity_format(diversity_calc(49, [48, 47, 51])).each{.say} diversity_format(diversity_calc(49, [48, 47, 51, 42])).each{.say}</lang>

Output:

average-error:  3.000
  crowd-error:  0.111
    diversity:  2.889
average-error: 14.500
  crowd-error:  4.000
    diversity: 10.500

TypeScript

<lang TypeScript> function sum(array: Array<number>): number {

   return array.reduce((a, b) => a + b)

}

function square(x : number) :number {

   return x * x

}

function mean(array: Array<number>): number {

   return sum(array) / array.length

}

function averageSquareDiff(a: number, predictions: Array<number>): number {

   return mean(predictions.map(x => square(x - a)))

}

function diversityTheorem(truth: number, predictions: Array<number>): Object {

   const average: number = mean(predictions)
   return {
       "average-error": averageSquareDiff(truth, predictions),
       "crowd-error": square(truth - average),
       "diversity": averageSquareDiff(average, predictions)
   }

}

console.log(diversityTheorem(49, [48,47,51])) console.log(diversityTheorem(49, [48,47,51,42])) </lang>

Output:

{ 'average-error': 3,
  'crowd-error': 0.11111111111111269,
  diversity: 2.888888888888889 }
{ 'average-error': 14.5, 'crowd-error': 4, diversity: 10.5 }

zkl

Translation of: Sidef

<lang zkl>fcn avgError(m,v){ v.apply('wrap(n){ (n - m).pow(2) }).sum(0.0)/v.len() }

fcn diversityCalc(truth,pred){ //(Float,List of Float)

  ae,cp := avgError(truth,pred), pred.sum(0.0)/pred.len();
  ce,pd := (cp - truth).pow(2),  avgError(cp, pred);
  return(ae,ce,pd)

}

fcn diversityFormat(stats){ // ( (averageError,crowdError,diversity) )

  T("average-error","crowd-error","diversity").zip(stats)
  .pump(String,Void.Xplode,"%13s :%7.3f\n".fmt)

}</lang> <lang zkl>diversityCalc(49.0, T(48.0,47.0,51.0)) : diversityFormat(_).println(); diversityCalc(49.0, T(48.0,47.0,51.0,42.0)) : diversityFormat(_).println();</lang>

Output:

average-error :  3.000
  crowd-error :  0.111
    diversity :  2.889

average-error : 14.500
  crowd-error :  4.000
    diversity : 10.500