Diversity prediction theorem

From Rosetta Code
Diversity prediction theorem is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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[edit]

Accepts inputs from command line, prints out usage on incorrect invocation.

 
/*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;
}
 

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++[edit]

 
#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;
}
 
Output:
average-error: 3
crowd-error: 0.11111
diversity: 2.88889
average-error: 14.5
crowd-error: 4
diversity: 10.5

C#[edit]

 
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});
}
}
Output:
average-error: 3
crowd-error: 0.11111
diversity: 2.88889
average-error: 14.5
crowd-error: 4
diversity: 10.5

Clojure[edit]

John Lawrence Aspden's code posted on Diversity Prediction Theorem.

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

JavaScript[edit]

ES5[edit]

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

ES6[edit]

(() => {
'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
);
}));
})();
Output:
[
  {
    "mean-error": "3.00",
    "crowd-error": "0.111",
    "diversity": "2.89"
  },
  {
    "mean-error": "14.5",
    "crowd-error": "4.00",
    "diversity": "10.5"
  }
]

Julia[edit]

Works with: Julia version 0.6
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
Output:
average-error : 3.0
crowd-error   : 0.11111111111111269
diversity     : 2.888888888888889

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

Kotlin[edit]

Translation of: TypeScript
// 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)))
}
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[edit]

sub diversity-calc($truth, @pred) {
my $ae = avg-error($truth, @pred); # average individual error
my $cp = ([+] @pred)/[email protected]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>);
Output:
average-error:  3.000
  crowd-error:  0.111
    diversity:  2.889
average-error: 14.500
  crowd-error:  4.000
    diversity: 10.500

REXX[edit]

version 1[edit]

/* 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 */
Output:
average-error=3
crowd-error=0.11111111111111111089
diversity=2.8888888888888888889
--------------------------------------
average-error=14.5
crowd-error=4
diversity=10.5

version 2[edit]

/*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*/
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[edit]

Translation of: Perl 6
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}
Output:
average-error:  3.000
  crowd-error:  0.111
    diversity:  2.889
average-error: 14.500
  crowd-error:  4.000
    diversity: 10.500

TypeScript[edit]

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

zkl[edit]

Translation of: Sidef
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)
}
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();
Output:
average-error :  3.000
  crowd-error :  0.111
    diversity :  2.889

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