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Diversity prediction theorem

From Rosetta Code
Task
Diversity prediction theorem
You are encouraged to solve this task according to the task description, using any language you may know.

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.


Definitions
  •   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
  •   Diversity Prediction Theorem:   Given a crowd of predictive models,     then
  Collective Error   =   Average Individual Error   ─   Prediction Diversity
Task

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

  •   the true value   and   the crowd estimates
  •   the average error
  •   the crowd error
  •   the prediction diversity


Use   (at least)   these two examples:

  •   a true value of   49   with crowd estimates of:   48   47   51
  •   a true value of   49   with crowd estimates of:   48   47   51   42


Also see
  •   Wikipedia entry:   Wisdom of the crowd
  •   University of Michigan: PDF paper         (exists on a web archive,   the Wayback Machine).



11l[edit]

Translation of: C++
F average_square_diff(a, predictions)
R sum(predictions.map(x -> (x - @a) ^ 2)) / predictions.len
 
F diversity_theorem(truth, predictions)
V average = sum(predictions) / predictions.len
print(‘average-error: ’average_square_diff(truth, predictions)"\n"‘’
‘crowd-error: ’((truth - average) ^ 2)"\n"‘’
‘diversity: ’average_square_diff(average, predictions))
 
diversity_theorem(49.0, [Float(48), 47, 51])
diversity_theorem(49.0, [Float(48), 47, 51, 42])
Output:
average-error: 3
crowd-error:   0.111111111
diversity:     2.888888889
average-error: 14.5
crowd-error:   4
diversity:     10.5

C[edit]

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

 
 
#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]

 
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

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

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}

D[edit]

Translation of: C#
import std.algorithm;
import std.stdio;
 
auto square = (real x) => x * x;
 
auto meanSquareDiff(R)(real a, R predictions) {
return predictions.map!(x => square(x - a)).mean;
}
 
void diversityTheorem(R)(real truth, R predictions) {
auto average = predictions.mean;
writeln("average-error: ", meanSquareDiff(truth, predictions));
writeln("crowd-error: ", square(truth - average));
writeln("diversity: ", meanSquareDiff(average, predictions));
writeln;
}
 
void main() {
diversityTheorem(49.0, [48.0, 47.0, 51.0]);
diversityTheorem(49.0, [48.0, 47.0, 51.0, 42.0]);
}
Output:
average-error: 3
crowd-error: 0.111111
diversity: 2.88889

average-error: 14.5
crowd-error: 4
diversity: 10.5

Factor[edit]

Works with: Factor version 0.99 2020-01-23
USING: kernel math math.statistics math.vectors prettyprint ;
 
TUPLE: div avg-err crowd-err diversity ;
 
: diversity ( x seq -- obj )
[ n-v dup v* mean ] [ mean swap - sq ]
[ nip dup mean v-n dup v* mean ] 2tri div boa ;
 
49 { 48 47 51 } diversity .
49 { 48 47 51 42 } diversity .
Output:
T{ div { avg-err 3 } { crowd-err 1/9 } { diversity 2+8/9 } }
T{ div { avg-err 14+1/2 } { crowd-err 4 } { diversity 10+1/2 } }

Fōrmulæ[edit]

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

In this page you can see the program(s) related to this task and their results.

Go[edit]

package main
 
import "fmt"
 
func averageSquareDiff(f float64, preds []float64) (av float64) {
for _, pred := range preds {
av += (pred - f) * (pred - f)
}
av /= float64(len(preds))
return
}
 
func diversityTheorem(truth float64, preds []float64) (float64, float64, float64) {
av := 0.0
for _, pred := range preds {
av += pred
}
av /= float64(len(preds))
avErr := averageSquareDiff(truth, preds)
crowdErr := (truth - av) * (truth - av)
div := averageSquareDiff(av, preds)
return avErr, crowdErr, div
}
 
func main() {
predsArray := [2][]float64{{48, 47, 51}, {48, 47, 51, 42}}
truth := 49.0
for _, preds := range predsArray {
avErr, crowdErr, div := diversityTheorem(truth, preds)
fmt.Printf("Average-error : %6.3f\n", avErr)
fmt.Printf("Crowd-error  : %6.3f\n", crowdErr)
fmt.Printf("Diversity  : %6.3f\n\n", div)
}
}
Output:
Average-error :  3.000
Crowd-error   :  0.111
Diversity     :  2.889

Average-error : 14.500
Crowd-error   :  4.000
Diversity     : 10.500

Groovy[edit]

Translation of: Java
class DiversityPredictionTheorem {
private static double square(double d) {
return d * d
}
 
private static double averageSquareDiff(double d, double[] predictions) {
return Arrays.stream(predictions)
.map({ it -> square(it - d) })
.average()
.orElseThrow()
}
 
private static String diversityTheorem(double truth, double[] predictions) {
double average = Arrays.stream(predictions)
.average()
.orElseThrow()
return String.format("average-error : %6.3f%n", averageSquareDiff(truth, predictions)) + String.format("crowd-error  : %6.3f%n", square(truth - average)) + String.format("diversity  : %6.3f%n", averageSquareDiff(average, predictions))
}
 
static void main(String[] args) {
println(diversityTheorem(49.0, [48.0, 47.0, 51.0] as double[]))
println(diversityTheorem(49.0, [48.0, 47.0, 51.0, 42.0] as double[]))
}
}
Output:
average-error :  3.000
crowd-error   :  0.111
diversity     :  2.889

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

J[edit]

Accepts inputs from command line, prints out usage on incorrect invocation. Were this compressed adaptation from C the content of file d.ijs

 
echo 'Use: ' , (;:inv 2 {. ARGV) , ' <reference value> <observations>'
 
data=: ([: ". [: ;:inv 2&}.) ::([: exit 1:) ARGV
 
([: exit (1: [email protected]('insufficient data'"
_)))^:(2 > #) data
 
mean=: +/ % #
variance=: [: mean [: *: -
 
averageError=: ({. variance }.)@:]
crowdError=: variance {.
diversity=: variance }.
 
echo (<;._2'average error;crowd error;diversity;') ,: ;/ (averageError`crowdError`diversity`:0~ [email protected]:}.) data
 
exit 0
 

example uses follow

$ ijconsole d.ijs bad data
Use: ijconsole d.ijs <reference value>  <observations>
insufficient data
$ ijconsole d.ijs 1
Use: ijconsole d.ijs <reference value>  <observations>
insufficient data
$ ijconsole d.ijs 1 2
Use: ijconsole d.ijs <reference value>  <observations>
┌─────────────┬───────────┬─────────┐
│average error│crowd error│diversity│
├─────────────┼───────────┼─────────┤
│1            │1          │0        │
└─────────────┴───────────┴─────────┘
$ ijconsole d.ijs a 3
Use: ijconsole d.ijs <reference value>  <observations>
$ ijconsole d.ijs 49 48,47,51,42
Use: ijconsole d.ijs <reference value>  <observations>
┌─────────────┬───────────┬─────────┐
│average error│crowd error│diversity│
├─────────────┼───────────┼─────────┤
│14.5         │4          │10.5     │
└─────────────┴───────────┴─────────┘
$ ijconsole d.ijs 49 48,47,51
Use: ijconsole d.ijs <reference value>  <observations>
┌─────────────┬───────────┬─────────┐
│average error│crowd error│diversity│
├─────────────┼───────────┼─────────┤
│3            │0.111111   │2.88889  │
└─────────────┴───────────┴─────────┘
$ ijconsole d.ijs 49 48 47 51         # commas don't interfere
Use: ijconsole d.ijs <reference value>  <observations>
┌─────────────┬───────────┬─────────┐
│average error│crowd error│diversity│
├─────────────┼───────────┼─────────┤
│3            │0.111111   │2.88889  │
└─────────────┴───────────┴─────────┘

Java[edit]

Translation of: Kotlin
import java.util.Arrays;
 
public class DiversityPredictionTheorem {
private static double square(double d) {
return d * d;
}
 
private static double averageSquareDiff(double d, double[] predictions) {
return Arrays.stream(predictions)
.map(it -> square(it - d))
.average()
.orElseThrow();
}
 
private static String diversityTheorem(double truth, double[] predictions) {
double average = Arrays.stream(predictions)
.average()
.orElseThrow();
return String.format("average-error : %6.3f%n", averageSquareDiff(truth, predictions))
+ String.format("crowd-error  : %6.3f%n", square(truth - average))
+ String.format("diversity  : %6.3f%n", averageSquareDiff(average, predictions));
}
 
public static void main(String[] args) {
System.out.println(diversityTheorem(49.0, new double[]{48.0, 47.0, 51.0}));
System.out.println(diversityTheorem(49.0, new double[]{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

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';
 
// diversityValues :: [Num] -> {
// mean-error :: Float,
// crowd-error :: Float,
// diversity :: Float
// }
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
)
};
};
 
// meanErrorSquared :: Num a => a -> [a] -> b
const meanErrorSquared = observed =>
predictions => mean(
predictions.map(x => Math.pow(x - observed, 2))
);
 
// mean :: Num a => [a] -> b
const mean = xs => {
const lng = xs.length;
return lng > 0 ? (
xs.reduce((a, b) => a + b, 0) / lng
) : undefined;
};
 
 
// ----------------------- TEST ------------------------
const main = () =>
JSON.stringify([{
observed: 49,
predictions: [48, 47, 51]
}, {
observed: 49,
predictions: [48, 47, 51, 42]
}].map(x => dictionaryAtPrecision(3)(
diversityValues(x.observed)(
x.predictions
)
)), null, 2);
 
 
// ---------------------- GENERIC ----------------------
 
// dictionaryAtPrecision :: Int -> Dict -> Dict
const dictionaryAtPrecision = n =>
// A dictionary of Float values, with
// all Floats adjusted to a given precision.
dct => Object.keys(dct).reduce(
(a, k) => Object.assign(
a, {
[k]: dct[k].toPrecision(n)
}
), {}
);
 
// MAIN ---
return main()
})();
Output:
[
  {
    "mean-error": "3.00",
    "crowd-error": "0.111",
    "diversity": "2.89"
  },
  {
    "mean-error": "14.5",
    "crowd-error": "4.00",
    "diversity": "10.5"
  }
]

Jsish[edit]

From Typescript entry.

/* Diverisity Prediction Theorem, in Jsish */
"use strict";
 
function sum(arr:array):number {
return arr.reduce(function(acc, cur, idx, arr) { return acc + cur; });
}
 
function square(x:number):number {
return x * x;
}
 
function mean(arr:array):number {
return sum(arr) / arr.length;
}
 
function averageSquareDiff(a:number, predictions:array):number {
return mean(predictions.map(function(x:number):number { return square(x - a); }));
}
 
function diversityTheorem(truth:number, predictions:array):object {
var average = mean(predictions);
return {
"average-error": averageSquareDiff(truth, predictions),
"crowd-error": square(truth - average),
"diversity": averageSquareDiff(average, predictions)
};
}
 
;diversityTheorem(49, [48,47,51]);
;diversityTheorem(49, [48,47,51,42]);
 
/*
=!EXPECTSTART!=
diversityTheorem(49, [48,47,51]) ==> { "average-error":3, "crowd-error":0.1111111111111127, diversity:2.888888888888889 }
diversityTheorem(49, [48,47,51,42]) ==> { "average-error":14.5, "crowd-error":4, diversity:10.5 }
=!EXPECTEND!=
*/
Output:
prompt$ jsish -u diversityPrediction.jsi
[PASS] diversityPrediction.jsi

jq[edit]

Works with: jq

Works with gojq, the Go implementation of jq

def diversitytheorem($actual; $predicted):
def mean: add/length;
 
($predicted | mean) as $mean
| { avgerr: ($predicted | map(. - $actual) | map(pow(.; 2)) | mean),
crderr: pow($mean - $actual; 2),
divers: ($predicted | map(. - $mean) | map(pow(.;2)) | mean) } ;
# The task:
([49, [48, 47, 51]],
[49, [48, 47, 51, 42]
])
| . as [$actual, $predicted]
| diversitytheorem($actual; $predicted)
Output:
{
  "avgerr": 3,
  "crderr": 0.11111111111111269,
  "divers": 2.888888888888889
}
{
  "avgerr": 14.5,
  "crderr": 4,
  "divers": 10.5
}

Julia[edit]

Works with: Julia version 1.2
import Statistics: mean
 
function diversitytheorem(truth::T, pred::Vector{T}) where T<:Number
μ = mean(pred)
avgerr = mean((pred .- truth) .^ 2)
crderr = (μ - truth) ^ 2
divers = mean((pred .- μ) .^ 2)
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

Lua[edit]

Translation of: C++
function square(x)
return x * x
end
 
function mean(a)
local s = 0
local c = 0
for i,v in pairs(a) do
s = s + v
c = c + 1
end
return s / c
end
 
function averageSquareDiff(a, predictions)
local results = {}
for i,x in pairs(predictions) do
table.insert(results, square(x - a))
end
return mean(results)
end
 
function diversityTheorem(truth, predictions)
local average = mean(predictions)
print("average-error: " .. averageSquareDiff(truth, predictions))
print("crowd-error: " .. square(truth - average))
print("diversity: " .. averageSquareDiff(average, predictions))
end
 
function main()
diversityTheorem(49, {48, 47, 51})
diversityTheorem(49, {48, 47, 51, 42})
end
 
main()
Output:
average-error: 3
crowd-error: 0.11111111111111
diversity: 2.8888888888889
average-error: 14.5
crowd-error: 4
diversity: 10.5

Mathematica/Wolfram Language[edit]

ClearAll[DiversityPredictionTheorem]
DiversityPredictionTheorem[trueval_?NumericQ, estimates_List] :=
Module[{avg, avgerr, crowderr, diversity},
avg = Mean[estimates];
avgerr = Mean[(estimates - trueval)^2];
crowderr = (trueval - avg)^2;
diversity = Mean[(estimates - avg)^2];
<|
"TrueValue" -> trueval,
"CrowdEstimates" -> estimates,
"AverageError" -> avgerr,
"CrowdError" -> crowderr,
"Diversity" -> diversity
|>
]
DiversityPredictionTheorem[49, {48, 47, 51}] // Dataset
DiversityPredictionTheorem[49, {48, 47, 51, 42}] // Dataset
Output:
TrueValue	49
CrowdEstimates	{48,47,51}
AverageError	3
CrowdError	1/9
Diversity	26/9

TrueValue	49
CrowdEstimates	{48,47,51,42}
AverageError	29/2
CrowdError	4
Diversity	21/2

Nim[edit]

import strutils, math, stats
 
func meanSquareDiff(refValue: float; estimates: seq[float]): float =
## Compute the mean of the squares of the differences
## between estimated values and a reference value.
for estimate in estimates:
result += (estimate - refValue)^2
result /= estimates.len.toFloat
 
 
const Samples = [(trueValue: 49.0, estimates: @[48.0, 47.0, 51.0]),
(trueValue: 49.0, estimates: @[48.0, 47.0, 51.0, 42.0])]
 
for (trueValue, estimates, ) in Samples:
let m = mean(estimates)
echo "True value: ", trueValue
echo "Estimates: ", estimates.join(", ")
echo "Average error: ", meanSquareDiff(trueValue, estimates)
echo "Crowd error: ", (m - trueValue)^2
echo "Prediction diversity: ", meanSquareDiff(m, estimates)
echo ""
Output:
True value:           49.0
Estimates:            48.0, 47.0, 51.0
Average error:        3.0
Crowd error:          0.1111111111111127
Prediction diversity: 2.888888888888889

True value:           49.0
Estimates:            48.0, 47.0, 51.0, 42.0
Average error:        14.5
Crowd error:          4.0
Prediction diversity: 10.5

Perl[edit]

sub diversity {
my($truth, @pred) = @_;
my($ae,$ce,$cp,$pd,$stats);
 
$cp += $_/@pred for @pred; # collective prediction
$ae = avg_error($truth, @pred); # average individual error
$ce = ($cp - $truth)**2; # collective error
$pd = avg_error($cp, @pred); # prediction diversity
 
my $fmt = "%13s: %6.3f\n";
$stats = sprintf $fmt, 'average-error', $ae;
$stats .= sprintf $fmt, 'crowd-error', $ce;
$stats .= sprintf $fmt, 'diversity', $pd;
}
 
sub avg_error {
my($m, @v) = @_;
my($avg_err);
$avg_err += ($_ - $m)**2 for @v;
$avg_err/@v;
}
 
print diversity(49, qw<48 47 51>) . "\n";
print diversity(49, qw<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

Phix[edit]

function mean(sequence s)
return sum(s)/length(s)
end function
 
function variance(sequence s, atom d)
return mean(sq_power(sq_sub(s,d),2))
end function
 
function diversity_theorem(atom reference, sequence observations)
atom average_error = variance(observations,reference),
average = mean(observations),
crowd_error = power(reference-average,2),
diversity = variance(observations,average)
return {{"average_error",average_error},
{"crowd_error",crowd_error},
{"diversity",diversity}}
end function
 
procedure test(atom reference, sequence observations)
sequence res = diversity_theorem(reference, observations)
for i=1 to length(res) do
printf(1," %14s : %g\n",res[i])
end for
end procedure
test(49, {48, 47, 51})
test(49, {48, 47, 51, 42})
Output:
  average_error : 3
    crowd_error : 0.111111
      diversity : 2.88889
  average_error : 14.5
    crowd_error : 4
      diversity : 10.5

PureBasic[edit]

Define.f ref=49.0, mea
NewList argV.f()
 
Macro put
Print(~"\n["+StrF(ref)+"]"+#TAB$)
ForEach argV() : Print(StrF(argV())+#TAB$) : Next
PrintN(~"\nAverage Error : "+StrF(vari(argV(),ref),5))
PrintN("Crowd Error  : "+StrF((ref-mea)*(ref-mea),5))
PrintN("Diversity  : "+StrF(vari(argV(),mea),5))
EndMacro
 
Macro LetArgV(v)
AddElement(argV()) : argV()=v
EndMacro
 
Procedure.f mean(List x.f())
Define.f m
ForEach x() : m+x() : Next
ProcedureReturn m/ListSize(x())
EndProcedure
 
Procedure.f vari(List x.f(),r.f)
NewList nx.f()
ForEach x() : AddElement(nx()) : nx()=(r-x())*(r-x()) : Next
ProcedureReturn mean(nx())
EndProcedure
 
If OpenConsole()=0 : End 1 : EndIf
Gosub SetA : ClearList(argV())
Gosub SetB : Input()
End
 
SetA:
LetArgV(48.0) : LetArgV(47.0) : LetArgV(51.0)
mea=mean(argV()) : put
Return
 
SetB:
LetArgV(48.0) : LetArgV(47.0) : LetArgV(51.0) : LetArgV(42.0)
mea=mean(argV()) : put
Return
Output:
[49]	48	47	51	
Average Error : 3.00000
Crowd Error   : 0.11111
Diversity     : 2.88889

[49]	48	47	51	42	
Average Error : 14.50000
Crowd Error   : 4.00000
Diversity     : 10.50000

Python[edit]

By composition of pure functions:

Works with: Python version 3.7
'''Diversity prediction theorem'''
 
from itertools import chain
from functools import reduce
 
 
# diversityValues :: Num a => a -> [a] ->
# { mean-Error :: a, crowd-error :: a, diversity :: a }
def diversityValues(x):
'''The mean error, crowd error and
diversity, for a given observation x
and a non-empty list of predictions ps.
'''

def go(ps):
mp = mean(ps)
return {
'mean-error': meanErrorSquared(x)(ps),
'crowd-error': pow(x - mp, 2),
'diversity': meanErrorSquared(mp)(ps)
}
return go
 
 
# meanErrorSquared :: Num -> [Num] -> Num
def meanErrorSquared(x):
'''The mean of the squared differences
between the observed value x and
a non-empty list of predictions ps.
'''

def go(ps):
return mean([
pow(p - x, 2) for p in ps
])
return go
 
 
# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Observed value: 49,
prediction lists: various.
'''

 
print(unlines(map(
showDiversityValues(49),
[
[48, 47, 51],
[48, 47, 51, 42],
[50, '?', 50, {}, 50], # Non-numeric values.
[] # Missing predictions.
]
)))
print(unlines(map(
showDiversityValues('49'), # String in place of number.
[
[50, 50, 50],
[40, 35, 40],
]
)))
 
 
# ---------------------- FORMATTING ----------------------
 
# showDiversityValues :: Num -> [Num] -> Either String String
def showDiversityValues(x):
'''Formatted string representation
of diversity values for a given
observation x and a non-empty
list of predictions p.
'''

def go(ps):
def showDict(dct):
w = 4 + max(map(len, dct.keys()))
 
def showKV(a, kv):
k, v = kv
return a + k.rjust(w, ' ') + (
' : ' + showPrecision(3)(v) + '\n'
)
return 'Predictions: ' + showList(ps) + ' ->\n' + (
reduce(showKV, dct.items(), '')
)
 
def showProblem(e):
return (
unlines(map(indented(1), e)) if (
isinstance(e, list)
) else indented(1)(repr(e))
) + '\n'
 
return 'Observation: ' + repr(x) + '\n' + (
either(showProblem)(showDict)(
bindLR(numLR(x))(
lambda n: bindLR(numsLR(ps))(
compose(Right, diversityValues(n))
)
)
)
)
return go
 
 
# ------------------ GENERIC FUNCTIONS -------------------
 
# Left :: a -> Either a b
def Left(x):
'''Constructor for an empty Either (option type) value
with an associated string.
'''

return {'type': 'Either', 'Right': None, 'Left': x}
 
 
# Right :: b -> Either a b
def Right(x):
'''Constructor for a populated Either (option type) value'''
return {'type': 'Either', 'Left': None, 'Right': x}
 
 
# bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
def bindLR(m):
'''Either monad injection operator.
Two computations sequentially composed,
with any value produced by the first
passed as an argument to the second.
'''

def go(mf):
return (
mf(m.get('Right')) if None is m.get('Left') else m
)
return go
 
 
# compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
'''Composition, from right to left,
of a series of functions.
'''

def go(f, g):
def fg(x):
return f(g(x))
return fg
return reduce(go, fs, identity)
 
 
# concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
'''A concatenated list over which a function has been mapped.
The list monad can be derived by using a function f which
wraps its output in a list,
(using an empty list to represent computational failure).
'''

def go(xs):
return chain.from_iterable(map(f, xs))
return go
 
 
# either :: (a -> c) -> (b -> c) -> Either a b -> c
def either(fl):
'''The application of fl to e if e is a Left value,
or the application of fr to e if e is a Right value.
'''

return lambda fr: lambda e: fl(e['Left']) if (
None is e['Right']
) else fr(e['Right'])
 
 
# identity :: a -> a
def identity(x):
'''The identity function.'''
return x
 
 
# indented :: Int -> String -> String
def indented(n):
'''String indented by n multiples
of four spaces.
'''

return lambda s: (4 * ' ' * n) + s
 
# mean :: [Num] -> Float
def mean(xs):
'''Arithmetic mean of a list
of numeric values.
'''

return sum(xs) / float(len(xs))
 
 
# numLR :: a -> Either String Num
def numLR(x):
'''Either Right x if x is a float or int,
or a Left explanatory message.'''

return Right(x) if (
isinstance(x, (float, int))
) else Left(
'Expected number, saw: ' + (
str(type(x)) + ' ' + repr(x)
)
)
 
 
# numsLR :: [a] -> Either String [Num]
def numsLR(xs):
'''Either Right xs if all xs are float or int,
or a Left explanatory message.'''

def go(ns):
ls, rs = partitionEithers(map(numLR, ns))
return Left(ls) if ls else Right(rs)
return bindLR(
Right(xs) if (
bool(xs) and isinstance(xs, list)
) else Left(
'Expected a non-empty list, saw: ' + (
str(type(xs)) + ' ' + repr(xs)
)
)
)(go)
 
 
# partitionEithers :: [Either a b] -> ([a],[b])
def partitionEithers(lrs):
'''A list of Either values partitioned into a tuple
of two lists, with all Left elements extracted
into the first list, and Right elements
extracted into the second list.
'''

def go(a, x):
ls, rs = a
r = x.get('Right')
return (ls + [x.get('Left')], rs) if None is r else (
ls, rs + [r]
)
return reduce(go, lrs, ([], []))
 
 
# showList :: [a] -> String
def showList(xs):
'''Compact string representation of a list'''
return '[' + ','.join(str(x) for x in xs) + ']'
 
 
# showPrecision :: Int -> Float -> String
def showPrecision(n):
'''A string showing a floating point number
at a given degree of precision.'''

def go(x):
return str(round(x, n))
return go
 
 
# unlines :: [String] -> String
def unlines(xs):
'''A single string derived by the intercalation
of a list of strings with the newline character.'''

return '\n'.join(xs)
 
 
# MAIN ---
if __name__ == '__main__':
main()
Output:
Observation:  49
Predictions: [48,47,51] ->
     mean-error : 3.0
    crowd-error : 0.111
      diversity : 2.889

Observation:  49
Predictions: [48,47,51,42] ->
     mean-error : 14.5
    crowd-error : 4.0
      diversity : 10.5

Observation:  49
    Expected number, saw: <class 'str'> '?'
    Expected number, saw: <class 'dict'> {}

Observation:  49
    "Expected a non-empty list, saw: <class 'list'> []"

Observation:  '49'
    "Expected number, saw: <class 'str'> '49'"

Observation:  '49'
    "Expected number, saw: <class 'str'> '49'"

R[edit]

R's vectorisation shines here. The hardest part of this task was giving each estimate its own numbered column, which is little more than a printing luxury. The actual mathematics was trivial, with each part done in essentially one line.

diversityStats<-function(trueValue, estimates)
{
collectivePrediction<-mean(estimates)
data.frame("True Value" = trueValue,
as.list(setNames(estimates, paste("Guess", seq_along(estimates)))), #Guesses, each with a title and column.
"Average Error" = mean((trueValue-estimates)^2),
"Crowd Error" = (trueValue-collectivePrediction)^2,
"Prediction Diversity" = mean((estimates-collectivePrediction)^2))
}
diversityStats(49, c(48, 47, 51))
diversityStats(49, c(48, 47, 51, 42))
Output:
> diversityStats(49, c(48, 47, 51))
  True.Value Guess.1 Guess.2 Guess.3 Average.Error Crowd.Error Prediction.Diversity
1         49      48      47      51             3   0.1111111             2.888889

> diversityStats(49, c(48, 47, 51, 42))
  True.Value Guess.1 Guess.2 Guess.3 Guess.4 Average.Error Crowd.Error Prediction.Diversity
1         49      48      47      51      42          14.5           4                 10.5

Raku[edit]

(formerly Perl 6)

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]

Uses greater precision, but rounds the output to six decimal digits past the decimal point   (see the last comment in the program).

/*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.*/
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

Ruby[edit]

Translation of: D
def mean(a) = a.sum(0.0) / a.size
def mean_square_diff(a, predictions) = mean(predictions.map { |x| square(x - a)**2 })
 
def diversity_theorem(truth, predictions)
average = mean(predictions)
puts "truth: #{truth}, predictions #{predictions}",
"average-error: #{mean_square_diff(truth, predictions)}",
"crowd-error: #{(truth - average)**2}",
"diversity: #{mean_square_diff(average, predictions)}",""
end
 
diversity_theorem(49.0, [48.0, 47.0, 51.0])
diversity_theorem(49.0, [48.0, 47.0, 51.0, 42.0])
Output:
truth: 49.0, predictions [48.0, 47.0, 51.0]
average-error: 3.0
crowd-error: 0.11111111111111269
diversity: 2.888888888888889

truth: 49.0, predictions [48.0, 47.0, 51.0, 42.0]
average-error: 14.5
crowd-error: 4.0
diversity: 10.5

Scala[edit]

Translation of: Kotlin
object DiversityPredictionTheorem {
def square(d: Double): Double
= d * d
 
def average(a: Array[Double]): Double
= a.sum / a.length
 
def averageSquareDiff(d: Double, predictions: Array[Double]): Double
= average(predictions.map(it => square(it - d)))
 
def diversityTheorem(truth: Double, predictions: Array[Double]): String = {
val avg = average(predictions)
f"average-error : ${averageSquareDiff(truth, predictions)}%6.3f\n" +
f"crowd-error  : ${square(truth - avg)}%6.3f\n"+
f"diversity  : ${averageSquareDiff(avg, predictions)}%6.3f\n"
}
 
def main(args: Array[String]): Unit = {
println(diversityTheorem(49.0, Array(48.0, 47.0, 51.0)))
println(diversityTheorem(49.0, Array(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

Sidef[edit]

Translation of: Raku
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 }

Visual Basic .NET[edit]

Translation of: C#
Module Module1
 
Function Square(x As Double) As Double
Return x * x
End Function
 
Function AverageSquareDiff(a As Double, predictions As IEnumerable(Of Double)) As Double
Return predictions.Select(Function(x) Square(x - a)).Average()
End Function
 
Sub DiversityTheorem(truth As Double, predictions As IEnumerable(Of Double))
Dim average = predictions.Average()
Console.WriteLine("average-error: {0}", AverageSquareDiff(truth, predictions))
Console.WriteLine("crowd-error: {0}", Square(truth - average))
Console.WriteLine("diversity: {0}", AverageSquareDiff(average, predictions))
End Sub
 
Sub Main()
DiversityTheorem(49.0, {48.0, 47.0, 51.0})
DiversityTheorem(49.0, {48.0, 47.0, 51.0, 42.0})
End Sub
 
End Module
Output:
average-error: 3
crowd-error: 0.111111111111113
diversity: 2.88888888888889
average-error: 14.5
crowd-error: 4
diversity: 10.5

Wren[edit]

Translation of: Go
import "/fmt" for Fmt
 
var averageSquareDiff = Fn.new { |f, preds|
var av = 0
for (pred in preds) av = av + (pred-f)*(pred-f)
return av/preds.count
}
 
var diversityTheorem = Fn.new { |truth, preds|
var av = (preds.reduce { |sum, pred| sum + pred }) / preds.count
var avErr = averageSquareDiff.call(truth, preds)
var crowdErr = (truth-av) * (truth-av)
var div = averageSquareDiff.call(av, preds)
return [avErr, crowdErr, div]
}
 
var predsList = [ [48, 47, 51], [48, 47, 51, 42] ]
var truth = 49
for (preds in predsList) {
var res = diversityTheorem.call(truth, preds)
Fmt.print("Average-error : $6.3f", res[0])
Fmt.print("Crowd-error  : $6.3f", res[1])
Fmt.print("Diversity  : $6.3f\n", res[2])
}
Output:
Average-error :  3.000
Crowd-error   :  0.111
Diversity     :  2.889

Average-error : 14.500
Crowd-error   :  4.000
Diversity     : 10.500

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