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
<lang 11l>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])</lang>
- Output:
average-error: 3 crowd-error: 0.111111111 diversity: 2.888888889 average-error: 14.5 crowd-error: 4 diversity: 10.5
C
Accepts inputs from command line, prints out usage on incorrect invocation. <lang C>
- 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#
<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
C++
<lang Cpp>
- 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
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}
D
<lang d>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]);
}</lang>
- Output:
average-error: 3 crowd-error: 0.111111 diversity: 2.88889 average-error: 14.5 crowd-error: 4 diversity: 10.5
Factor
<lang factor>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 .</lang>
- 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
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
<lang go>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) }
}</lang>
- Output:
Average-error : 3.000 Crowd-error : 0.111 Diversity : 2.889 Average-error : 14.500 Crowd-error : 4.000 Diversity : 10.500
Groovy
<lang groovy>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[])) }
}</lang>
- Output:
average-error : 3.000 crowd-error : 0.111 diversity : 2.889 average-error : 14.500 crowd-error : 4.000 diversity : 10.500
Haskell
<lang haskell>mean :: (Fractional a, Foldable t) => t a -> a mean lst = sum lst / fromIntegral (length lst)
meanSq :: Fractional c => c -> [c] -> c meanSq x = mean . map (\y -> (x-y)^^2)
diversityPrediction x estimates = do
putStrLn $ "TrueValue:\t" ++ show x putStrLn $ "CrowdEstimates:\t" ++ show estimates let avg = mean estimates let avgerr = meanSq x estimates putStrLn $ "AverageError:\t" ++ show avgerr let crowderr = (x - avg)^^2 putStrLn $ "CrowdError:\t" ++ show crowderr let diversity = meanSq avg estimates putStrLn $ "Diversity:\t" ++ show diversity</lang>
λ> diversityPrediction 49 [48,47,51] TrueValue: 49.0 CrowdEstimates: [48.0,47.0,51.0] AverageError: 3.0 CrowdError: 0.11111111111111269 Diversity: 2.888888888888889 λ> diversityPrediction 49 [48,47,51,42] TrueValue: 49.0 CrowdEstimates: [48.0,47.0,51.0,42.0] AverageError: 14.5 CrowdError: 4.0 Diversity: 10.5
J
Accepts inputs from command line, prints out usage on incorrect invocation. Were this compressed adaptation from C the content of file d.ijs <lang C> echo 'Use: ' , (;:inv 2 {. ARGV) , ' <reference value> <observations>'
data=: ([: ". [: ;:inv 2&}.) ::([: exit 1:) ARGV
([: exit (1: echo@('insufficient data'"_)))^:(2 > #) data
mean=: +/ % # variance=: [: mean [: *: -
averageError=: ({. variance }.)@:] crowdError=: variance {. diversity=: variance }.
echo (<;._2'average error;crowd error;diversity;') ,: ;/ (averageError`crowdError`diversity`:0~ mean@:}.) data
exit 0 </lang> 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
<lang java>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})); }
}</lang>
- Output:
average-error : 3.000 crowd-error : 0.111 diversity : 2.889 average-error : 14.500 crowd-error : 4.000 diversity : 10.500
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';
// 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()
})();</lang>
- Output:
[ { "mean-error": "3.00", "crowd-error": "0.111", "diversity": "2.89" }, { "mean-error": "14.5", "crowd-error": "4.00", "diversity": "10.5" } ]
Jsish
From Typescript entry. <lang javascript>/* 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!
- /</lang>
- Output:
prompt$ jsish -u diversityPrediction.jsi [PASS] diversityPrediction.jsi
jq
Works with gojq, the Go implementation of jq <lang 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(. - $mean) | map(pow(.;2)) | mean) } ;</lang><lang jq># The task:
([49, [48, 47, 51]], [49, [48, 47, 51, 42] ]) | . as [$actual, $predicted] | diversitytheorem($actual; $predicted)</lang>
- Output:
{ "avgerr": 3, "crderr": 0.11111111111111269, "divers": 2.888888888888889 } { "avgerr": 14.5, "crderr": 4, "divers": 10.5 }
Julia
<lang julia>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</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
<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
Lua
<lang lua>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()</lang>
- Output:
average-error: 3 crowd-error: 0.11111111111111 diversity: 2.8888888888889 average-error: 14.5 crowd-error: 4 diversity: 10.5
Mathematica/Wolfram Language
<lang Mathematica>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</lang>
- 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
<lang Nim>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 ""</lang>
- 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
<lang perl>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; $avg_err/@v;
}
print diversity(49, qw<48 47 51>) . "\n"; print diversity(49, qw<48 47 51 42>);</lang>
- Output:
average-error: 3.000 crowd-error: 0.111 diversity: 2.889 average-error: 14.500 crowd-error: 4.000 diversity: 10.500
Phix
with javascript_semantics function mean(sequence s) return sum(s)/length(s) end function function variance(sequence s, atom d) return mean(sq_power(sq_sub(s,d),2)) end function function diversity_theorem(atom reference, sequence observations) atom average_error = variance(observations,reference), average = mean(observations), crowd_error = power(reference-average,2), diversity = variance(observations,average) return {{"average_error",average_error}, {"crowd_error",crowd_error}, {"diversity",diversity}} end function procedure test(atom reference, sequence observations) sequence res = diversity_theorem(reference, observations) for i=1 to length(res) do printf(1," %14s : %g\n",res[i]) end for end procedure test(49, {48, 47, 51}) test(49, {48, 47, 51, 42})
- Output:
average_error : 3 crowd_error : 0.111111 diversity : 2.88889 average_error : 14.5 crowd_error : 4 diversity : 10.5
PureBasic
<lang PureBasic>Define.f ref=49.0, mea NewList argV.f()
Macro put
Print(~"\n["+StrF(ref)+"]"+#TAB$) 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</lang>
- 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
By composition of pure functions:
<lang python>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()</lang>
- 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
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. <lang rsplus>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))</lang>
- 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
Racket
<lang racket>#lang racket
(define (mean l)
(/ (apply + l) (length l)))
(define (diversity-theorem truth predictions)
(define μ (mean predictions)) (define (avg-sq-diff a) (mean (map (λ (p) (sqr (- p a))) predictions))) (hash 'average-error (avg-sq-diff truth) 'crowd-error (sqr (- truth μ)) 'diversity (avg-sq-diff μ)))
(println (diversity-theorem 49 '(48 47 51))) (println (diversity-theorem 49 '(48 47 51 42)))</lang>
- Output:
'#hash((average-error . 3) (crowd-error . 1/9) (diversity . 2 8/9)) '#hash((average-error . 14 1/2) (crowd-error . 4) (diversity . 10 1/2))
Raku
(formerly 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
Uses greater precision, but rounds the output to six decimal digits past the decimal point (see the last comment in the program). <lang rexx>/*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.*/</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
Ruby
<lang ruby>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])</lang>
- 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
<lang scala>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))) }
}</lang>
- Output:
average-error : 3.000 crowd-error : 0.111 diversity : 2.889 average-error : 14.500 crowd-error : 4.000 diversity : 10.500
Sidef
<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 }
Visual Basic .NET
<lang vbnet>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</lang>
- Output:
average-error: 3 crowd-error: 0.111111111111113 diversity: 2.88888888888889 average-error: 14.5 crowd-error: 4 diversity: 10.5
Wren
<lang ecmascript>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])
}</lang>
- Output:
Average-error : 3.000 Crowd-error : 0.111 Diversity : 2.889 Average-error : 14.500 Crowd-error : 4.000 Diversity : 10.500
zkl
<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