Verify distribution uniformity/Naive
You are encouraged to solve this task according to the task description, using any language you may know.
This task is an adjunct to Seven-sided dice from five-sided dice.
- Task
Create a function to check that the random integers returned from a small-integer generator function have uniform distribution.
The function should take as arguments:
- The function (or object) producing random integers.
- The number of times to call the integer generator.
- A 'delta' value of some sort that indicates how close to a flat distribution is close enough.
The function should produce:
- Some indication of the distribution achieved.
- An 'error' if the distribution is not flat enough.
Show the distribution checker working when the produced distribution is flat enough and when it is not. (Use a generator from Seven-sided dice from five-sided dice).
See also:
11l
<lang 11l>F dice5()
R random:(1..5)
F distcheck(func, repeats, delta)
V bin = DefaultDict[Int, Int]()
L 1..repeats
bin[func()]++
V target = repeats I/ bin.len
V deltacount = Int(delta / 100.0 * target)
assert(all(bin.values().map(count -> abs(@target - count) < @deltacount)), ‘Bin distribution skewed from #. +/- #.: #.’.format(target, deltacount, sorted(bin.items()).map((key, count) -> (key, @target - count))))
print(bin)
distcheck(dice5, 1000000, 1)</lang>
- Output:
DefaultDict([1 = 199586, 2 = 200094, 3 = 198933, 4 = 200824, 5 = 200563])
Ada
<lang Ada>with Ada.Numerics.Discrete_Random, Ada.Text_IO;
procedure Naive_Random is
type M_1000 is mod 1000; package Rand is new Ada.Numerics.Discrete_Random(M_1000); Gen: Rand.Generator;
procedure Perform(Modulus: Positive; Expected, Margin: Natural;
Passed: out boolean) is
Low: Natural := (100-Margin) * Expected/100;
High: Natural := (100+Margin) * Expected/100;
Buckets: array(0 .. Modulus-1) of Natural := (others => 0);
Index: Natural;
begin
for I in 1 .. Expected * Modulus loop
Index := Integer(Rand.Random(Gen)) mod Modulus;
Buckets(Index) := Buckets(Index) + 1;
end loop;
Passed := True;
for I in Buckets'Range loop
Ada.Text_IO.Put("Bucket" & Integer'Image(I+1) & ":" &
Integer'Image(Buckets(I)));
if Buckets(I) < Low or else Buckets(I) > High then
Ada.Text_IO.Put_Line(" (failed)");
Passed := False;
else
Ada.Text_IO.Put_Line(" (passed)");
end if;
end loop;
Ada.Text_IO.New_Line;
end Perform;
Number_Of_Buckets: Positive := Natural'Value(Ada.Text_IO.Get_Line); Expected_Per_Bucket: Natural := Natural'Value(Ada.Text_IO.Get_Line); Margin_In_Percent: Natural := Natural'Value(Ada.Text_IO.Get_Line); OK: Boolean;
begin
Ada.Text_IO.Put_Line( "Buckets:" & Integer'Image(Number_Of_Buckets) &
", Expected:" & Integer'Image(Expected_Per_Bucket) &
", Margin:" & Integer'Image(Margin_In_Percent));
Rand.Reset(Gen);
Perform(Modulus => Number_Of_Buckets,
Expected => Expected_Per_Bucket,
Margin => Margin_In_Percent,
Passed => OK);
Ada.Text_IO.Put_Line("Test Passed? (" & Boolean'Image(OK) & ")");
end Naive_Random;</lang>
Sample run 1 (all buckets good):
7 1000 3 Buckets: 7, Expected: 1000, Margin: 3 Bucket 1: 1006 (passed) Bucket 2: 1030 (passed) Bucket 3: 997 (passed) Bucket 4: 985 (passed) Bucket 5: 976 (passed) Bucket 6: 1024 (passed) Bucket 7: 982 (passed) Test Passed? (TRUE)
Sample run 2 (some buckets too large / to small):
7 1000 3 Buckets: 7, Expected: 1000, Margin: 3 Bucket 1: 1034 (failed) Bucket 2: 985 (passed) Bucket 3: 1025 (passed) Bucket 4: 933 (failed) Bucket 5: 1000 (passed) Bucket 6: 1016 (passed) Bucket 7: 1007 (passed) Test Passed? (FALSE)
AutoHotkey
<lang AutoHotkey>MsgBox, % DistCheck("dice7",10000,3)
DistCheck(function, repetitions, delta) { Loop, % 7 ; initialize array
{ bucket%A_Index% := 0
}
Loop, % repetitions ; populate buckets
{ v := %function%()
bucket%v% += 1
}
lbnd := round((repetitions/7)*(100-delta)/100)
ubnd := round((repetitions/7)*(100+delta)/100)
text := "Distribution check:`n`nTotal elements = " repetitions
. "`n`nMargin = " delta "% --> Lbound = " lbnd ", Ubound = " ubnd "`n"
Loop, % 7
{ text := text "`nBucket " A_Index " contains " bucket%A_Index% " elements."
If bucket%A_Index% not between %lbnd% and %ubnd%
text := text " Skewed."
}
Return, text
}</lang>
Distribution check: Total elements = 10000 Margin = 3% --> Lbound = 1386, Ubound = 1471 Bucket 1 contains 1450 elements. Bucket 2 contains 1374 elements. Skewed. Bucket 3 contains 1412 elements. Bucket 4 contains 1465 elements. Bucket 5 contains 1370 elements. Skewed. Bucket 6 contains 1485 elements. Skewed. Bucket 7 contains 1444 elements.
BBC BASIC
<lang bbcbasic> MAXRND = 7
FOR r% = 2 TO 5
check% = FNdistcheck(FNdice5, 10^r%, 0.05)
PRINT "Over "; 10^r% " runs dice5 ";
IF check% THEN
PRINT "failed distribution check with "; check% " bin(s) out of range"
ELSE
PRINT "passed distribution check"
ENDIF
NEXT
END
DEF FNdistcheck(RETURN func%, repet%, delta)
LOCAL i%, m%, r%, s%, bins%()
DIM bins%(MAXRND)
FOR i% = 1 TO repet%
r% = FN(^func%)
bins%(r%) += 1
IF r%>m% m% = r%
NEXT
FOR i% = 1 TO m%
IF bins%(i%)/(repet%/m%) > 1+delta s% += 1
IF bins%(i%)/(repet%/m%) < 1-delta s% += 1
NEXT
= s%
DEF FNdice5 = RND(5)</lang>
Output:
Over 100 runs dice5 failed distribution check with 3 bin(s) out of range Over 1000 runs dice5 failed distribution check with 1 bin(s) out of range Over 10000 runs dice5 passed distribution check Over 100000 runs dice5 passed distribution check
C
<lang c>#include <stdlib.h>
- include <stdio.h>
- include <math.h>
inline int rand5() { int r, rand_max = RAND_MAX - (RAND_MAX % 5); while ((r = rand()) >= rand_max); return r / (rand_max / 5) + 1; }
inline int rand5_7() { int r; while ((r = rand5() * 5 + rand5()) >= 27); return r / 3 - 1; }
/* assumes gen() returns a value from 1 to n */ int check(int (*gen)(), int n, int cnt, double delta) /* delta is relative */ { int i = cnt, *bins = calloc(sizeof(int), n); double ratio; while (i--) bins[gen() - 1]++; for (i = 0; i < n; i++) { ratio = bins[i] * n / (double)cnt - 1; if (ratio > -delta && ratio < delta) continue;
printf("bin %d out of range: %d (%g%% vs %g%%), ", i + 1, bins[i], ratio * 100, delta * 100); break; } free(bins); return i == n; }
int main() { int cnt = 1; while ((cnt *= 10) <= 1000000) { printf("Count = %d: ", cnt); printf(check(rand5_7, 7, cnt, 0.03) ? "flat\n" : "NOT flat\n"); }
return 0;
}</lang>output
Count = 10: bin 1 out of range: 1 (-30% vs 3%), NOT flat Count = 100: bin 1 out of range: 11 (-23% vs 3%), NOT flat Count = 1000: bin 1 out of range: 150 (5% vs 3%), NOT flat Count = 10000: bin 3 out of range: 1477 (3.39% vs 3%), NOT flat Count = 100000: flat Count = 1000000: flat
C++
<lang cpp>#include <map>
- include <iostream>
- include <cmath>
template<typename F>
bool test_distribution(F f, int calls, double delta)
{
typedef std::map<int, int> distmap; distmap dist;
for (int i = 0; i < calls; ++i) ++dist[f()];
double mean = 1.0/dist.size();
bool good = true;
for (distmap::iterator i = dist.begin(); i != dist.end(); ++i)
{
if (std::abs((1.0 * i->second)/calls - mean) > delta)
{
std::cout << "Relative frequency " << i->second/(1.0*calls)
<< " of result " << i->first
<< " deviates by more than " << delta
<< " from the expected value " << mean << "\n";
good = false;
}
}
return good;
}</lang>
Clojure
The code could be shortened if the verify function did the output itself, but the "Clojure way" is to have functions, whenever possible, avoid side effects (like printing) and just return a result. Then the "application-level" code uses doseq and println to display the output to the user. The built-in (rand-int) function is used for all three runs of the verify function, but first with small N to simulate experimental error in a small sample size, then with larger N to show it working properly on large N. <lang clojure>(defn verify [rand n & [delta]]
(let [rands (frequencies (repeatedly n rand))
avg (/ (reduce + (map val rands)) (count rands))
max-delta (* avg (or delta 1/10))
acceptable? #(<= (- avg max-delta) % (+ avg max-delta))]
(for [[num count] (sort rands)]
[num count (acceptable? count)])))
(doseq [n [100 1000 10000]
[num count okay?] (verify #(rand-int 7) n)]
(println "Saw" num count "times:"
(if okay? "that's" " not") "acceptable"))</lang>
Saw 1 13 times: that's acceptable Saw 2 15 times: that's acceptable Saw 3 11 times: not acceptable Saw 4 10 times: not acceptable Saw 5 19 times: not acceptable Saw 6 17 times: not acceptable Saw 0 121 times: not acceptable Saw 1 128 times: not acceptable Saw 2 161 times: not acceptable Saw 3 146 times: that's acceptable Saw 4 134 times: that's acceptable Saw 5 170 times: not acceptable Saw 6 140 times: that's acceptable Saw 0 1480 times: that's acceptable Saw 1 1372 times: that's acceptable Saw 2 1411 times: that's acceptable Saw 3 1442 times: that's acceptable Saw 4 1395 times: that's acceptable Saw 5 1432 times: that's acceptable Saw 6 1468 times: that's acceptable
Common Lisp
<lang lisp>(defun check-distribution (function n &optional (delta 1.0))
(let ((bins (make-hash-table)))
(loop repeat n do (incf (gethash (funcall function) bins 0)))
(loop with target = (/ n (hash-table-count bins))
for key being each hash-key of bins using (hash-value value)
when (> (abs (- value target)) (* 0.01 delta n))
do (format t "~&Distribution potentially skewed for ~w:~
expected around ~w got ~w." key target value)
finally (return bins))))</lang>
> (check-distribution 'd7 1000)
Distribution potentially skewed for 1: expected around 1000/7 got 153.
Distribution potentially skewed for 2: expected around 1000/7 got 119.
Distribution potentially skewed for 3: expected around 1000/7 got 125.
Distribution potentially skewed for 7: expected around 1000/7 got 156.
T
#<EQL Hash Table{7} 200B5A53>
> (check-distribution 'd7 10000)
NIL
#<EQL Hash Table{7} 200CB5BB>
D
<lang d>import std.stdio, std.string, std.math, std.algorithm, std.traits;
/** Bin the answers to fn() and check bin counts are within +/- delta % of repeats/bincount.
- /
void distCheck(TF)(in TF func, in int nRepeats, in double delta) /*@safe*/ if (isCallable!TF) {
int[int] counts;
foreach (immutable i; 0 .. nRepeats)
counts[func()]++;
immutable double target = nRepeats / double(counts.length);
immutable int deltaCount = cast(int)(delta / 100.0 * target);
foreach (immutable k, const count; counts)
if (abs(target - count) >= deltaCount)
throw new Exception(format(
"distribution potentially skewed for '%s': '%d'\n",
k, count));
foreach (immutable k; counts.keys.sort())
writeln(k, " ", counts[k]);
writeln;
}
version (verify_distribution_uniformity_naive_main) {
void main() {
import std.random;
distCheck(() => uniform(1, 6), 1_000_000, 1);
}
}</lang> If compiled with -version=verify_distribution_uniformity_naive_main:
- Output:
1 199389 2 2 4 200016 5 200424
Elixir
<lang elixir>defmodule VerifyDistribution do
def naive( generator, times, delta_percent ) do
dict = Enum.reduce( List.duplicate(generator, times), Map.new, &update_counter/2 )
values = Map.values(dict)
average = Enum.sum( values ) / map_size( dict )
delta = average * (delta_percent / 100)
fun = fn {_key, value} -> abs(value - average) > delta end
too_large_dict = Enum.filter( dict, fun )
return( length(too_large_dict), too_large_dict, average, delta_percent )
end
def return( 0, _too_large_dict, _average, _delta ), do: :ok
def return( _n, too_large_dict, average, delta ) do
{:error, {too_large_dict, :failed_expected_average, average, 'with_delta_%', delta}}
end
def update_counter( fun, dict ), do: Map.update( dict, fun.(), 1, &(&1+1) )
end
fun = fn -> Dice.dice7 end IO.inspect VerifyDistribution.naive( fun, 100000, 3 ) IO.inspect VerifyDistribution.naive( fun, 100, 3 )</lang>
- Output:
:ok
{:error,
{[{1, 16}, {2, 10}, {4, 15}, {5, 8}, {6, 20}, {7, 17}],
:failed_expected_average, 14.285714285714286, 'with_delta_%', 3}}
Erlang
<lang Erlang> -module( verify_distribution_uniformity ).
-export( [naive/3] ).
naive( Generator, Times, Delta_percent ) ->
Dict = lists:foldl( fun update_counter/2, dict:new(), lists:duplicate(Times, Generator) ), Values = [dict:fetch(X, Dict) || X <- dict:fetch_keys(Dict)], Average = lists:sum( Values ) / dict:size( Dict ), Delta = Average * (Delta_percent / 100), Fun = fun(_Key, Value) -> erlang:abs(Value - Average) > Delta end, Too_large_dict = dict:filter( Fun, Dict ), return( dict:size(Too_large_dict), Too_large_dict, Average, Delta_percent ).
return( 0, _Too_large_dict, _Average, _Delta ) -> ok; return( _N, Too_large_dict, Average, Delta ) -> {error, {dict:to_list(Too_large_dict), failed_expected_average, Average, 'with_delta_%', Delta}}.
update_counter( Fun, Dict ) -> dict:update_counter( Fun(), 1, Dict ). </lang>
- Output:
Calling dice:dice7/0 few times shows skewed distribution.
61> Fun = fun dice:dice7/0.
62> verify_distribution_uniformity:naive( Fun, 100000, 3).
ok
63> verify_distribution_uniformity:naive( Fun, 100, 3).
{error,{[{3,15},{6,15},{5,13},{1,20},{4,11},{7,12}],
failed_expected_average,14.285714285714286,'with_delta_%',
3}}
Euler Math Toolbox
Following the task verbatim.
<lang> >function checkrandom (frand$, n:index, delta:positive real) ... $ v=zeros(1,n); $ loop 1 to n; v{#}=frand$(); end; $ K=max(v); $ fr=getfrequencies(v,1:K); $ return max(fr/n-1/K)<delta/sqrt(n); $ endfunction >function dice () := intrandom(1,1,6); >checkrandom("dice",1000000,1)
1
>wd = 0|((1:6)+[-0.01,0.01,0,0,0,0])/6
[ 0 0.165 0.335 0.5 0.666666666667 0.833333333333 1 ]
>function wrongdice () := find(wd,random()) >checkrandom("wrongdice",1000000,1)
0
</lang>
Checking the dice7 from dice5 generator.
<lang> >function dice5 () := intrandom(1,1,5); >function dice7 () ... $ repeat $ k=(dice5()-1)*5+dice5(); $ if k<=21 then return ceil(k/3); endif; $ end; $ endfunction >checkrandom("dice7",1000000,1)
1
</lang>
Faster implementation with the matrix language.
<lang> >function dice5(n) := intrandom(1,n,5)-1; >function dice7(n) ... $ v=dice5(2*n)*5+dice5(2*n); $ return v[nonzeros(v<=21)][1:n]; $ endfunction >test=dice7(1000000); >function checkrandom (v, delta=1) ... $ K=max(v); n=cols(v); $ fr=getfrequencies(v,1:K); $ return max(fr/n-1/K)<delta/sqrt(n); $ endfunction >checkrandom(test)
1
>wd = 0|((1:6)+[-0.01,0.01,0,0,0,0])/6
[ 0 0.165 0.335 0.5 0.666666666667 0.833333333333 1 ]
>function wrongdice (n) := find(wd,random(1,n)) >checkrandom(wrongdice(1000000))
0
</lang>
Factor
<lang factor>USING: kernel random sequences assocs locals sorting prettyprint
math math.functions math.statistics math.vectors math.ranges ;
IN: rosetta-code.dice7
! Output a random integer 1..5.
- dice5 ( -- x )
5 [1,b] random
! Output a random integer 1..7 using dice5 as randomness source.
- dice7 ( -- x )
0 [ dup 21 < ] [ drop dice5 5 * dice5 + 6 - ] do until 7 rem 1 +
! Roll the die by calling the quotation the given number of times and return ! an array with roll results. ! Sample call: 1000 [ dice7 ] roll
- roll ( times quot: ( -- x ) -- array )
[ call( -- x ) ] curry replicate
! Input array contains outcomes of a number of die throws. Each die result is ! an integer in the range 1..X. Calculate and return the number of each ! of the results in the array so that in the first position of the result ! there is the number of ones in the input array, in the second position ! of the result there is the number of twos in the input array, etc.
- count-dice-outcomes ( X array -- array )
histogram swap [1,b] [ over [ 0 or ] change-at ] each sort-keys values
! Verify distribution uniformity/Naive. Delta is the acceptable deviation ! from the ideal number of items in each bucket, expressed as a fraction of ! the total count. Sides is the number of die sides. Die-func is a word that ! produces a random number on stack in the range [1..sides], times is the ! number of times to call it. ! Sample call: 0.02 7 [ dice7 ] 100000 verify
- verify ( delta sides die-func: ( -- random ) times -- )
sides times die-func roll count-dice-outcomes dup . times sides / :> ideal-count ideal-count v-n vabs times v/n delta [ < ] curry all? [ "Random enough" . ] [ "Not random enough" . ] if
! Call verify with 1, 10, 100, ... 1000000 rolls of 7-sided die.
- verify-all ( -- )
{ 1 10 100 1000 10000 100000 1000000 }
[| times | 0.02 7 [ dice7 ] times verify ] each
- </lang>
Output:
USE: rosetta-code.dice7 verify-all
{ 0 0 0 1 0 0 0 }
"Not random enough"
{ 0 2 3 1 1 1 2 }
"Not random enough"
{ 17 12 15 11 13 13 19 }
"Not random enough"
{ 140 130 141 148 143 155 143 }
"Random enough"
{ 1457 1373 1427 1433 1443 1382 1485 }
"Random enough"
{ 14225 14320 14216 14326 14415 14084 14414 }
"Random enough"
{ 142599 141910 142524 143029 143353 142696 143889 }
"Random enough"
Forth
requires Forth200x locals <lang forth>: .bounds ( u1 u2 -- ) ." lower bound = " . ." upper bound = " 1- . cr ;
- init-bins ( n -- addr )
cells dup allocate throw tuck swap erase ;
- expected ( u1 cnt -- u2 ) over 2/ + swap / ;
- calc-limits ( n cnt pct -- low high )
>r expected r> over 100 */ 2dup + 1+ >r - r> ;
- make-histogram ( bins xt cnt -- )
0 ?do 2dup execute 1- cells + 1 swap +! loop 2drop ;
- valid-bin? ( addr n low high -- f )
2>r cells + @ dup . 2r> within ;
- check-distribution {: xt cnt n pct -- f :}
\ assumes xt generates numbers from 1 to n
n init-bins {: bins :}
n cnt pct calc-limits {: low high :}
high low .bounds
bins xt cnt make-histogram
true \ result flag
n 0 ?do
i 1+ . ." : " bins i low high valid-bin?
dup 0= if ." not " then ." ok" cr
and
loop
bins free throw ;</lang>
- Output:
cr ' d7 1000000 7 1 check-distribution . lower bound = 141429 upper bound = 144285 1 : 143241 ok 2 : 142397 ok 3 : 143522 ok 4 : 142909 ok 5 : 142001 ok 6 : 142844 ok 7 : 143086 ok -1 cr ' d7 10000 7 1 check-distribution . lower bound = 1415 upper bound = 1443 1 : 1431 ok 2 : 1426 ok 3 : 1413 not ok 4 : 1427 ok 5 : 1437 ok 6 : 1450 not ok 7 : 1416 ok 0
Fortran
<lang fortran>subroutine distcheck(randgen, n, delta)
interface
function randgen
integer :: randgen
end function randgen
end interface
real, intent(in) :: delta
integer, intent(in) :: n
integer :: i, mval, lolim, hilim
integer, allocatable :: buckets(:)
integer, allocatable :: rnums(:)
logical :: skewed = .false.
allocate(rnums(n))
do i = 1, n
rnums(i) = randgen()
end do
mval = maxval(rnums) allocate(buckets(mval)) buckets = 0 do i = 1, n buckets(rnums(i)) = buckets(rnums(i)) + 1 end do
lolim = n/mval - n/mval*delta
hilim = n/mval + n/mval*delta
do i = 1, mval
if(buckets(i) < lolim .or. buckets(i) > hilim) then
write(*,"(a,i0,a,i0,a,i0)") "Distribution potentially skewed for bucket ", i, " Expected: ", &
n/mval, " Actual: ", buckets(i)
skewed = .true.
end if
end do
if (.not. skewed) write(*,"(a)") "Distribution uniform" deallocate(rnums) deallocate(buckets)
end subroutine</lang>
FreeBASIC
<lang freebasic> Randomize Timer Function dice5() As Integer
Return Int(Rnd * 5) + 1
End Function
Function dice7() As Integer
Dim As Integer temp
Do
temp = dice5() * 5 + dice5() -6
Loop Until temp < 21
Return (temp Mod 7) +1
End Function
Function distCheck(n As Ulongint, delta As Double) As Ulongint
Dim As Ulongint a(n)
Dim As Ulongint maxBucket = 0
Dim As Ulongint minBucket = 1000000
For i As Ulongint = 1 To n
a(i) = dice5()
If a(i) > maxBucket Then maxBucket = a(i)
If a(i) < minBucket Then minBucket = a(i)
Next i
Dim As Ulongint nBuckets = maxBucket + 1
Dim As Ulongint buckets(maxBucket)
For i As Ulongint = 1 To n
buckets(a(i)) += 1
Next i
'check buckets
Dim As Ulongint expected = n / (maxBucket-minBucket+1)
Dim As Ulongint minVal = Int(expected*(1-delta))
Dim As Ulongint maxVal = Int(expected*(1+delta))
expected = Int(expected)
Print "minVal", "Expected", "maxVal"
Print minVal, expected, maxVal
Print "Bucket", "Counter", "pass/fail"
distCheck = true
For i As Ulongint = minBucket To maxBucket
Print i, buckets(i), Iif((minVal > buckets(i)) Or (buckets(i) > maxVal),"fail","")
If (minVal > buckets(i)) Or (buckets(i) > maxVal) Then Return false
Next i
End Function
Dim Shared As Ulongint n = 1000 Print "Testing ";n;" times" If Not(distCheck(n, 0.05)) Then Print "Test failed" Else Print "Test passed" Print
n = 10000 Print "Testing ";n;" times" If Not(distCheck(n, 0.05)) Then Print "Test failed" Else Print "Test passed" Print
n = 50000 Print "Testing ";n;" times" If Not(distCheck(n, 0.05)) Then Print "Test failed" Else Print "Test passed" Print Sleep </lang>
- Output:
Igual que la entrada de Liberty BASIC.
Go
<lang go>package main
import (
"fmt" "math" "math/rand" "time"
)
// "given" func dice5() int {
return rand.Intn(5) + 1
}
// function specified by task "Seven-sided dice from five-sided dice" func dice7() (i int) {
for {
i = 5*dice5() + dice5()
if i < 27 {
break
}
}
return (i / 3) - 1
}
// function specified by task "Verify distribution uniformity/Naive" // // Parameter "f" is expected to return a random integer in the range 1..n. // (Values out of range will cause an unceremonious crash.) // "Max" is returned as an "indication of distribution achieved." // It is the maximum delta observed from the count representing a perfectly // uniform distribution. // Also returned is a boolean, true if "max" is less than threshold // parameter "delta." func distCheck(f func() int, n int,
repeats int, delta float64) (max float64, flatEnough bool) {
count := make([]int, n)
for i := 0; i < repeats; i++ {
count[f()-1]++
}
expected := float64(repeats) / float64(n)
for _, c := range count {
max = math.Max(max, math.Abs(float64(c)-expected))
}
return max, max < delta
}
// Driver, produces output satisfying both tasks. func main() {
rand.Seed(time.Now().UnixNano())
const calls = 1000000
max, flatEnough := distCheck(dice7, 7, calls, 500)
fmt.Println("Max delta:", max, "Flat enough:", flatEnough)
max, flatEnough = distCheck(dice7, 7, calls, 500)
fmt.Println("Max delta:", max, "Flat enough:", flatEnough)
}</lang> Output:
Max delta: 356.1428571428696 Flat enough: true Max delta: 787.8571428571304 Flat enough: false
Haskell
<lang haskell>import System.Random import Data.List import Control.Monad import Control.Arrow
distribCheck :: IO Int -> Int -> Int -> IO [(Int,(Int,Bool))] distribCheck f n d = do
nrs <- replicateM n f
let group = takeWhile (not.null) $ unfoldr (Just. (partition =<< (==). head)) nrs
avg = (fromIntegral n) / fromIntegral (length group)
ul = round $ (100 + fromIntegral d)/100 * avg
ll = round $ (100 - fromIntegral d)/100 * avg
return $ map (head &&& (id &&& liftM2 (&&) (>ll)(<ul)).length) group</lang>
Example: <lang haskell>*Main> mapM_ print .sort =<< distribCheck (randomRIO(1,6)) 100000 3 (1,(16911,True)) (2,(16599,True)) (3,(16670,True)) (4,(16624,True)) (5,(16526,True)) (6,(16670,True))</lang>
Hy
<lang lisp>(import [collections [Counter]]) (import [random [randint]])
(defn uniform? [f repeats delta]
- Call 'f' 'repeats' times, then check if the proportion of each
- value seen is within 'delta' of the reciprocal of the count
- of distinct values.
(setv bins (Counter (list-comp (f) [i (range repeats)]))) (setv target (/ 1 (len bins))) (all (list-comp (<= (- target delta) (/ n repeats) (+ target delta)) [n (.values bins)])))</lang>
Examples of use:
<lang lisp>(for [f [
(fn [] (randint 1 10)) (fn [] (if (randint 0 1) (randint 1 9) (randint 1 10)))]] (print (uniform? f 5000 .02)))</lang>
Icon and Unicon
This example assumes the random number generator is passed to verify_uniform as a co-expression. The co-expression rnd is prompted for its next value using @rnd. The co-expression is created using create (|?10) where the vertical bar means generate an infinite sequence of what is to its right (in this case, ?10 generates a random integer in the range [1,10]).
<lang Icon># rnd : a co-expression, which will generate the random numbers
- n : the number of numbers to test
- delta: tolerance in non-uniformity
- This procedure fails if after the sampling the difference
- in uniformity exceeds delta, a proportion of n / number-of-alternatives
procedure verify_uniform (rnd, n, delta)
# generate a table counting the outcome of the generator
results := table (0)
every (1 to n) do results[@rnd] +:= 1
# retrieve the statistics
smallest := n
largest := 0
every num := key(results) do { # record result and limits
write (num || " " || results[num])
if results[num] < smallest then smallest := results[num]
if results[num] > largest then largest := results[num]
}
# decide if difference is within bounds defined by delta
return largest-smallest < delta * n / *results
end
procedure main ()
gen_1 := create (|?10) # uniform integers, 1 to 10
if verify_uniform (gen_1, 1000000, 0.01)
then write ("uniform")
else write ("skewed")
gen_2 := create (|(if ?2 = 1 then 6 else ?10)) # skewed integers, 1 to 10
if verify_uniform (gen_2, 1000000, 0.01)
then write ("uniform")
else write ("skewed")
end</lang> Output:
5 99988 2 99998 10 99894 7 99948 4 100271 1 99917 9 99846 6 99943 3 99824 8 100371 uniform 5 49940 2 50324 10 50243 7 49982 4 50295 1 50162 9 49800 6 549190 3 50137 8 49927 skewed
J
This solution defines the checker as an adverb. Adverbs combine with the verb immediately to their left to create a new verb. So for a verb generateDistribution and an adverb checkUniform, the new verb might be thought of as checkGeneratedDistributionisUniform.
The delta is given as an optional left argument (x), defaulting to 5%. The right argument (y) is any valid argument to the distribution generating verb.
<lang j>checkUniform=: adverb define
0.05 u checkUniform y
:
n=. */y
delta=. x
sample=. u n NB. the "u" refers to the verb to left of adverb
freqtable=. /:~ (~. sample) ,. #/.~ sample
expected=. n % # freqtable
errmsg=. 'Distribution is potentially skewed'
errmsg assert (delta * expected) > | expected - {:"1 freqtable
freqtable
)</lang> It is possible to use tacit expressions within an explicit definition enabling a more functional and concise style: <lang j>checkUniformT=: adverb define
0.05 u checkUniformT y
:
freqtable=. /:~ (~. ,. #/.~) u n=. */y
errmsg=. 'Distribution is potentially skewed'
errmsg assert ((n % #) (x&*@[ > |@:-) {:"1) freqtable
freqtable
)</lang>
Show usage using rollD7t given in Seven-dice from Five-dice:
<lang j> 0.05 rollD7t checkUniform 1e5
1 14082
2 14337
3 14242
4 14470
5 14067
6 14428
7 14374
0.05 rollD7t checkUniform 1e2
|Distribution is potentially skewed: assert | errmsg assert(delta*expected)>|expected-{:"1 freqtable</lang>
Java
<lang java>import static java.lang.Math.abs; import java.util.*; import java.util.function.IntSupplier;
public class Test {
static void distCheck(IntSupplier f, int nRepeats, double delta) {
Map<Integer, Integer> counts = new HashMap<>();
for (int i = 0; i < nRepeats; i++)
counts.compute(f.getAsInt(), (k, v) -> v == null ? 1 : v + 1);
double target = nRepeats / (double) counts.size();
int deltaCount = (int) (delta / 100.0 * target);
counts.forEach((k, v) -> {
if (abs(target - v) >= deltaCount)
System.out.printf("distribution potentially skewed "
+ "for '%s': '%d'%n", k, v);
});
counts.keySet().stream().sorted().forEach(k
-> System.out.printf("%d %d%n", k, counts.get(k)));
}
public static void main(String[] a) {
distCheck(() -> (int) (Math.random() * 5) + 1, 1_000_000, 1);
}
}</lang>
1 200439 2 201016 3 199406 4 199869 5 199270
JavaScript
<lang javascript>function distcheck(random_func, times, opts) {
if (opts === undefined) opts = {}
opts['delta'] = opts['delta'] || 2;
var count = {}, vals = [];
for (var i = 0; i < times; i++) {
var val = random_func();
if (! has_property(count, val)) {
count[val] = 1;
vals.push(val);
}
else
count[val] ++;
}
vals.sort(function(a,b) {return a-b});
var target = times / vals.length; var tolerance = target * opts['delta'] / 100;
for (var i = 0; i < vals.length; i++) {
var val = vals[i];
if (Math.abs(count[val] - target) > tolerance)
throw "distribution potentially skewed for " + val +
": expected result around " + target + ", got " +count[val];
else
print(val + "\t" + count[val]);
}
}
function has_property(obj, propname) {
return typeof(obj[propname]) == "undefined" ? false : true;
}
try {
distcheck(function() {return Math.floor(10 * Math.random())}, 100000);
print();
distcheck(function() {return (Math.random() > 0.95 ? 1 : 0)}, 100000);
} catch (e) {
print(e);
}</lang> Output:
0 9945 1 9862 2 9954 3 10104 4 9861 5 10140 6 10066 7 10001 8 10101 9 9966 distribution potentially skewed for 0: expected result around 50000, got 95040
Julia
<lang julia>using Printf
function distcheck(f::Function, rep::Int=10000, Δ::Int=3)
smpl = f(rep) counts = Dict(k => count(smpl .== k) for k in unique(smpl)) expected = rep / length(counts) lbound = expected * (1 - 0.01Δ) ubound = expected * (1 + 0.01Δ) noobs = count(x -> !(lbound ≤ x ≤ ubound), values(counts)) if noobs > 0 warn(@sprintf "%2.4f%% values out of bounds" noobs / rep) end return counts
end
- Dice5 check
distcheck(x -> rand(1:5, x))
- Dice7 check
distcheck(dice7)</lang>
Kotlin
<lang scala>// version 1.1.3
import java.util.Random
val r = Random()
fun dice5() = 1 + r.nextInt(5)
fun checkDist(gen: () -> Int, nRepeats: Int, tolerance: Double = 0.5) {
val occurs = mutableMapOf<Int, Int>()
for (i in 1..nRepeats) {
val d = gen()
if (occurs.containsKey(d))
occurs[d] = occurs[d]!! + 1
else
occurs.put(d, 1)
}
val expected = (nRepeats.toDouble()/ occurs.size).toInt()
val maxError = (expected * tolerance / 100.0).toInt()
println("Repetitions = $nRepeats, Expected = $expected")
println("Tolerance = $tolerance%, Max Error = $maxError\n")
println("Integer Occurrences Error Acceptable")
val f = " %d %5d %5d %s"
var allAcceptable = true
for ((k,v) in occurs.toSortedMap()) {
val error = Math.abs(v - expected)
val acceptable = if (error <= maxError) "Yes" else "No"
if (acceptable == "No") allAcceptable = false
println(f.format(k, v, error, acceptable))
}
println("\nAcceptable overall: ${if (allAcceptable) "Yes" else "No"}")
}
fun main(args: Array<String>) {
checkDist(::dice5, 1_000_000) println() checkDist(::dice5, 100_000)
}</lang>
Sample output:
Repetitions = 1000000, Expected = 200000 Tolerance = 0.5%, Max Error = 1000 Integer Occurrences Error Acceptable 1 200074 74 Yes 2 200497 497 Yes 3 199295 705 Yes 4 199822 178 Yes 5 200312 312 Yes Acceptable overall: Yes Repetitions = 100000, Expected = 20000 Tolerance = 0.5%, Max Error = 100 Integer Occurrences Error Acceptable 1 20265 265 No 2 20229 229 No 3 19836 164 No 4 19931 69 Yes 5 19739 261 No Acceptable overall: No
Liberty BASIC
LB cannot pass user-defined function by name, so we use predefined function name - GENERATOR <lang lb> n=1000 print "Testing ";n;" times" if not(check(n, 0.05)) then print "Test failed" else print "Test passed" print
n=10000 print "Testing ";n;" times" if not(check(n, 0.05)) then print "Test failed" else print "Test passed" print
n=50000 print "Testing ";n;" times" if not(check(n, 0.05)) then print "Test failed" else print "Test passed" print
end
function check(n, delta)
'fill randoms
dim a(n)
maxBucket=0
minBucket=1e10
for i = 1 to n
a(i) = GENERATOR()
if a(i)>maxBucket then maxBucket=a(i)
if a(i)<minBucket then minBucket=a(i)
next
'fill buckets
nBuckets = maxBucket+1 'from 0
dim buckets(maxBucket)
for i = 1 to n
buckets(a(i)) = buckets(a(i))+1
next
'check buckets
expected=n/(maxBucket-minBucket+1)
minVal=int(expected*(1-delta))
maxVal=int(expected*(1+delta))
expected=int(expected)
print "minVal", "Expected", "maxVal"
print minVal, expected, maxVal
print "Bucket", "Counter", "pass/fail"
check = 1
for i = minBucket to maxBucket
print i, buckets(i), _
iif$((minVal > buckets(i)) OR (buckets(i) > maxVal) ,"fail","")
if (minVal > buckets(i)) OR (buckets(i) > maxVal) then check = 0
next
end function
function iif$(test, valYes$, valNo$)
iif$ = valNo$ if test then iif$ = valYes$
end function
function GENERATOR()
'GENERATOR = int(rnd(0)*10) '0..9 GENERATOR = 1+int(rnd(0)*5) '1..5: dice5
end function </lang>
- Output:
Testing 1000 times minVal Expected maxVal 190 200 210 Bucket Counter pass/fail 1 213 fail 2 204 3 192 4 188 fail 5 203 Test failed Testing 10000 times minVal Expected maxVal 1900 2000 2100 Bucket Counter pass/fail 1 2041 2 1952 3 1975 4 2026 5 2006 Test passed Testing 50000 times minVal Expected maxVal 9500 10000 10500 Bucket Counter pass/fail 1 10012 2 10207 3 10009 4 9911 5 9861 Test passed
Mathematica/Wolfram Language
<lang Mathematica>SetAttributes[CheckDistribution, HoldFirst] CheckDistribution[function_,number_,delta_] :=(Print["Expected: ", N[number/7], ", Generated :", Transpose[Tally[Table[function, {number}]]]2 // Sort]; If[(Max[#]-Min[#])&
[Transpose[Tally[Table[function, {number}]]]2] < delta*number/700, "Flat", "Skewed"])</lang>
Example usage:
CheckDistribution[RandomInteger[{1, 7}], 10000, 5]
->Expected: 1428.57, Generated :{1372,1420,1429,1431,1433,1450,1465}
->"Skewed"
CheckDistribution[RandomInteger[{1, 7}], 100000, 5]
->Expected: 14285.7, Generated :{14182,14186,14240,14242,14319,14407,14424}
->"Flat"
Nim
<lang Nim>import tables
proc checkDist(f: proc(): int; repeat: Positive; tolerance: float) =
var counts: CountTable[int] for _ in 1..repeat: counts.inc f()
let expected = (repeat / counts.len).toInt # Rounded to nearest. let allowedDelta = (expected.toFloat * tolerance / 100).toInt var maxDelta = 0 for val, count in counts.pairs: let d = abs(count - expected) if d > maxDelta: maxDelta = d let status = if maxDelta <= allowedDelta: "passed" else: "failed" echo "Checking ", repeat, " values with a tolerance of ", tolerance, "%." echo "Random generator ", status, " the uniformity test." echo "Max delta encountered = ", maxDelta, " Allowed delta = ", allowedDelta
when isMainModule:
import random randomize() proc rand5(): int = rand(1..5) checkDist(rand5, 1_000_000, 0.5)
</lang>
- Output:
Checking 1000000 values with a tolerance of 0.5%. Random generator passed the uniformity test. Max delta encountered = 659 Allowed delta = 1000
OCaml
<lang ocaml>let distcheck fn n ?(delta=1.0) () =
let h = Hashtbl.create 5 in
for i = 1 to n do
let v = fn() in
let n =
try Hashtbl.find h v
with Not_found -> 0
in
Hashtbl.replace h v (n+1)
done;
Hashtbl.iter (fun v n -> Printf.printf "%d => %d\n%!" v n) h;
let target = (float n) /. float (Hashtbl.length h) in
Hashtbl.iter (fun key value ->
if abs_float(float value -. target) > 0.01 *. delta *. (float n)
then (Printf.eprintf
"distribution potentially skewed for '%d': expected around %f, got %d\n%!"
key target value)
) h;
- </lang>
PARI/GP
This tests the purportedly random 7-sided die with a slightly biased 1000-sided die. <lang parigp>dice5()=random(5)+1;
dice7()={
my(t); while((t=dice5()*5+dice5()) > 26,); t\3-1
};
cumChi2(chi2,dof)={ my(g=gamma(dof/2)); incgam(dof/2,chi2/2,g)/g };
test(f,n,alpha=.05)={ v=vector(n,i,f()); my(s,ave,dof,chi2,p); s=sum(i=1,n,v[i],0.); ave=s/n; dof=n-1; chi2=sum(i=1,n,(v[i]-ave)^2)/ave; p=cumChi2(chi2,dof); if(p<alpha, error("Not flat enough, significance only "p) , print("Flat with significance "p); ) };
test(dice7, 10^5) test(()->if(random(1000),random(1000),1), 10^5)</lang> Output:
Flat with significance 0.2931867820813680387842134664085280183 ### user error: Not flat enough, significance only 5.391077606003910233 E-3500006
Perl
Testing two 'types' of 7-sided dice. Both appear to be fair.
<lang perl>sub roll7 { 1+int rand(7) } sub roll5 { 1+int rand(5) } sub roll7_5 {
while(1) {
my $d7 = (5*&roll5 + &roll5 - 6) % 8;
return $d7 if $d7;
}
}
my $threshold = 5;
print dist( $_, $threshold, \&roll7 ) for <1001 1000006>; print dist( $_, $threshold, \&roll7_5 ) for <1001 1000006>;
sub dist { my($n, $threshold, $producer) = @_;
my @dist; my $result; my $expect = $n / 7; $result .= sprintf "%10d expected\n", $expect;
for (1..$n) { @dist[&$producer]++; }
for my $i (1..7) {
my $v = @dist[$i];
my $pct = ($v - $expect)/$expect*100;
$result .= sprintf "%d %8d %6.1f%%%s\n", $i, $v, $pct, (abs($pct) > $threshold ? ' - skewed' : );
}
return $result . "\n";
}</lang>
- Output:
143 expected
1 144 0.7%
2 137 -4.2%
3 121 -15.4% - skewed
4 163 14.0% - skewed
5 150 4.9%
6 138 -3.5%
7 148 3.5%
142858 expected
1 142332 -0.4%
2 142648 -0.1%
3 143615 0.5%
4 142305 -0.4%
5 142703 -0.1%
6 142821 -0.0%
7 143582 0.5%
143 expected
1 149 4.2%
2 159 11.2% - skewed
3 154 7.7% - skewed
4 130 -9.1% - skewed
5 143 0.0%
6 138 -3.5%
7 128 -10.5% - skewed
142858 expected
1 142574 -0.2%
2 143043 0.1%
3 142446 -0.3%
4 143325 0.3%
5 142949 0.1%
6 142990 0.1%
7 142679 -0.1%
Phix
with javascript_semantics function check(integer fid, range, iterations, atom delta) -- -- fid: routine_id of function that yields integer 1..range -- range: the maximum value that is returned from fid -- iterations: number of iterations to test -- delta: variance, for example 0.005 means 0.5% -- -- returns -1/0/1 for impossible/not flat/flat. -- atom av = iterations/range -- average/expected value if floor(av)<av-delta*av or ceil(av)>av+delta*av then return -1 -- impossible end if sequence counts = repeat(0,range) for i=1 to iterations do integer cdx = fid() counts[cdx] += 1 end for atom max_delta = max(sq_abs(sq_sub(counts,av))) return max_delta<delta*av end function function rand7() return rand(7) end function constant flats = {"impossible","not flat","flat"} for p=2 to 7 do integer n = power(10,p) -- n = n+7-remainder(n,7) integer flat = check(rand7, 7, n, 0.005) printf(1,"%d iterations: %s\n",{n,flats[flat+2]}) end for
- Output:
100 iterations: impossible 1000 iterations: impossible 10000 iterations: not flat 100000 iterations: not flat 1000000 iterations: flat 10000000 iterations: flat
At the specified 0.5%, 1000000 iterations is occasionally not flat, and 10000 is sometimes flat at 3%.
As shown above, it is not mathematically possible to distribute 1000 over 7 bins with <= 0.5% variance.
At 100 iterations, the permitted range is ~14.21..14.36, so you could not get even one bin right.
At 1000 iterations, 142 is too low (and 144 too high), they would all have to be 143, but 7*143=1001.
The commented-out adjustment to n (as Raku) changes the "1000 impossible" result to "1001 not flat",
except of course for the one-in-however-many-gazillion chance of getting exactly 143 of each.
PicoLisp
The following function takes a count, and allowed deviation in per mill (one-tenth of a percent), and a 'prg' code body (i.e. an arbitrary number of executable expressions). <lang PicoLisp>(de checkDistribution (Cnt Pm . Prg)
(let Res NIL
(do Cnt (accu 'Res (run Prg 1) 1))
(let
(N (/ Cnt (length Res))
Min (*/ N (- 1000 Pm) 1000)
Max (*/ N (+ 1000 Pm) 1000) )
(for R Res
(prinl (cdr R) " " (if (>= Max (cdr R) Min) "Good" "Bad")) ) ) ) )</lang>
Output:
: (checkDistribution 100000 5 (rand 1 7)) 14299 Good 14394 Bad 14147 Bad 14418 Bad 14159 Bad 14271 Good 14312 Good
PureBasic
<lang PureBasic>Prototype RandNum_prt()
Procedure.s distcheck(*function.RandNum_prt, repetitions, delta.d)
Protected text.s, maxIndex = 0
Dim bucket(maxIndex) ;array will be resized as needed
For i = 1 To repetitions ;populate buckets
v = *function()
If v > maxIndex
maxIndex = v
Redim bucket(maxIndex)
EndIf
bucket(v) + 1
Next
lbnd = Round((repetitions / maxIndex) * (100 - delta) / 100, #PB_Round_Up)
ubnd = Round((repetitions / maxIndex) * (100 + delta) / 100, #PB_Round_Down)
text = "Distribution check:" + #crlf$ + #crlf$
text + "Total elements = " + Str(repetitions) + #crlf$ + #crlf$
text + "Margin = " + StrF(delta, 2) + "% --> Lbound = " + Str(lbnd) + ", Ubound = " + Str(ubnd) + #crlf$
For i = 1 To maxIndex
If bucket(i) < lbnd Or bucket(i) > ubnd
text + #crlf$ + "Bucket " + Str(i) + " contains " + Str(bucket(i)) + " elements. Skewed."
EndIf
Next
ProcedureReturn text
EndProcedure
MessageRequester("Results", distcheck(@dice7(), 1000000, 0.20))</lang> A small delta was chosen to increase the chance of a skewed result in the sample output:
Distribution check: Total elements = 1000000 Margin = 0.20% --> Lbound = 142572, Ubound = 143142 Bucket 1 contains 141977 elements. Skewed. Bucket 6 contains 143860 elements. Skewed.
Python
<lang python>from collections import Counter from pprint import pprint as pp
def distcheck(fn, repeats, delta):
\
Bin the answers to fn() and check bin counts are within +/- delta %
of repeats/bincount
bin = Counter(fn() for i in range(repeats))
target = repeats // len(bin)
deltacount = int(delta / 100. * target)
assert all( abs(target - count) < deltacount
for count in bin.values() ), "Bin distribution skewed from %i +/- %i: %s" % (
target, deltacount, [ (key, target - count)
for key, count in sorted(bin.items()) ]
)
pp(dict(bin))</lang>
Sample output:
>>> distcheck(dice5, 1000000, 1)
{1: 200244, 2: 199831, 3: 199548, 4: 199853, 5: 200524}
>>> distcheck(dice5, 1000, 1)
Traceback (most recent call last):
File "<pyshell#30>", line 1, in <module>
distcheck(dice5, 1000, 1)
File "C://Paddys/rand7fromrand5.py", line 54, in distcheck
for key, count in sorted(bin.items()) ]
AssertionError: Bin distribution skewed from 200 +/- 2: [(1, 4), (2, -33), (3, 6), (4, 11), (5, 12)]
Quackery
The word distribution tests a specified word (Quackery function) which should return numbers in the range 1 to 7 inclusive. The word dice7, which satisfies this requirement, is defined at Seven-sided dice from five-sided dice#Quackery.
<lang Quackery> [ stack [ 0 0 0 0 0 0 0 ] ] is bins ( --> s )
[ 7 times
[ 0 bins take
i poke
bins put ] ] is emptybins ( --> )
[ bins share over peek
1+ bins take rot poke
bins put ] is bincrement ( n --> )
[ emptybins
over 7 / temp put
swap times
[ over do 1 -
bincrement ]
bins share dup echo cr
witheach
[ temp share - abs
over > if
[ say "Number of "
i^ 1+ echo
say "s is sketchy."
cr ] ]
2drop temp release ] is distribution ( x n n --> )</lang>
- Output:
Testing in the Quackery shell.
/O> ' dice7 1000 20 distribution ... [ 131 123 160 144 156 145 141 ] Stack empty. /O> ' dice7 1000 10 distribution ... [ 137 138 130 160 143 150 142 ] Number of 3s is sketchy. Number of 4s is sketchy.
R
<lang r>distcheck <- function(fn, repetitions=1e4, delta=3) {
if(is.character(fn))
{
fn <- get(fn)
}
if(!is.function(fn))
{
stop("fn is not a function")
}
samp <- fn(n=repetitions)
counts <- table(samp)
expected <- repetitions/length(counts)
lbound <- expected * (1 - 0.01*delta)
ubound <- expected * (1 + 0.01*delta)
status <- ifelse(counts < lbound, "under",
ifelse(counts > ubound, "over", "okay"))
data.frame(value=names(counts), counts=as.vector(counts), status=status)
} distcheck(dice7.vec)</lang>
Racket
Returns a pair of a boolean stating uniformity and either the "uniform" distribution or a report of the first skew number found.
<lang racket>#lang racket (define (pretty-fraction f)
(if (integer? f) f
(let* ((d (denominator f)) (n (numerator f)) (q (quotient n d)) (r (remainder n d)))
(format "~a ~a" q (/ r d)))))
(define (test-uniformity/naive r n δ)
(define observation (make-hash))
(for ((_ (in-range n))) (hash-update! observation (r) add1 0))
(define target (/ n (hash-count observation)))
(define max-skew (* n δ 1/100))
(define (skewed? v)
(> (abs (- v target)) max-skew))
(let/ec ek
(cons
#t
(for/list ((k (sort (hash-keys observation) <)))
(define v (hash-ref observation k))
(when (skewed? v)
(ek (cons
#f
(format "~a distribution of ~s potentially skewed for ~a. expected ~a got ~a"
'test-uniformity/naive r k (pretty-fraction target) v))))
(cons k v)))))
(define (straight-die)
(min 6 (add1 (random 6))))
(define (crooked-die)
(min 6 (add1 (random 7))))
- Test whether the builtin generator is uniform
(test-uniformity/naive (curry random 10) 1000 5)
- Test whether a straight die is uniform
(test-uniformity/naive straight-die 1000 5)
- Test whether a biased die fails
(test-uniformity/naive crooked-die 1000 5)</lang>
- Output:
'(#t (0 . 96) (1 . 100) (2 . 103) (3 . 86) (4 . 94) (5 . 111) (6 . 106) (7 . 99) (8 . 108) (9 . 97)) '(#t (1 . 169) (2 . 185) (3 . 184) (4 . 163) (5 . 144) (6 . 155)) '(#f . "test-uniformity/naive distribution of #<procedure:crooked-die> potentially skewed for 6. expected 166 2/3 got 262")
Raku
(formerly Perl 6) Since the tested function is rolls of a 7 sided die, the test numbers are magnitudes of 10x bumped up to the closest multiple of 7. This reduces spurious error from there not being an integer expected value. <lang perl6>my $d7 = 1..7; sub roll7 { $d7.roll };
my $threshold = 3;
for 14, 105, 1001, 10003, 100002, 1000006 -> $n
{ dist( $n, $threshold, &roll7 ) };
sub dist ( $n is copy, $threshold, &producer ) {
my @dist;
my $expect = $n / 7;
say "Expect\t",$expect.fmt("%.3f");
loop ($_ = $n; $n; --$n) { @dist[&producer()]++; }
for @dist.kv -> $i, $v is copy {
next unless $i;
$v //= 0;
my $pct = ($v - $expect)/$expect*100;
printf "%d\t%d\t%+.2f%% %s\n", $i, $v, $pct,
($pct.abs > $threshold ?? '- skewed' !! );
}
say ;
}</lang> Sample output:
Expect 2.000 1 2 +0.00% 2 3 +50.00% - skewed 3 2 +0.00% 4 2 +0.00% 5 3 +50.00% - skewed 6 0 -100.00% - skewed 7 2 +0.00% Expect 15.000 1 15 +0.00% 2 17 +13.33% - skewed 3 13 -13.33% - skewed 4 16 +6.67% - skewed 5 14 -6.67% - skewed 6 16 +6.67% - skewed 7 14 -6.67% - skewed Expect 143.000 1 134 -6.29% - skewed 2 142 -0.70% 3 141 -1.40% 4 137 -4.20% - skewed 5 142 -0.70% 6 170 +18.88% - skewed 7 135 -5.59% - skewed Expect 1429.000 1 1396 -2.31% 2 1468 +2.73% 3 1405 -1.68% 4 1442 +0.91% 5 1453 +1.68% 6 1417 -0.84% 7 1422 -0.49% Expect 14286.000 1 14222 -0.45% 2 14320 +0.24% 3 14326 +0.28% 4 14425 +0.97% 5 14140 -1.02% 6 14275 -0.08% 7 14294 +0.06% Expect 142858.000 1 142510 -0.24% 2 142436 -0.30% 3 142438 -0.29% 4 143152 +0.21% 5 142905 +0.03% 6 143232 +0.26% 7 143333 +0.33%
REXX
<lang rexx>/*REXX program simulates a number of trials of a random digit and show it's skew %. */ parse arg func times delta seed . /*obtain arguments (options) from C.L. */ if func== | func=="," then func= 'RANDOM' /*function not specified? Use default.*/ if times== | times=="," then times= 1000000 /*times " " " " */ if delta== | delta=="," then delta= 1/2 /*delta% " " " " */ if datatype(seed, 'W') then call random ,,seed /*use some RAND seed for repeatability.*/ highDig= 9 /*use this var for the highest digit. */ !.= 0 /*initialize all possible random trials*/
do times /* [↓] perform a bunch of trials. */
if func=='RANDOM' then ?= random(highDig) /*use RANDOM function.*/
else interpret '?=' func "(0,"highDig')' /* " specified " */
!.?= !.? + 1 /*bump the invocation counter.*/
end /*times*/ /* [↑] store trials ───► pigeonholes. */
/* [↓] compute the digit's skewness. */
g= times / (1 + highDig) /*calculate number of each digit throw.*/ w= max(9, length( commas(times) ) ) /*maximum length of number of trials.*/ pad= left(, 9) /*this is used for output indentation. */ say pad 'digit' center(" hits", w) ' skew ' "skew %" 'result' /*header. */ say sep /*display a separator line. */
/** [↑] show header and the separator.*/
do k=0 to highDig /*process each of the possible digits. */
skew= g - !.k /*calculate the skew for the digit. */
skewPC= (1 - (g - abs(skew)) / g) * 100 /* " " " percentage for dig*/
say pad center(k, 5) right( commas(!.k), w) right(skew, 6) ,
right( format(skewPC, , 3), 6) center( word('ok skewed', 1+(skewPC>delta)), 6)
end /*k*/
say sep /*display a separator line. */ y= 5+1+w+1+6+1+6+1+6 /*width + seps*/ say pad center(" (with " commas(times) ' trials)' , y) /*# trials. */ say pad center(" (skewed when exceeds " delta'%)' , y) /*skewed note.*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ sep: say pad '─────' center(, w, '─') '──────' "──────" '──────'; return</lang>
- output when using the default inputs:
digit hits skew skew % result
───── ───────── ────── ────── ──────
0 99,739 261 0.261 ok
1 99,819 181 0.181 ok
2 100,463 -463 0.463 ok
3 99,787 213 0.213 ok
4 99,632 368 0.368 ok
5 100,787 -787 0.787 skewed
6 99,704 296 0.296 ok
7 99,605 395 0.395 ok
8 100,488 -488 0.488 ok
9 99,976 24 0.024 ok
───── ───────── ────── ───── ──────
(with 1,000,000 trials)
(skewed when exceeds 0.5%)
Ring
<lang ring>
- Project : Verify distribution uniformity/Naive
maxrnd = 7 for r = 2 to 5
check = distcheck(pow(10,r), 0.05)
see "over " + pow(10,r) + " runs dice5 " + nl
if check
see "failed distribution check with " + check + " bin(s) out of range" + nl
else
see "passed distribution check" + nl
ok
next
func distcheck(repet, delta) m = 1 s = 0 bins = list(maxrnd) for i = 1 to repet
r = dice5() + 1 bins[r] = bins[r] + 1 if r>m m = r ok
next for i = 1 to m
if bins[i]/(repet/m) > 1+delta s = s + 1 ok if bins[i]/(repet/m) < 1-delta s = s + 1 ok
next return s
func dice5
return random(5)
</lang> Output:
Over 100 runs dice5 failed distribution check with 3 bin(s) out of range Over 1000 runs dice5 failed distribution check with 1 bin(s) out of range Over 10000 runs dice5 passed distribution check Over 100000 runs dice5 passed distribution check
Ruby
<lang ruby>def distcheck(n, delta=1)
unless block_given?
raise ArgumentError, "pass a block to this method"
end
h = Hash.new(0)
n.times {h[ yield ] += 1}
target = 1.0 * n / h.length
h.each do |key, value|
if (value - target).abs > 0.01 * delta * n
raise StandardError,
"distribution potentially skewed for '#{key}': expected around #{target}, got #{value}"
end
end
puts h.sort.map{|k, v| "#{k} #{v}"}
end
if __FILE__ == $0
begin
distcheck(100_000) {rand(10)}
distcheck(100_000) {rand > 0.95}
rescue StandardError => e
p e
end
end</lang>
- Output:
0 10048 1 9949 2 9920 3 9919 4 9957 5 10087 6 9835 7 10026 8 10069 9 10190 #<StandardError: distribution potentially skewed for 'false': expected around 50000.0, got 95040>
Run BASIC
<lang runbasic>s$ = "#########################" dim num(100) for i = 1 to 1000
n = (rnd(1) * 10) + 1 num(n) = num(n) + 1
next i
for i = 1 to 10
print using("###",i);" "; using("#####",num(i));" ";left$(s$,num(i) / 5)
next i</lang>
1 90 ################## 2 110 ###################### 3 105 ##################### 4 100 #################### 5 107 ##################### 6 133 ######################### 7 85 ################# 8 96 ################### 9 82 ################ 10 92 ##################*
Scala
Imperative, ugly, mutable data
<lang Scala>object DistrubCheck1 extends App {
private def distCheck(f: () => Int, nRepeats: Int, delta: Double): Unit = {
val counts = scala.collection.mutable.Map[Int, Int]()
for (_ <- 0 until nRepeats)
counts.updateWith(f()) {
case Some(count) => Some(count + 1)
case None => Some(1)
}
val target: Double = nRepeats.toDouble / counts.size
val deltaCount: Int = (delta / 100.0 * target).toInt
counts.foreach {
case (k, v) =>
if (math.abs(target - v) >= deltaCount)
println(f"distribution potentially skewed for $k%s: $v%d")
}
counts.toIndexedSeq.foreach(entry => println(f"${entry._1}%d ${entry._2}%d"))
}
distCheck(() => 1 + util.Random.nextInt(5), 1_000_000, 1)
}</lang>
Functional Style
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
<lang Scala>object DistrubCheck2 extends App {
private def distCheck(f: () => Int, nRepeats: Int, delta: Double): Unit = {
val counts: Map[Int, Int] =
(0 until nRepeats).map(_ => f()).groupBy(identity).map { case (k, v) => (k, v.size) }
val target = nRepeats / counts.size.toDouble
counts.withFilter { case (_, v) => math.abs(target - v) >= (delta / 100.0 * target) }
.foreach { case (k, v) => println(f"distribution potentially skewed for $k%s: $v%d") }
counts.toIndexedSeq.foreach(entry => println(f"${entry._1}%d ${entry._2}%d"))
}
distCheck(() => 1 + util.Random.nextInt(5), 1_000_000, 1)
}</lang>
Tcl
<lang tcl>proc distcheck {random times {delta 1}} {
for {set i 0} {$i<$times} {incr i} {incr vals([uplevel 1 $random])}
set target [expr {$times / [array size vals]}]
foreach {k v} [array get vals] {
if {abs($v - $target) > $times * $delta / 100.0} {
error "distribution potentially skewed for $k: expected around $target, got $v"
}
}
foreach k [lsort -integer [array names vals]] {lappend result $k $vals($k)}
return $result
}</lang> Demonstration: <lang tcl># First, a uniformly distributed random variable puts [distcheck {expr {int(10*rand())}} 100000]
- Now, one that definitely isn't!
puts [distcheck {expr {rand()>0.95}} 100000]</lang> Which produces this output (error in red):
0 10003 1 9851 2 10058 3 10193 4 10126 5 10002 6 9852 7 9964 8 9957 9 9994
distribution potentially skewed for 0: expected around 50000, got 94873
VBScript
<lang vb>Option Explicit
sub verifydistribution(calledfunction, samples, delta) Dim i, n, maxdiff 'We could cheat via Dim d(7), but "7" wasn't mentioned in the Task. Heh. Dim d : Set d = CreateObject("Scripting.Dictionary") wscript.echo "Running """ & calledfunction & """ " & samples & " times..." for i = 1 to samples Execute "n = " & calledfunction d(n) = d(n) + 1 next n = d.Count maxdiff = 0 wscript.echo "Expected average count is " & Int(samples/n) & " across " & n & " buckets." for each i in d.Keys dim diff : diff = abs(1 - d(i) / (samples/n)) if diff > maxdiff then maxdiff = diff wscript.echo "Bucket " & i & " had " & d(i) & " occurences" _ & vbTab & " difference from expected=" & FormatPercent(diff, 2) next wscript.echo "Maximum found variation is " & FormatPercent(maxdiff, 2) _ & ", desired limit is " & FormatPercent(delta, 2) & "." if maxdiff > delta then wscript.echo "Skewed!" else wscript.echo "Smooth!" end sub</lang> Demonstration with included Seven-sided dice from five-sided dice#VBScript code: <lang vb>verifydistribution "dice7", 1000, 0.03 verifydistribution "dice7", 100000, 0.03</lang> Which produces this output:
Running "dice7" 1000 times... Expected average count is 142 across 7 buckets. Bucket 2 had 150 occurences difference from expected=5.00% Bucket 7 had 147 occurences difference from expected=2.90% Bucket 6 had 146 occurences difference from expected=2.20% Bucket 5 had 141 occurences difference from expected=1.30% Bucket 1 had 152 occurences difference from expected=6.40% Bucket 4 had 115 occurences difference from expected=19.50% Bucket 3 had 149 occurences difference from expected=4.30% Maximum found variation is 19.50%, desired limit is 3.00%. Skewed! Running "dice7" 100000 times... Expected average count is 14285 across 7 buckets. Bucket 5 had 14420 occurences difference from expected=0.94% Bucket 4 had 14298 occurences difference from expected=0.09% Bucket 2 had 14202 occurences difference from expected=0.59% Bucket 7 had 14201 occurences difference from expected=0.59% Bucket 6 had 14237 occurences difference from expected=0.34% Bucket 3 had 14263 occurences difference from expected=0.16% Bucket 1 had 14379 occurences difference from expected=0.65% Maximum found variation is 0.94%, desired limit is 3.00%. Smooth!
Vlang
<lang vlang>import rand import rand.seed import math // "given" fn dice5() int {
return rand.intn(5) or {0} + 1
}
// fntion specified by task "Seven-sided dice from five-sided dice" fn dice7() int {
mut i := 0
for {
i = 5*dice5() + dice5()
if i < 27 {
break
}
}
return (i / 3) - 1
}
// fntion specified by task "Verify distribution uniformity/Naive" // // Parameter "f" is expected to return a random integer in the range 1..n. // (Values out of range will cause an unceremonious crash.) // "Max" is returned as an "indication of distribution achieved." // It is the maximum delta observed from the count representing a perfectly // uniform distribution. // Also returned is a boolean, true if "max" is less than threshold // parameter "delta." fn dist_check(f fn() int, n int,
repeats int, delta f64) (f64, bool) {
mut max := 0.0
mut count := []int{len: n}
for _ in 0..repeats {
count[f()-1]++
}
expected := f64(repeats) / f64(n)
for c in count {
max = math.max(max, math.abs(f64(c)-expected))
}
return max, max < delta
}
// Driver, produces output satisfying both tasks. fn main() {
rand.seed(seed.time_seed_array(2))
calls := 1000000
mut max, mut flat_enough := dist_check(dice7, 7, calls, 500)
println("Max delta: $max Flat enough: $flat_enough")
max, flat_enough = dist_check(dice7, 7, calls, 500)
println("Max delta: $max Flat enough: $flat_enough")
}</lang>
- Output:
Max delta: 723.8571428571304 Flat enough: false Max delta: 435.1428571428696 Flat enough: true
Wren
<lang ecmascript>import "random" for Random import "/fmt" for Fmt import "/sort" for Sort
var r = Random.new()
var dice5 = Fn.new { 1 + r.int(5) }
var checkDist = Fn.new { |gen, nRepeats, tolerance|
var occurs = {}
for (i in 1..nRepeats) {
var d = gen.call()
occurs[d] = occurs.containsKey(d) ? occurs[d] + 1 : 1
}
var expected = (nRepeats/occurs.count).floor
var maxError = (expected*tolerance/100).floor
System.print("Repetitions = %(nRepeats), Expected = %(expected)")
System.print("Tolerance = %(tolerance)\%, Max Error = %(maxError)\n")
System.print("Integer Occurrences Error Acceptable")
var f = " $d $5d $5d $s"
var allAcceptable = true
occurs = occurs.toList
Sort.quick(occurs)
for (me in occurs) {
var error = (me.value - expected).abs
var acceptable = (error <= maxError) ? "Yes" : "No"
if (acceptable == "No") allAcceptable = false
Fmt.print(f, me.key, me.value, error, acceptable)
}
System.print("\nAcceptable overall: %(allAcceptable ? "Yes" : "No")")
}
checkDist.call(dice5, 1e6, 0.5) System.print() checkDist.call(dice5, 1e5, 0.5)</lang>
- Output:
Sample run:
Repetitions = 1000000, Expected = 200000 Tolerance = 0.5%, Max Error = 1000 Integer Occurrences Error Acceptable 1 199599 401 Yes 2 199676 324 Yes 3 200561 561 Yes 4 200647 647 Yes 5 199517 483 Yes Acceptable overall: Yes Repetitions = 100000, Expected = 20000 Tolerance = 0.5%, Max Error = 100 Integer Occurrences Error Acceptable 1 19780 220 No 2 20005 5 Yes 3 20206 206 No 4 19920 80 Yes 5 20089 89 Yes Acceptable overall: No
zkl
This tests the random spread over 0..9. It starts at 10 samples and doubles the sample size until the spread is within 0.1% of 10% for each bucket. <lang zkl>fcn rtest(N){
dist:=L(0,0,0,0,0,0,0,0,0,0);
do(N){n:=(0).random(10); dist[n]=dist[n]+1}
sum:=dist.sum();
dist=dist.apply('wrap(n){n.toFloat()/sum*100});
if (dist.filter((10.0).closeTo.fp1(0.1)).len() == 10)
{ "Good enough at %,d: %s".fmt(N,dist).println(); return(True); }
False
}
n:=10; while(not rtest(n)) {n*=2}</lang>
- Output:
Reported numbers is the percent that bucket has of all samples.
Good enough at 163,840: L(10.0665,9.94019,10.0146,9.99939,10.0775,10.0201,9.93713,10.0775,9.9054,9.96155)
- Programming Tasks
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