Verify distribution uniformity/Chi-squared test
(Redirected from Verify Distribution Uniformity with Chi-Squared Test)
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Write a function to determine whether a given set of frequency counts could plausibly have come from a uniform distribution by using the test with a significance level of 5%.
The function should return a boolean that is true if and only if the distribution is one that a uniform distribution (with appropriate number of degrees of freedom) may be expected to produce.
Note: normally a two-tailed test would be used for this kind of problem.
- Reference
-
- an entry at the MathWorld website: chi-squared distribution.
- Related task
11l
V a = 12
V k1_factrl = 1.0
[Float] c
c.append(sqrt(2.0 * math:pi))
L(k) 1 .< a
c.append(exp(a - k) * (a - k) ^ (k - 0.5) / k1_factrl)
k1_factrl *= -k
F gamma_spounge(z)
V accm = :c[0]
L(k) 1 .< :a
accm += :c[k] / (z + k)
accm *= exp(-(z + :a)) * (z + :a) ^ (z + 0.5)
R accm / z
F GammaInc_Q(a, x)
V a1 = a - 1
V a2 = a - 2
F f0(t)
R t ^ @a1 * exp(-t)
F df0(t)
R (@a1 - t) * t ^ @a2 * exp(-t)
V y = a1
L f0(y) * (x - y) > 2.0e-8 & y < x
y += 0.3
I y > x
y = x
V h = 3.0e-4
V n = Int(y / h)
h = y / n
V hh = 0.5 * h
V gamax = h * sum(((n - 1 .< -1).step(-1).map(j -> @h * j)).map(t -> @f0(t) + @hh * @df0(t)))
R gamax / gamma_spounge(a)
F chi2UniformDistance(dataSet)
V expected = sum(dataSet) * 1.0 / dataSet.len
V cntrd = (dataSet.map(d -> d - @expected))
R sum(cntrd.map(x -> x * x)) / expected
F chi2Probability(dof, distance)
R 1.0 - GammaInc_Q(0.5 * dof, 0.5 * distance)
F chi2IsUniform(dataSet, significance)
V dof = dataSet.len - 1
V dist = chi2UniformDistance(dataSet)
R chi2Probability(dof, dist) > significance
V dset1 = [199809, 200665, 199607, 200270, 199649]
V dset2 = [522573, 244456, 139979, 71531, 21461]
L(ds) (dset1, dset2)
print(‘Data set: ’ds)
V dof = ds.len - 1
V distance = chi2UniformDistance(ds)
print(‘dof: #. distance: #.4’.format(dof, distance), end' ‘ ’)
V prob = chi2Probability(dof, distance)
print(‘probability: #.4’.format(prob), end' ‘ ’)
print(‘uniform? ’(I chi2IsUniform(ds, 0.05) {‘Yes’} E ‘No’))- Output:
Data set: [199809, 200665, 199607, 200270, 199649] dof: 4 distance: 4.1463 probability: 0.3866 uniform? Yes Data set: [522573, 244456, 139979, 71531, 21461] dof: 4 distance: 790063.2759 probability: -1.5002e-8 uniform? No
Ada
First, we specify a simple package to compute the Chi-Square Distance from the uniform distribution:
package Chi_Square is
type Flt is digits 18;
type Bins_Type is array(Positive range <>) of Natural;
function Distance(Bins: Bins_Type) return Flt;
end Chi_Square;
Next, we implement that package:
package body Chi_Square is
function Distance(Bins: Bins_Type) return Flt is
Bad_Bins: Natural := 0;
Sum: Natural := 0;
Expected: Flt;
Result: Flt;
begin
for I in Bins'Range loop
if Bins(I) < 5 then
Bad_Bins := Bad_Bins + 1;
end if;
Sum := Sum + Bins(I);
end loop;
if 5*Bad_Bins > Bins'Length then
raise Program_Error with "too many (almost) empty bins";
end if;
Expected := Flt(Sum) / Flt(Bins'Length);
Result := 0.0;
for I in Bins'Range loop
Result := Result + ((Flt(Bins(I)) - Expected)**2) / Expected;
end loop;
return Result;
end Distance;
end Chi_Square;
Finally, we actually implement the Chi-square test. We do not actually compute the Chi-square probability; rather we hardcode a table of values for 5% significance level, which has been picked from Wikipedia [1]:
with Ada.Text_IO, Ada.Command_Line, Chi_Square; use Ada.Text_IO;
procedure Test_Chi_Square is
package Ch2 renames Chi_Square; use Ch2;
package FIO is new Float_IO(Flt);
B: Bins_Type(1 .. Ada.Command_Line.Argument_Count);
Bound_For_5_Per_Cent: constant array(Positive range <>) of Flt :=
( 1 => 3.84, 2 => 5.99, 3 => 7.82, 4 => 9.49, 5 => 11.07,
6 => 12.59, 7 => 14.07, 8 => 15.51, 9 => 16.92, 10 => 18.31);
-- picked from http://en.wikipedia.org/wiki/Chi-squared_distribution
Dist: Flt;
begin
for I in B'Range loop
B(I) := Natural'Value(Ada.Command_Line.Argument(I));
end loop;
Dist := Distance(B);
Put("Degrees of Freedom:" & Integer'Image(B'Length-1) & ", Distance: ");
FIO.Put(Dist, Fore => 6, Aft => 2, Exp => 0);
if Dist <= Bound_For_5_Per_Cent(B'Length-1) then
Put_Line("; (apparently uniform)");
else
Put_Line("; (deviates significantly from uniform)");
end if;
end;
- Output:
$ ./Test_Chi_Square 199809 200665 199607 200270 199649 Degrees of Freedom: 4, Distance: 4.15; (apparently uniform) $ ./Test_Chi_Square 522573 244456 139979 71531 21461 Degrees of Freedom: 4, Distance: 790063.28; (deviates significantly from uniform)
C
This first sections contains the functions required to compute the Chi-Squared probability. These are not needed if a library containing the necessary function is availabile (e.g. see Numerical Integration, Gamma function).
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#ifndef M_PI
#define M_PI 3.14159265358979323846
#endif
typedef double (* Ifctn)( double t);
/* Numerical integration method */
double Simpson3_8( Ifctn f, double a, double b, int N)
{
int j;
double l1;
double h = (b-a)/N;
double h1 = h/3.0;
double sum = f(a) + f(b);
for (j=3*N-1; j>0; j--) {
l1 = (j%3)? 3.0 : 2.0;
sum += l1*f(a+h1*j) ;
}
return h*sum/8.0;
}
#define A 12
double Gamma_Spouge( double z )
{
int k;
static double cspace[A];
static double *coefs = NULL;
double accum;
double a = A;
if (!coefs) {
double k1_factrl = 1.0;
coefs = cspace;
coefs[0] = sqrt(2.0*M_PI);
for(k=1; k<A; k++) {
coefs[k] = exp(a-k) * pow(a-k,k-0.5) / k1_factrl;
k1_factrl *= -k;
}
}
accum = coefs[0];
for (k=1; k<A; k++) {
accum += coefs[k]/(z+k);
}
accum *= exp(-(z+a)) * pow(z+a, z+0.5);
return accum/z;
}
double aa1;
double f0( double t)
{
return pow(t, aa1)*exp(-t);
}
double GammaIncomplete_Q( double a, double x)
{
double y, h = 1.5e-2; /* approximate integration step size */
/* this cuts off the tail of the integration to speed things up */
y = aa1 = a-1;
while((f0(y) * (x-y) > 2.0e-8) && (y < x)) y += .4;
if (y>x) y=x;
return 1.0 - Simpson3_8( &f0, 0, y, (int)(y/h))/Gamma_Spouge(a);
}
This section contains the functions specific to the task.
double chi2UniformDistance( double *ds, int dslen)
{
double expected = 0.0;
double sum = 0.0;
int k;
for (k=0; k<dslen; k++)
expected += ds[k];
expected /= k;
for (k=0; k<dslen; k++) {
double x = ds[k] - expected;
sum += x*x;
}
return sum/expected;
}
double chi2Probability( int dof, double distance)
{
return GammaIncomplete_Q( 0.5*dof, 0.5*distance);
}
int chiIsUniform( double *dset, int dslen, double significance)
{
int dof = dslen -1;
double dist = chi2UniformDistance( dset, dslen);
return chi2Probability( dof, dist ) > significance;
}
Testing
int main(int argc, char **argv)
{
double dset1[] = { 199809., 200665., 199607., 200270., 199649. };
double dset2[] = { 522573., 244456., 139979., 71531., 21461. };
double *dsets[] = { dset1, dset2 };
int dslens[] = { 5, 5 };
int k, l;
double dist, prob;
int dof;
for (k=0; k<2; k++) {
printf("Dataset: [ ");
for(l=0;l<dslens[k]; l++)
printf("%.0f, ", dsets[k][l]);
printf("]\n");
dist = chi2UniformDistance(dsets[k], dslens[k]);
dof = dslens[k]-1;
printf("dof: %d distance: %.4f", dof, dist);
prob = chi2Probability( dof, dist );
printf(" probability: %.6f", prob);
printf(" uniform? %s\n", chiIsUniform(dsets[k], dslens[k], 0.05)? "Yes":"No");
}
return 0;
}
C#
using System;
class Program
{
public delegate double Func(double x);
public static double Simpson38(Func f, double a, double b, int n)
{
double h = (b - a) / n;
double h1 = h / 3;
double sum = f(a) + f(b);
for (int j = 3 * n - 1; j > 0; j--)
{
if (j % 3 == 0)
{
sum += 2 * f(a + h1 * j);
}
else
{
sum += 3 * f(a + h1 * j);
}
}
return h * sum / 8;
}
// Lanczos Approximation for Gamma Function
private static double SpecialFunctionGamma(double z)
{
double[] p =
{
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
};
if (z < 0.5)
return Math.PI / (Math.Sin(Math.PI * z) * SpecialFunctionGamma(1 - z));
z -= 1;
double x = 0.99999999999980993;
for (int i = 0; i < p.Length; i++)
{
x += p[i] / (z + i + 1);
}
double t = z + p.Length - 0.5;
return Math.Sqrt(2 * Math.PI) * Math.Pow(t, z + 0.5) * Math.Exp(-t) * x;
}
public static double GammaIncQ(double a, double x)
{
double aa1 = a - 1;
Func f = t => Math.Pow(t, aa1) * Math.Exp(-t);
double y = aa1;
double h = 1.5e-2;
while (f(y) * (x - y) > 2e-8 && y < x)
{
y += .4;
}
if (y > x)
{
y = x;
}
return 1 - Simpson38(f, 0, y, (int)(y / h / SpecialFunctionGamma(a)));
}
public static double Chi2Ud(int[] ds)
{
double sum = 0, expected = 0;
foreach (var d in ds)
{
expected += d;
}
expected /= ds.Length;
foreach (var d in ds)
{
double x = d - expected;
sum += x * x;
}
return sum / expected;
}
public static double Chi2P(int dof, double distance)
{
return GammaIncQ(.5 * dof, .5 * distance);
}
const double SigLevel = .05;
static void Main(string[] args)
{
int[][] datasets = new int[][]
{
new int[]
{
199809,
200665,
199607,
200270,
199649
},
new int[]
{
522573,
244456,
139979,
71531,
21461
},
};
foreach (var dset in datasets)
{
UTest(dset);
}
}
static void UTest(int[] dset)
{
Console.WriteLine("Uniform distribution test");
int sum = 0;
foreach (var c in dset)
{
sum += c;
}
Console.WriteLine($" dataset: {string.Join(", ", dset)}");
Console.WriteLine($" samples: {sum}");
Console.WriteLine($" categories: {dset.Length}");
int dof = dset.Length - 1;
Console.WriteLine($" degrees of freedom: {dof}");
double dist = Chi2Ud(dset);
Console.WriteLine($" chi square test statistic: {dist}");
double p = Chi2P(dof, dist);
Console.WriteLine($" p-value of test statistic: {p}");
bool sig = p < SigLevel;
Console.WriteLine($" significant at {SigLevel * 100}% level? {sig}");
Console.WriteLine($" uniform? {!sig}\n");
}
}
- Output:
Uniform distribution test dataset: 199809, 200665, 199607, 200270, 199649 samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 4.14628 p-value of test statistic: 0.386570833082767 significant at 5% level? False uniform? True Uniform distribution test dataset: 522573, 244456, 139979, 71531, 21461 samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 790063.27594 p-value of test statistic: 2.35282904270662E-11 significant at 5% level? True uniform? False
C++
#include <iostream>
#include <vector>
#include <cmath>
#include <iomanip>
void print_vector(const std::vector<int32_t>& list) {
std::cout << "[";
for ( uint64_t i = 0; i < list.size() - 1; ++i ) {
std::cout << list[i] << ", ";
}
std::cout << list.back() << "]" << std::endl;
}
bool is_significant(const double p_value, const double significance_level) {
return p_value > significance_level;
}
// The normalised lower incomplete gamma function.
double gamma_cdf(const double aX, const double aK) {
double result = 0.0;
for ( uint32_t m = 0; m <= 99; ++m ) {
result += pow(aX, m) / tgamma(aK + m + 1);
}
result *= pow(aX, aK) * exp(-aX);
return std::isnan(result) ? 1.0 : result;
}
// The cumulative probability function of the Chi-squared distribution.
double cdf(const double aX, const double aK) {
if ( aX > 1'000 && aK < 100 ) {
return 1.0;
}
return ( aX > 0.0 && aK > 0.0 ) ? gamma_cdf(aX / 2, aK / 2) : 0.0;
}
void chi_squared_test(const std::vector<int32_t>& observed) {
double sum = 0.0;
for ( uint64_t i = 0; i < observed.size(); ++i ) {
sum += observed[i];
}
const double expected = sum / observed.size();
const int32_t degree_freedom = observed.size() - 1;
double test_statistic = 0.0;
for ( uint64_t i = 0; i < observed.size(); ++i ) {
test_statistic += pow(observed[i] - expected, 2) / expected;
}
const double p_value = 1.0 - cdf(test_statistic, degree_freedom);
std::cout << "\nUniform distribution test" << std::setprecision(6) << std::endl;
std::cout << " observed values : "; print_vector(observed);
std::cout << " expected value : " << expected << std::endl;
std::cout << " degrees of freedom: " << degree_freedom << std::endl;
std::cout << " test statistic : " << test_statistic << std::endl;
std::cout.setf(std::ios::fixed);
std::cout << " p-value : " << p_value << std::endl;
std::cout.unsetf(std::ios::fixed);
std::cout << " is 5% significant?: " << std::boolalpha << is_significant(p_value, 0.05) << std::endl;
}
int main() {
const std::vector<std::vector<int32_t>> datasets = { { 199809, 200665, 199607, 200270, 199649 },
{ 522573, 244456, 139979, 71531, 21461 } };
for ( std::vector<int32_t> dataset : datasets ) {
chi_squared_test(dataset);
}
}
- Output:
Uniform distribution test
observed values : [199809, 200665, 199607, 200270, 199649]
expected value : 200000
degrees of freedom: 4
test statistic : 4.14628
p-value : 0.386571
is 5% significant?: true
Uniform distribution test
observed values : [522573, 244456, 139979, 71531, 21461]
expected value : 200000
degrees of freedom: 4
test statistic : 790063
p-value : 0.000000
is 5% significant?: false
D
import std.stdio, std.algorithm, std.mathspecial;
real x2Dist(T)(in T[] data) pure nothrow @safe @nogc {
immutable avg = data.sum / data.length;
immutable sqs = reduce!((a, b) => a + (b - avg) ^^ 2)(0.0L, data);
return sqs / avg;
}
real x2Prob(in real dof, in real distance) pure nothrow @safe @nogc {
return gammaIncompleteCompl(dof / 2, distance / 2);
}
bool x2IsUniform(T)(in T[] data, in real significance=0.05L)
pure nothrow @safe @nogc {
return x2Prob(data.length - 1.0L, x2Dist(data)) > significance;
}
void main() {
immutable dataSets = [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461]];
writefln(" %4s %12s %12s %8s %s",
"dof", "distance", "probability", "Uniform?", "dataset");
foreach (immutable ds; dataSets) {
immutable dof = ds.length - 1;
immutable dist = ds.x2Dist;
immutable prob = x2Prob(dof, dist);
writefln("%4d %12.3f %12.8f %5s %6s",
dof, dist, prob, ds.x2IsUniform ? "YES" : "NO", ds);
}
}
- Output:
dof distance probability Uniform? dataset 4 4.146 0.38657083 YES [199809, 200665, 199607, 200270, 199649] 4 790063.276 0.00000000 NO [522573, 244456, 139979, 71531, 21461]
Elixir
defmodule Verify do
defp gammaInc_Q(a, x) do
a1 = a-1
f0 = fn t -> :math.pow(t, a1) * :math.exp(-t) end
df0 = fn t -> (a1-t) * :math.pow(t, a-2) * :math.exp(-t) end
y = while_loop(f0, x, a1)
n = trunc(y / 3.0e-4)
h = y / n
hh = 0.5 * h
sum = Enum.reduce(n-1 .. 0, 0, fn j,sum ->
t = h * j
sum + f0.(t) + hh * df0.(t)
end)
h * sum / gamma_spounge(a, make_coef)
end
defp while_loop(f, x, y) do
if f.(y)*(x-y) > 2.0e-8 and y < x, do: while_loop(f, x, y+0.3), else: min(x, y)
end
@a 12
defp make_coef do
coef0 = [:math.sqrt(2.0 * :math.pi)]
{_, coef} = Enum.reduce(1..@a-1, {1.0, coef0}, fn k,{k1_factrl,c} ->
h = :math.exp(@a-k) * :math.pow(@a-k, k-0.5) / k1_factrl
{-k1_factrl*k, [h | c]}
end)
Enum.reverse(coef) |> List.to_tuple
end
defp gamma_spounge(z, coef) do
accm = Enum.reduce(1..@a-1, elem(coef,0), fn k,res -> res + elem(coef,k) / (z+k) end)
accm * :math.exp(-(z+@a)) * :math.pow(z+@a, z+0.5) / z
end
def chi2UniformDistance(dataSet) do
expected = Enum.sum(dataSet) / length(dataSet)
Enum.reduce(dataSet, 0, fn d,sum -> sum + (d-expected)*(d-expected) end) / expected
end
def chi2Probability(dof, distance) do
1.0 - gammaInc_Q(0.5*dof, 0.5*distance)
end
def chi2IsUniform(dataSet, significance\\0.05) do
dof = length(dataSet) - 1
dist = chi2UniformDistance(dataSet)
chi2Probability(dof, dist) > significance
end
end
dsets = [ [ 199809, 200665, 199607, 200270, 199649 ],
[ 522573, 244456, 139979, 71531, 21461 ] ]
Enum.each(dsets, fn ds ->
IO.puts "Data set:#{inspect ds}"
dof = length(ds) - 1
IO.puts " degrees of freedom: #{dof}"
distance = Verify.chi2UniformDistance(ds)
:io.fwrite " distance: ~.4f~n", [distance]
:io.fwrite " probability: ~.4f~n", [Verify.chi2Probability(dof, distance)]
:io.fwrite " uniform? ~s~n", [(if Verify.chi2IsUniform(ds), do: "Yes", else: "No")]
end)
- Output:
Data set:[199809, 200665, 199607, 200270, 199649] degrees of freedom: 4 distance: 4.1463 probability: 0.3866 uniform? Yes Data set:[522573, 244456, 139979, 71531, 21461] degrees of freedom: 4 distance: 790063.2759 probability: -0.0000 uniform? No
Fortran
Instead of implementing the chi-squared distribution by ourselves, we bind to GNU Scientific Library; so we need a module to interface to the function we need (gsl_cdf_chisq_Q)
module gsl_mini_bind_m
use iso_c_binding
implicit none
private
public :: p_value
interface
function gsl_cdf_chisq_q(x, nu) bind(c, name='gsl_cdf_chisq_Q')
import
real(c_double), value :: x
real(c_double), value :: nu
real(c_double) :: gsl_cdf_chisq_q
end function gsl_cdf_chisq_q
end interface
contains
!> Get p-value from chi-square distribution
real function p_value(x, df)
real, intent(in) :: x
integer, intent(in) :: df
p_value = real(gsl_cdf_chisq_q(real(x, c_double), real(df, c_double)))
end function p_value
end module gsl_mini_bind_m
Now we're ready to complete the task.
program chi2test
use gsl_mini_bind_m, only: p_value
implicit none
real :: dset1(5) = [199809., 200665., 199607., 200270., 199649.]
real :: dset2(5) = [522573., 244456., 139979., 71531., 21461.]
real :: dist, prob
integer :: dof
write (*, '(A)', advance='no') "Dataset 1:"
write (*, '(5(F12.4,:,1x))') dset1
dist = chisq(dset1)
dof = size(dset1) - 1
write (*, '(A,I4,A,F12.4)') 'dof: ', dof, ' chisq: ', dist
prob = p_value(dist, dof)
write (*, '(A,F12.4)') 'probability: ', prob
write (*, '(A,L)') 'uniform? ', prob > 0.05
! Lazy copy/past :|
write (*, '(/A)', advance='no') "Dataset 2:"
write (*, '(5(F12.4,:,1x))') dset2
dist = chisq(dset2)
dof = size(dset2) - 1
write (*, '(A,I4,A,F12.4)') 'dof: ', dof, ' chisq: ', dist
prob = p_value(dist, dof)
write (*, '(A,F12.4)') 'probability: ', prob
write (*, '(A,L)') 'uniform? ', prob > 0.05
contains
!> Get chi-square value for a set of data `ds`
real function chisq(ds)
real, intent(in) :: ds(:)
real :: expected, summa
expected = sum(ds)/size(ds)
summa = sum((ds - expected)**2)
chisq = summa/expected
end function chisq
end program chi2test
Output:
Dataset 1: 199809.0000 200665.0000 199607.0000 200270.0000 199649.0000
dof: 4 chisq: 4.1463
probability: 0.3866
uniform? T
Dataset 2: 522573.0000 244456.0000 139979.0000 71531.0000 21461.0000
dof: 4 chisq: 790063.2500
probability: 0.0000
uniform? FFreeBASIC
Function integrate(a As Double, b As Double, n As Integer, f As Function(x As Double, a As Double) As Double, param As Double) As Double
Dim As Double h = (b - a) / n
Dim As Double sum = 0
For i As Integer = 0 To n - 1
Dim As Double x = a + i * h
sum += (f(x, param) + 4 * f(x + h/2, param) + f(x + h, param)) / 6
Next
Return sum * h
End Function
Function f0(t As Double, a As Double) As Double
Return (t ^ (a - 1)) * Exp(-t)
End Function
' Simple implementation of the gamma function
Function gammaFunc(x As Double) As Double
If x = 1 Or x = 2 Then
Return 1
Elseif x > 0 And x < 1 Then
Return gammaFunc(x + 1) / x
Elseif x > 1 Then
Return (x - 1) * gammaFunc(x - 1)
Else
Return 0 ' Undefined for x <= 0
End If
End Function
Function gammaIncomplete(a As Double, x As Double) As Double
Dim As Double am1 = a - 1
Dim As Double h = 1.5e-2
Dim As Double y = am1
While (f0(y, a) * (x - y) > 2e-8 And y < x)
y += 0.4
Wend
If y > x Then y = x
Dim As Double integralResult = integrate(0, y, Int(y/h), @f0, a)
Return 1 - integralResult / gammaFunc(a)
End Function
Function chi2UniformDistance(ds() As Double) As Double
Dim As Double expected = 0
Dim As Integer i, n = Ubound(ds) - Lbound(ds) + 1
For i = Lbound(ds) To Ubound(ds)
expected += ds(i)
Next
expected /= n
Dim As Double sum = 0
For i = Lbound(ds) To Ubound(ds)
sum += (ds(i) - expected) ^ 2
Next
Return sum / expected
End Function
Function chi2Probability(dof As Integer, dist As Double) As Double
Return gammaIncomplete(0.5 * dof, 0.5 * dist)
End Function
Function chiIsUniform(ds() As Double, significance As Double) As Boolean
Dim As Integer dof = Ubound(ds) - Lbound(ds)
Dim As Double dist = chi2UniformDistance(ds())
Return chi2Probability(dof, dist) > significance
End Function
' Main program
Dim As Double dsets(1, 4) = { _
{199809, 200665, 199607, 200270, 199649}, _
{522573, 244456, 139979, 71531, 21461} }
For i As Integer = 0 To 1
Dim As Double ds(4)
For j As Integer = 0 To 4
ds(j) = dsets(i, j)
Next
Print "Dataset: ";
For j As Integer = 0 To 4
Print Using "###### "; ds(j);
Next
Print
Dim As Integer dof = Ubound(ds) - Lbound(ds)
Dim As Double dist = chi2UniformDistance(ds())
Print Using "DOF: # Distance: &"; dof; dist;
Dim As Double prob = chi2Probability(dof, dist)
Print Using " Probability: ##.#####"; prob;
Dim As String uniform = Iif(chiIsUniform(ds(), 0.05), "Yes", "No")
Print " Uniform? "; uniform
Print
Next
Sleep
- Output:
Dataset: 199809 200665 199607 200270 199649 DOF: 4 Distance: 4.14628 Probability: 0.38657 Uniform? Yes Dataset: 522573 244456 139979 71531 21461 DOF: 4 Distance: 790063.27594 Probability: 0.00000 Uniform? No
Go
Go has a nice gamma function in the library. Otherwise, it's mostly a port from C. Note, this implementation of the incomplete gamma function works for these two test cases, but, I believe, has serious limitations. See talk page.
package main
import (
"fmt"
"math"
)
type ifctn func(float64) float64
func simpson38(f ifctn, a, b float64, n int) float64 {
h := (b - a) / float64(n)
h1 := h / 3
sum := f(a) + f(b)
for j := 3*n - 1; j > 0; j-- {
if j%3 == 0 {
sum += 2 * f(a+h1*float64(j))
} else {
sum += 3 * f(a+h1*float64(j))
}
}
return h * sum / 8
}
func gammaIncQ(a, x float64) float64 {
aa1 := a - 1
var f ifctn = func(t float64) float64 {
return math.Pow(t, aa1) * math.Exp(-t)
}
y := aa1
h := 1.5e-2
for f(y)*(x-y) > 2e-8 && y < x {
y += .4
}
if y > x {
y = x
}
return 1 - simpson38(f, 0, y, int(y/h/math.Gamma(a)))
}
func chi2ud(ds []int) float64 {
var sum, expected float64
for _, d := range ds {
expected += float64(d)
}
expected /= float64(len(ds))
for _, d := range ds {
x := float64(d) - expected
sum += x * x
}
return sum / expected
}
func chi2p(dof int, distance float64) float64 {
return gammaIncQ(.5*float64(dof), .5*distance)
}
const sigLevel = .05
func main() {
for _, dset := range [][]int{
{199809, 200665, 199607, 200270, 199649},
{522573, 244456, 139979, 71531, 21461},
} {
utest(dset)
}
}
func utest(dset []int) {
fmt.Println("Uniform distribution test")
var sum int
for _, c := range dset {
sum += c
}
fmt.Println(" dataset:", dset)
fmt.Println(" samples: ", sum)
fmt.Println(" categories: ", len(dset))
dof := len(dset) - 1
fmt.Println(" degrees of freedom: ", dof)
dist := chi2ud(dset)
fmt.Println(" chi square test statistic: ", dist)
p := chi2p(dof, dist)
fmt.Println(" p-value of test statistic: ", p)
sig := p < sigLevel
fmt.Printf(" significant at %2.0f%% level? %t\n", sigLevel*100, sig)
fmt.Println(" uniform? ", !sig, "\n")
}
Output:
Uniform distribution test dataset: [199809 200665 199607 200270 199649] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 4.14628 p-value of test statistic: 0.3865708330827673 significant at 5% level? false uniform? true Uniform distribution test dataset: [522573 244456 139979 71531 21461] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 790063.27594 p-value of test statistic: 2.3528290427066167e-11 significant at 5% level? true uniform? false
Hy
(import
[scipy.stats [chisquare]]
[collections [Counter]])
(defn uniform? [f repeats &optional [alpha .05]]
"Call 'f' 'repeats' times and do a chi-squared test for uniformity
of the resulting discrete distribution. Return false iff the
null hypothesis of uniformity is rejected for the test with
size 'alpha'."
(<= alpha (second (chisquare
(.values (Counter (take repeats (repeatedly f))))))))
Examples of use:
(import [random [randint]])
(for [f [
(fn [] (randint 1 10))
(fn [] (if (randint 0 1) (randint 1 9) (randint 1 10)))]]
(print (uniform? f 5000)))
J
Solution (Tacit):
require 'stats/base'
countCats=: #@~. NB. counts the number of unique items
getExpected=: #@] % [ NB. divides no of items by category count
getObserved=: #/.~@] NB. counts frequency for each category
calcX2=: [: +/ *:@(getObserved - getExpected) % getExpected NB. calculates test statistic
calcDf=: <:@[ NB. calculates degrees of freedom for uniform distribution
NB.*isUniform v Tests (5%) whether y is uniformly distributed
NB. result is: boolean describing if distribution y is uniform
NB. y is: distribution to test
NB. x is: optionally specify number of categories possible
isUniform=: (countCats $: ]) : (0.95 > calcDf chisqcdf :: 1: calcX2)
Solution (Explicit):
require 'stats/base'
NB.*isUniformX v Tests (5%) whether y is uniformly distributed
NB. result is: boolean describing if distribution y is uniform
NB. y is: distribution to test
NB. x is: optionally specify number of categories possible
isUniformX=: verb define
(#~. y) isUniformX y
:
signif=. 0.95 NB. set significance level
expected=. (#y) % x NB. number of items divided by the category count
observed=. #/.~ y NB. frequency count for each category
X2=. +/ (*: observed - expected) % expected NB. the test statistic
degfreedom=. <: x NB. degrees of freedom
signif > degfreedom chisqcdf :: 1: X2
)
Example Usage:
FairDistrib=: 1e6 ?@$ 5
UnfairDistrib=: (9.5e5 ?@$ 5) , (5e4 ?@$ 4)
isUniformX FairDistrib
1
isUniformX UnfairDistrib
0
isUniform 4 4 4 5 5 5 5 5 5 5 NB. uniform if only 2 categories possible
1
4 isUniform 4 4 4 5 5 5 5 5 5 5 NB. not uniform if 4 categories possible
0
Java
import static java.lang.Math.pow;
import java.util.Arrays;
import static java.util.Arrays.stream;
import org.apache.commons.math3.special.Gamma;
public class Test {
static double x2Dist(double[] data) {
double avg = stream(data).sum() / data.length;
double sqs = stream(data).reduce(0, (a, b) -> a + pow((b - avg), 2));
return sqs / avg;
}
static double x2Prob(double dof, double distance) {
return Gamma.regularizedGammaQ(dof / 2, distance / 2);
}
static boolean x2IsUniform(double[] data, double significance) {
return x2Prob(data.length - 1.0, x2Dist(data)) > significance;
}
public static void main(String[] a) {
double[][] dataSets = {{199809, 200665, 199607, 200270, 199649},
{522573, 244456, 139979, 71531, 21461}};
System.out.printf(" %4s %12s %12s %8s %s%n",
"dof", "distance", "probability", "Uniform?", "dataset");
for (double[] ds : dataSets) {
int dof = ds.length - 1;
double dist = x2Dist(ds);
double prob = x2Prob(dof, dist);
System.out.printf("%4d %12.3f %12.8f %5s %6s%n",
dof, dist, prob, x2IsUniform(ds, 0.05) ? "YES" : "NO",
Arrays.toString(ds));
}
}
}
dof distance probability Uniform? dataset 4 4,146 0,38657083 YES [199809.0, 200665.0, 199607.0, 200270.0, 199649.0] 4 790063,276 0,00000000 NO [522573.0, 244456.0, 139979.0, 71531.0, 21461.0]
jq
Also works with gojq, the Go implementation of jq.
This entry uses a two-tailed test, as is appropriate for this type of problem as illustrated by the last example. The test is based on the assumption that the sample size is large enough for the χ2 distribution to be used.
The implementation of `Chi2_cdf` here uses the recursion relation for the gamma function and should be both fast, accurate and quite robust. For an industrial-strength algorithm, see e.g. https://people.sc.fsu.edu/~jburkardt/c_src/asa239/asa239.c
Generic Functions
def round($dec):
if type == "string" then .
else pow(10;$dec) as $m
| . * $m | floor / $m
end;
# sum of squares
def ss(s): reduce s as $x (0; . + ($x * $x));
# Cumulative density function of the chi-squared distribution with $k
# degrees of freedom
# The recursion formula for gamma is used for efficiency and robustness.
def Chi2_cdf($x; $k):
if $x == 0 then 0
elif $x > (1e3 * $k) then 1
else 1e-15 as $tol # for example
| { s: 0, m: 0, term: (1 / ((($k/2)+1)|gamma)) }
| until (.term|length < $tol; # length here is abs
.s += .term
| .m += 1
| .term *= (($x/2) / (($k/2) + .m )) )
| .s * ( ((-$x/2) + ($k/2)*(($x/2)|log)) | exp)
end ;Tasks
# Input: array of frequencies
def chi2UniformDistance:
(add / length) as $expected
| ss(.[] - $expected) / $expected;
# Input: a number
# Output: an indication of the probability of observing this value or higher
# assuming the value is drawn from a chi-squared distribution with $dof degrees
# of freedom
def chi2Probability($dof):
(1 - Chi2_cdf(.; $dof))
| if . < 1e-10 then "< 1e-10"
else .
end;
# Input: array of frequencies
# Output: result of a two-tailed test based on the chi-squared statistic
# assuming the sample size is large enough
def chiIsUniform($significance):
(length - 1) as $dof
| chi2UniformDistance
| Chi2_cdf(.; $dof) as $cdf
| if $cdf
then ($significance/2) as $s
| $cdf > $s and $cdf < (1-$s)
else false
end;
def dsets: [
[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461],
[19,14,6,18,7,5,1], # low entropy
[9,11,9,10,15,11,5], # high entropy
[20,20,20] # made-up
];
def task:
dsets[]
| "Dataset: \(.)",
( chi2UniformDistance as $dist
| (length - 1) as $dof
| "DOF: \($dof) D (Distance): \($dist)",
" Estimated probability of observing a value >= D: \($dist|chi2Probability($dof)|round(2))",
" Uniform? \( (select(chiIsUniform(0.05)) | "Yes") // "No" )\n" ) ;
taskDataset: [199809,200665,199607,200270,199649] DOF: 4 D (Distance): 4.14628 Estimated probability of observing a value >= D: 0.38 Uniform? Yes Dataset: [522573,244456,139979,71531,21461] DOF: 4 D (Distance): 790063.27594 Estimated probability of observing a value >= D: < 1e-10 Uniform? No Dataset: [19,14,6,18,7,5,1] DOF: 6 D (Distance): 29.2 Estimated probability of observing a value >= D: 0 Uniform? No Dataset: [9,11,9,10,15,11,5] DOF: 6 D (Distance): 5.4 Estimated probability of observing a value >= D: 0.49 Uniform? Yes Dataset: [20,20,20] DOF: 2 D (Distance): 0 Estimated probability of observing a value >= D: 1 Uniform? No
Julia
# v0.6
using Distributions
function eqdist(data::Vector{T}, α::Float64=0.05)::Bool where T <: Real
if ! (0 ≤ α ≤ 1); error("α must be in [0, 1]") end
exp = mean(data)
chisqval = sum((x - exp) ^ 2 for x in data) / exp
pval = ccdf(Chisq(2), chisqval)
return pval > α
end
data1 = [199809, 200665, 199607, 200270, 199649]
data2 = [522573, 244456, 139979, 71531, 21461]
for data in (data1, data2)
println("Data:\n$data")
println("Hypothesis test: the original population is ", (eqdist(data) ? "" : "not "), "uniform.\n")
end
- Output:
Data: [199809, 200665, 199607, 200270, 199649] Hypothesis test: the original population is uniform. Data: [522573, 244456, 139979, 71531, 21461] Hypothesis test: the original population is not uniform.
Kotlin
This program reuses Kotlin code from the Gamma function and Numerical Integration tasks but otherwise is a translation of the C entry for this task.
// version 1.1.51
typealias Func = (Double) -> Double
fun gammaLanczos(x: Double): Double {
var xx = x
val p = doubleArrayOf(
0.99999999999980993,
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
)
val g = 7
if (xx < 0.5) return Math.PI / (Math.sin(Math.PI * xx) * gammaLanczos(1.0 - xx))
xx--
var a = p[0]
val t = xx + g + 0.5
for (i in 1 until p.size) a += p[i] / (xx + i)
return Math.sqrt(2.0 * Math.PI) * Math.pow(t, xx + 0.5) * Math.exp(-t) * a
}
fun integrate(a: Double, b: Double, n: Int, f: Func): Double {
val h = (b - a) / n
var sum = 0.0
for (i in 0 until n) {
val x = a + i * h
sum += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0
}
return sum * h
}
fun gammaIncompleteQ(a: Double, x: Double): Double {
val aa1 = a - 1.0
fun f0(t: Double) = Math.pow(t, aa1) * Math.exp(-t)
val h = 1.5e-2
var y = aa1
while ((f0(y) * (x - y) > 2.0e-8) && y < x) y += 0.4
if (y > x) y = x
return 1.0 - integrate(0.0, y, (y / h).toInt(), ::f0) / gammaLanczos(a)
}
fun chi2UniformDistance(ds: DoubleArray): Double {
val expected = ds.average()
val sum = ds.map { val x = it - expected; x * x }.sum()
return sum / expected
}
fun chi2Probability(dof: Int, distance: Double) =
gammaIncompleteQ(0.5 * dof, 0.5 * distance)
fun chiIsUniform(ds: DoubleArray, significance: Double):Boolean {
val dof = ds.size - 1
val dist = chi2UniformDistance(ds)
return chi2Probability(dof, dist) > significance
}
fun main(args: Array<String>) {
val dsets = listOf(
doubleArrayOf(199809.0, 200665.0, 199607.0, 200270.0, 199649.0),
doubleArrayOf(522573.0, 244456.0, 139979.0, 71531.0, 21461.0)
)
for (ds in dsets) {
println("Dataset: ${ds.asList()}")
val dist = chi2UniformDistance(ds)
val dof = ds.size - 1
print("DOF: $dof Distance: ${"%.4f".format(dist)}")
val prob = chi2Probability(dof, dist)
print(" Probability: ${"%.6f".format(prob)}")
val uniform = if (chiIsUniform(ds, 0.05)) "Yes" else "No"
println(" Uniform? $uniform\n")
}
}
- Output:
Dataset: [199809.0, 200665.0, 199607.0, 200270.0, 199649.0] DOF: 4 Distance: 4.1463 Probability: 0.386571 Uniform? Yes Dataset: [522573.0, 244456.0, 139979.0, 71531.0, 21461.0] DOF: 4 Distance: 790063.2759 Probability: 0.000000 Uniform? No
Mathematica /Wolfram Language
This code explicity assumes a discrete uniform distribution since the chi square test is a poor test choice for continuous distributions and requires Mathematica version 2 or later
discreteUniformDistributionQ[data_, {min_Integer, max_Integer}, confLevel_: .05] :=
If[$VersionNumber >= 8,
confLevel <= PearsonChiSquareTest[data, DiscreteUniformDistribution[{min, max}]],
Block[{v, k = max - min, n = Length@data},
v = (k + 1) (Plus @@ (((Length /@ Split[Sort@data]))^2))/n - n;
GammaRegularized[k/2, 0, v/2] <= 1 - confLevel]]
discreteUniformDistributionQ[data_] :=discreteUniformDistributionQ[data, data[[Ordering[data][[{1, -1}]]]]]
code used to create test data requires Mathematica version 6 or later
uniformData = RandomInteger[10, 100];
nonUniformData = Total@RandomInteger[10, {5, 100}];
{discreteUniformDistributionQ[uniformData],discreteUniformDistributionQ[nonUniformData]}
- Output:
{True,False}
Nim
We use the gamma function from the “math” module. To simplify the code, we use also the “lenientops” module which provides mixed operations between floats ane integers.
import lenientops, math, stats, strformat, sugar
func simpson38(f: (float) -> float; a, b: float; n: int): float =
let h = (b - a) / n
let h1 = h / 3
var sum = f(a) + f(b)
for i in countdown(3 * n - 1, 1):
if i mod 3 == 0:
sum += 2 * f(a + h1 * i)
else:
sum += 3 * f(a + h1 * i)
result = h * sum / 8
func gammaIncQ(a, x: float): float =
let aa1 = a - 1
func f(t: float): float = pow(t, aa1) * exp(-t)
var y = aa1
let h = 1.5e-2
while f(y) * (x - y) > 2e-8 and y < x:
y += 0.4
if y > x: y = x
result = 1 - simpson38(f, 0, y, (y / h / gamma(a)).toInt)
func chi2ud(ds: openArray[int]): float =
let expected = mean(ds)
var s = 0.0
for d in ds:
let x = d.toFloat - expected
s += x * x
result = s / expected
func chi2p(dof: int; distance: float): float =
gammaIncQ(0.5 * dof, 0.5 * distance)
const SigLevel = 0.05
proc utest(dset: openArray[int]) =
echo "Uniform distribution test"
let s = sum(dset)
echo " dataset:", dset
echo " samples: ", s
echo " categories: ", dset.len
let dof = dset.len - 1
echo " degrees of freedom: ", dof
let dist = chi2ud(dset)
echo " chi square test statistic: ", dist
let p = chi2p(dof, dist)
echo " p-value of test statistic: ", p
let sig = p < SigLevel
echo &" significant at {int(SigLevel * 100)}% level? {sig}"
echo &" uniform? {not sig}\n"
for dset in [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461]]:
utest(dset)
- Output:
Uniform distribution test dataset:[199809, 200665, 199607, 200270, 199649] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 4.14628 p-value of test statistic: 0.3865708330827673 significant at 5% level? false uniform? true Uniform distribution test dataset:[522573, 244456, 139979, 71531, 21461] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 790063.27594 p-value of test statistic: 2.34864350190378e-11 significant at 5% level? true uniform? false
OCaml
This code needs to be compiled with library gsl.cma.
let sqr x = x *. x
let chi2UniformDistance distrib =
let count, len = Array.fold_left (fun (s, c) e -> s + e, succ c)
(0, 0) distrib in
let expected = float count /. float len in
let distance = Array.fold_left (fun s e ->
s +. sqr (float e -. expected) /. expected
) 0. distrib in
let dof = float (pred len) in
dof, distance
let chi2Proba dof distance =
Gsl_sf.gamma_inc_Q (0.5 *. dof) (0.5 *. distance)
let chi2IsUniform distrib significance =
let dof, distance = chi2UniformDistance distrib in
let likelihoodOfRandom = chi2Proba dof distance in
likelihoodOfRandom > significance
let _ =
List.iter (fun distrib ->
let dof, distance = chi2UniformDistance distrib in
Printf.printf "distribution ";
Array.iter (Printf.printf "\t%d") distrib;
Printf.printf "\tdistance %g" distance;
Printf.printf "\t[%g > 0.05]" (chi2Proba dof distance);
if chi2IsUniform distrib 0.05 then Printf.printf " fair\n"
else Printf.printf " unfair\n"
)
[
[| 199809; 200665; 199607; 200270; 199649 |];
[| 522573; 244456; 139979; 71531; 21461 |]
]
Output
distribution 199809 200665 199607 200270 199649 distance 4.14628 [0.386571 > 0.05] fair distribution 522573 244456 139979 71531 21461 distance 790063 [0 > 0.05] unfair
PARI/GP
The solution is very easy in GP since PARI includes a good incomplete gamma implementation; the sum function is also useful for clarity. Most of the code is just for displaying results.
The sample data for the test was taken from Go.
cumChi2(chi2,dof)={
my(g=gamma(dof/2));
incgam(dof/2,chi2/2,g)/g
};
test(v,alpha=.05)={
my(chi2,p,s=sum(i=1,#v,v[i]),ave=s/#v);
print("chi^2 statistic: ",chi2=sum(i=1,#v,(v[i]-ave)^2)/ave);
print("p-value: ",p=cumChi2(chi2,#v-1));
if(p<alpha,
print("Significant at the alpha = "alpha" level: not uniform");
,
print("Not significant at the alpha = "alpha" level: uniform");
)
};
test([199809, 200665, 199607, 200270, 199649])
test([522573, 244456, 139979, 71531, 21461])Perl
use List::Util qw(sum reduce);
use constant pi => 3.14159265;
sub incomplete_G_series {
my($s, $z) = @_;
my $n = 10;
push @numers, $z**$_ for 1..$n;
my @denoms = $s+1;
push @denoms, $denoms[-1]*($s+$_) for 2..$n;
my $M = 1;
$M += $numers[$_-1]/$denoms[$_-1] for 1..$n;
$z**$s / $s * exp(-$z) * $M;
}
sub G_of_half {
my($n) = @_;
if ($n % 2) { f(2*$_) / (4**$_ * f($_)) * sqrt(pi) for int ($n-1) / 2 }
else { f(($n/2)-1) }
}
sub f { reduce { $a * $b } 1, 1 .. $_[0] } # factorial
sub chi_squared_cdf {
my($k, $x) = @_;
my $f = $k < 20 ? 20 : 10;
if ($x == 0) { 0.0 }
elsif ($x < $k + $f*sqrt($k)) { incomplete_G_series($k/2, $x/2) / G_of_half($k) }
else { 1.0 }
}
sub chi_squared_test {
my(@bins) = @_;
$significance = 0.05;
my $n = @bins;
my $N = sum @bins;
my $expected = $N / $n;
my $chi_squared = sum map { ($_ - $expected)**2 / $expected } @bins;
my $p_value = 1 - chi_squared_cdf($n-1, $chi_squared);
return $chi_squared, $p_value, $p_value > $significance ? 'True' : 'False';
}
for $dataset ([199809, 200665, 199607, 200270, 199649], [522573, 244456, 139979, 71531, 21461]) {
printf "C2 = %10.3f, p-value = %.3f, uniform = %s\n", chi_squared_test(@$dataset);
}
- Output:
C2 = 4.146, p-value = 0.387, uniform = True C2 = 790063.276, p-value = 0.000, uniform = False
Phix
with javascript_semantics function f(atom aa1, t) return power(t, aa1) * exp(-t) end function function simpson38(atom aa1, a, b, integer n) atom h := (b-a)/n, h1 := h/3, tot := f(aa1,a) + f(aa1,b) for j=3*n-1 to 1 by -1 do tot += (3-(mod(j,3)=0)) * f(aa1,a+h1*j) end for return h*tot/8 end function --<copy of gamma from Gamma_function#Phix> sequence c = repeat(0,12) function gamma(atom z) atom accm = c[1] if accm=0 then accm = sqrt(2*PI) c[1] = accm atom k1_factrl = 1 -- (k - 1)!*(-1)^k with 0!==1 for k=2 to 12 do c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl k1_factrl *= -(k-1) end for end if for k=2 to 12 do accm += c[k]/(z+k-1) end for accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1) return accm/z end function --</copy of gamma> function gammaIncQ(atom a, x) atom aa1 := a-1, y := aa1, h := 1.5e-2 while f(aa1,y)*(x-y) > 2e-8 and y < x do y += 0.4 end while if y > x then y = x end if return 1 - simpson38(aa1,0,y,floor(y/h/gamma(a))) end function function chi2ud(sequence ds) atom expected = sum(ds)/length(ds), tot = sum(sq_power(sq_sub(ds,expected),2)) return tot/expected end function function chi2p(integer dof, atom distance) return gammaIncQ(0.5*dof,0.5*distance) end function constant sigLevel = 0.05 procedure utest(sequence dset) printf(1,"Uniform distribution test\n") integer tot = sum(dset), dof := length(dset)-1 atom dist := chi2ud(dset), p := chi2p(dof, dist) bool sig := p < sigLevel printf(1," dataset: %v\n",{dset}) printf(1," samples: %d\n", tot) printf(1," categories: %d\n", length(dset)) printf(1," degrees of freedom: %d\n", dof) printf(1," chi square test statistic: %g\n", dist) printf(1," p-value of test statistic: %g\n", p) printf(1," significant at %.0f%% level? %t\n", {sigLevel*100, sig}) printf(1," uniform? %t\n",not sig) end procedure utest({199809, 200665, 199607, 200270, 199649}) utest({522573, 244456, 139979, 71531, 21461})
- Output:
Uniform distribution test
dataset: {199809,200665,199607,200270,199649}
samples: 1000000
categories: 5
degrees of freedom: 4
chi square test statistic: 4.14628
p-value of test statistic: 0.386571
significant at 5% level? false
uniform? true
Uniform distribution test
dataset: {522573,244456,139979,71531,21461}
samples: 1000000
categories: 5
degrees of freedom: 4
chi square test statistic: 790063
p-value of test statistic: 2.35282e-11
significant at 5% level? true
uniform? false
Python
Python: Using only standard libraries
Implements the Chi Square Probability function with an integration. I'm sure there are better ways to do this. Compare to OCaml implementation.
import math
import random
def GammaInc_Q( a, x):
a1 = a-1
a2 = a-2
def f0( t ):
return t**a1*math.exp(-t)
def df0(t):
return (a1-t)*t**a2*math.exp(-t)
y = a1
while f0(y)*(x-y) >2.0e-8 and y < x: y += .3
if y > x: y = x
h = 3.0e-4
n = int(y/h)
h = y/n
hh = 0.5*h
gamax = h * sum( f0(t)+hh*df0(t) for t in ( h*j for j in xrange(n-1, -1, -1)))
return gamax/gamma_spounge(a)
c = None
def gamma_spounge( z):
global c
a = 12
if c is None:
k1_factrl = 1.0
c = []
c.append(math.sqrt(2.0*math.pi))
for k in range(1,a):
c.append( math.exp(a-k) * (a-k)**(k-0.5) / k1_factrl )
k1_factrl *= -k
accm = c[0]
for k in range(1,a):
accm += c[k] / (z+k)
accm *= math.exp( -(z+a)) * (z+a)**(z+0.5)
return accm/z;
def chi2UniformDistance( dataSet ):
expected = sum(dataSet)*1.0/len(dataSet)
cntrd = (d-expected for d in dataSet)
return sum(x*x for x in cntrd)/expected
def chi2Probability(dof, distance):
return 1.0 - GammaInc_Q( 0.5*dof, 0.5*distance)
def chi2IsUniform(dataSet, significance):
dof = len(dataSet)-1
dist = chi2UniformDistance(dataSet)
return chi2Probability( dof, dist ) > significance
dset1 = [ 199809, 200665, 199607, 200270, 199649 ]
dset2 = [ 522573, 244456, 139979, 71531, 21461 ]
for ds in (dset1, dset2):
print "Data set:", ds
dof = len(ds)-1
distance =chi2UniformDistance(ds)
print "dof: %d distance: %.4f" % (dof, distance),
prob = chi2Probability( dof, distance)
print "probability: %.4f"%prob,
print "uniform? ", "Yes"if chi2IsUniform(ds,0.05) else "No"
Output:
Data set: [199809, 200665, 199607, 200270, 199649] dof: 4 distance: 4.146280 probability: 0.3866 uniform? Yes Data set: [522573, 244456, 139979, 71531, 21461] dof: 4 distance: 790063.275940 probability: 0.0000 uniform? No
Python: Using scipy
This uses the library routine scipy.stats.chisquare.
from scipy.stats import chisquare
if __name__ == '__main__':
dataSets = [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461]]
print(f"{'Distance':^12} {'pvalue':^12} {'Uniform?':^8} {'Dataset'}")
for ds in dataSets:
dist, pvalue = chisquare(ds)
uni = 'YES' if pvalue > 0.05 else 'NO'
print(f"{dist:12.3f} {pvalue:12.8f} {uni:^8} {ds}")
- Output:
Distance pvalue Uniform? Dataset
4.146 0.38657083 YES [199809, 200665, 199607, 200270, 199649]
790063.276 0.00000000 NO [522573, 244456, 139979, 71531, 21461]
R
R being a statistical computating language, the chi-squared test is built in with the function "chisq.test"
dset1=c(199809,200665,199607,200270,199649)
dset2=c(522573,244456,139979,71531,21461)
chi2IsUniform<-function(dataset,significance=0.05){
chi2IsUniform=(chisq.test(dataset)$p.value>significance)
}
for (ds in list(dset1,dset2)){
print(c("Data set:",ds))
print(chisq.test(ds))
print(paste("uniform?",chi2IsUniform(ds)))
}
Output:
[1] "Data set:" "199809" "200665" "199607" "200270" "199649"
Chi-squared test for given probabilities
data: ds
X-squared = 4.1463, df = 4, p-value = 0.3866
[1] "uniform? TRUE"
[1] "Data set:" "522573" "244456" "139979" "71531" "21461"
Chi-squared test for given probabilities
data: ds
X-squared = 790063.3, df = 4, p-value < 2.2e-16
[1] "uniform? FALSE"
Racket
#lang racket
(require
racket/flonum (planet williams/science:4:5/science)
(only-in (planet williams/science/unsafe-ops-utils) real->float))
; (chi^2-goodness-of-fit-test observed expected df)
; Given: observed, a sequence of observed frequencies
; expected, a sequence of expected frequencies
; df, the degrees of freedom
; Result: P-value = 1-chi^2cdf(X^2,df) , the p-value
(define (chi^2-goodness-of-fit-test observed expected df)
(define X^2 (for/sum ([o observed] [e expected])
(/ (sqr (- o e)) e)))
(- 1.0 (chi-squared-cdf X^2 df)))
(define (is-uniform? rand n α)
; Use significance level α to test whether
; n small random numbers generated by rand
; have a uniform distribution.
; Observed values:
(define o (make-vector 10 0))
; generate n random integers from 0 to 9.
(for ([_ (+ n 1)])
(define r (rand 10))
(vector-set! o r (+ (vector-ref o r) 1)))
; Expected values:
(define ex (make-vector 10 (/ n 10)))
; Calculate the P-value:
(define P (chi^2-goodness-of-fit-test o ex (- n 1)))
; If the P-value is larger than α we accept the
; hypothesis that the numbers are distributed uniformly.
(> P α))
; Test whether the builtin generator is uniform:
(is-uniform? random 1000 0.05)
; Test whether the constant generator fails:
(is-uniform? (λ(_) 5) 1000 0.05)
Output:
#t
#f
Raku
(formerly Perl 6)
For the incomplete gamma function we use a series expansion related to Kummer's confluent hypergeometric function (see http://en.wikipedia.org/wiki/Incomplete_gamma_function#Evaluation_formulae). The gamma function is calculated in closed form, as we only need its value at integers and half integers.
sub incomplete-γ-series($s, $z) {
my \numers = $z X** 1..*;
my \denoms = [\*] $s X+ 1..*;
my $M = 1 + [+] (numers Z/ denoms) ... * < 1e-6;
$z**$s / $s * exp(-$z) * $M;
}
sub postfix:<!>(Int $n) { [*] 2..$n }
sub Γ-of-half(Int $n where * > 0) {
($n %% 2) ?? (($_-1)! given $n div 2)
!! ((2*$_)! / (4**$_ * $_!) * sqrt(pi) given ($n-1) div 2);
}
# degrees of freedom constrained due to numerical limitations
sub chi-squared-cdf(Int $k where 1..200, $x where * >= 0) {
my $f = $k < 20 ?? 20 !! 10;
given $x {
when 0 { 0.0 }
when * < $k + $f*sqrt($k) { incomplete-γ-series($k/2, $x/2) / Γ-of-half($k) }
default { 1.0 }
}
}
sub chi-squared-test(@bins, :$significance = 0.05) {
my $n = +@bins;
my $N = [+] @bins;
my $expected = $N / $n;
my $chi-squared = [+] @bins.map: { ($^bin - $expected)**2 / $expected }
my $p-value = 1 - chi-squared-cdf($n-1, $chi-squared);
return (:$chi-squared, :$p-value, :uniform($p-value > $significance));
}
for [< 199809 200665 199607 200270 199649 >],
[< 522573 244456 139979 71531 21461 >]
-> $dataset
{
my %t = chi-squared-test($dataset);
say 'data: ', $dataset;
say "χ² = {%t<chi-squared>}, p-value = {%t<p-value>.fmt('%.4f')}, uniform = {%t<uniform>}";
}
- Output:
data: 199809 200665 199607 200270 199649 χ² = 4.14628, p-value = 0.3866, uniform = True data: 522573 244456 139979 71531 21461 χ² = 790063.27594, p-value = 0.0000, uniform = False
REXX
Programming notes:
The use of the pow was elided as it can just be replaced with t**(a-1).
The gamma was replaced with a simple version. The argument
for gamma is (in the cases used herein) always
positive, and is
either an integer, or a number which is a multiple of 1/2, both of these cases can be calculated with
a straight─forward calculation.
/*REXX program performs a chi─squared test to verify a given distribution is uniform. */
numeric digits length( pi() ) - length(.) /*enough decimal digs for calculations.*/
@.=; @.1= 199809 200665 199607 200270 199649
@.2= 522573 244456 139979 71531 21461
do s=1 while @.s\==''; call uTest @.s /*invoke uTest with a data set of #'s.*/
end /*s*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!: procedure; parse arg x; p=1; do j=2 to x; p= p*j; end /*j*/; return p
chi2p: procedure; parse arg dof, distance; return gammaI( dof/2, distance/2 )
f: parse arg t; if t=0 then return 0; return t ** (a-1) * exp(-t)
e: e =2.718281828459045235360287471352662497757247093699959574966967627724; return e
pi: pi=3.141592653589793238462643383279502884197169399375105820974944592308; return pi
/*──────────────────────────────────────────────────────────────────────────────────────*/
!!: procedure; parse arg x; if x<2 then return 1; p= x
do k=2+x//2 to x-1 by 2; p= p*k; end /*k*/; return p
/*──────────────────────────────────────────────────────────────────────────────────────*/
chi2ud: procedure: parse arg ds; sum=0; expect= 0
do j=1 for words(ds); expect= expect + word(ds, j)
end /*j*/
expect = expect / words(ds)
do k=1 for words(ds)
sum= sum + (word(ds, k) - expect) **2
end /*k*/
return sum / expect
/*──────────────────────────────────────────────────────────────────────────────────────*/
exp: procedure; parse arg x; ix= x%1; if abs(x-ix)>.5 then ix= ix + sign(x); x= x-ix
z=1; _=1; w=z; do j=1; _= _*x/j; z= (z + _)/1; if z==w then leave; w=z
end /*j*/; if z\==0 then z= z * e()**ix; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/
gamma: procedure; parse arg x; if datatype(x, 'W') then return !(x-1) /*Int? Use fact*/
n= trunc(x) /*at this point, X is pos and a multiple of 1/2.*/
d= !!(n+n - 1) /*compute the double factorial of: 2*n - 1. */
if n//2 then p= -1 /*if N is odd, then use a negative unity. */
else p= 1 /*if N is even, then use a positive unity. */
if x>0 then return p * d * sqrt(pi()) / (2**n)
return p * (2**n) * sqrt(pi()) / d
/*──────────────────────────────────────────────────────────────────────────────────────*/
gammaI: procedure; parse arg a,x; y= a-1; do while f(y)*(x-y) > 2e-8 & y<x; y= y + .4
end /*while*/
y= min(x, y)
return 1 - simp38(0, y, y / 0.015 / gamma(a-1) % 1)
/*──────────────────────────────────────────────────────────────────────────────────────*/
simp38: procedure; parse arg a, b, n; h= (b-a) / n; h1= h / 3
sum= f(a) + f(b)
do j=3*n-1 by -1 while j>0
if j//3 == 0 then sum= sum + 2 * f(a + h1*j)
else sum= sum + 3 * f(a + h1*j)
end /*j*/
return h * sum / 8
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits; h= d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .;g=g *.5'e'_%2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g
/*──────────────────────────────────────────────────────────────────────────────────────*/
uTest: procedure; parse arg dset; sum= 0; pad= left('', 11); sigLev= 1/20 /*5%*/
say; say ' ' center(" Uniform distribution test ", 75, '═')
#= words(dset); sigPC= sigLev*100/1
do j=1 for #; sum= sum + word(dset, j)
end /*j*/
say pad " dataset: " dset
say pad " samples: " sum
say pad " categories: " #
say pad " degrees of freedom: " # - 1
dist= chi2ud(dset)
P= chi2p(# - 1, dist)
sig = (abs(P) < dist * sigLev)
say pad "significant at " sigPC'% level? ' word('no yes', sig + 1)
say pad " is the dataset uniform? " word('no yes', (\(sig))+ 1)
return
- output when using the default inputs:
════════════════════════ Uniform distribution test ════════════════════════
dataset: 199809 200665 199607 200270 199649
samples: 1000000
categories: 5
degrees of freedom: 4
significant at 5% level? no
is the dataset uniform? yes
════════════════════════ Uniform distribution test ════════════════════════
dataset: 522573 244456 139979 71531 21461
samples: 1000000
categories: 5
degrees of freedom: 4
significant at 5% level? yes
is the dataset uniform? no
Ruby
def gammaInc_Q(a, x)
a1, a2 = a-1, a-2
f0 = lambda {|t| t**a1 * Math.exp(-t)}
df0 = lambda {|t| (a1-t) * t**a2 * Math.exp(-t)}
y = a1
y += 0.3 while f0[y]*(x-y) > 2.0e-8 and y < x
y = x if y > x
h = 3.0e-4
n = (y/h).to_i
h = y/n
hh = 0.5 * h
sum = 0
(n-1).step(0, -1) do |j|
t = h * j
sum += f0[t] + hh * df0[t]
end
h * sum / gamma_spounge(a)
end
A = 12
k1_factrl = 1.0
coef = [Math.sqrt(2.0*Math::PI)]
COEF = (1...A).each_with_object(coef) do |k,c|
c << Math.exp(A-k) * (A-k)**(k-0.5) / k1_factrl
k1_factrl *= -k
end
def gamma_spounge(z)
accm = (1...A).inject(COEF[0]){|res,k| res += COEF[k] / (z+k)}
accm * Math.exp(-(z+A)) * (z+A)**(z+0.5) / z
end
def chi2UniformDistance(dataSet)
expected = dataSet.inject(:+).to_f / dataSet.size
dataSet.map{|d|(d-expected)**2}.inject(:+) / expected
end
def chi2Probability(dof, distance)
1.0 - gammaInc_Q(0.5*dof, 0.5*distance)
end
def chi2IsUniform(dataSet, significance=0.05)
dof = dataSet.size - 1
dist = chi2UniformDistance(dataSet)
chi2Probability(dof, dist) > significance
end
dsets = [ [ 199809, 200665, 199607, 200270, 199649 ],
[ 522573, 244456, 139979, 71531, 21461 ] ]
for ds in dsets
puts "Data set:#{ds}"
dof = ds.size - 1
puts " degrees of freedom: %d" % dof
distance = chi2UniformDistance(ds)
puts " distance: %.4f" % distance
puts " probability: %.4f" % chi2Probability(dof, distance)
puts " uniform? %s" % (chi2IsUniform(ds) ? "Yes" : "No")
end
- Output:
Data set:[199809, 200665, 199607, 200270, 199649] degrees of freedom: 4 distance: 4.1463 probability: 0.3866 uniform? Yes Data set:[522573, 244456, 139979, 71531, 21461] degrees of freedom: 4 distance: 790063.2759 probability: -0.0000 uniform? No
Rust
use statrs::function::gamma::gamma_li;
fn chi_distance(dataset: &[u32]) -> f64 {
let expected = f64::from(dataset.iter().sum::<u32>()) / dataset.len() as f64;
dataset
.iter()
.fold(0., |acc, &elt| acc + (elt as f64 - expected).powf(2.))
/ expected
}
fn chi2_probability(dof: f64, distance: f64) -> f64 {
1. - gamma_li(dof * 0.5, distance * 0.5)
}
fn chi2_uniform(dataset: &[u32], significance: f64) -> bool {
let d = chi_distance(&dataset);
chi2_probability(dataset.len() as f64 - 1., d) > significance
}
fn main() {
let dsets = vec![
vec![199809, 200665, 199607, 200270, 199649],
vec![522573, 244456, 139979, 71531, 21461],
];
for ds in dsets {
println!("Data set: {:?}", ds);
let d = chi_distance(&ds);
print!("Distance: {:.6} ", d);
print!(
"Chi2 probability: {:.6} ",
chi2_probability(ds.len() as f64 - 1., d)
);
print!("Uniform? {}\n", chi2_uniform(&ds, 0.05));
}
}
- Output:
Data set: [199809, 200665, 199607, 200270, 199649] Distance: 4.146280 Chi2 probability: 0.386571 Uniform? true Data set: [522573, 244456, 139979, 71531, 21461] Distance: 790063.275940 Chi2 probability: 0.000000 Uniform? false
Scala
- Output:
See it yourself by running in your browser Scastie (remote JVM).
import org.apache.commons.math3.special.Gamma.regularizedGammaQ
object ChiSquare extends App {
private val dataSets: Seq[Seq[Double]] =
Seq(
Seq(199809, 200665, 199607, 200270, 199649),
Seq(522573, 244456, 139979, 71531, 21461)
)
private def χ2IsUniform(data: Seq[Double], significance: Double) =
χ2Prob(data.size - 1.0, χ2Dist(data)) > significance
private def χ2Dist(data: Seq[Double]) = {
val avg = data.sum / data.size
data.reduce((a, b) => a + math.pow(b - avg, 2)) / avg
}
private def χ2Prob(dof: Double, distance: Double) =
regularizedGammaQ(dof / 2, distance / 2)
printf(" %4s %10s %12s %8s %s%n",
"d.f.", "χ²distance", "χ²probability", "Uniform?", "dataset")
dataSets.foreach { ds =>
val (dist, dof) = (χ2Dist(ds), ds.size - 1)
printf("%4d %11.3f %13.8f %5s %6s%n",
dof, dist, χ2Prob(dof.toDouble, dist), if (χ2IsUniform(ds, 0.05)) "YES" else "NO", ds.mkString(", "))
}
}
Sidef
# Confluent hypergeometric function of the first kind F_1(a;b;z)
func F1(a, b, z, limit=100) {
sum(0..limit, {|k|
rising_factorial(a, k) / rising_factorial(b, k) * z**k / k!
})
}
func γ(a,x) { # lower incomplete gamma function γ(a,x)
#a**(-1) * x**a * F1(a, a+1, -x) # simpler formula
a**(-1) * x**a * exp(-x) * F1(1, a+1, x) # slightly better convergence
}
func P(a,z) { # regularized gamma function P(a,z)
γ(a,z) / Γ(a)
}
func chi_squared_cdf (k, x) {
var f = (k<20 ? 20 : 10)
given(x) {
when (0) { 0 }
case (. < (k + f*sqrt(k))) { P(k/2, x/2) }
else { 1 }
}
}
func chi_squared_test(arr, significance = 0.05) {
var n = arr.len
var N = arr.sum
var expected = N/n
var χ_squared = arr.sum_by {|v| (v-expected)**2 / expected }
var p_value = (1 - chi_squared_cdf(n-1, χ_squared))
[χ_squared, p_value, p_value > significance]
}
[
%n< 199809 200665 199607 200270 199649 >,
%n< 522573 244456 139979 71531 21461 >,
].each {|dataset|
var r = chi_squared_test(dataset)
say "data: #{dataset}"
say "χ² = #{r[0]}, p-value = #{r[1].round(-4)}, uniform = #{r[2]}\n"
}
- Output:
data: [199809, 200665, 199607, 200270, 199649] χ² = 4.14628, p-value = 0.3866, uniform = true data: [522573, 244456, 139979, 71531, 21461] χ² = 790063.27594, p-value = 0, uniform = false
Tcl
package require Tcl 8.5
package require math::statistics
proc isUniform {distribution {significance 0.05}} {
set count [tcl::mathop::+ {*}[dict values $distribution]]
set expected [expr {double($count) / [dict size $distribution]}]
set X2 0.0
foreach value [dict values $distribution] {
set X2 [expr {$X2 + ($value - $expected)**2 / $expected}]
}
set degreesOfFreedom [expr {[dict size $distribution] - 1}]
set likelihoodOfRandom [::math::statistics::incompleteGamma \
[expr {$degreesOfFreedom / 2.0}] [expr {$X2 / 2.0}]]
expr {$likelihoodOfRandom > $significance}
}
Testing:
proc makeDistribution {operation {count 1000000}} {
for {set i 0} {$i<$count} {incr i} {incr distribution([uplevel 1 $operation])}
return [array get distribution]
}
set distFair [makeDistribution {expr int(rand()*5)}]
puts "distribution \"$distFair\" assessed as [expr [isUniform $distFair]?{fair}:{unfair}]"
set distUnfair [makeDistribution {expr int(rand()*rand()*5)}]
puts "distribution \"$distUnfair\" assessed as [expr [isUniform $distUnfair]?{fair}:{unfair}]"
Output:
distribution "0 199809 4 199649 1 200665 2 199607 3 200270" assessed as fair distribution "4 21461 0 522573 1 244456 2 139979 3 71531" assessed as unfair
VBA
The built in worksheetfunction ChiSq_Dist of Excel VBA is used. Output formatted like R.
Private Function Test4DiscreteUniformDistribution(ObservationFrequencies() As Variant, Significance As Single) As Boolean
'Returns true if the observed frequencies pass the Pearson Chi-squared test at the required significance level.
Dim Total As Long, Ei As Long, i As Integer
Dim ChiSquared As Double, DegreesOfFreedom As Integer, p_value As Double
Debug.Print "[1] ""Data set:"" ";
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
Total = Total + ObservationFrequencies(i)
Debug.Print ObservationFrequencies(i); " ";
Next i
DegreesOfFreedom = UBound(ObservationFrequencies) - LBound(ObservationFrequencies)
'This is exactly the number of different categories minus 1
Ei = Total / (DegreesOfFreedom + 1)
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
ChiSquared = ChiSquared + (ObservationFrequencies(i) - Ei) ^ 2 / Ei
Next i
p_value = 1 - WorksheetFunction.ChiSq_Dist(ChiSquared, DegreesOfFreedom, True)
Debug.Print
Debug.Print " Chi-squared test for given frequencies"
Debug.Print "X-squared ="; ChiSquared; ", ";
Debug.Print "df ="; DegreesOfFreedom; ", ";
Debug.Print "p-value = "; Format(p_value, "0.0000")
Test4DiscreteUniformDistribution = p_value > Significance
End Function
Public Sub test()
Dim O() As Variant
O = [{199809,200665,199607,200270,199649}]
Debug.Print "[1] ""Uniform? "; Test4DiscreteUniformDistribution(O, 0.05); """"
O = [{522573,244456,139979,71531,21461}]
Debug.Print "[1] ""Uniform? "; Test4DiscreteUniformDistribution(O, 0.05); """"
End Sub
{{out}
[1] "Data set:" 199809 200665 199607 200270 199649 Chi-squared test for given frequencies X-squared = 4.14628 , df = 4 , p-value = 0.3866 [1] "Uniform? True" [1] "Data set:" 522573 244456 139979 71531 21461 Chi-squared test for given frequencies X-squared = 790063.27594 , df = 4 , p-value = 0.0000 [1] "Uniform? False"
V (Vlang)
import math
type Ifctn = fn(f64) f64
fn simpson38(f Ifctn, a f64, b f64, n int) f64 {
h := (b - a) / f64(n)
h1 := h / 3
mut sum := f(a) + f(b)
for j := 3*n - 1; j > 0; j-- {
if j%3 == 0 {
sum += 2 * f(a+h1*f64(j))
} else {
sum += 3 * f(a+h1*f64(j))
}
}
return h * sum / 8
}
fn gamma_inc_q(a f64, x f64) f64 {
aa1 := a - 1
f := Ifctn(fn[aa1](t f64) f64 {
return math.pow(t, aa1) * math.exp(-t)
})
mut y := aa1
h := 1.5e-2
for f(y)*(x-y) > 2e-8 && y < x {
y += .4
}
if y > x {
y = x
}
return 1 - simpson38(f, 0, y, int(y/h/math.gamma(a)))
}
fn chi2ud(ds []int) f64 {
mut sum, mut expected := 0.0,0.0
for d in ds {
expected += f64(d)
}
expected /= f64(ds.len)
for d in ds {
x := f64(d) - expected
sum += x * x
}
return sum / expected
}
fn chi2p(dof int, distance f64) f64 {
return gamma_inc_q(.5*f64(dof), .5*distance)
}
const sig_level = .05
fn main() {
for dset in [
[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461],
] {
utest(dset)
}
}
fn utest(dset []int) {
println("Uniform distribution test")
mut sum := 0
for c in dset {
sum += c
}
println(" dataset: $dset")
println(" samples: $sum")
println(" categories: $dset.len")
dof := dset.len - 1
println(" degrees of freedom: $dof")
dist := chi2ud(dset)
println(" chi square test statistic: $dist")
p := chi2p(dof, dist)
println(" p-value of test statistic: $p")
sig := p < sig_level
println(" significant at ${sig_level*100:2.0f}% level? $sig")
println(" uniform? ${!sig}\n")
}- Output:
Uniform distribution test dataset: [199809 200665 199607 200270 199649] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 4.14628 p-value of test statistic: 0.3865708330827673 significant at 5% level? false uniform? true Uniform distribution test dataset: [522573 244456 139979 71531 21461] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 790063.27594 p-value of test statistic: 2.3528290427066167e-11 significant at 5% level? true uniform? false
Wren
import "./math" for Math, Nums
import "./fmt" for Fmt
var integrate = Fn.new { |a, b, n, f|
var h = (b - a) / n
var sum = 0
for (i in 0...n) {
var x = a + i*h
sum = sum + (f.call(x) + 4 * f.call(x + h/2) + f.call(x + h)) / 6
}
return sum * h
}
var gammaIncomplete = Fn.new { |a, x|
var am1 = a - 1
var f0 = Fn.new { |t| t.pow(am1) * (-t).exp }
var h = 1.5e-2
var y = am1
while ((f0.call(y) * (x - y) > 2e-8) && y < x) y = y + 0.4
if (y > x) y = x
return 1 - integrate.call(0, y, (y/h).truncate, f0) / Math.gamma(a)
}
var chi2UniformDistance = Fn.new { |ds|
var expected = Nums.mean(ds)
var sum = Nums.sum(ds.map { |d| (d - expected).pow(2) }.toList)
return sum / expected
}
var chi2Probability = Fn.new { |dof, dist| gammaIncomplete.call(0.5*dof, 0.5*dist) }
var chiIsUniform = Fn.new { |ds, significance|
var dof = ds.count - 1
var dist = chi2UniformDistance.call(ds)
return chi2Probability.call(dof, dist) > significance
}
var dsets = [
[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461]
]
for (ds in dsets) {
System.print("Dataset: %(ds)")
var dist = chi2UniformDistance.call(ds)
var dof = ds.count - 1
Fmt.write("DOF: $d Distance: $.4f", dof, dist)
var prob = chi2Probability.call(dof, dist)
Fmt.write(" Probability: $.6f", prob)
var uniform = chiIsUniform.call(ds, 0.05) ? "Yes" : "No"
System.print(" Uniform? %(uniform)\n")
}
- Output:
Dataset: [199809, 200665, 199607, 200270, 199649] DOF: 4 Distance: 4.1463 Probability: 0.386571 Uniform? Yes Dataset: [522573, 244456, 139979, 71531, 21461] DOF: 4 Distance: 790063.2759 Probability: 0.000000 Uniform? No
zkl
/* Numerical integration method */
fcn Simpson3_8(f,a,b,N){ // fcn,double,double,Int --> double
h,h1:=(b - a)/N, h/3.0;
h*[1..3*N - 1].reduce('wrap(sum,j){
l1:=(if(j%3) 3.0 else 2.0);
sum + l1*f(a + h1*j);
},f(a) + f(b))/8.0;
}
const A=12;
fcn Gamma_Spouge(z){ // double --> double
var coefs=fcn{ // this runs only once, at construction time
a,coefs:=A.toFloat(),(A).pump(List(),0.0);
k1_factrl:=1.0;
coefs[0]=(2.0*(0.0).pi).sqrt();
foreach k in ([1.0..A-1]){
coefs[k]=(a - k).exp() * (a - k).pow(k - 0.5) / k1_factrl;
k1_factrl*=-k;
}
coefs
}();
( [1..A-1].reduce('wrap(accum,k){ accum + coefs[k]/(z + k) },coefs[0])
* (-(z + A)).exp()*(z + A).pow(z + 0.5) )
/ z;
}
fcn f0(t,aa1){ t.pow(aa1)*(-t).exp() }
fcn GammaIncomplete_Q(a,x){ // double,double --> double
h:=1.5e-2; /* approximate integration step size */
/* this cuts off the tail of the integration to speed things up */
y:=a - 1; f:=f0.fp1(y);
while((f(y)*(x - y)>2.0e-8) and (y<x)){ y+=0.4; }
if(y>x) y=x;
1.0 - Simpson3_8(f,0.0,y,(y/h).toInt())/Gamma_Spouge(a);
}fcn chi2UniformDistance(ds){ // --> double
dslen :=ds.len();
expected:=dslen.reduce('wrap(sum,k){ sum + ds[k] },0.0)/dslen;
sum := dslen.reduce('wrap(sum,k){ x:=ds[k] - expected; sum + x*x },0.0);
sum/expected
}
fcn chi2Probability(dof,distance){ GammaIncomplete_Q(0.5*dof, 0.5*distance) }
fcn chiIsUniform(dset,significance=0.05){
significance < chi2Probability(-1.0 + dset.len(),chi2UniformDistance(dset))
}datasets:=T( T(199809.0, 200665.0, 199607.0, 200270.0, 199649.0),
T(522573.0, 244456.0, 139979.0, 71531.0, 21461.0) );
println(" %4s %12s %12s %8s %s".fmt(
"dof", "distance", "probability", "Uniform?", "dataset"));
foreach ds in (datasets){
dof :=ds.len() - 1;
dist:=chi2UniformDistance(ds);
prob:=chi2Probability(dof,dist);
println("%4d %12.3f %12.8f %5s %6s".fmt(
dof, dist, prob, chiIsUniform(ds) and "YES" or "NO",
ds.concat(",")));
}- Output:
dof distance probability Uniform? dataset 4 4.146 0.38657083 YES 199809,200665,199607,200270,199649 4 790063.276 0.00000000 NO 522573,244456,139979,71531,21461
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