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Calculate the information entropy (Shannon entropy) of a given input string.

Entropy is the expected value of the measure of information content in a system. In general, the Shannon entropy of a variable $X$ is defined as:

H(X)=\sum _{x\in \Omega }P(x)I(x)

where the information content $I(x)=-\log _{b}P(x)$ . If the base of the logarithm $b=2$ , the result is expressed in bits, a unit of information. Therefore, given a string $S$ of length $n$ where $P(s_{i})$ is the relative frequency of each character, the entropy of a string in bits is:

H(S)=-\sum _{i=0}^{n}P(s_{i})\log _{2}(P(s_{i}))

For this task, use "1223334444" as an example. The result should be around 1.84644 bits.

Related Tasks:

Ada

Uses Ada 2012. <lang Ada>with Ada.Text_IO, Ada.Float_Text_IO, Ada.Numerics.Elementary_Functions;

procedure Count_Entropy is

  package TIO renames Ada.Text_IO;

  Count: array(Character) of Natural := (others => 0);
  Sum:   Natural := 0;
  Line: String := "1223334444";

begin

  for I in Line'Range loop   -- count the characters
     Count(Line(I)) := Count(Line(I))+1;
     Sum := Sum + 1;
  end loop;

  declare   -- compute the entropy and print it
     function P(C: Character) return Float is (Float(Count(C)) / Float(Sum));
     use Ada.Numerics.Elementary_Functions, Ada.Float_Text_IO;
     Result: Float := 0.0;
  begin
     for Ch in Character loop
        Result := Result -
         (if P(Ch)=0.0 then 0.0 else P(Ch) * Log(P(Ch), Base => 2.0));
     end loop;
     Put(Result, Fore => 1, Aft => 5, Exp => 0);
  end;

end Count_Entropy;</lang>

Aime

<lang aime>integer i, l; record r; real h, x; text s;

s = argv(1); l = length(s);

i = l; while (i) {

   i -= 1;
   rn_a_integer(r, cut(s, i, 1), 1);

}

h = 0; if (r_first(r, s)) {

   do {
       x = r_q_integer(r, s);
       x /= l;
       h -= x * log2(x);
   } while (r_greater(r, s, s));

}

o_real(6, h); o_newline();</lang> Examples:

$ aime -a tmp/entr 1223334444
1.846439
$ aime -a tmp/entr 'Rosetta Code is the best site in the world!'
3.646513
$ aime -a tmp/entr 1234567890abcdefghijklmnopqrstuvwxyz
5.169925

ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32

<lang algol68># calculate the shannon entropy of a string #

PROC shannon entropy = ( STRING s )REAL:

   BEGIN

       INT string length = ( UPB s - LWB s ) + 1;

       # count the occurances of each character #

       [ 0 : max abs char ]INT char count;

       FOR char pos FROM LWB char count TO UPB char count DO
           char count[ char pos ] := 0
       OD;

       FOR char pos FROM LWB s TO UPB s DO
           char count[ ABS s[ char pos ] ] +:= 1
       OD;

       # calculate the entropy, we use log base 10 and then convert #
       # to log base 2 after calculating the sum                    #

       REAL entropy := 0;

       FOR char pos FROM LWB char count TO UPB char count DO
           IF char count[ char pos ] /= 0
           THEN
               # have a character that occurs in the string #
               REAL probability = char count[ char pos ] / string length;
               entropy -:= probability * log( probability )
           FI
       OD;

       entropy / log( 2 )
   END; # shannon entropy #

main: (

   # test the shannon entropy routine #
   print( ( shannon entropy( "1223334444" ), newline ) )

) </lang>

Output:

+1.84643934467102e  +0

ALGOL W

Translation of: ALGOL 68

<lang algolw>begin

   % calculates the shannon entropy of a string                          %
   % strings are fixed length in algol W and the length is part of the   %
   % type, so we declare the string parameter to be the longest possible %
   % string length (256 characters) and have a second parameter to       %
   % specify how much is actually used                                   %
   real procedure shannon_entropy ( string(256) value s
                                  ; integer     value stringLength
                                  );
   begin

       real    probability, entropy;

       % algol W assumes there are 256 possible characters %
       integer MAX_CHAR;
               MAX_CHAR := 256;

       % declarations must preceed statements, so we start a new         %
       % block here so we can use MAX_CHAR as an array bound             %
       begin

           % increment an integer variable                               %
           procedure incI ( integer value result a ) ; a := a + 1;

           integer array charCount( 1 :: MAX_CHAR );

           % count the occurances of each character in s                 %
           for charPos := 1 until MAX_CHAR do charCount( charPos ) := 0;
           for sPos := 0 until stringLength - 1 do incI( charCount( decode( s( sPos | 1 ) ) ) );

           % calculate the entropy, we use log base 10 and then convert  %
           % to log base 2 after calculating the sum                     %

           entropy := 0.0;
           for charPos := 1 until MAX_CHAR do
           begin
               if charCount( charPos ) not = 0
               then begin
                   % have a character that occurs in the string          %
                   probability := charCount( charPos ) / stringLength;
                   entropy     := entropy - ( probability * log( probability ) )
               end 
           end charPos

       end;

       entropy / log( 2 )
   end shannon_entropy ;

   % test the shannon entropy routine                                    %
   r_format := "A"; r_w := 12; r_d := 6; % set output to fixed format    %
   write( shannon_entropy( "1223334444", 10 ) )

end.</lang>

Output:

    1.846439

AutoHotkey

<lang AutoHotkey>MsgBox, % Entropy(1223334444)

Entropy(n) {

   a := [], len := StrLen(n), m := n
   while StrLen(m)
   {
       s := SubStr(m, 1, 1)
       m := RegExReplace(m, s, "", c)
       a[s] := c
   }
   for key, val in a
   {
       m := Log(p := val / len)
       e -= p * m / Log(2)
   }
   return, e

}</lang>

Output:

1.846440

AWK

<lang awk>#!/usr/bin/awk -f { for (i=1; i<= length($0); i++) { H[substr($0,i,1)]++; N++; } }

END { for (i in H) { p = H[i]/N; E -= p * log(p); } print E/log(2); }</lang>

Usage:

<lang bash> echo 1223334444 |./entropy.awk 1.84644 </lang>

Burlesque

C

<lang c>#include <stdio.h>

include <stdlib.h>
include <stdbool.h>
include <string.h>
include <math.h>

define MAXLEN 100 //maximum string length

int makehist(char *S,int *hist,int len){ int wherechar[256]; int i,histlen; histlen=0; for(i=0;i<256;i++)wherechar[i]=-1; for(i=0;i<len;i++){ if(wherechar[(int)S[i]]==-1){ wherechar[(int)S[i]]=histlen; histlen++; } hist[wherechar[(int)S[i]]]++; } return histlen; }

double entropy(int *hist,int histlen,int len){ int i; double H; H=0; for(i=0;i<histlen;i++){ H-=(double)hist[i]/len*log2((double)hist[i]/len); } return H; }

int main(void){ char S[MAXLEN]; int len,*hist,histlen; double H; scanf("%[^\n]",S); len=strlen(S); hist=(int*)calloc(len,sizeof(int)); histlen=makehist(S,hist,len); //hist now has no order (known to the program) but that doesn't matter H=entropy(hist,histlen,len); printf("%lf\n",H); return 0; }</lang> Examples: <lang>$ ./entropy 1223334444 1.846439 $ ./entropy Rosetta Code is the best site in the world! 3.646513</lang>

C++

<lang cpp>#include <string>

include <map>
include <iostream>
include <algorithm>
include <cmath>

double log2( double number ) {

  return log( number ) / log( 2 ) ;

}

int main( int argc , char *argv[ ] ) {

  std::string teststring( argv[ 1 ] ) ;
  std::map<char , int> frequencies ;
  for ( char c : teststring )
    frequencies[ c ] ++ ;
  int numlen = teststring.length( ) ;
  double infocontent = 0 ;
  for ( std::pair<char , int> p : frequencies ) {
     double freq = static_cast<double>( p.second ) / numlen ;
     infocontent += freq * log2( freq ) ;
  }
  infocontent *= -1 ;
  std::cout << "The information content of " << teststring 
     << " is " << infocontent << " !\n" ;
  return 0 ;

}</lang>

Output:

The information content of 1223334444 is 1.84644 !

Clojure

<lang Clojure>(defn entropy [s]

 (let [len (count s), log-2 (Math/log 2)]
   (->> (frequencies s)
        (map (fn _ v
               (let [rf (/ v len)]
                 (-> (Math/log rf) (/ log-2) (* rf) Math/abs))))
        (reduce +))))</lang>

Output:

<lang Clojure>(entropy "1223334444") 1.8464393446710154</lang>

C#

Translation of C++. <lang csharp> using System; using System.Collections.Generic; namespace Entropy { class Program { public static double logtwo(double num) { return Math.Log(num)/Math.Log(2); } public static void Main(string[] args) { label1: string input = Console.ReadLine(); double infoC=0; Dictionary<char,double> table = new Dictionary<char, double>();

foreach (char c in input) { if (table.ContainsKey(c)) table[c]++; else table.Add(c,1);

} double freq; foreach (KeyValuePair<char,double> letter in table) { freq=letter.Value/input.Length; infoC+=freq*logtwo(freq); } infoC*=-1; Console.WriteLine("The Entropy of {0} is {1}",input,infoC); goto label1;

} } } </lang>

Output:

The Entropy of 1223334444 is 1.84643934467102

Without using Hashtables or Dictionaries: <lang csharp>using System; namespace Entropy { class Program { public static double logtwo(double num) { return Math.Log(num)/Math.Log(2); } static double Contain(string x,char k) { double count=0; foreach (char Y in x) { if(Y.Equals(k)) count++; } return count; } public static void Main(string[] args) { label1: string input = Console.ReadLine(); double infoC=0; double freq; string k=""; foreach (char c1 in input) { if (!(k.Contains(c1.ToString()))) k+=c1; } foreach (char c in k) { freq=Contain(input,c)/(double)input.Length; infoC+=freq*logtwo(freq); } infoC/=-1; Console.WriteLine("The Entropy of {0} is {1}",input,infoC); goto label1;

} } }</lang>

CoffeeScript

<lang coffeescript>entropy = (s) ->

   freq = (s) ->
       result = {}
       for ch in s.split ""
           result[ch] ?= 0
           result[ch]++
       return result

   frq = freq s
   n = s.length
   ((frq[f]/n for f of frq).reduce ((e, p) -> e - p * Math.log(p)), 0) * Math.LOG2E

console.log "The entropy of the string '1223334444' is #{entropy '1223334444'}"</lang>

Output:

The entropy of the string '1223334444' is 1.8464393446710157

Common Lisp

<lang lisp>(defun entropy (string)

 (let ((table (make-hash-table :test 'equal))
       (entropy 0))
   (mapc (lambda (c) (setf (gethash c table) (+ (gethash c table 0) 1)))
         (coerce string 'list))
   (maphash (lambda (k v) (decf entropy (* (/ v (length input-string)) (log (/ v (length input-string)) 2))))
            table)
   entropy))

</lang>

D

<lang d>import std.stdio, std.algorithm, std.math;

double entropy(T)(T[] s) pure nothrow if (__traits(compiles, s.sort())) {

   immutable sLen = s.length;
   return s
          .sort()
          .group
          .map!(g => g[1] / double(sLen))
          .map!(p => -p * p.log2)
          .sum;

}

void main() {

   "1223334444"d.dup.entropy.writeln;

}</lang>

Output:

1.84644

Elixir

Works with: Erlang/OTP version 18

:math.log2 was added in OTP 18. <lang elixir>defmodule RC do

 def entropy(str) do
   leng = String.length(str)
   String.split(str, "", trim: true)
   |> Enum.group_by(&(&1))
   |> Enum.map(fn{_,value} -> length(value) end)
   |> Enum.reduce(0, fn count, entropy ->
        freq = count / leng
        entropy - freq * :math.log2(freq)
      end)
 end

end

IO.inspect RC.entropy("1223334444")</lang>

Output:

1.8464393446710154

Emacs Lisp

<lang lisp>(defun shannon-entropy (input)

 (let ((freq-table (make-hash-table))

(entropy 0) (length (+ (length input) 0.0)))

   (mapcar (lambda (x)

(puthash x (+ 1 (gethash x freq-table 0)) freq-table)) input)

   (maphash (lambda (k v)

(set 'entropy (+ entropy (* (/ v length) (log (/ v length) 2))))) freq-table)

 (- entropy)))</lang>

Output:

After adding the above to the emacs runtime, you can run the function interactively in the scratch buffer as shown below (type ctrl-j at the end of the first line and the output will be placed by emacs on the second line). <lang lisp>(shannon-entropy "1223334444") 1.8464393446710154</lang>

Erlang

<lang Erlang> -module( entropy ).

-export( [shannon/1, task/0] ).

shannon( String ) -> shannon_information_content( lists:foldl(fun count/2, dict:new(), String), erlang:length(String) ).

task() -> shannon( "1223334444" ).

count( Character, Dict ) -> dict:update_counter( Character, 1, Dict ).

shannon_information_content( Dict, String_length ) -> {_String_length, Acc} = dict:fold( fun shannon_information_content/3, {String_length, 0.0}, Dict ), Acc / math:log( 2 ).

shannon_information_content( _Character, How_many, {String_length, Acc} ) ->

       Frequency = How_many / String_length,

{String_length, Acc - (Frequency * math:log(Frequency))}. </lang>

Output:

24> entropy:task().
1.8464393446710157

Euler Math Toolbox

<lang EulerMathToolbox>>function entropy (s) ... $ v=strtochar(s); $ m=getmultiplicities(unique(v),v); $ m=m/sum(m); $ return sum(-m*logbase(m,2)) $endfunction >entropy("1223334444")

1.84643934467</lang>

F#

<lang fsharp>open System

let ld x = Math.Log x / Math.Log 2.

let entropy (s : string) =

   let n = float s.Length
   Seq.groupBy id s
   |> Seq.map (fun (_, vals) -> float (Seq.length vals) / n)
   |> Seq.fold (fun e p -> e - p * ld p) 0.

printfn "%f" (entropy "1223334444")</lang>

Output:

1.846439

Forth

<lang forth>: flog2 ( f -- f ) fln 2e fln f/ ;

create freq 256 cells allot

entropy ( str len -- f )

 freq 256 cells erase
 tuck
 bounds do
   i c@ cells freq +
   1 swap +!
 loop
 0e
 256 0 do
   i cells freq + @ ?dup if
     s>f dup s>f f/
     fdup flog2 f* f-
   then
 loop
 drop ;

s" 1223334444" entropy f. \ 1.84643934467102 ok </lang>

Fortran

Please find the GNU/linux compilation instructions along with sample run among the comments at the start of the FORTRAN 2008 source. This program acquires input from the command line argument, thereby demonstrating the fairly new get_command_argument intrinsic subroutine. The expression of the algorithm is a rough translated of the j solution. Thank you. <lang FORTRAN> !-*- mode: compilation; default-directory: "/tmp/" -*- !Compilation started at Tue May 21 21:43:12 ! !a=./f && make $a && OMP_NUM_THREADS=2 $a 1223334444 !gfortran -std=f2008 -Wall -ffree-form -fall-intrinsics f.f08 -o f ! Shannon entropy of 1223334444 is 1.84643936 ! !Compilation finished at Tue May 21 21:43:12

program shannonEntropy

 implicit none
 integer :: num, L, status
 character(len=2048) :: s
 num = 1
 call get_command_argument(num, s, L, status)
 if ((0 /= status) .or. (L .eq. 0)) then
   write(0,*)'Expected a command line argument with some length.'
 else
   write(6,*)'Shannon entropy of '//(s(1:L))//' is ', se(s(1:L))
 endif

contains

 !     algebra
 !
 ! 2**x = y
 ! x*log(2) = log(y)
 ! x = log(y)/log(2)

 !   NB. The j solution
 !   entropy=:  +/@:-@(* 2&^.)@(#/.~ % #)
 !   entropy '1223334444'
 !1.84644
 
 real function se(s)
   implicit none
   character(len=*), intent(in) :: s
   integer, dimension(256) :: tallies
   real, dimension(256) :: norm
   tallies = 0
   call TallyKey(s, tallies)
   ! J's #/. works with the set of items in the input.
   ! TallyKey is sufficiently close that, with the merge, gets the correct result.
   norm = tallies / real(len(s))
   se = sum(-(norm*log(merge(1.0, norm, norm .eq. 0))/log(2.0)))
 end function se

 subroutine TallyKey(s, counts)
   character(len=*), intent(in) :: s
   integer, dimension(256), intent(out) :: counts
   integer :: i, j
   counts = 0
   do i=1,len(s)
     j = iachar(s(i:i))
     counts(j) = counts(j) + 1
   end do
 end subroutine TallyKey

end program shannonEntropy </lang>

FreeBASIC

<lang FreeBASIC>' version 25-06-2015 ' compile with: fbc -s console

Sub calc_entropy(source As String, base_ As Integer)

   Dim As Integer i, sourcelen = Len(source), totalchar(255)
   Dim As Double prop, entropy

   For i = 0 To sourcelen -1
       totalchar(source[i]) += 1
   Next

   Print "Char    count"
   For i = 0 To 255
       If totalchar(i) = 0 Then Continue For
       Print "   "; Chr(i); Using "   ######"; totalchar(i)
       prop = totalchar(i) / sourcelen
       entropy = entropy - (prop * Log (prop) / Log(base_))
   Next

   Print : Print "The Entropy of "; Chr(34); source; Chr(34); " is"; entropy

End Sub

' ------=< MAIN >=------

calc_entropy("1223334444", 2) Print

' empty keyboard buffer While InKey <> "" : Var _key_ = InKey : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:

Char    count
   1        1
   2        2
   3        3
   4        4

The Entropy of "1223334444" is 1.846439344671015

friendly interactive shell

Sort of hacky, but friendly interactive shell isn't really optimized for mathematic tasks (in fact, it doesn't even have associative arrays).

<lang fishshell>function entropy

   for arg in $argv
       set name count_$arg
       if not count $$name > /dev/null
           set $name 0
           set values $values $arg
       end
       set $name (math $$name + 1)
   end
   set entropy 0
   for value in $values
       set name count_$value
       set entropy (echo "
           scale = 50
           p = "$$name" / "(count $argv)"
           $entropy - p * l(p)
       " | bc -l)
   end
   echo "$entropy / l(2)" | bc -l

end entropy (echo 1223334444 | fold -w1)</lang>

Output:

1.84643934467101549345

Go

<lang go>package main

import (

   "fmt"
   "math"

)

const s = "1223334444"

func main() {

   m := map[rune]float64{}
   for _, r := range s {
       m[r]++
   }
   hm := 0.
   for _, c := range m {
       hm += c * math.Log2(c)
   }
   const l = float64(len(s))
   fmt.Println(math.Log2(l) - hm/l)

}</lang>

Output:

1.8464393446710152

Groovy

<lang groovy>String.metaClass.getShannonEntrophy = {

   -delegate.inject([:]) { map, v -> map[v] = (map[v] ?: 0) + 1; map }.values().inject(0.0) { sum, v ->
       def p = (BigDecimal)v / delegate.size()
       sum + p * Math.log(p) / Math.log(2)
   }

}</lang> Testing <lang groovy>[ '1223334444': '1.846439344671',

 '1223334444555555555': '1.969811065121',
 '122333': '1.459147917061',
 '1227774444': '1.846439344671',
 aaBBcccDDDD: '1.936260027482',
 '1234567890abcdefghijklmnopqrstuvwxyz': '5.169925004424',
 'Rosetta Code': '3.084962500407' ].each { s, expected ->

   println "Checking $s has a shannon entrophy of $expected"
   assert sprintf('%.12f', s.shannonEntrophy) == expected

}</lang>

Output:

Checking 1223334444 has a shannon entrophy of 1.846439344671
Checking 1223334444555555555 has a shannon entrophy of 1.969811065121
Checking 122333 has a shannon entrophy of 1.459147917061
Checking 1227774444 has a shannon entrophy of 1.846439344671
Checking aaBBcccDDDD has a shannon entrophy of 1.936260027482
Checking 1234567890abcdefghijklmnopqrstuvwxyz has a shannon entrophy of 5.169925004424
Checking Rosetta Code has a shannon entrophy of 3.084962500407

Haskell

<lang haskell>import Data.List

main = print $ entropy "1223334444"

entropy s =

sum . map lg' . fq' . map (fromIntegral.length) . group . sort $ s
 where lg' c = (c * ) . logBase 2 $ 1.0 / c
       fq' c = let sc = sum c in map (/ sc) c </lang>

Icon and Unicon

Hmmm, the 2nd equation sums across the length of the string (for the example, that would be the sum of 10 terms). However, the answer cited is for summing across the different characters in the string (sum of 4 terms). The code shown here assumes the latter and works in Icon and Unicon. This assumption is consistent with the Wikipedia description.

<lang unicon>procedure main(a)

   s := !a | "1223334444"
   write(H(s))

end

procedure H(s)

   P := table(0.0)
   every P[!s] +:= 1.0/*s
   every (h := 0.0) -:= P[c := key(P)] * log(P[c],2)
   return h

end</lang>

Output:

->en
1.846439344671015
->

J

Solution:<lang j> entropy=: +/@(-@* 2&^.)@(#/.~ % #)</lang>

Example:

<lang j> entropy '1223334444' 1.84644

  entropy i.256

8

  entropy 256$9

0

  entropy 256$0 1

1

  entropy 256$0 1 2 3

2</lang>

So it looks like entropy is roughly the number of bits which would be needed to distinguish between each item in the argument (for example, with perfect compression). Note that in some contexts this might not be the same thing as information because the choice of the items themselves might matter. But it's good enough in contexts with a fixed set of symbols.

Java

Translation of: NetRexx

Translation of: REXX

Works with: Java version 7+

<lang java5>import java.lang.Math; import java.util.Map; import java.util.HashMap;

public class REntropy {

 @SuppressWarnings("boxing")
 public static double getShannonEntropy(String s) {
   int n = 0;
   Map<Character, Integer> occ = new HashMap<>();

   for (int c_ = 0; c_ < s.length(); ++c_) {
     char cx = s.charAt(c_);
     if (occ.containsKey(cx)) {
       occ.put(cx, occ.get(cx) + 1);
     } else {
       occ.put(cx, 1);
     }
     ++n;
   }

   double e = 0.0;
   for (Map.Entry<Character, Integer> entry : occ.entrySet()) {
     char cx = entry.getKey();
     double p = (double) entry.getValue() / n;
     e += p * log2(p);
   }
   return -e;
 }

 private static double log2(double a) {
   return Math.log(a) / Math.log(2);
 }
 public static void main(String[] args) {
   String[] sstr = {
     "1223334444",
     "1223334444555555555", 
     "122333", 
     "1227774444",
     "aaBBcccDDDD",
     "1234567890abcdefghijklmnopqrstuvwxyz",
     "Rosetta Code",
   };

   for (String ss : sstr) {
     double entropy = REntropy.getShannonEntropy(ss);
     System.out.printf("Shannon entropy of %40s: %.12f%n", "\"" + ss + "\"", entropy);
   }
   return;
 }

}</lang>

Output:

Shannon entropy of                             "1223334444": 1.846439344671
Shannon entropy of                    "1223334444555555555": 1.969811065278
Shannon entropy of                                 "122333": 1.459147917027
Shannon entropy of                             "1227774444": 1.846439344671
Shannon entropy of                            "aaBBcccDDDD": 1.936260027532
Shannon entropy of   "1234567890abcdefghijklmnopqrstuvwxyz": 5.169925001442
Shannon entropy of                           "Rosetta Code": 3.084962500721

JavaScript

Works with: ECMA-262 (5.1)

The proces function builds a histogram of character frequencies then iterates over it.

The entropy function calls into process and evaluates the frequencies as they're passed back. <lang JavaScript>(function(shannon) {

 // Create a dictionary of character frequencies and iterate over it.
 function process(s, evaluator) {
   var h = Object.create(null), k;
   s.split().forEach(function(c) {
     h[c] && h[c]++ || (h[c] = 1); });
   if (evaluator) for (k in h) evaluator(k, h[k]);
   return h;
 };
 // Measure the entropy of a string in bits per symbol.
 shannon.entropy = function(s) {
   var sum = 0,len = s.length;
   process(s, function(k, f) {
     var p = f/len;
     sum -= p * Math.log(p) / Math.log(2);
   });
   return sum;
 };

})(window.shannon = window.shannon || {});

// Log the Shannon entropy of a string. function logEntropy(s) {

 console.log('Entropy of "' + s + '" in bits per symbol:', shannon.entropy(s));

}

logEntropy('1223334444'); logEntropy('0'); logEntropy('01'); logEntropy('0123'); logEntropy('01234567'); logEntropy('0123456789abcdef');</lang>

Output:

Entropy of "1223334444" in bits per symbol: 1.8464393446710154
Entropy of "0" in bits per symbol: 0
Entropy of "01" in bits per symbol: 1
Entropy of "0123" in bits per symbol: 2
Entropy of "01234567" in bits per symbol: 3
Entropy of "0123456789abcdef" in bits per symbol: 4

jq

For efficiency with long strings, we use a hash (a JSON object) to compute the frequencies.

The helper function, counter, could be defined as an inner function of entropy, but for the sake of clarity and because it is independently useful, it is defined separately. <lang jq># Input: an array of strings.

Output: an object with the strings as keys, the values of which are the corresponding frequencies.

def counter:

 reduce .[] as $item ( {}; .[$item] += 1 ) ;

entropy in bits of the input string

def entropy:

 (explode | map( [.] | implode ) | counter
   | [ .[] | . * log ] | add) as $sum
 | ((length|log) - ($sum / length)) / (2|log) ;</lang>

Example:

<lang jq>"1223334444" | entropy # => 1.8464393446710154</lang>

Julia

A oneliner, probably not efficient on very long strings. <lang Julia>entropy(s)=-sum(x->x*log(2,x), [count(x->x==c,s)/length(s) for c in unique(s)])</lang>

Output:

julia> entropy("1223334444")
1.8464393446710154

Liberty BASIC

<lang lb> dim countOfChar( 255) ' all possible one-byte ASCII chars

   source$    ="1223334444"
   charCount  =len( source$)
   usedChar$  =""

   for i =1 to len( source$)   '   count which chars are used in source
       ch$             =mid$( source$, i, 1)
       if not( instr( usedChar$, ch$)) then usedChar$ =usedChar$ +ch$
       'currentCh$      =mid$(
       j               =instr( usedChar$, ch$)
       countOfChar( j) =countOfChar( j) +1
   next i

   l =len( usedChar$)
   for i =1 to l
       probability =countOfChar( i) /charCount
       entropy     =entropy -( probability *logBase( probability, 2))
   next i

   print " Characters used and the number of occurrences of each "
   for i =1 to l
       print " '"; mid$( usedChar$, i, 1); "'", countOfChar( i)
   next i

   print " Entropy of '"; source$; "' is  "; entropy; " bits."
   print " The result should be around 1.84644 bits."

   end
   function logBase( x, b) '   in LB log() is base 'e'.
       logBase =log( x) /log( 2)
   end function

</lang>

Output:

 Characters used and the number of occurrences of each
 '1'          1
 '2'          2
 '3'          3
 '4'          4
 Entropy of '1223334444' is  1.84643934 bits.
 The result should be around 1.84644 bits.

Lang5

<lang lang5>: -rot rot rot ; [] '__A set : dip swap __A swap 1 compress append '__A set execute __A -1 extract nip ; : nip swap drop ; : sum '+ reduce ;

2array 2 compress ; : comb "" split ; : lensize length nip ;

<group> #( a -- 'a )

   grade subscript dup 's dress distinct strip
   length 1 2array reshape swap
   'A set
   : `filter(*)  A in A swap select ;
   '`filter apply
   ;

elements(*) lensize ;

entropy #( s -- n )

   length "<group> 'elements apply" dip /
   dup neg swap log * 2 log / sum ;

"1223334444" comb entropy . # 1.84643934467102</lang>

Mathematica

<lang Mathematica>shE[s_String] := -Plus @@ ((# Log[2., #]) & /@ ((Length /@ Gather[#])/

        Length[#]) &[Characters[s]])</lang>

Example:

<lang Mathematica> shE["1223334444"]

1.84644 shE["Rosetta Code"] 3.08496</lang>

MATLAB / Octave

This version allows for any input vectors, including strings, floats, negative integers, etc. <lang MATLAB>function E = entropy(d) if ischar(d), d=abs(d); end;

       [Y,I,J] = unique(d);

H = sparse(J,1,1); p = full(H(H>0))/length(d); E = -sum(p.*log2(p)); end; </lang>

Usage:

<lang MATLAB>> entropy('1223334444') ans = 1.8464</lang>

NetRexx

Translation of: REXX

<lang NetRexx>/* NetRexx */ options replace format comments java crossref savelog symbols

runSample(Arg) return

-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ /* REXX ***************************************************************

* 28.02.2013 Walter Pachl
**********************************************************************/

method getShannonEntropy(s = "1223334444") public static --trace var occ c chars n cn i e p pl

 Numeric Digits 30
 occ = 0
 chars = 
 n = 0
 cn = 0
 Loop i = 1 To s.length()
   c = s.substr(i, 1)
   If chars.pos(c) = 0 Then Do
     cn = cn + 1
     chars = chars || c
     End
   occ[c] = occ[c] + 1
   n = n + 1
   End i
 p = 
 Loop ci = 1 To cn
   c = chars.substr(ci, 1)
   p[c] = occ[c] / n
   End ci
 e = 0
 Loop ci = 1 To cn
   c = chars.substr(ci, 1)
   pl = log2(p[c])
   e = e + p[c] * pl
   End ci
 Return -e

-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ method log2(a = double) public static binary returns double

 return Math.log(a) / Math.log(2)

-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ method runSample(Arg) public static

 parse Arg sstr
 if sstr =  then
   sstr = '1223334444' -
          '1223334444555555555' -
          '122333' -
          '1227774444' -
          'aaBBcccDDDD' -
          '1234567890abcdefghijklmnopqrstuvwxyz' -
          'Rosetta_Code'
 say 'Calculating Shannons entropy for the following list:'
 say '['(sstr.space(1, ',')).changestr(',', ', ')']'
 say
 entropies = 0
 ssMax = 0
 -- This crude sample substitutes a '_' character for a space in the input strings
 loop w_ = 1 to sstr.words()
   ss = sstr.word(w_)
   ssMax = ssMax.max(ss.length())
   ss_ = ss.changestr('_', ' ')
   entropy = getShannonEntropy(ss_)
   entropies[ss] = entropy
   end w_
 loop report = 1 to sstr.words()
   ss = sstr.word(report)
   ss_ = ss.changestr('_', ' ')
   Say 'Shannon entropy of' ('"'ss_'"').right(ssMax + 2)':' entropies[ss].format(null, 12)
   end report
 return

</lang>

Output:

Calculating Shannon's entropy for the following list:
[1223334444, 1223334444555555555, 122333, 1227774444, aaBBcccDDDD, 1234567890abcdefghijklmnopqrstuvwxyz, Rosetta_Code]

Shannon entropy of                           "1223334444": 1.846439344671
Shannon entropy of                  "1223334444555555555": 1.969811065278
Shannon entropy of                               "122333": 1.459147917027
Shannon entropy of                           "1227774444": 1.846439344671
Shannon entropy of                          "aaBBcccDDDD": 1.936260027532
Shannon entropy of "1234567890abcdefghijklmnopqrstuvwxyz": 5.169925001442
Shannon entropy of                         "Rosetta Code": 3.084962500721

Nim

<lang nim>import tables, math

proc entropy(s): float =

 var t = initCountTable[char]()
 for c in s: t.inc(c)
 for x in t.values: result -= x/s.len * log2(x/s.len)

echo entropy("1223334444")</lang>

OCaml

<lang ocaml>(* generic OCaml, using a mutable Hashtbl *)

(* pre-bake & return an inner-loop function to bin & assemble a character frequency map *) let get_fproc (m: (char, int) Hashtbl.t) :(char -> unit) =

 (fun (c:char) -> try
                    Hashtbl.replace m c ( (Hashtbl.find m c) + 1) 
                  with Not_found -> Hashtbl.add m c 1)

(* pre-bake and return an inner-loop function to do the actual entropy calculation *) let get_calc (slen:int) :(float -> float) =

 let slen_float = float_of_int slen in
 let log_2 = log 2.0 in

 (fun v -> let pt = v /. slen_float in
               pt *. ((log pt) /. log_2) )

(* main function, given a string argument it:

      builds a (mutable) frequency map (initial alphabet size of 255, but it's auto-expanding), 
      extracts the relative probability values into a list, 
      folds-in the basic entropy calculation and returns the result. *)

let shannon (s:string) :float =

 let freq_hash = Hashtbl.create 255 in
 String.iter (get_fproc freq_hash) s;

 let relative_probs = Hashtbl.fold (fun k v b -> (float v)::b) freq_hash [] in
 let calc = get_calc (String.length s) in

  -1.0 *. List.fold_left (fun b x -> b +. calc x) 0.0 relative_probs

</lang>

output:

 1.84643934467

Oforth

<lang Oforth>func: entropy(s) { | freq sz |

  s size dup ifZero: [ return ] asFloat ->sz
  ListBuffer newSize(255) ->freq
  s apply(#[ dup freq at dup ifNull: [ drop 0 ] 1 + swap freq put ])
  0.0 freq applyIf(#notNull, #[ sz / dup ln * - ]) Ln2 /

}

entropy("1223334444") println</lang>

Output:

1.84643934467102

Pascal

Free Pascal (http://freepascal.org).

<lang Pascal> PROGRAM entropytest;

USES StrUtils, Math;

TYPE FArray = ARRAY of CARDINAL;

VAR strng: STRING = '1223334444';

// list unique characters in a string FUNCTION uniquechars(str: STRING): STRING; VAR n: CARDINAL; BEGIN uniquechars := ; FOR n := 1 TO length(str) DO IF (PosEx(str[n],str,n)>0) AND (PosEx(str[n],uniquechars,1)=0) THEN uniquechars += str[n]; END;

// obtain a list of character-frequencies for a string // given a string containing its unique characters FUNCTION frequencies(str,ustr: STRING): FArray; VAR u,s,p,o: CARDINAL; BEGIN SetLength(frequencies, Length(ustr)+1); p := 0; FOR u := 1 TO length(ustr) DO FOR s := 1 TO length(str) DO BEGIN o := p; p := PosEx(ustr[u],str,s); IF (p>o) THEN INC(frequencies[u]); END; END;

// Obtain the Shannon entropy of a string FUNCTION entropy(s: STRING): EXTENDED; VAR pf : FArray; us : STRING; i,l: CARDINAL; BEGIN us := uniquechars(s); pf := frequencies(s,us); l := length(s); entropy := 0.0; FOR i := 1 TO length(us) DO entropy -= pf[i]/l * log2(pf[i]/l); END;

BEGIN Writeln('Entropy of "',strng,'" is ',entropy(strng):2:5, ' bits.'); END. </lang>

Output:

Entropy of "1223334444" is 1.84644 bits.

PARI/GP

<lang parigp>entropy(s)=s=Vec(s);my(v=vecsort(s,,8));-sum(i=1,#v,(x->x*log(x))(sum(j=1,#s,v[i]==s[j])/#s))/log(2)</lang>

>entropy("1223334444")
%1 = 1.8464393446710154934341977463050452232

Perl

<lang Perl>sub entropy {

   my %count; $count{$_}++ for @_;
   my $entropy = 0;
   for (values %count) {
       my $p = $_/@_;
       $entropy -= $p * log $p;
   }
   $entropy / log 2

}

print entropy split //, "1223334444";</lang>

Perl 6

<lang Perl 6>sub entropy(@a) {

   [+] map -> \p { p * -log p }, bag(@a).values »/» +@a;

}

say log(2) R/ entropy '1223334444'.comb;</lang>

Output:

1.84643934467102

In case we would like to add this function to Perl 6's core, here is one way it could be done:

<lang Perl 6>use MONKEY_TYPING; augment class Bag {

   method entropy {

[+] map -> \p { - p * log p }, self.values »/» +self;

}

say '1223334444'.comb.Bag.entropy / log 2;</lang>

PL/I

<lang pli>*process source xref attributes or(!);

/*--------------------------------------------------------------------
* 08.08.2014 Walter Pachl  translated from REXX version 1
*-------------------------------------------------------------------*/
ent: Proc Options(main);
Dcl (index,length,log2,substr) Builtin;
Dcl sysprint Print;
Dcl occ(100) Bin fixed(31) Init((100)0);
Dcl (n,cn,ci,i,pos) Bin fixed(31) Init(0);
Dcl chars Char(100) Var Init();
Dcl s Char(100) Var Init('1223334444');
Dcl c Char(1);
Dcl (occf,p(100)) Dec Float(18);
Dcl e Dec Float(18) Init(0);
Do i=1 To length(s);
  c=substr(s,i,1);
  pos=index(chars,c);
  If pos=0 Then Do;
    pos=length(chars)+1;
    cn+=1;
    chars=chars!!c;
    End;
  occ(pos)+=1;
  n+=1;
  End;
 do ci=1 To cn;
   occf=occ(ci);
   p(ci)=occf/n;
   End;
 Do ci=1 To cn;
   e=e+p(ci)*log2(p(ci));
   End;
 Put Edit('s=!!s!! Entropy=',-e)(Skip,a,f(15,12));
 End;</lang>

Output:

s='1223334444' Entropy= 1.846439344671

PowerShell

<lang PowerShell> function entropy ($string) {

   $n = $string.Length
   $string.ToCharArray() | group | foreach{
       $p = $_.Count/$n
       $i = [Math]::Log($p,2)
       -$p*$i
   } | measure -Sum | foreach Sum

} entropy "1223334444" </lang> Output:

1.84643934467102

Prolog

This solution calculates the run-length encoding of the input string to get the relative frequencies of its characters.

-module(shannon_entropy, [shannon_entropy/2]).

%! shannon_entropy(+String, -Entropy) is det. % % Calculate the shannon Entropy of String. % % Example query: % == % ?- shannon_entropy(1223334444, H). % H = 1.8464393446710154. % == % shannon_entropy(String, Entropy):- atom_chars(String, Cs) ,relative_frequencies(Cs, Frequencies) ,findall(CI ,(member(_C-F, Frequencies) ,log2(F, L) ,CI is F * L ) ,CIs) ,foldl(sum, CIs, 0, E) ,Entropy is -E.

%! frequencies(+Characters,-Frequencies) is det. % % Calculates the relative frequencies of elements in the list of % Characters. % % Frequencies is a key-value list with elements of the form: % C-F, where C a character in the list and F its relative % frequency in the list. % % Example query: % == % ?- relative_frequencies([a,a,a,b,b,b,b,b,b,c,c,c,a,a,f], Fs). % Fs = [a-0.3333333333333333, b-0.4, c-0.2,f-0.06666666666666667]. % == % relative_frequencies(List, Frequencies):- run_length_encoding(List, Rle) ,keysort(Rle, Sorted_Rle) ,group_pairs_by_key(Sorted_Rle, Elements_Run_lengths) ,length(List, Elements_in_list) ,findall(E-Frequency_of_E ,(member(E-RLs, Elements_Run_lengths) ,foldl(plus, RLs, 0, Occurences_of_E) ,Frequency_of_E is Occurences_of_E / Elements_in_list ) ,Frequencies).

%! run_length_encoding(+List, -Run_length_encoding) is det. % % Converts a list to its run-length encoded form where each "run" % of contiguous repeats of the same element is replaced by that % element and the length of the run. % % Run_length_encoding is a key-value list, where each element is a % term: % % Element:term-Repetitions:number. % % Example query: % == % ?- run_length_encoding([a,a,a,b,b,b,b,b,b,c,c,c,a,a,f], RLE). % RLE = [a-3, b-6, c-3, a-2, f-1]. % == % run_length_encoding([], []-0):- !. % No more results needed.

run_length_encoding([Head|List], Run_length_encoded_list):- run_length_encoding(List, [Head-1], Reversed_list) % The resulting list is in reverse order due to the head-to-tail processing ,reverse(Reversed_list, Run_length_encoded_list).

%! run_length_encoding(+List,+Initialiser,-Accumulator) is det. % % Business end of run_length_encoding/3. Calculates the run-length % encoded form of a list and binds the result to the Accumulator. % Initialiser is a list [H-1] where H is the first element of the % input list. % run_length_encoding([], Fs, Fs).

% Run of N consecutive identical elements run_length_encoding([C|Cs],[C-F|Fs], Acc):-

       % Backtracking would produce successive counts

% of runs of C at different indices in the list. ! ,F_ is F + 1 ,run_length_encoding(Cs, [C-F_| Fs], Acc).

% End of a run of N consecutive identical elements. run_length_encoding([C|Cs], Fs, Acc):- run_length_encoding(Cs,[C-1|Fs], Acc).

/* Arithmetic helper predicates */

%! log2(N, L2_N) is det. % % L2_N is the logarithm with base 2 of N. % log2(N, L2_N):- L_10 is log10(N) ,L_2 is log10(2) ,L2_N is L_10 / L_2.

%! sum(+A,+B,?Sum) is det. % % True when Sum is the sum of numbers A and B. % % Helper predicate to allow foldl/4 to do addition. The following % call will raise an error (because there is no predicate +/3): % == % foldl(+, [1,2,3], 0, Result). % == % % This will not raise an error: % == % foldl(sum, [1,2,3], 0, Result). % == % sum(A, B, Sum):- must_be(number, A) ,must_be(number, B) ,Sum is A + B. </lang>

Example query:

?- shannon_entropy(1223334444, H).
H = 1.8464393446710154.

PureBasic

<lang purebasic>#TESTSTR="1223334444" NewMap uchar.i() : Define.d e

Procedure.d nlog2(x.d) : ProcedureReturn Log(x)/Log(2) : EndProcedure

Procedure countchar(s$, Map uchar())

 If Len(s$)
   uchar(Left(s$,1))=CountString(s$,Left(s$,1))
   s$=RemoveString(s$,Left(s$,1))
   ProcedureReturn countchar(s$, uchar())
 EndIf

EndProcedure

countchar(#TESTSTR,uchar())

ForEach uchar()

 e-uchar()/Len(#TESTSTR)*nlog2(uchar()/Len(#TESTSTR))

Output:

Entropy of [1223334444] = 1.846439344671015

Python

Python: Longer version

<lang python>from __future__ import division import math

def hist(source):

   hist = {}; l = 0;
   for e in source:
       l += 1
       if e not in hist:
           hist[e] = 0
       hist[e] += 1
   return (l,hist)

def entropy(hist,l):

   elist = []
   for v in hist.values():
       c = v / l
       elist.append(-c * math.log(c ,2))
   return sum(elist)

def printHist(h):

   flip = lambda (k,v) : (v,k)
   h = sorted(h.iteritems(), key = flip)
   print 'Sym\thi\tfi\tInf'
   for (k,v) in h:
       print '%s\t%f\t%f\t%f'%(k,v,v/l,-math.log(v/l, 2))

source = "1223334444" (l,h) = hist(source); print '.[Results].' print 'Length',l print 'Entropy:', entropy(h, l) printHist(h)</lang>

Output:

.[Results].
Length 10
Entropy: 1.84643934467
Sym	hi	fi	Inf
1	1.000000	0.100000	3.321928
2	2.000000	0.200000	2.321928
3	3.000000	0.300000	1.736966
4	4.000000	0.400000	1.321928

Python: More succinct version

The Counter module is only available in Python >= 2.7.

<lang python>>>> import math >>> from collections import Counter >>> >>> def entropy(s): ... p, lns = Counter(s), float(len(s)) ... return -sum( count/lns * math.log(count/lns, 2) for count in p.values()) ... >>> entropy("1223334444") 1.8464393446710154 >>> </lang>

Uses Python 2

<lang python>def Entropy(text):

   import math
   log2=lambda x:math.log(x)/math.log(2)
   exr={}
   infoc=0
   for each in text:
       try:
           exr[each]+=1
       except:
           exr[each]=1
   textlen=len(text)
   for k,v in exr.items():
       freq  =  1.0*v/textlen
       infoc+=freq*log2(freq)
   infoc*=-1
   return infoc

while True:

   print Entropy(raw_input('>>>'))</lang>

R

<lang r>entropy = function(s)

  {freq = prop.table(table(strsplit(s, )[1]))
   -sum(freq * log(freq, base = 2))}

print(entropy("1223334444")) # 1.846439</lang>

Racket

<lang racket>#lang racket (require math) (provide entropy hash-entropy list-entropy digital-entropy)

(define (hash-entropy h)

 (define (log2 x) (/ (log x) (log 2)))
 (define n (for/sum [(c (in-hash-values h))] c))
 (- (for/sum ([c (in-hash-values h)] #:unless (zero? c))
      (* (/ c n) (log2 (/ c n))))))

(define (list-entropy x) (hash-entropy (samples->hash x)))

(define entropy (compose list-entropy string->list)) (define digital-entropy (compose entropy number->string))

(module+ test

 (require rackunit)
 (check-= (entropy "1223334444") 1.8464393446710154 1E-8)
 (check-= (digital-entropy 1223334444) (entropy "1223334444") 1E-8)
 (check-= (digital-entropy 1223334444) 1.8464393446710154 1E-8)
 (check-= (entropy "xggooopppp") 1.8464393446710154 1E-8))

(module+ main (entropy "1223334444"))</lang>

Output:

 1.8464393446710154

REXX

version 1

<lang rexx>/* REXX ***************************************************************

28.02.2013 Walter Pachl
12.03.2013 Walter Pachl typo in log corrected. thanx for testing
22.05.2013 -"- extended the logic to accept other strings
25.05.2013 -"- 'my' log routine is apparently incorrect
25.05.2013 -"- problem identified & corrected
- - - - /

Numeric Digits 30 Parse Arg s If s= Then

 s="1223334444"

occ.=0 chars= n=0 cn=0 Do i=1 To length(s)

 c=substr(s,i,1)
 If pos(c,chars)=0 Then Do
   cn=cn+1
   chars=chars||c
   End
 occ.c=occ.c+1
 n=n+1
 End

do ci=1 To cn

 c=substr(chars,ci,1)
 p.c=occ.c/n
 /* say c p.c */
 End

e=0 Do ci=1 To cn

 c=substr(chars,ci,1)
 e=e+p.c*log(p.c,30,2)
 End

Say 'Version 1:' s 'Entropy' format(-e,,12) Exit

log: Procedure /***********************************************************************

Return log(x) -- with specified precision and a specified base
Three different series are used for the ranges 0 to 0.5
0.5 to 1.5
1.5 to infinity
03.09.1992 Walter Pachl
25.05.2013 -"- 'my' log routine is apparently incorrect
25.05.2013 -"- problem identified & corrected
- - - - /

 Parse Arg x,prec,b
 If prec= Then prec=9
 Numeric Digits (2*prec)
 Numeric Fuzz   3
 Select
   When x<=0 Then r='*** invalid argument ***'
   When x<0.5 Then Do
     z=(x-1)/(x+1)
     o=z
     r=z
     k=1
     Do i=3 By 2
       ra=r
       k=k+1
       o=o*z*z
       r=r+o/i
       If r=ra Then Leave
       End
     r=2*r
     End
   When x<1.5 Then Do
     z=(x-1)
     o=z
     r=z
     k=1
     Do i=2 By 1
       ra=r
       k=k+1
       o=-o*z
       r=r+o/i
       If r=ra Then Leave
       End
     End
   Otherwise /* 1.5<=x */ Do
     z=(x+1)/(x-1)
     o=1/z
     r=o
     k=1
     Do i=3 By 2
       ra=r
       k=k+1
       o=o/(z*z)
       r=r+o/i
       If r=ra Then Leave
       End
     r=2*r
     End
   End
 If b<> Then
   r=r/log(b,prec)
 Numeric Digits (prec)
 r=r+0
 Return r </lang>

<lang rexx>/* REXX ***************************************************************

Test program to compare Versions 1 and 2
(the latter tweaked to be acceptable by my (oo)Rexx
and to give the same output.)
version 1 was extended to accept the strings of the incorrect flag
22.05.2013 Walter Pachl (I won't analyze the minor differences)
25.05.2013 I did now analyze and had to discover that
'my' log routine is apparently incorrect
25.05.2013 problem identified & corrected
- - - - /

Call both '1223334444' Call both '1223334444555555555' Call both '122333' Call both '1227774444' Call both 'aaBBcccDDDD' Call both '1234567890abcdefghijklmnopqrstuvwxyz' Exit both:

 Parse Arg s
 Call entropy  s
 Call entropy2 s
 Say ' '
 Return

</lang>

Output:

Version 1: 1223334444 Entropy 1.846439344671
Version 2: 1223334444 Entropy 1.846439344671

Version 1: 1223334444555555555 Entropy 1.969811065278
Version 2: 1223334444555555555 Entropy 1.969811065278

Version 1: 122333 Entropy 1.459147917027
Version 2: 122333 Entropy 1.459147917027

Version 1: 1227774444 Entropy 1.846439344671
Version 2: 1227774444 Entropy 1.846439344671

Version 1: 1234567890abcdefghijklmnopqrstuvwxyz Entropy 5.169925001442
Version 2: 1234567890abcdefghijklmnopqrstuvwxyz Entropy 5.169925001442

version 2

REXX doesn't have a BIF for LOG or LN, so the subroutine (function) LOG2 is included herein.

The LOG2 subroutine is only included here for functionality, not to document how to calculate LOG₂ using REXX. <lang rexx>/*REXX program calculates the information entropy for a given character string*/ numeric digits 50 /*use 50 decimal digits for precision. */ parse arg $; if $= then $=1223334444 /*obtain the optional input from the CL*/

=0; @.=0; L=length($); $$= /*define handy-dandy REXX variables. */

     do j=1  for L;  _=substr($,j,1)  /*process each character in  $  string.*/
     if @._==0  then do;  #=#+1       /*Unique?  Yes, bump character counter.*/
                          $$=$$ || _  /*add this character to the  $$  list. */
                     end
     @._=@._+1                        /*keep track of this character's count.*/
     end   /*j*/

sum=0 /*calculate info entropy for each char.*/

     do i=1  for #;  _=substr($$,i,1) /*obtain a character from unique list. */
     sum=sum  -  @._/L  * log2(@._/L) /*add (negatively) the char entropies. */
     end   /*i*/

say ' input string: ' $ say 'string length: ' L say ' unique chars: ' # ; say say 'the information entropy of the string ──► ' format(sum,,12) " bits." exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────LOG2 subroutine───────────────────────────*/ log2: procedure; parse arg x 1 ox; ig= x>1.5; is=1-2*(ig\==1); ii=0 numeric digits digits()+5 /* [↓] precision of E must be ≥ digits().*/ e=2.7182818284590452353602874713526624977572470936999595749669676277240766303535

  do  while  ig & ox>1.5 | \ig&ox<.5;      _=e;      do k=-1;  iz=ox* _**-is
  if k>=0 & (ig & iz<1 | \ig&iz>.5)  then leave;     _=_*_;    izz=iz;   end
  ox=izz;  ii=ii+is*2**k;  end;            x=x* e** -ii-1;  z=0;    _=-1;   p=z
    do k=1;   _=-_*x;   z=z+_/k;     if z=p  then leave;  p=z;   end  /*k*/
  r=z+ii;  if arg()==2  then return r;       return r/log2(2,0)</lang>

Output:

when using the default input of

1223334444

 input string:  1223334444
string length:  10
 unique chars:  4

the information entropy of the string ──►  1.846439344671  bits.

Output:

when using the input of

Rosetta Code

 input string:  Rosetta Code
string length:  12
 unique chars:  9

the information entropy of the string ──►  3.084962500721  bits.

Ruby

Works with: Ruby version 1.9

<lang ruby>def entropy(s)

 counts = Hash.new(0.0)
 s.each_char { |c| counts[c] += 1 }
 leng = s.length
 
 counts.values.reduce(0) do |entropy, count|
   freq = count / leng
   entropy - freq * Math.log2(freq)
 end

end

p entropy("1223334444")</lang>

Output:

1.8464393446710154

One-liner, same performance (or better): <lang ruby>def entropy2(s)

 s.each_char.group_by(&:to_s).values.map { |x| x.length / s.length.to_f }.reduce(0) { |e, x| e - x*Math.log2(x) }

end</lang>

Rust

<lang Rust>// works for Rust 0.9 fn entropy(s: &str) -> f32 { let mut entropy: f32 = 0.0; let mut histogram = [0, ..256]; let len = s.len();

for i in range(0, len) { histogram[s[i]] += 1; } for i in range(0, 256) { if histogram[i] > 0 { let ratio = (histogram[i] as f32 / len as f32) as f32; entropy -= (ratio * log2(ratio)) as f32; } }

entropy }</lang>

Scala

<lang scala>import scala.math._

def entropy( v:String ) = { v

 .groupBy (a => a)
 .values
 .map( i => i.length.toDouble / v.length )
 .map( p => -p * log10(p) / log10(2))
 .sum

}

// Confirm that "1223334444" has an entropy of about 1.84644 assert( math.round( entropy("1223334444") * 100000 ) * 0.00001 == 1.84644 )</lang>

scheme

A version capable of calculating multidimensional entropy. <lang scheme> (define (entropy input)

 (define (close? a b)
   (define (norm x y)
     (define (infinite_norm m n)
       (define (absminus p q)
            (cond ((null? p) '())
               (else (cons (abs (- (car p) (car q))) (absminus (cdr p) (cdr q))))))
       (define (mm l)
            (cond ((null? (cdr l)) (car l))
                  ((> (car l) (cadr l)) (mm (cons (car l) (cddr l))))
                  (else (mm (cdr l)))))
       (mm (absminus m n)))
     (if (pair? x) (infinite_norm x y) (abs (- x y))))
   (let ((epsilon 0.2))
     (< (norm a b) epsilon)))
 (define (freq-list x)
   (define (f x)
     (define (count a b)
       (cond ((null? b) 1)
             (else (+ (if (close? a (car b)) 1 0) (count a (cdr b))))))
     (let ((t (car x)) (tt (cdr x)))
       (count t tt)))
   (define (g x)
     (define (filter a b)
       (cond ((null? b) '())
             ((close? a (car b)) (filter a (cdr b)))
             (else (cons (car b) (filter a (cdr b))))))
     (let ((t (car x)) (tt (cdr x)))
       (filter t tt)))  
   (cond ((null? x) '())
         (else (cons (f x) (freq-list (g x))))))
 (define (scale x)
   (define (sum x)
     (if (null? x) 0.0 (+ (car x) (sum (cdr x)))))
   (let ((z (sum x)))
     (map (lambda(m) (/ m z)) x)))
 (define (cal x)
   (if (null? x) 0 (+ (* (car x) (/ (log (car x)) (log 2))) (cal (cdr x)))))
 (- (cal (scale (freq-list input)))))

(entropy (list 1 2 2 3 3 3 4 4 4 4)) (entropy (list (list 1 1) (list 1.1 1.1) (list 1.2 1.2) (list 1.3 1.3) (list 1.5 1.5) (list 1.6 1.6))) </lang>

Output:

1.8464393446710154 bits

1.4591479170272448 bits

Seed7

<lang seed7>$ include "seed7_05.s7i";

 include "float.s7i";
 include "math.s7i";

const func float: entropy (in string: stri) is func

 result
   var float: entropy is 0.0;
 local
   var hash [char] integer: count is (hash [char] integer).value;
   var char: ch is ' ';
   var float: p is 0.0;
 begin
   for ch range stri do
     if ch in count then
       incr(count[ch]);
     else
       count @:= [ch] 1;
     end if;
   end for;
   for key ch range count do
     p := flt(count[ch]) / flt(length(stri));
     entropy -:= p * log(p) / log(2.0);
   end for;
 end func ;

const proc: main is func

 begin
   writeln(entropy("1223334444") digits 5);
 end func;</lang>

Output:

1.84644

Sidef

<lang ruby>func entropy(s) {

 var counts = Hash.new();
 counts.default(0);
 s.each { |c| counts[c]++ };
 var len = s.len;
 [0, counts.values.map {|count| 
   var freq = count/len; freq * freq.log2 }...
 ]«-»;

}

say entropy("1223334444");</lang>

Output:

1.846439344671015493434197746305045223237

Tcl

<lang tcl>proc entropy {str} {

   set log2 [expr log(2)]
   foreach char [split $str ""] {dict incr counts $char}
   set entropy 0.0
   foreach count [dict values $counts] {

set freq [expr {$count / double([string length $str])}] set entropy [expr {$entropy - $freq * log($freq)/$log2}]

   }
   return $entropy

}</lang> Demonstration: <lang tcl>puts [format "entropy = %.5f" [entropy "1223334444"]] puts [format "entropy = %.5f" [entropy "Rosetta Code"]]</lang>

Output:

entropy = 1.84644
entropy = 3.08496

XPL0

<lang XPL0>code real RlOut=48, Ln=54; \intrinsic routines string 0; \use zero-terminated strings

func StrLen(A); \Return number of characters in an ASCIIZ string char A; int I; for I:= 0, -1>>1-1 do

   if A(I) = 0 then return I;

func real Entropy(Str); \Return Shannon entropy of string char Str; int Len, I, Count(128); real Sum, Prob; [Len:= StrLen(Str); for I:= 0 to 127 do Count(I):= 0; for I:= 0 to Len-1 do \count number of each character in string

   Count(Str(I)):= Count(Str(I)) + 1;

Sum:= 0.0; for I:= 0 to 127 do

   if Count(I) # 0 then        \(avoid Ln(0.0) error)
       [Prob:= float(Count(I)) / float(Len);   \probability of char in string
       Sum:= Sum + Prob*Ln(Prob);
       ];

return -Sum/Ln(2.0); ];

RlOut(0, Entropy("1223334444"))</lang>

Output:

    1.84644

zkl

Translation of: D

<lang zkl>fcn entropy(text){

  text.pump(Void,fcn(c,freq){ c=c.toAsc(); freq[c]+=1; freq }
      .fp1( (0).pump(256,List,0.0).copy() )) // array[256] of 0.0
  .filter()		      // remove all zero entries from array
  .apply('/(text.len()))     // (num of char)/len
  .apply(fcn(p){-p*p.log()}) // |p*ln(p)|
  .sum(0.0)/(2.0).log();     // sum * ln(e)/ln(2) to convert to log2

}

entropy("1223334444").println(" bits");</lang>

Output:

1.84644 bits