Entropy
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Calculate the Shannon entropy H of a given input string.
Given the discrete random variable that is a string of "symbols" (total characters) consisting of different characters (n=2 for binary), the Shannon entropy of X in bits/symbol is :
where is the count of character .
For this task, use X="1223334444" as an example. The result should be 1.84644... bits/symbol. This assumes X was a random variable, which may not be the case, or it may depend on the observer.
This coding problem calculates the "specific" or "intensive" entropy that finds its parallel in physics with "specific entropy" S0 which is entropy per kg or per mole, not like physical entropy S and therefore not the "information" content of a file. It comes from Boltzmann's H-theorem where where N=number of molecules. Boltzmann's H is the same equation as Shannon's H, and it gives the specific entropy H on a "per molecule" basis.
The "total", "absolute", or "extensive" information entropy is
- bits
This is not the entropy being coded here, but it is the closest to physical entropy and a measure of the information content of a string. But it does not look for any patterns that might be available for compression, so it is a very restricted, basic, and certain measure of "information". Every binary file with an equal number of 1's and 0's will have S=N bits. All hex files with equal symbol frequencies will have bits of entropy. The total entropy in bits of the example above is S= 10*18.4644 = 18.4644 bits.
The H function does not look for any patterns in data or check if X was a random variable. For example, X=000000111111 gives the same calculated entropy in all senses as Y=010011100101. For most purposes it is usually more relevant to divide the gzip length by the length of the original data to get an informal measure of how much "order" was in the data.
Two other "entropies" are useful:
Normalized specific entropy:
which varies from 0 to 1 and it has units of "entropy/symbol" or just 1/symbol. For this example, Hn<\sub>= 0.923.
Normalized total (extensive) entropy:
which varies from 0 to N and does not have units. It is simply the "entropy", but it needs to be called "total normalized extensive entropy" so that it is not confused with Shannon's (specific) entropy or physical entropy. For this example, Sn<\sub>= 9.23.
Shannon himself is the reason his "entropy/symbol" H function is very confusingly called "entropy". That's like calling a function that returns a speed a "meter". See section 1.7 of his classic A Mathematical Theory of Communication and search on "per symbol" and "units" to see he always stated his entropy H has units of "bits/symbol" or "entropy/symbol" or "information/symbol". So it is legitimate to say entropy NH is "information".
In keeping with Landauer's limit, the physics entropy generated from erasing N bits is if the bit storage device is perfectly efficient. This can be solved for H2*N to (arguably) get the number of bits of information that a physical entropy represents.
- Related tasks
11l
F entropy(source)
DefaultDict[Char, Int] hist
L(c) source
hist[c]++
V r = 0.0
L(v) hist.values()
V c = Float(v) / source.len
r -= c * log2(c)
R r
print(entropy(‘1223334444’))- Output:
1.84644
Ada
Uses Ada 2012.
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;
Aime
integer c;
real h, v;
index x;
data s;
for (, c in (s = argv(1))) {
x[c] += 1r;
}
h = 0;
for (, v in x) {
v /= ~s;
h -= v * log2(v);
}
o_form("/d6/\n", h);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
BEGIN
# calculate the shannon entropy of a string #
PROC shannon entropy = ( STRING s )REAL:
BEGIN
INT string length = ( UPB s - LWB s ) + 1;
# count the occurences 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 #
# test the shannon entropy routine #
print( ( shannon entropy( "1223334444" ), newline ) )
END- Output:
+1.84643934467102e +0
ALGOL W
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.- Output:
1.846439
APL
ENTROPY←{-+/R×2⍟R←(+⌿⍵∘.=∪⍵)÷⍴⍵}
⍝ How it works:
⎕←UNIQUE←∪X←'1223334444'
1234
⎕←TABLE_OF_OCCURENCES←X∘.=UNIQUE
1 0 0 0
0 1 0 0
0 1 0 0
0 0 1 0
0 0 1 0
0 0 1 0
0 0 0 1
0 0 0 1
0 0 0 1
0 0 0 1
⎕←COUNT←+⌿TABLE_OF_OCCURENCES
1 2 3 4
⎕←N←⍴X
10
⎕←RATIO←COUNT÷N
0.1 0.2 0.3 0.4
-+/RATIO×2⍟RATIO
1.846439345
- Output:
ENTROPY X 1.846439345
Arturo
entropy: function [s][
t: #[]
loop s 'c [
unless key? t c -> t\[c]: 0
t\[c]: t\[c] + 1
]
result: new 0
loop values t 'x ->
'result - (x//(size s)) * log x//(size s) 2
return result
]
print entropy "1223334444"
- Output:
1.846439344671015
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
}
- Output:
1.846440
AWK
#!/usr/bin/awk -f
{
N = length
for (i = 1; i <= N; ++i)
++H[substr($0, i, 1)]
}
END {
for (i in H)
S += H[i] * log(H[i])
print (log(N) - S / N) / log(2)
}
- Usage:
echo 1223334444 |./entropy.awk
1.84644
BASIC
Works with older (unstructured) Microsoft-style BASIC.
10 DEF FN L(X)=LOG(X)/LOG(2)
20 S$="1223334444"
30 U$=""
40 FOR I=1 TO LEN(S$)
50 K=0
60 FOR J=1 TO LEN(U$)
70 IF MID$(U$,J,1)=MID$(S$,I,1) THEN K=1
80 NEXT J
90 IF K=0 THEN U$=U$+MID$(S$,I,1)
100 NEXT I
110 DIM R(LEN(U$)-1)
120 FOR I=1 TO LEN(U$)
130 C=0
140 FOR J=1 TO LEN(S$)
150 IF MID$(U$,I,1)=MID$(S$,J,1) THEN C=C+1
160 NEXT J
170 R(I-1)=(C/LEN(S$))*FN L(C/LEN(S$))
180 NEXT I
190 E=0
200 FOR I=0 TO LEN(U$)-1
210 E=E-R(I)
220 NEXT I
230 PRINT E
- Output:
1.84643935
QBasic
FUNCTION L (X)
L = LOG(X) / LOG(2)
END FUNCTION
S$ = "1223334444"
U$ = ""
FOR I = 1 TO LEN(S$)
K = 0
FOR J = 1 TO LEN(U$)
IF MID$(U$, J, 1) = MID$(S$, I, 1) THEN K = 1
NEXT J
IF K = 0 THEN U$ = U$ + MID$(S$, I, 1)
NEXT I
DIM R(LEN(U$) - 1)
FOR I = 1 TO LEN(U$)
C = 0
FOR J = 1 TO LEN(S$)
IF MID$(U$, I, 1) = MID$(S$, J, 1) THEN C = C + 1
NEXT J
R(I - 1) = (C / LEN(S$)) * L(C / LEN(S$))
NEXT I
E = 0
FOR I = 0 TO LEN(U$) - 1
E = E - R(I)
NEXT I
PRINT E
END
Sinclair ZX81 BASIC
Works with 1k of RAM.
10 LET X$="1223334444"
20 LET U$=""
30 FOR I=1 TO LEN X$
40 LET K=0
50 FOR J=1 TO LEN U$
60 IF U$(J)=X$(I) THEN LET K=K+1
70 NEXT J
80 IF K=0 THEN LET U$=U$+X$(I)
90 NEXT I
100 DIM R(LEN U$)
110 FOR I=1 TO LEN U$
120 LET C=0
130 FOR J=1 TO LEN X$
140 IF U$(I)=X$(J) THEN LET C=C+1
150 NEXT J
160 LET R(I)=C/LEN X$*LN (C/LEN X$)/LN 2
170 NEXT I
180 LET E=0
190 FOR I=1 TO LEN U$
200 LET E=E-R(I)
210 NEXT I
220 PRINT E
- Output:
1.8464393
uBasic/4tH
uBasic/4tH is an integer BASIC only. So, fixed point arithmetic is required go fulfill this task. Some loss of precision is unavoidable.
If Info("wordsize") < 64 Then Print "This program requires a 64-bit uBasic" : End
s := "1223334444"
u := ""
x := FUNC(_Fln(FUNC(_Ntof(2)))) ' calculate LN(2)
For i = 0 TO Len(s)-1
k = 0
For j = 0 TO Len(u)-1
If Peek(u, j) = Peek(s, i) Then k = 1
Next
If k = 0 THEN u = Join(u, Char (Peek (s, i)))
Next
Dim @r(Len(u)-1)
For i = 0 TO Len(u)-1
c = 0
For J = 0 TO Len(s)-1
If Peek(u, i) = Peek (s, j) Then c = c + 1
Next
q = FUNC(_Fdiv(c, Len(s)))
@r(i) = FUNC(_Fmul(q, FUNC(_Fdiv(FUNC(_Fln(q)), x))))
Next
e = 0
For i = 0 To Len(u) - 1
e = e - @r(i)
Next
Print Using "+?.####"; FUNC(_Ftoi(e))
End
_Fln Param (1) : Return (FUNC(_Ln(a@*4))/4)
_Fmul Param (2) : Return ((a@*b@)/16384)
_Fdiv Param (2) : Return ((a@*16384)/b@)
_Ntof Param (1) : Return (a@*16384)
_Ftoi Param (1) : Return ((10000*a@)/16384)
_Ln
Param (1)
Local (2)
c@=681391
If (a@<32768) Then a@=SHL(a@, 16) : c@=c@-726817
If (a@<8388608) Then a@=SHL(a@, 8) : c@=c@-363409
If (a@<134217728) Then a@=SHL(a@, 4) : c@=c@-181704
If (a@<536870912) Then a@=SHL(a@, 2) : c@=c@-90852
If (a@<1073741824) Then a@=SHL(a@, 1) : c@=c@-45426
b@=a@+SHL(a@, -1) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-26573
b@=a@+SHL(a@, -2) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-14624
b@=a@+SHL(a@, -3) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-7719
b@=a@+SHL(a@, -4) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-3973
b@=a@+SHL(a@, -5) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-2017
b@=a@+SHL(a@, -6) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-1016
b@=a@+SHL(a@, -7) : If (AND(b@, 2147483648)) = 0 Then a@=b@ : c@=c@-510
a@=2147483648-a@;
c@=c@-SHL(a@, -15)
Return (c@)
- Output:
1.8461 0 OK, 0:638
BBC BASIC
REM >entropy
PRINT FNentropy("1223334444")
END
:
DEF FNentropy(x$)
LOCAL unique$, count%, n%, ratio(), u%, i%, j%
unique$ = ""
n% = LEN x$
FOR i% = 1 TO n%
IF INSTR(unique$, MID$(x$, i%, 1)) = 0 THEN unique$ += MID$(x$, i%, 1)
NEXT
u% = LEN unique$
DIM ratio(u% - 1)
FOR i% = 1 TO u%
count% = 0
FOR j% = 1 TO n%
IF MID$(unique$, i%, 1) = MID$(x$, j%, 1) THEN count% += 1
NEXT
ratio(i% - 1) = (count% / n%) * FNlogtwo(count% / n%)
NEXT
= -SUM(ratio())
:
DEF FNlogtwo(n)
= LN n / LN 2
- Output:
1.84643934
BQN
H ← -∘(+´⊢×2⋆⁼⊢)∘((+˝⊢=⌜⍷)÷≠)
H "1223334444"
- Output:
1.8464393446710154
Burlesque
blsq ) "1223334444"F:u[vv^^{1\/?/2\/LG}m[?*++
1.8464393446710157C
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <string.h>
#include <math.h>
#define MAXLEN 100 //maximum string length
int makehist(unsigned 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){
unsigned 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;
}
Examples:
$ ./entropy
1223334444
1.846439
$ ./entropy
Rosetta Code is the best site in the world!
3.646513
C#
Translation of C++.
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;
}
}
}
- Output:
The Entropy of 1223334444 is 1.84643934467102
Without using Hashtables or Dictionaries:
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;
}
}
}
C++
#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 ) ;
}
std::cout << "The information content of " << teststring
<< " is " << infocontent << std::endl ;
return 0 ;
}
- Output:
(entropy "1223334444") The information content of 1223334444 is 1.84644
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 +))))
- Output:
(entropy "1223334444")
1.8464393446710154
CLU
% NOTE: when compiling with Portable CLU,
% this program needs to be merged with 'useful.lib' to get log()
%
% pclu -merge $CLUHOME/lib/useful.lib -compile entropy.clu
shannon = proc (s: string) returns (real)
% find the frequency of each character
freq: array[int] := array[int]$fill(0, 256, 0)
for c: char in string$chars(s) do
i: int := char$c2i(c)
freq[i] := freq[i] + 1
end
% calculate the component for each character
h: real := 0.0
rlen: real := real$i2r(string$size(s))
for i: int in array[int]$indexes(freq) do
if freq[i] ~= 0 then
f: real := real$i2r(freq[i]) / rlen
h := h - f * log(f) / log(2.0)
end
end
return (h)
end shannon
start_up = proc ()
po: stream := stream$primary_output()
stream$putl(po, f_form(shannon("1223334444"), 1, 6))
end start_up- Output:
1.846439
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'}"
- Output:
The entropy of the string '1223334444' is 1.8464393446710157
Common Lisp
Not very Common Lisp-y version:
(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))
More like Common Lisp version:
(defun entropy (string &aux (length (length string)))
(declare (type string string))
(let ((table (make-hash-table)))
(loop for char across string
do (incf (gethash char table 0)))
(- (loop for freq being each hash-value in table
for freq/length = (/ freq length)
sum (* freq/length (log freq/length 2))))))
Crystal
# Method to calculate sum of Float64 array
def sum(array : Array(Float64))
res = 0
array.each do |n|
res += n
end
res
end
# Method to calculate which char appears how often
def histogram(source : String)
hist = {} of Char => Int32
l = 0
source.each_char do |e|
if !hist.has_key? e
hist[e] = 0
end
hist[e] += 1
end
return Tuple.new(source.size, hist)
end
# Method to calculate entropy from histogram
def entropy(hist : Hash(Char, Int32), l : Int32)
elist = [] of Float64
hist.each do |el|
v = el[1]
c = v / l
elist << (-c * Math.log(c, 2))
end
return sum elist
end
source = "1223334444"
hist_res = histogram source
l = hist_res[0]
h = hist_res[1]
puts ".[Results]."
puts "Length: " + l.to_s
puts "Entropy: " + (entropy h, l).to_s
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;
}
- Output:
1.84644
Delphi
Just fix Pascal code to run in Delphi.
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
Result := '';
for n := 1 to length(str) do
if (PosEx(str[n], str, n) > 0) and (PosEx(str[n], Result, 1) = 0) then
Result := Result + 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(Result, 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(Result[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);
Result := 0.0;
for i := 1 to length(us) do
Result := Result - pf[i] / l * log2(pf[i] / l);
end;
begin
Writeln('Entropy of "', strng, '" is ', entropy(strng): 2: 5, ' bits.');
readln;
end.
EasyLang
func entropy s$ .
len d[] 255
for c$ in strchars s$
d[strcode c$] += 1
.
for cnt in d[]
if cnt > 0
prop = cnt / len s$
entr -= (prop * log10 prop / log10 2)
.
.
return entr
.
print entropy "1223334444"
EchoLisp
(lib 'hash)
;; counter: hash-table[key]++
(define (count++ ht k )
(hash-set ht k (1+ (hash-ref! ht k 0))))
(define (hi count n )
(define pi (// count n))
(- (* pi (log2 pi))))
;; (H [string|list]) → entropy (bits)
(define (H info)
(define S (if(string? info) (string->list info) info))
(define ht (make-hash))
(define n (length S))
(for ((s S)) (count++ ht s))
(for/sum ((s (make-set S))) (hi (hash-ref ht s) n)))
- Output:
;; by increasing entropy
(H "🔴") → 0
(H "🔵🔴") → 1
(H "1223334444") → 1.8464393446710154
(H "♖♘♗♕♔♗♘♖♙♙♙♙♙♙♙♙♙") → 2.05632607578088
(H "EchoLisp") → 3
(H "Longtemps je me suis couché de bonne heure") → 3.860828877124944
(H "azertyuiopmlkjhgfdsqwxcvbn") → 4.700439718141092
(H (for/list ((i 1000)) (random 1000))) → 9.13772704467521
(H (for/list ((i 100_000)) (random 100_000))) → 15.777516877140766
(H (for/list ((i 1000_000)) (random 1000_000))) → 19.104028424596976
Elena
ELENA 6.x :
import system'math;
import system'collections;
import system'routines;
import extensions;
extension op
{
logTwo()
= self.ln() / 2.ln();
}
public program()
{
var input := console.readLine();
var infoC := 0.0r;
var table := Dictionary.new();
input.forEach::(ch)
{
var n := table[ch];
if (nil == n)
{
table[ch] := 1
}
else
{
table[ch] := n + 1
}
};
var freq := 0;
table.forEach::(letter)
{
freq := letter.toInt().realDiv(input.Length);
infoC += (freq * freq.logTwo())
};
infoC *= -1;
console.printLine("The Entropy of ", input, " is ", infoC)
}- Output:
The Entropy of 1223334444 is 1.846439344671
Elixir
:math.log2 was added in OTP 18.
defmodule RC do
def entropy(str) do
leng = String.length(str)
String.graphemes(str)
|> 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")
- Output:
1.8464393446710154
Emacs 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)))
- 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).
(shannon-entropy "1223334444")
1.8464393446710154
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))}.
- Output:
24> entropy:task(). 1.8464393446710157
Euler Math Toolbox
>function entropy (s) ...
$ v=strtochar(s);
$ m=getmultiplicities(unique(v),v);
$ m=m/sum(m);
$ return sum(-m*logbase(m,2))
$endfunction
>entropy("1223334444")
1.84643934467Excel
This solution uses the LAMBDA, LET, and MAP functions introduced into the Microsoft 365 version of Excel in 2021. The LET function is able to use functions as first class citizens. Taking advantage of this makes the solution much simpler. The solution below looks for the string in cell A1.
=LET(
_MainS,A1,
_N,LEN(_MainS),
_Chars,UNIQUE(MID(_MainS,SEQUENCE(LEN(_MainS),1,1,1),1)),
calcH,LAMBDA(_c,(_c/_N)*LOG(_c/_N,2)),
getCount,LAMBDA(_i,LEN(_MainS)-LEN(SUBSTITUTE(_MainS,_i,""))),
_CharMap,MAP(_Chars,LAMBDA(a, calcH(getCount(a)))),
-SUM(_CharMap)
)_Chars uses the SEQUENCE function to split the text into an array. The UNIQUE function then returns a list of unique characters in the string.
calcH applies the calculation described at the top of the page that will then be summed for each character
getCount uses the SUBSTITUTE method to count the occurrences of a character within the string.
If you needed to re-use this calculation then you could wrap it in a LAMBDA function within the name manager, changing A1 to a variable name (e.g. String):
ShannonEntropyH2=LAMBDA(String,LET(_MainS,String,_N,LEN(_MainS),_Chars,UNIQUE(MID(_MainS,SEQUENCE(LEN(_MainS),1,1,1),1)),calcH,LAMBDA(_c,(_c/_N)*LOG(_c/_N,2)),getCount,LAMBDA(_i,LEN(_MainS)-LEN(SUBSTITUTE(_MainS,_i,""))),_CharMap,MAP(_Chars,LAMBDA(a, calcH(getCount(a)))),-SUM(_CharMap)))Then you can just use the named lambda. E.g. If A1 = 1223334444 then:
=ShannonEntropyH2(A1)Returns 1.846439345
F#
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")
- Output:
1.846439
Factor
USING: assocs kernel math math.functions math.statistics
prettyprint sequences ;
IN: rosetta-code.entropy
: shannon-entropy ( str -- entropy )
[ length ] [ histogram >alist [ second ] map ] bi
[ swap / ] with map
[ dup log 2 log / * ] map-sum neg ;
"1223334444" shannon-entropy .
"Factor is my favorite programming language." shannon-entropy .
- Output:
1.846439344671015 4.04291723248433
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
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.
!-*- 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
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 <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
- 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).
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)
- Output:
1.84643934467101549345
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
Test case
Go
Go: Slice version
package main
import (
"fmt"
"math"
"strings"
)
func main(){
fmt.Println(H("1223334444"))
}
// for ASCII strings
func H(data string) (entropy float64) {
if data == "" {
return 0
}
for i := 0; i < 256; i++ {
px := float64(strings.Count(data, string(byte(i)))) / float64(len(data))
if px > 0 {
entropy += -px * math.Log2(px)
}
}
return entropy
}
- Output:
1.8464393446710154
Go: Map version
package main
import (
"fmt"
"math"
)
func main() {
const s = "1223334444"
l := float64(0)
m := map[rune]float64{}
for _, r := range s {
m[r]++
l++
}
var hm float64
for _, c := range m {
hm += c * math.Log2(c)
}
fmt.Println(math.Log2(l) - hm/l)
}
- Output:
1.8464393446710152
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)
}
}
Testing
[ '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
}
- 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
import Data.List
main = print $ entropy "1223334444"
entropy :: (Ord a, Floating c) => [a] -> c
entropy = sum . map lg . fq . map genericLength . group . sort
where lg c = -c * logBase 2 c
fq c = let sc = sum c in map (/ sc) c
Or, inlining with an applicative expression (turns out to be fractionally faster):
import Data.List (genericLength, group, sort)
entropy
:: (Ord a, Floating c)
=> [a] -> c
entropy =
sum .
map (negate . ((*) <*> logBase 2)) .
(map =<< flip (/) . sum) . map genericLength . group . sort
main :: IO ()
main = print $ entropy "1223334444"
- Output:
1.8464393446710154
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.
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
- Output:
->en 1.846439344671015 ->
J
Solution:
entropy=: +/@(-@* 2&^.)@(#/.~ % #)
- Example:
entropy '1223334444'
1.84644
entropy i.256
8
entropy 256$9
0
entropy 256$0 1
1
entropy 256$0 1 2 3
2
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
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;
}
}
- 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
Calculate the entropy of a string by determining the frequency of each character, then summing each character's probability multiplied by the log base 2 of that same probability, taking the negative of the sum.
// Shannon entropy in bits per symbol.
function entropy(str) {
const len = str.length
// Build a frequency map from the string.
const frequencies = Array.from(str)
.reduce((freq, c) => (freq[c] = (freq[c] || 0) + 1) && freq, {})
// Sum the frequency of each character.
return Object.values(frequencies)
.reduce((sum, f) => sum - f/len * Math.log2(f/len), 0)
}
console.log(entropy('1223334444')) // 1.8464393446710154
console.log(entropy('0')) // 0
console.log(entropy('01')) // 1
console.log(entropy('0123')) // 2
console.log(entropy('01234567')) // 3
console.log(entropy('0123456789abcdef')) // 4
- Output:
1.8464393446710154 0 1 2 3 4
- Another variant
const entropy = (s) => {
const split = s.split('');
const counter = {};
split.forEach(ch => {
if (!counter[ch]) counter[ch] = 1;
else counter[ch]++;
});
const lengthf = s.length * 1.0;
const counts = Object.values(counter);
return -1 * counts
.map(count => count / lengthf * Math.log2(count / lengthf))
.reduce((a, b) => a + b);
};
- Output:
console.log(entropy("1223334444")); // 1.8464393446710154
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.
# 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) ;- Example:
"1223334444" | entropy # => 1.8464393446710154Jsish
From Javascript entry.
/* Shannon entropy, in Jsish */
function values(obj:object):array {
var vals = [];
for (var key in obj) vals.push(obj[key]);
return vals;
}
function entropy(s) {
var split = s.split('');
var counter = {};
split.forEach(function(ch) {
if (!counter[ch]) counter[ch] = 1;
else counter[ch]++;
});
var lengthf = s.length * 1.0;
var counts = values(counter);
return -1 * counts.map(function(count) {
return count / lengthf * (Math.log(count / lengthf) / Math.log(2));
})
.reduce(function(a, b) { return a + b; }
);
};
if (Interp.conf('unitTest')) {
; entropy('1223334444');
; entropy('Rosetta Code');
; entropy('password');
}
- Output:
prompt$ jsish --U entropy.jsi
entropy('1223334444') ==> 1.84643934467102
entropy('Rosetta Code') ==> 3.08496250072116
entropy('password') ==> 2.75
Julia
entropy(s) = -sum(x -> x * log(2, x), count(x -> x == c, s) / length(s) for c in unique(s))
@show entropy("1223334444")
@show entropy([1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5])
- Output:
entropy("1223334444") = 1.8464393446710154
entropy([1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5]) = 2.103909910282364
K
entropy: {(`ln[#x]-(+/{x*`ln@x}@+/{x=\:?x}x)%#x)%`ln@2}
entropy "1223334444"
- Output:
1.8464393446710161
Kotlin
// version 1.0.6
fun log2(d: Double) = Math.log(d) / Math.log(2.0)
fun shannon(s: String): Double {
val counters = mutableMapOf<Char, Int>()
for (c in s) {
if (counters.containsKey(c)) counters[c] = counters[c]!! + 1
else counters.put(c, 1)
}
val nn = s.length.toDouble()
var sum = 0.0
for (key in counters.keys) {
val term = counters[key]!! / nn
sum += term * log2(term)
}
return -sum
}
fun main(args: Array<String>) {
val samples = arrayOf(
"1223334444",
"1223334444555555555",
"122333",
"1227774444",
"aaBBcccDDDD",
"1234567890abcdefghijklmnopqrstuvwxyz",
"Rosetta Code"
)
println(" String Entropy")
println("------------------------------------ ------------------")
for (sample in samples) println("${sample.padEnd(36)} -> ${"%18.16f".format(shannon(sample))}")
}
- Output:
String Entropy ------------------------------------ ------------------ 1223334444 -> 1.8464393446710154 1223334444555555555 -> 1.9698110652780971 122333 -> 1.4591479170272448 1227774444 -> 1.8464393446710154 aaBBcccDDDD -> 1.9362600275315274 1234567890abcdefghijklmnopqrstuvwxyz -> 5.1699250014423095 Rosetta Code -> 3.0849625007211556
Ksh
function entropy {
typeset -i i len=${#1}
typeset -X13 r=0
typeset -Ai counts
for ((i = 0; i < len; ++i))
do
counts[${1:i:1}]+=1
done
for i in "${counts[@]}"
do
r+='i * log2(i)'
done
r='log2(len) - r / len'
print -r -- "$r"
}
printf '%g\n' "$(entropy '1223334444')"
- Output:
1.84644
Lambdatalk
{def entropy
{def entropy.count
{lambda {:s :c :i}
{let { {:c {/ {A.get :i :c} {A.length :s}}}
} {* :c {log2 :c}}}}}
{def entropy.sum
{lambda {:s :c}
{- {+ {S.map {entropy.count :s :c}
{S.serie 0 {- {A.length :c} 1}}}}}}}
{lambda {:s}
{entropy.sum {A.split :s} {cdr {W.frequency :s}}}}}
-> entropy
The W.frequency function is explained in rosettacode.org/wiki/Letter_frequency#Lambdatalk
{def txt 1223334444}
-> txt
{def F {W.frequency {txt}}}
-> F
characters: {car {F}} -> [1,2,3,4]
frequencies: {cdr {F}} -> [1,2,3,4]
{entropy {txt}}
-> 1.8464393446710154
{entropy 0}
-> 0
{entropy 00000000000000}
-> 0
{entropy 11111111111111}
-> 0
{entropy 01}
-> 1
{entropy Lambdatalk}
-> 2.8464393446710154
{entropy entropy}
-> 2.807354922057604
{entropy abcdefgh}
-> 3
{entropy Rosetta Code}
-> 3.084962500721156
{entropy Longtemps je me suis couché de bonne heure}
-> 3.8608288771249444
{entropy abcdefghijklmnopqrstuvwxyz}
-> 4.70043971814109
{entropy abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz}
-> 4.70043971814109
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 . # 1.84643934467102Liberty BASIC
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- 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.
Lua
function log2 (x) return math.log(x) / math.log(2) end
function entropy (X)
local N, count, sum, i = X:len(), {}, 0
for char = 1, N do
i = X:sub(char, char)
if count[i] then
count[i] = count[i] + 1
else
count[i] = 1
end
end
for n_i, count_i in pairs(count) do
sum = sum + count_i / N * log2(count_i / N)
end
return -sum
end
print(entropy("1223334444"))
Mathematica / Wolfram Language
shE[s_String] := -Plus @@ ((# Log[2., #]) & /@ ((Length /@ Gather[#])/
Length[#]) &[Characters[s]])
- Example:
shE["1223334444"]
1.84644
shE["Rosetta Code"]
3.08496
MATLAB / Octave
This version allows for any input vectors, including strings, floats, negative integers, etc.
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;
- Usage:
> entropy('1223334444')
ans = 1.8464
MiniScript
entropy = function(s)
count = {}
for c in s
if count.hasIndex(c) then count[c] = count[c]+1 else count[c] = 1
end for
sum = 0
for x in count.values
countOverN = x / s.len
sum = sum + countOverN * log(countOverN, 2)
end for
return -sum
end function
print entropy("1223334444")
- Output:
1.846439
Modula-2
MODULE Entropy;
FROM InOut IMPORT WriteString, WriteLn;
FROM RealInOut IMPORT WriteReal;
FROM Strings IMPORT Length;
FROM MathLib IMPORT ln;
PROCEDURE entropy(s: ARRAY OF CHAR): REAL;
VAR freq: ARRAY [0..255] OF CARDINAL;
i, length: CARDINAL;
h, f: REAL;
BEGIN
(* the entropy of the empty string is zero *)
length := Length(s);
IF length = 0 THEN RETURN 0.0; END;
(* find the frequency of each character *)
FOR i := 0 TO 255 DO freq[i] := 0; END;
FOR i := 0 TO length-1 DO
INC(freq[ORD(s[i])]);
END;
(* calculate the component for each character *)
h := 0.0;
FOR i := 0 TO 255 DO
IF freq[i] # 0 THEN
f := FLOAT(freq[i]) / FLOAT(length);
h := h - f * (ln(f) / ln(2.0));
END;
END;
RETURN h;
END entropy;
BEGIN
WriteReal(entropy("1223334444"), 14);
WriteLn;
END Entropy.
- Output:
1.8464394E+00
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 Shannon''s 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
- 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
import tables, math
proc entropy(s: string): 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")
Nu
def 'str entropy' [] {
split chars | do {|len|
(uniq -c).count
| each { ($in | math log 2) * $in }
| math sum
| ($len | math log 2) - $in / $len
} ($in | length)
}
'1223334444' | str entropy- Output:
1.8464393446710154
Objeck
use Collection;
class Entropy {
function : native : GetShannonEntropy(result : String) ~ Float {
frequencies := IntMap->New();
each(i : result) {
c := result->Get(i);
if(frequencies->Has(c)) {
count := frequencies->Find(c)->As(IntHolder);
count->Set(count->Get() + 1);
}
else {
frequencies->Insert(c, IntHolder->New(1));
};
};
length := result->Size();
entropy := 0.0;
counts := frequencies->GetValues();
each(i : counts) {
count := counts->Get(i)->As(IntHolder)->Get();
freq := count->As(Float) / length;
entropy += freq * (freq->Log() / 2.0->Log());
};
return -1 * entropy;
}
function : Main(args : String[]) ~ Nil {
inputs := [
"1223334444",
"1223334444555555555",
"122333",
"1227774444",
"aaBBcccDDDD",
"1234567890abcdefghijklmnopqrstuvwxyz",
"Rosetta Code"];
each(i : inputs) {
input := inputs[i];
"Shannon entropy of '{$input}': "->Print();
GetShannonEntropy(inputs[i])->PrintLine();
};
}
}- Output:
Shannon entropy of '1223334444': 1.84644 Shannon entropy of '1223334444555555555': 1.96981 Shannon entropy of '122333': 1.45915 Shannon entropy of '1227774444': 1.84644 Shannon entropy of 'aaBBcccDDDD': 1.93626 Shannon entropy of '1234567890abcdefghijklmnopqrstuvwxyz': 5.16993 Shannon entropy of 'Rosetta Code': 3.08496
OCaml
- By using a map, purely functional
module CharMap = Map.Make(Char)
let entropy s =
let count map c =
CharMap.update c (function Some n -> Some (n +. 1.) | None -> Some 1.) map
and calc _ n sum =
sum +. n *. Float.log2 n
in
let sum = CharMap.fold calc (String.fold_left count CharMap.empty s) 0.
and len = float (String.length s) in
Float.log2 len -. sum /. len
let () =
entropy "1223334444" |> string_of_float |> print_endline
- By 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
- Output:
1.84643934467
Oforth
: entropy(s) -- f
| freq sz |
s size dup ifZero: [ return ] asFloat ->sz
ListBuffer initValue(255, 0) ->freq
s apply( #[ dup freq at 1+ freq put ] )
0.0 freq applyIf( #[ 0 <> ], #[ sz / dup ln * - ] ) Ln2 / ;
entropy("1223334444") .- Output:
1.84643934467102
ooRexx
/* REXX */
Numeric Digits 16
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*rxcalclog(p.c)/rxcalclog(2)
End
Say s 'Entropy' format(-e,,12)
Exit
::requires 'rxmath' LIBRARY
- Output:
1223334444 Entropy 1.846439344671
PARI/GP
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)>entropy("1223334444")
%1 = 1.8464393446710154934341977463050452232
Pascal
Free Pascal (http://freepascal.org).
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.
- Output:
Entropy of "1223334444" is 1.84644 bits.
PascalABC.NET
function entropy(s: string): real;
begin
result := 0.0;
foreach var c in s.distinct do
begin
var freq := s.count(x -> x = c) / s.length;
result += -freq * log2(freq);
end;
end;
begin
entropy('1223334444').println;
end
- Output:
1.84643934467102
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";
Phix
with javascript_semantics function entropy(sequence s) sequence symbols = {}, counts = {} integer N = length(s) for i=1 to N do object si = s[i] integer k = find(si,symbols) if k=0 then symbols = append(symbols,si) counts = append(counts,1) else counts[k] += 1 end if end for atom H = 0 integer n = length(counts) for i=1 to n do atom ci = counts[i]/N H -= ci*log2(ci) end for return H end function ?entropy("1223334444")
- Output:
1.846439345
PHP
<?php
function shannonEntropy($string) {
$h = 0.0;
$len = strlen($string);
foreach (count_chars($string, 1) as $count) {
$h -= (double) ($count / $len) * log((double) ($count / $len), 2);
}
return $h;
}
$strings = array(
'1223334444',
'1225554444',
'aaBBcccDDDD',
'122333444455555',
'Rosetta Code',
'1234567890abcdefghijklmnopqrstuvwxyz',
);
foreach ($strings AS $string) {
printf(
'%36s : %s' . PHP_EOL,
$string,
number_format(shannonEntropy($string), 6)
);
}
- Output:
1223334444 : 1.846439
1225554444 : 1.846439
aaBBcccDDDD : 1.936260
122333444455555 : 2.149255
Rosetta Code : 3.084963
1234567890abcdefghijklmnopqrstuvwxyz : 5.169925
Picat
go =>
["1223334444",
"Rosetta Code is the best site in the world!",
"1234567890abcdefghijklmnopqrstuvwxyz",
"Picat is fun"].map(entropy).println(),
nl.
% probabilities of each element/character in L
entropy(L) = Entropy =>
Len = L.length,
Occ = new_map(), % # of occurrences
foreach(E in L)
Occ.put(E, Occ.get(E,0) + 1)
end,
Entropy = -sum([P2*log2(P2) : _C=P in Occ, P2 = P/Len]).- Output:
[1.846439344671016,3.646513010214172,5.169925001442309,3.251629167387823]
PicoLisp
PicoLisp only supports fixed point arithmetic, but it does have the ability to call libc transcendental functions (for log)
(scl 8)
(load "@lib/math.l")
(setq LN2 0.693147180559945309417)
(de tabulate-chars (Str)
(let Map NIL
(for Ch (chop Str)
(if (assoc Ch Map)
(con @ (inc (cdr @)))
(setq Map (cons (cons Ch 1) Map))))
Map))
(de entropy (Str)
(let (
Sz (length Str)
Hist (tabulate-chars Str)
)
(*/
(sum
'((Pair)
(let R (*/ (cdr Pair) 1. Sz)
(- (*/ R (log R) 1.))))
Hist)
1. LN2)))- Output:
: (format (entropy "1223334444") *Scl) -> "1.84643934"
PL/I
*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;- Output:
s='1223334444' Entropy= 1.846439344671
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"
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)
% Sort Run-length encoded list and aggregate lengths by element
,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)
% Sum the list of lengths of runs of E
,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 F consecutive occurrences of C
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 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.
Example query:
?- shannon_entropy(1223334444, H). H = 1.8464393446710154.
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))
Next
OpenConsole()
Print("Entropy of ["+#TESTSTR+"] = "+StrD(e,15))
Input()
- Output:
Entropy of [1223334444] = 1.846439344671015
Python
Python: Longer version
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)
- 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.
from math import log2
from collections import Counter
def entropy(s):
p, lns = Counter(s), float(len(s))
return log2(lns) - sum(count * log2(count) for count in p.values()) / lns
print(entropy("1223334444"))
- Output:
1.8464393446710154
Uses Python 2
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('>>>'))
R
entropy <- function(str) {
vec <- strsplit(str, "")[[1]]
N <- length(vec)
p_xi <- table(vec) / N
-sum(p_xi * log(p_xi, 2))
}
- Output:
> entropy("1223334444")
[1] 1.846439
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"))
- Output:
1.8464393446710154
Raku
(formerly Perl 6)
sub entropy(@a) {
[+] map -> \p { p * -log p }, bag(@a).values »/» +@a;
}
say log(2) R/ entropy '1223334444'.comb;
- Output:
1.84643934467102
In case we would like to add this function to Raku's core, here is one way it could be done:
use MONKEY-TYPING;
augment class Bag {
method entropy {
[+] map -> \p { - p * log p },
self.values »/» +self;
}
}
say '1223334444'.comb.Bag.entropy / log 2;
REXX
version 1
/* 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
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.
/*REXX program calculates the information entropy for a specified character string. */
numeric digits length( e() ) % 2 - length(.) /*use 1/2 of the decimal digits of E. */
parse arg $; if $='' then $= 1223334444 /*obtain the optional input from the CL*/
#=0; @.= 0; L= length($) /*define handy-dandy REXX variables. */
$$= /*initialize the $$ list. */
do j=1 for L; _= substr($, j, 1) /*process each character in $ string.*/
if @._==0 then do; #= # + 1 /*Unique? Yes, bump character counter.*/
$$= $$ || _ /*add this character to the $$ 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. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
e: e= 2.718281828459045235360287471352662497757247093699959574966967627724076630; return e
/*──────────────────────────────────────────────────────────────────────────────────────*/
log2: procedure; parse arg x 1 ox; ig= x>1.5; ii= 0; is= 1 - 2 * (ig\==1)
numeric digits digits()+5; call e /*the precision of E must be≥digits(). */
do while ig & ox>1.5 | \ig&ox<.5; _= e; do j=-1; iz= ox * _ ** -is
if j>=0 & (ig & iz<1 | \ig&iz>.5) then leave; _= _ * _; izz= iz; end /*j*/
ox=izz; ii=ii+is*2**j; end /*while*/; 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, .)
- 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.
Ring
decimals(8)
entropy = 0
countOfChar = list(255)
source="1223334444"
charCount =len( source)
usedChar =""
for i =1 to len( source)
ch =substr(source, i, 1)
if not(substr( usedChar, ch)) usedChar =usedChar +ch ok
j =substr( usedChar, ch)
countOfChar[j] =countOfChar[j] +1
next
l =len(usedChar)
for i =1 to l
probability =countOfChar[i] /charCount
entropy =entropy - (probability *logBase(probability, 2))
next
see "Characters used and the number of occurrences of each " + nl
for i =1 to l
see "'" + substr(usedChar, i, 1) + "' " + countOfChar[i] + nl
next
see " Entropy of " + source + " is " + entropy + " bits." + nl
see " The result should be around 1.84644 bits." + nl
func logBase (x, b)
logBase =log( x) /log( 2)
return logBaseOutput:
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.
RPL
| Code | Comments |
|---|---|
≪
DUP SIZE 2 LN → str len log2
≪ { 255 } 0 CON
1 len FOR j
str j DUP SUB
NUM DUP2 GET 1 + PUT
NEXT
0 1 255 FOR j
IF OVER j GET
THEN LAST len / DUP LN log2 / * + END
NEXT
NEG SWAP DROP
≫ ≫ 'NTROP' STO
|
NTROP ( "string" -- entropy )
Initialize local variables
Initialize a vector with 255 counters
For each character in the string...
... increase the counter according to ASCII code
For each non-zero counter
calculate term
Change sign and forget the vector
|
The following line of code delivers what is required:
"1223334444" NTROP
- Output:
1: 1.84643934467
Ruby
def entropy(s)
counts = s.chars.tally
leng = s.length.to_f
counts.values.reduce(0) do |entropy, count|
freq = count / leng
entropy - freq * Math.log2(freq)
end
end
p entropy("1223334444")
- Output:
1.8464393446710154
Run BASIC
dim chrCnt( 255) ' possible ASCII chars
source$ = "1223334444"
numChar = len(source$)
for i = 1 to len(source$) ' count which chars are used in source
ch$ = mid$(source$,i,1)
if not( instr(chrUsed$, ch$)) then chrUsed$ = chrUsed$ + ch$
j = instr(chrUsed$, ch$)
chrCnt(j) =chrCnt(j) +1
next i
lc = len(chrUsed$)
for i = 1 to lc
odds = chrCnt(i) /numChar
entropy = entropy - (odds * (log(odds) / log(2)))
next i
print " Characters used and times used of each "
for i = 1 to lc
print " '"; mid$(chrUsed$,i,1); "'";chr$(9);chrCnt(i)
next i
print " Entropy of '"; source$; "' is "; entropy; " bits."
endCharacters used and times used of each '1' 1 '2' 2 '3' 3 '4' 4 Entropy of '1223334444' is 1.84643939 bits.
Rust
fn entropy(s: &[u8]) -> f32 {
let mut histogram = [0u64; 256];
for &b in s {
histogram[b as usize] += 1;
}
histogram
.iter()
.cloned()
.filter(|&h| h != 0)
.map(|h| h as f32 / s.len() as f32)
.map(|ratio| -ratio * ratio.log2())
.sum()
}
fn main() {
let arg = std::env::args().nth(1).expect("Need a string.");
println!("Entropy of {} is {}.", arg, entropy(arg.as_bytes()));
}
- Output:
$ ./entropy 1223334444 Entropy of 1223334444 is 1.8464394.
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 )
scheme
A version capable of calculating multidimensional entropy.
(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)))
- Output:
1.8464393446710154 bits 1.4591479170272448 bits
Scilab
function E = entropy(d)
d=strsplit(d);
n=unique(string(d));
N=size(d,'r');
count=zeros(n);
n_size = size(n,'r');
for i = 1:n_size
count(i) = sum ( d == n(i) );
end
E=0;
for i=1:length(count)
E = E - count(i)/N * log(count(i)/N) / log(2);
end
endfunction
word ='1223334444';
E = entropy(word);
disp('The entropy of '+word+' is '+string(E)+'.');
- Output:
The entropy of 1223334444 is 1.8464393.
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;- Output:
1.84644
SETL
program shannon_entropy;
print(entropy "1223334444");
op entropy(symbols);
hist := {};
loop for symbol in symbols do
hist(symbol) +:= 1;
end loop;
h := 0.0;
loop for count = hist(symbol) do
f := count / #symbols;
h -:= f * log f / log 2;
end loop;
return h;
end op;
end program;- Output:
1.84643934467102
Sidef
func entropy(s) {
var counts = Hash.new;
s.each { |c| counts{c} := 0 ++ };
var len = s.len;
[0, counts.values.map {|count|
var freq = count/len; freq * freq.log2 }...
]«-»;
}
say entropy("1223334444");
- Output:
1.846439344671015493434197746305045223237
Standard ML
val Entropy = fn input =>
let
val N = Real.fromInt (String.size input) ;
val term = fn a => Math.ln (a/N) * a / ( Math.ln 2.0 * N ) ;
val v0 = Vector.tabulate (255,fn i=>0) ;
val freq = Vector.map Real.fromInt (* List.foldr: count occurrences *)
(List.foldr (fn (i,v) => Vector.update( v, ord i, Vector.sub(v,ord i) + 1) ) v0 (explode input) )
in
~ (Vector.foldr (fn (a,s) => if a > 0.0 then term a + s else s) 0.0 freq )
end ;Entropy "1223334444" ; val it = 1.846439345: real
Swift
import Foundation
func entropy(of x: String) -> Double {
return x
.reduce(into: [String: Int](), {cur, char in
cur[String(char), default: 0] += 1
})
.values
.map({i in Double(i) / Double(x.count) } as (Int) -> Double)
.map({p in -p * log2(p) } as (Double) -> Double)
.reduce(0.0, +)
}
print(entropy(of: "1223334444"))
- Output:
1.8464393446710154
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
}
Demonstration:
puts [format "entropy = %.5f" [entropy "1223334444"]]
puts [format "entropy = %.5f" [entropy "Rosetta Code"]]
- Output:
entropy = 1.84644 entropy = 3.08496
V (Vlang)
Vlang: Map version
import math
import arrays
fn hist(source string) map[string]int {
mut hist := map[string]int{}
for e in source.split('') {
if e !in hist {
hist[e] = 0
}
hist[e]+=1
}
return hist
}
fn entropy(hist map[string]int, l int) f64 {
mut elist := []f64{}
for _,v in hist {
c := f64(v) / f64(l)
elist << -c * math.log2(c)
}
return arrays.sum<f64>(elist) or {-1}
}
fn main(){
input := "1223334444"
h := hist(input)
e := entropy(h, input.len)
println(e)
}- Output:
1.8464393446710152
Wren
var s = "1223334444"
var m = {}
for (c in s) {
var d = m[c]
m[c] = (d) ? d + 1 : 1
}
var hm = 0
for (k in m.keys) {
var c = m[k]
hm = hm + c * c.log2
}
var l = s.count
System.print(l.log2 - hm/l)
- Output:
1.846439344671
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"))- Output:
1.84644
Zig
const std = @import("std");
const math = std.math;
pub fn main() !void {
const stdout = std.io.getStdOut().outStream();
try stdout.print("{d:.12}\n", .{H("1223334444")});
}
fn H(s: []const u8) f64 {
var counts = [_]u16{0} ** 256;
for (s) |ch|
counts[ch] += 1;
var h: f64 = 0;
for (counts) |c|
if (c != 0) {
const p = @intToFloat(f64, c) / @intToFloat(f64, s.len);
h -= p * math.log2(p);
};
return h;
}
- Output:
1.846439344671
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");- Output:
1.84644 bits
ZX Spectrum Basic
10 LET s$="1223334444": LET base=2: LET entropy=0
20 LET sourcelen=LEN s$
30 DIM t(255)
40 FOR i=1 TO sourcelen
50 LET number= CODE s$(i)
60 LET t(number)=t(number)+1
70 NEXT i
80 PRINT "Char";TAB (6);"Count"
90 FOR i=1 TO 255
100 IF t(i)<>0 THEN PRINT CHR$ i;TAB (6);t(i): LET prop=t(i)/sourcelen: LET entropy=entropy-(prop*(LN prop)/(LN base))
110 NEXT i
120 PRINT '"The Entropy of """;s$;""" is ";entropy
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- Tcl
- V (Vlang)
- Wren
- XPL0
- Zig
- Zkl
- ZX Spectrum Basic