Cumulative standard deviation: Difference between revisions

Content added Content deleted

Inline

Revision as of 14:58, 8 August 2009

Write a stateful function, class, generator or coroutine that takes a series of floating point numbers, one at a time, and returns the running standard deviation of the series. The task implementation should use the most natural programming style of those listed for the function in the implementation language; the task must state which is being used. Do not apply Bessel's correction; the returned standard deviation should always be computed as if the sample seen so far is the entire population.

Use this to compute the standard deviation of this demonstration set, $\{2,4,4,4,5,5,7,9\}$ , which is $2$ .

Ada

<lang ada> with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions; with Ada.Text_IO; use Ada.Text_IO;

procedure Test_Deviation is

  type Sample is record
     N       : Natural := 0;
     Mean    : Float := 0.0;
     Squares : Float := 0.0;
  end record;
  procedure Add (Data : in out Sample; Point : Float) is
  begin
     Data.N       := Data.N + 1;
     Data.Mean    := Data.Mean    + Point;
     Data.Squares := Data.Squares + Point ** 2;
  end Add;
  function Deviation (Data : Sample) return Float is
  begin
     return Sqrt (Data.Squares / Float (Data.N) - (Data.Mean / Float (Data.N)) ** 2);
  end Deviation;

  Data : Sample;
  Test : array (1..8) of Float := (2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0);

begin

  for Item in Test'Range loop
     Add (Data, Test (Item));
  end loop;
  Put_Line ("Deviation" & Float'Image (Deviation (Data)));

end Test_Deviation; </lang> Sample output:

Deviation 2.00000E+00

AutoHotkey

ahk forum: discussion <lang AutoHotkey>std(2),std(4),std(4),std(4),std(5),std(5),std(7) MsgBox % std(9) ; 2

std(x="") {

 Static sum:=0, sqr:=0, n:=0
 If (x="")                    ; blank parameter: reset
    sum := 0, sqr := 0, n := 0
 Else
    sum += x, sqr += x*x, n++ ; update state
 Return sqrt((sqr-sum*sum/n)/n)

}</lang>

C

Of course this function is not thread-safe nor it can be used to compute the standard deviation just for a set of values per time.

<lang c>#include <stdio.h>

include <stdlib.h>
include <math.h>

enum Action { VALUE, STDDEV, MEAN, VAR, COUNT, RESET }; double stat_object(double v, enum Action action) {

 static double sum = 0.0;
 static double sum2 = 0.0;
 static size_t num = 0;

 double m;

 switch(action) {
 case VALUE:
   num++;
   sum += v;
   sum2 += v*v;
   return stat_object(0.0, STDDEV);
 case STDDEV:
   return sqrt(stat_object(0.0, VAR));
 case MEAN:
   return (num>0) ? sum/(double)num : 0.0;
 case VAR:
   m = stat_object(0.0, MEAN);
   return (num>0) ? (sum2/(double)num - m*m) : 0.0;
 case COUNT:
   return num;
 case RESET:
   sum = sum2 = 0.0; num = 0;
   return 0.0;
 }

}</lang>

<lang c>double v[] = { 2,4,4,4,5,5,7,9 };

int main() {

 int i;
 double sd;

 for(i=0; i < sizeof(v)/sizeof(double) ; i++)
   sd = stat_object(v[i], VALUE);

 printf("standard dev = %lf\n", sd);

 return 0;

}</lang>

E

This implementation produces two (function) objects sharing state. It is idiomatic in E to separate input from output (read from write) rather than combining them into one object.

The algorithm is

Translation of: Perl

and the results were checked against #Python.

<lang e>def makeRunningStdDev() {

   var sum := 0.0
   var sumSquares := 0.0
   var count := 0.0
   
   def insert(v) {
       sum += v
       sumSquares += v ** 2
       count += 1
   }
   
   /** Returns the standard deviation of the inputs so far, or null if there
       have been no inputs. */
   def stddev() {
       if (count > 0) {
           def meanSquares := sumSquares/count
           def mean := sum/count
           def variance := meanSquares - mean**2
           return variance.sqrt()
       }
   }
   
   return [insert, stddev]

}</lang>

<lang e>? def [insert, stddev] := makeRunningStdDev()

value: <insert>, <stddev>

? [stddev()]

value: [null]

? for value in [2,4,4,4,5,5,7,9] { > insert(value) > println(stddev()) > } 0.0 1.0 0.9428090415820626 0.8660254037844386 0.9797958971132716 1.0 1.3997084244475297 2.0</lang>

Forth

<lang forth>: f+! ( x addr -- ) dup f@ f+ f! ;

st-count ( stats -- n ) f@ ;

st-sum ( stats -- sum ) float+ f@ ;

st-sumsq ( stats -- sum*sum ) 2 floats + f@ ;

st-add ( fnum stats -- )

   1e dup f+!  float+
 fdup dup f+!  float+
 fdup f*  f+! ;

st-mean ( stats -- mean )

 dup st-sum st-count f/ ;

st-variance ( stats -- var )

 dup st-sumsq
 dup st-mean fdup f* dup st-count f*  f-
 st-count f/ ;

st-stddev ( stats -- stddev )

 st-variance fsqrt ;</lang>

This variation is more numerically stable when there are large numbers of samples or large sample ranges. <lang forth>: st-count ( stats -- n ) f@ ;

st-mean ( stats -- mean ) float+ f@ ;

st-nvar ( stats -- n*var ) 2 floats + f@ ;

st-variance ( stats -- var ) dup st-nvar st-count f/ ;

st-stddev ( stats -- stddev ) st-variance fsqrt ;

st-add ( x stats -- )

 1e dup f+!			\ update count
 fdup dup st-mean f- fswap
 ( delta x )
 fover dup st-count f/
 ( delta x delta/n )
 float+ dup f+!		\ update mean
 ( delta x )
 dup f@ f-  f*  float+ f+! ;	\ update nvar</lang>

<lang forth>create stats 0e f, 0e f, 0e f,

2e stats st-add 4e stats st-add 4e stats st-add 4e stats st-add 5e stats st-add 5e stats st-add 7e stats st-add 9e stats st-add

stats st-stddev f. \ 2.</lang>

Fortran

Translation of: C

Works with: Fortran version 95 and later

This one imitates C and suffers the same problems: the function is not thread-safe and must be used to compute the stddev for one set per time.

<lang fortran>program Test_Stddev

 implicit none

 real, dimension(8) :: v = (/ 2,4,4,4,5,5,7,9 /)
 integer :: i
 real :: sd

 do i = 1, size(v)
    sd = stat_object(v(i))
 end do

 print *, "std dev = ", sd

contains

 recursive function stat_object(a, cmd) result(stddev)
   real :: stddev
   real, intent(in) :: a
   character(len=*), intent(in), optional :: cmd

   real, save :: summa = 0.0, summa2 = 0.0
   integer, save :: num = 0

   real :: m

   if ( .not. present(cmd) ) then
      num = num + 1
      summa = summa + a
      summa2 = summa2 + a*a
      stddev = stat_object(0.0, "stddev")
   else
      select case(cmd)
      case("stddev")
         stddev = sqrt(stat_object(0.0, "variance"))
      case("variance")
         m = stat_object(0.0, "mean")
         if ( num > 0 ) then
            stddev = summa2/real(num) - m*m
         else
            stddev = 0.0
         end if
      case("count")
         stddev = real(num)
      case("mean")
         if ( num > 0 ) then
            stddev = summa/real(num)
         else
            stddev = 0.0
         end if
      case("reset")
         summa = 0.0
         summa2 = 0.0
         num = 0
      case default
         stddev = 0.0
      end select
   end if

 end function stat_object

end program Test_Stddev</lang>

J

Translation of: R

J is block-oriented; it expresses algorithms with the semantics of all the data being available at once. It does not have native lexical closure or coroutine semantics. It is possible to implement these semantics in J; see Moving Average for an example. We will not reprise that here.

Instead, we will take a cue from the R implementation and reverse a the Bessel correction to stddev:

   require'stats'
   (%:@:(%~<:)@:# * stddev)\ 2 4 4 4 5 5 7 9
0 1 0.942809 0.866025 0.979796 1 1.39971 2

Java

Works with: Java version 1.5+

<lang java5>import java.util.LinkedList; public class StdDev {

   LinkedList<Double> nums;
   public StdDev() {
       nums = new LinkedList<Double>();
   }
   public static void main(String[] args) {
       double[] testData = {2,4,4,4,5,5,7,9};
       StdDev sd = new StdDev();
       
       for (double x : testData) {
           sd.newNum(x);
           System.out.println(sd.getSD());
       }
   }

   public void newNum(double num) {
       nums.add(num);
   }

   public double getAvg() {
       if (nums.isEmpty()) return 0;
       double ret = 0;
       double sum = 0;
       for (double num : nums) {
          sum += num;
       }
       return sum / nums.size();
   }

   public double getSD() {
       double sqDiffs = 0;
       double avg = getAvg();
       for (double num : nums) {
          sqDiffs += (num - avg) * (num - avg);
       }
       return Math.sqrt(sqDiffs / nums.size());
   }

}</lang>

Objective-C

<lang objc>#import <stdio.h>

import <math.h>
import <objc/Object.h>

@interface SDAccum : Object {

 double sum, sum2;
 unsigned int num;

} -(id)init; -(double)value: (double)v; -(unsigned int)count; -(double)mean; -(double)variance; -(double)stddev; @end

@implementation SDAccum -(id)init {

 sum = sum2 = 0.0;
 num = 0;
 return self;

} -(double)value: (double)v {

 sum = sum + v;
 sum2 = sum2 + v*v;
 num++;
 return [self stddev];

} -(unsigned int)count {

 return num;

} -(double)mean {

 return (num>0) ? sum/(double)num : 0.0;

} -(double)variance {

 double m = [self mean];
 return (num>0) ? (sum2/(double)num - m*m) : 0.0;

} -(double)stddev {

 return sqrt([self variance]);

} @end</lang>

<lang objc>double v[] = { 2,4,4,4,5,5,7,9 };

int main() {

 int i;
 double sd;

 id sdacc = [SDAccum new];

 for(i=0; i < sizeof(v)/sizeof(double) ; i++)
   sd = [sdacc value: v[i]];

 printf("std dev = %lf\n", sd);

 return 0;

}</lang>

Ocaml

let sqr x = x *. x

let stddev l =

 let n, sx, sx2 = List.fold_left (fun (n, sx, sx2) x -> succ n, sx +. x, sx2 +. sqr x) (0, 0., 0.) l in
 sqrt ((sx2 -. sqr sx /. float n) /. float n)

let _ =

 let l = [ 2.;4.;4.;4.;5.;5.;7.;9. ] in
 Printf.printf "Liste: ";
 List.iter (Printf.printf "%g  ") l;
 Printf.printf "\nStandard deviation: %g\n" (stddev l)

</lang>

Sample output:

Liste: 2  4  4  4  5  5  7  9
Standard deviation: 2

Perl

<lang perl>{

   package SDAccum;
   sub new {

my $class = shift; my $self = {}; $self->{sum} = 0.0; $self->{sum2} = 0.0; $self->{num} = 0; bless $self, $class; return $self;

   }
   sub count {

my $self = shift; return $self->{num};

   }
   sub mean {

my $self = shift; return ($self->{num}>0) ? $self->{sum}/$self->{num} : 0.0;

   }
   sub variance {

my $self = shift; my $m = $self->mean; return ($self->{num}>0) ? $self->{sum2}/$self->{num} - $m * $m : 0.0;

   }
   sub stddev {

my $self = shift; return sqrt($self->variance);

   }
   sub value {

my $self = shift; my $v = shift; $self->{sum} += $v; $self->{sum2} += $v * $v; $self->{num}++; return $self->stddev;

}</lang>

<lang perl>my $sdacc = SDAccum->new; my $sd;

foreach my $v ( 2,4,4,4,5,5,7,9 ) {

   $sd = $sdacc->value($v);

} print "std dev = $sd\n";</lang>

Python

Using a function with attached properties

The program should work with Python 2.x and 3.x, although the output would not be a tuple in 3.x <lang python>>>> from math import sqrt >>> def sd(x):

   sd.sum  += x
   sd.sum2 += x*x
   sd.n    += 1.0
   sum, sum2, n = sd.sum, sd.sum2, sd.n
   return sqrt(sum2/n - sum*sum/n/n)

>>> sd.sum = sd.sum2 = sd.n = 0 >>> for value in (2,4,4,4,5,5,7,9):

   print (value, sd(value))

(2, 0.0) (4, 1.0) (4, 0.94280904158206258) (4, 0.8660254037844386) (5, 0.97979589711327075) (5, 1.0) (7, 1.3997084244475311) (9, 2.0) >>> </lang>

Using a class instance

<lang python>>>> class SD(object): # Plain () for python 3.x def __init__(self): self.sum, self.sum2, self.n = (0,0,0) def sd(self, x): self.sum += x self.sum2 += x*x self.n += 1.0 sum, sum2, n = self.sum, self.sum2, self.n return sqrt(sum2/n - sum*sum/n/n)

>>> sd_inst = SD() >>> for value in (2,4,4,4,5,5,7,9): print (value, sd_inst.sd(value))</lang>

Using a Closure

Works with: Python version 3.x

<lang python>>>> from math import sqrt >>> def sdcreator(): sum = sum2 = n = 0 def sd(x): nonlocal sum, sum2, n

sum += x sum2 += x*x n += 1.0 return sqrt(sum2/n - sum*sum/n/n) return sd

>>> sd = sdcreator() >>> for value in (2,4,4,4,5,5,7,9): print (value, sd(value))

2 0.0 4 1.0 4 0.942809041582 4 0.866025403784 5 0.979795897113 5 1.0 7 1.39970842445 9 2.0</lang>

R

Built-in Std Dev fn

The built-in standard deviation function applies the Bessel correction. To reverse this, we can apply an uncorrection.
If na.rm is true, missing data points (NA values) are removed.

reverseBesselCorrection <- function(x, na.rm=FALSE)
{
  if(na.rm) x <- x[!is.na(x)]
  len <- length(x)
  if(len < 2) stop("2 or more data points required")
  sqrt((len-1)/len)
}
testdata <- c(2,4,4,4,5,5,7,9)
reverseBesselCorrection(testdata)*sd(testdata) #2

</lang>

From scratch

Again, if na.rm is true, missing data points (NA values) are removed.

uncorrectedsd <- function(x, na.rm=FALSE)
{
  len <- length(x)
  if(len < 2) stop("2 or more data points required")
  mu <- mean(x, na.rm=na.rm)
  ssq <- sum((x - mu)^2, na.rm=na.rm)
  usd <- sqrt(ssq/len)
  usd
}
uncorrectedsd(testdata) #2

</lang>

REXX

Works with: oorexx

This is indeed Object REXX.

<lang orexx>sdacc = .SDAccum~new x = .array~of(2,4,4,4,5,5,7,9) sd = 0 do i = 1 to x~size

  sd = sdacc~value(x[i])

end

say "std dev = "sd

class SDAccum

method sum attribute

method sum2 attribute

method count attribute

method init

 self~sum = 0.0
 self~sum2 = 0.0
 self~count = 0

method value

 expose sum sum2 count
 parse arg x
 sum = sum + x
 sum2 = sum2 + x*x
 count = count + 1
 return self~stddev

method mean

 expose sum count
 return sum/count

method variance

 expose sum2  count
 m = self~mean
 return sum2/count - m*m

method stddev

 return self~sqrt(self~variance)

method sqrt

 arg n
 if n = 0 then return 0
 ans = n / 2
 prev = n
 do until prev = ans
   prev = ans
   ans = ( prev + ( n / prev ) ) / 2
 end
 return ans</lang>

Ruby

Object

Uses an object to keep state.

"Simplification of the formula [...] for standard deviation [...] can be memorized as taking the square root of (the average of the squares less the square of the average)." [1]

<lang ruby>class StdDevAccumulator

 def initialize
   @n, @sum, @sumofsquares = 0, 0.0, 0.0
 end
 
 def <<(num)
   # return self to make this possible:  sd << 1 << 2 << 3 # => 0.816496580927726
   @n += 1
   @sum += num
   @sumofsquares += num**2
   self
 end
 
 def stddev
   Math.sqrt( (@sumofsquares / @n) - (@sum / @n)**2 )
 end
 
 def to_s
   stddev.to_s
 end

end

sd = StdDevAccumulator.new i = 0 [2,4,4,4,5,5,7,9].each {|n| puts "adding #{n}: stddev of #{i+=1} samples is #{sd << n}" }</lang>

adding 2: stddev of 1 samples is 0.0
adding 4: stddev of 2 samples is 1.0
adding 4: stddev of 3 samples is 0.942809041582063
adding 4: stddev of 4 samples is 0.866025403784439
adding 5: stddev of 5 samples is 0.979795897113272
adding 5: stddev of 6 samples is 1.0
adding 7: stddev of 7 samples is 1.39970842444753
adding 9: stddev of 8 samples is 2.0

Closure

<lang ruby>def sdaccum

 n, sum, sum2 = 0, 0.0, 0.0
 lambda do |num|
   n += 1
   sum += num
   sum2 += num**2
   Math.sqrt( (sum2 / n) - (sum / n)**2 )
 end

end

sd = sdaccum [2,4,4,4,5,5,7,9].each {|n| print sd.call(n), ", "}</lang>

0.0, 1.0, 0.942809041582063, 0.866025403784439, 0.979795897113272, 1.0, 1.39970842444753, 2.0,

Smalltalk

Works with: GNU Smalltalk

<lang smalltalk>Object subclass: SDAccum [

   |sum sum2 num|
   SDAccum class >> new [  |o| 
       o := super basicNew.
       ^ o init.
   ]
   init [ sum := 0. sum2 := 0. num := 0 ]
   value: aValue [ 
     sum := sum + aValue.
     sum2 := sum2 + ( aValue * aValue ).
     num := num + 1.
     ^ self stddev
   ]
   count [ ^ num ]
   mean [ num>0 ifTrue: [^ sum / num] ifFalse: [ ^ 0.0 ] ]
   variance [ |m| m := self mean.
              num>0 ifTrue: [^ (sum2/num) - (m*m) ] ifFalse: [ ^ 0.0 ]
            ]
   stddev [ ^ (self variance) sqrt ]

].</lang>

<lang smalltalk>|sdacc sd| sdacc := SDAccum new.

( 2 4 4 4 5 5 7 9 ) do: [ :v | sd := sdacc value: v ].

('std dev = %1' % { sd }) displayNl.</lang>

Tcl

With a Class

Works with: Tcl version 8.6

<lang tcl>oo::class create SDAccum {

   variable sum sum2 num
   constructor {} {
       set sum 0.0
       set sum2 0.0
       set num 0
   }
   method value {x} {
       set sum2 [expr {$sum2 + $x**2}]
       set sum [expr {$sum + $x}]
       incr num
       return [my stddev]
   }
   method count {} {
       return $num
   }
   method mean {} {
       expr {$sum / $num}
   }
   method variance {} {
       expr {$sum2/$num - [my mean]**2}
   }
   method stddev {} {
       expr {sqrt([my variance])}
   }

}

Demonstration

set sdacc [SDAccum new] foreach val {2 4 4 4 5 5 7 9} {

   set sd [$sdacc value $val]

} puts "the standard deviation is: $sd"</lang> which produces the output:

the standard deviation is: 2.0

With a Coroutine

Works with: Tcl version 8.6

<lang tcl># Make a coroutine out of a lambda application coroutine sd apply {{} {

   set sum 0.0
   set sum2 0.0
   set sd {}
   # Keep processing argument values until told not to...
   while {[set val [yield $sd]] ne "stop"} {
       incr n
       set sum [expr {$sum + $val}]
       set sum2 [expr {$sum2 + $val**2}]
       set sd [expr {sqrt($sum2/$n - ($sum/$n)**2)}]
   }

}}

Demonstration

foreach val {2 4 4 4 5 5 7 9} {

   set sd [sd $val]

} sd stop puts "the standard deviation is: $sd"</lang>