Acme-Tools
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Percentile 50 is the same as the I<median>, percentile 25 is the first
quartile, 75 is the third quartile.
B<Input:>
First argument is your wanted percentile, or a refrence to a list of percentiles you want from the dataset.
If the first argument to percentile() is a scalar, this percentile is returned.
If the first argument is a reference to an array, then all those percentiles are returned as an array.
Second, third, fourth and so on argument are the numbers from which you want to find the percentile(s).
B<Examples:>
This finds the 50-percentile (the median) to the four numbers 1, 2, 3 and 4:
print "Median = " . percentile(50, 1,2,3,4); # 2.5
This:
@data=(11, 5, 3, 5, 7, 3, 1, 17, 4, 2, 6, 4, 12, 9, 0, 5);
@p = map percentile($_,@data), (25, 50, 75);
Is the same as this:
@p = percentile([25, 50, 75], @data);
But the latter is faster, especially if @data is large since it sorts
the numbers only once internally.
B<Example:>
Data: 1, 4, 6, 7, 8, 9, 22, 24, 39, 49, 555, 992
Average (or mean) is 143
Median is 15.5 (which is the average of 9 and 22 who both equally lays in the middle)
The 25-percentile is 6.25 which are between 6 and 7, but closer to 6.
The 75-percentile is 46.5, which are between 39 and 49 but close to 49.
Linear interpolation is used to find the 25- and 75-percentile and any
other x-percentile which doesn't fall exactly on one of the numbers in
the set.
B<Interpolation:>
As you saw, 6.25 are closer to 6 than to 7 because 25% along the set of
the twelve numbers is closer to the third number (6) than to he fourth
(7). The median (50-percentile) is also really interpolated, but it is
always in the middle of the two center numbers if there are an even count
of numbers.
However, there is two methods of interpolation:
Example, we have only three numbers: 5, 6 and 7.
Method 1: The most common is to say that 5 and 7 lays on the 25- and
75-percentile. This method is used in Acme::Tools.
Method 2: In Oracle databases the least and greatest numbers
always lay on the 0- and 100-percentile.
As an argument on why Oracles (and others?) definition is not the best way is to
look at your data as for instance temperature measurements. If you
place the highest temperature on the 100-percentile you are sort of
saying that there can never be a higher temperatures in future measurements.
A quick non-exhaustive Google survey suggests that method 1 here is most used.
The larger the data sets, the less difference there is between the two methods.
B<Extrapolation:>
In method one, when you want a percentile outside of any possible
interpolation, you use the smallest and second smallest to extrapolate
from. For instance in the data set C<5, 6, 7>, if you want an
x-percentile of x < 25, this is below 5.
If you feel tempted to go below 0 or above 100, C<percentile()> will
I<die> (or I<croak> to be more precise)
Another method could be to use "soft curves" instead of "straight
lines" in interpolation. Maybe B-splines or Bezier curves. This is not
used here.
For large sets of data Hoares algorithm would be faster than the
simple straightforward implementation used in C<percentile()>
here. Hoares don't sort all the numbers fully.
B<Differences between the two main methods described above:>
Data: 1, 4, 6, 7, 8, 9, 22, 24, 39, 49, 555, 992
Percentile Method 1 Method 2
(Acme::Tools::percentile (Oracle)
and others)
------------- ----------------------------- ---------
0 -2 1
1 -1.61 1.33
25 6.25 6.75
50 (median) 15.5 15.5
75 46.5 41.5
99 1372.19 943.93
100 1429 992
Found like this:
perl -MAcme::Tools -le 'print for percentile([0,1,25,50,75,99,100], 1,4,6,7,8,9,22,24,39,49,555,992)'
And like this in Oracle-databases:
select
percentile_cont(0.00) within group(order by n) per0,
percentile_cont(0.01) within group(order by n) per1,
percentile_cont(0.25) within group(order by n) per25,
percentile_cont(0.50) within group(order by n) per50,
percentile_cont(0.75) within group(order by n) per75,
Two persons: two ways (they can swap places)
Three persons: 6
Four persons: 24
Five persons: 120
Six persons: 720
The formula is C<x!> where the postfix unary operator C<!>, also known as I<faculty> is defined as:
C<x! = x * (x-1) * (x-2) ... * 1>. Example: C<5! = 5 * 4 * 3 * 2 * 1 = 120>.Run this to see the 100 first C<< n! >>
perl -MAcme::Tools -le'$i=big(1);print "$_!=",$i*=$_ for 1..100'
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
.
.
.
100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
C<permutations()> takes a list and return a list of arrayrefs for each
of the permutations of the input list:
permutations('a','b'); #returns (['a','b'],['b','a'])
permutations('a','b','c'); #returns (['a','b','c'],['a','c','b'],
# ['b','a','c'],['b','c','a'],
# ['c','a','b'],['c','b','a'])
Up to five input arguments C<permutations()> is probably as fast as it
can be in this pure perl implementation (see source). For more than
five, it could be faster. How fast is it now: Running with different
n, this many time took that many seconds:
n times seconds
-- ------- ---------
2 100000 0.32
3 10000 0.09
4 10000 0.33
5 1000 0.18
6 100 0.27
7 10 0.21
8 1 0.17
9 1 1.63
10 1 17.00
If the first argument is a coderef, that sub will be called for each permutation and the return from those calls with be the real return from C<permutations()>. For example this:
print for permutations(sub{join"",@_},1..3);
...will print the same as:
print for map join("",@$_), permutations(1..3);
...but the first of those two uses less RAM if 3 has been say 9.
Changing 3 with 10, and many computers hasn't enough memory
for the latter.
The examples prints:
123
132
213
231
312
321
If you just want to say calculate something on each permutation,
but is not interested in the list of them, you just don't
take the return. That is:
my $ant;
permutations(sub{$ant++ if $_[-1]>=$_[0]*2},1..9);
...is the same as:
$$_[-1]>=$$_[0]*2 and $ant++ for permutations(1..9);
...but the first uses next to nothing of memory compared to the latter. They have about the same speed.
(The examples just counts the permutations where the last number is at least twice as large as the first)
C<permutations()> was created to find all combinations of a persons
name. This is useful in "fuzzy" name searches with
L<String::Similarity> if you can not be certain what is first, middle
and last names. In foreign or unfamiliar names it can be difficult to
know that.
=cut
#TODO: see t/test_perm.pl and t/test_perm2.pl
sub permutations {
my $code=ref($_[0]) eq 'CODE' ? shift() : undef;
$code and @_<6 and return map &$code(@$_),permutations(@_);
return [@_] if @_<2;
return ([@_[0,1]],[@_[1,0]]) if @_==2;
return ([@_[0,1,2]],[@_[0,2,1]],[@_[1,0,2]],
[@_[1,2,0]],[@_[2,0,1]],[@_[2,1,0]]) if @_==3;
return ([@_[0,1,2,3]],[@_[0,1,3,2]],[@_[0,2,1,3]],[@_[0,2,3,1]],
[@_[0,3,1,2]],[@_[0,3,2,1]],[@_[1,0,2,3]],[@_[1,0,3,2]],
[@_[1,2,0,3]],[@_[1,2,3,0]],[@_[1,3,0,2]],[@_[1,3,2,0]],
[@_[2,0,1,3]],[@_[2,0,3,1]],[@_[2,1,0,3]],[@_[2,1,3,0]],
[@_[2,3,0,1]],[@_[2,3,1,0]],[@_[3,0,1,2]],[@_[3,0,2,1]],
[@_[3,1,0,2]],[@_[3,1,2,0]],[@_[3,2,0,1]],[@_[3,2,1,0]]) if @_==4;
return ([@_[0,1,2,3,4]],[@_[0,1,2,4,3]],[@_[0,1,3,2,4]],[@_[0,1,3,4,2]],[@_[0,1,4,2,3]],
[@_[0,1,4,3,2]],[@_[0,2,1,3,4]],[@_[0,2,1,4,3]],[@_[0,2,3,1,4]],[@_[0,2,3,4,1]],
[@_[0,2,4,1,3]],[@_[0,2,4,3,1]],[@_[0,3,1,2,4]],[@_[0,3,1,4,2]],[@_[0,3,2,1,4]],
[@_[0,3,2,4,1]],[@_[0,3,4,1,2]],[@_[0,3,4,2,1]],[@_[0,4,1,2,3]],[@_[0,4,1,3,2]],
[@_[0,4,2,1,3]],[@_[0,4,2,3,1]],[@_[0,4,3,1,2]],[@_[0,4,3,2,1]],[@_[1,0,2,3,4]],
[@_[1,0,2,4,3]],[@_[1,0,3,2,4]],[@_[1,0,3,4,2]],[@_[1,0,4,2,3]],[@_[1,0,4,3,2]],
[@_[1,2,0,3,4]],[@_[1,2,0,4,3]],[@_[1,2,3,0,4]],[@_[1,2,3,4,0]],[@_[1,2,4,0,3]],
[@_[1,2,4,3,0]],[@_[1,3,0,2,4]],[@_[1,3,0,4,2]],[@_[1,3,2,0,4]],[@_[1,3,2,4,0]],
[@_[1,3,4,0,2]],[@_[1,3,4,2,0]],[@_[1,4,0,2,3]],[@_[1,4,0,3,2]],[@_[1,4,2,0,3]],
[@_[1,4,2,3,0]],[@_[1,4,3,0,2]],[@_[1,4,3,2,0]],[@_[2,0,1,3,4]],[@_[2,0,1,4,3]],
[@_[2,0,3,1,4]],[@_[2,0,3,4,1]],[@_[2,0,4,1,3]],[@_[2,0,4,3,1]],[@_[2,1,0,3,4]],
[@_[2,1,0,4,3]],[@_[2,1,3,0,4]],[@_[2,1,3,4,0]],[@_[2,1,4,0,3]],[@_[2,1,4,3,0]],
[@_[2,3,0,1,4]],[@_[2,3,0,4,1]],[@_[2,3,1,0,4]],[@_[2,3,1,4,0]],[@_[2,3,4,0,1]],
[@_[2,3,4,1,0]],[@_[2,4,0,1,3]],[@_[2,4,0,3,1]],[@_[2,4,1,0,3]],[@_[2,4,1,3,0]],
[@_[2,4,3,0,1]],[@_[2,4,3,1,0]],[@_[3,0,1,2,4]],[@_[3,0,1,4,2]],[@_[3,0,2,1,4]],
[@_[3,0,2,4,1]],[@_[3,0,4,1,2]],[@_[3,0,4,2,1]],[@_[3,1,0,2,4]],[@_[3,1,0,4,2]],
[@_[3,1,2,0,4]],[@_[3,1,2,4,0]],[@_[3,1,4,0,2]],[@_[3,1,4,2,0]],[@_[3,2,0,1,4]],
[@_[3,2,0,4,1]],[@_[3,2,1,0,4]],[@_[3,2,1,4,0]],[@_[3,2,4,0,1]],[@_[3,2,4,1,0]],
[@_[3,4,0,1,2]],[@_[3,4,0,2,1]],[@_[3,4,1,0,2]],[@_[3,4,1,2,0]],[@_[3,4,2,0,1]],
( run in 0.950 second using v1.01-cache-2.11-cpan-b85c58fdc1d )