Scaling and normalizing data using logarithmic scale - php

Question

I want to create a rank system for users on my website. The ranks will be decided by a number of factors like how long they've been a member, how many posts they created etc. Every data item is also divided by a "weight" determined by me, so that it is more representative of the actual user activity - I don't want 1 post be as significant as 1 day as a member. After the weighing all stats are added together to a total.
Then, I have to normalize the totals so that they are assigned to the ranks which range from 1 to 20, since some members have just a few points of activity and some veteran members have thousands of points. I do this by normalizing the data and scaling it down to the 1-20 rank range with this function:
function normalize($userTotal, $minOriginalRange, $maxOriginalRange, $minNewRange, $maxNewRange){
return $minNewRange + ((($maxNewRange - $minNewRange) * ($originalValue - $minOriginalRange)) / ($maxOriginalRange - $minOriginalRange));
}
This is usually called like so:
normalize(getUserTotal(), 0, getHighestTotalOfAllMembers(), 1, 20);
And so I got this as a result, key is rank and value is number of members who would get that rank:
Array
(
[1] => 7418
[2] => 1918
[3] => 289
[4] => 102
[5] => 62
[6] => 28
[7] => 21
[8] => 14
[9] => 1
[10] => 8
[11] => 6
[12] => 5
[13] => 1
[14] => 1
[17] => 1
[20] => 1
)
As you can see there are tons of users who are ranked low and very few who get assigned the mid and high ranks. I'd like to fix this by calculating the rank assigned using a logarithmic scale, so that it is easy to climb the ranks in the lower tiers and gets harder and harder the higher you go. That way it should spread out more evenly and more users will have ranks in the middle.
I don't know how to approach this however, I have never used logarithmic scales and always resorted to simple arithmetic in my code.

Answer 1

You would use a php math logarithm function and map it over the final array, for example:
function logfunction($v){
return round(log1p($v),2);
}
$simple_array = [7418, 1918, 289, 102, 62, 28, 21, 14, 1, 8, 6, 5, 1, 1, 1];
$logarithmic_array = array_map(logfunction, $simple_array);
print_r($logarithmic_array);
In the above, I use the log1p() function whichreturns log(1 + number) computed in a way that is accurate even when the value of number is close to zero (See: http://php.net/manual/en/function.log1p.php). Then I round the result to 2 decimal places for the sake of readability. The resulting $logarithmic_array output is:
Array
(
[0] => 8.91
[1] => 7.56
[2] => 5.67
[3] => 4.63
[4] => 4.14
[5] => 3.37
[6] => 3.09
[7] => 2.71
[8] => 0.69
[9] => 2.2
[10] => 1.95
[11] => 1.79
[12] => 0.69
[13] => 0.69
[14] => 0.69
)