Transforming range of variable to 0 to 1 and custom ranges

Created:18 Jul 2020 00:39:07 , in  Maths

One useful skill when it comes to dealing with variables is to be able to change their range. One of the most useful and desirable ranges is [0,1]. Occasionally you might need a different range like [-1,1], too. This article describes a couple of methods of scaling and descaling a variable's range.

Quick and easy transformation to [0,1]

One way to scale range of variable x to range from 0 to 1 is based on the following function:

scaled variable x, called y, will be in the range between 0 and 1.

Scaling a to custom range

If a range from a to b, where a is the minimum and b is the maximum allowed value, is required, one of the three following formulas can be used:

1)

2)

where e is the Euler's number

3)

Scaling variable from [0,1] to a custom range

If a variable y is in range from 0 to 1, it can be scaled to a variable z which is in range between a and b as follows:

Min-max scaling

Min-max scaling, also known as min-max normalization, is another easy way, which is frequently used for transforming variable's range.

The formula for min-max scaling is as follows:

where a and b denote minimum and maximum values of x in a set of data respectively.

Descaling variable y back to x can be done using:

Min-max scaling in PHP

Two functions given below perform min-max scaling and descaling as described in the previous section.

function minMaxNormalize($x, $a, $b) {
  $y = false;
  try {
    $y = ($x - $a) / ($b - $a);
  }catch(DivisionByZeroError $e){ 
    echo $e -> getMessage(),"\n";
  }
  return $y;
}

function minMaxDenormalize($y, $a, $b) {
  $x = ($y * ($b - $a) + $a);
  return $x;
}

Here is how to use these functions

Suppose that the range of values of x is from -10 to 10, but we want it to be between 0 and 1. To transform the range, do the following:

minMaxNormalize(7, -10, 10);
// => 0.85

To transform variables to be back in the range [-10,10] :

minMaxDenormalize(0.85,-10,10);
// => 7

Conclusion

As can be seen in this article, there are quiet a few ways of transforming a range of a variable both to [0,1] or to a custom range. I hope you have found at least one of them sufficient for your purposes. If you have something to add, you are welcome to post a comment below.

This post was updated on 18 Jul 2020 01:30:45

Tags:  php