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Introduction
 Minmax normalization is an operation which rescales a set of data.

This can be useful when:
 Comparing data from two different scales
 Converting data to a new scale

In most situations, data is normalized to a fit a target range of [0, 1]
 The smallest value in the original set would be mapped to 0
 The largest value in the original set would be mapped to 1
 Every other value would be mapped to a value somewhere between these two bounds

It is also called:
 Feature Scaling
 MinMax Scaling
 Rescaling
 Normalization
Normalizing to [0, 1]

A set of numbers will have:

A smallest value:
 This is also called the lower bound or least element

It is denoted:
min(x)

A largest value:
 This is also called the upper bound or greatest element

It is denoted:
max(x)

A range:
 The difference between the smallest and largest values

It is denoted:
max(x)  min(x)

A smallest value:
 Normalization is the process of changing the lower and upper bounds to be 0 and 1 respectively
Algorithm

First we modify the data to have a lower bound of 0. To do this we subtract
min(x)
from each value: 
Then we modify the data to have an upper bound of 1. We do this by dividing each value by the original range:

Finally, if we combine these two steps we get:
Example
Normalize the following data:

First we calculate the lower bound, upper bound, and range:
 min(x) = 7
 max(x) = 21
 max(x)  min(x) = 14

Next we subtract the lower bound from each value:

Finally, we divide by the range:
Code
import numpy as np def normalize(x): min = np.min(x) max = np.max(x) range = max  min return [(a  min) / range for a in x] x = [7, 21, 13, 15] normalizedX = normalize(x) print(normalizedX) # prints: [0.0, 1.0, 0.42857142857142855, 0.5714285714285714]
Normalizing from [0, 1]
 If our numbers are in the range [0, 1] then we can scale them to have a different lower and upper bound

To achieve this, we simply do the reverse of normalization:
 Find the new range by subtracting the lower bound from the upper bound
 Multiply each value by the new range
 and add the new lower bound to each value:
Example
Normalize the following data to have a lower bound of 3 and an upper bound of 24:

first we calculate the range:

Then we multiply each value by the range:

Finally, we add the lower bound to each value:
Code
def normalize(normalizedX, newLowerBound, newUpperBound): range = newUpperBound  newLowerBound return [a * range + newLowerBound for a in normalizedX] normalizedX = [0.0, 1.0, 3/7, 4/7] x = normalize(normalizedX, 3, 24) print(x) # prints: [3.0, 24.0, 12.0, 15.0]
Normalizing from one range to another
 Sometimes we need to normalize data in which neither the source range nor the target range is [0, 1]
 In these situations, we first normalize the data to range of [0, 1], and then normalize it again to the true target range.

These two steps can be combined:
Code
import numpy as np def normalize(x, newLowerBound, newUpperBound): min = np.min(x) max = np.max(x) range = max  min newRange = newUpperBound  newLowerBound return [((a  min) / range) * newRange + newLowerBound for a in x] x = [7, 21, 13, 15] y = normalize(x, 3, 24) print(y) # prints: [3.0, 24.0, 12.0, 15.0]