Normalization › Mathematical Foundation

For a feature $j$ with $n$ observations, min-max normalization transforms each value $x_{ij}$ to $x'_{ij}$ using:

$$x'_{ij} = \frac{x_{ij} - \min(x_j)}{\max(x_j) - \min(x_j)}$$

where:

• $x'_{ij}$ is the normalized value of feature $j$ for observation $i$
• $\min(x_j)$ is the minimum value of feature $j$
• $\max(x_j)$ is the maximum value of feature $j$

After normalization, each feature in your dataset is transformed so that it has a minimum value of 0 and a maximum value of 1. This transformation ensures that all features, regardless of their original range or units, are directly comparable and contribute equally to the analysis. In practical terms, this means that no single feature will dominate the learning process simply because it has a larger numeric range. Instead, every feature is scaled to the same [0, 1] interval, allowing algorithms.especially those sensitive to feature magnitude, such as neural networks or distance-based methods.to perform optimally and fairly.

• Minimum: $\min(x'_j) = 0$ for all features
• Maximum: $\max(x'_j) = 1$ for all features
• Range: $\text{Range}(x'_j) = 1$ for all features

This ensures that all features contribute equally to distance calculations and optimization processes.