Min-Max Scaling

data-analysis

Definition

Min-Max Scaling

Min-Max-Scaling is a data preprocessing method used to rescale numerical data to a fixed range, e.g. $[0,1]$ or $[a,b]$.

Let $X$ be the data space. Each value of a data point $x\in X$ at dimension $j$, i.e., $x_j$, is scaled such that it fits into the specific range of $[\text{min},\text{max}]$.

For range $[0,1]$:

$$x_j\leftarrow x_j\frac{x_j-{\min }_{x\in X}(x_j)}{{\max }_{x\in x}(x_j)-{\min }_{x\in X}(x_j)}$$

For range $[a,b]$:

$$x_j\leftarrow a+\frac{\left[x_j-{\min }_{x\in X}(x_j)\right](b-a)}{{\max }_{x\in X}(x_j)-{\min }_{x\in X}(x_j)}$$

Transformation Characteristics

While min-max scaling preserves the relative distances between values, it is highly sensitive to outliers, as a single extreme value can significantly influence the $\min$ and $\max$ statistics, thereby compressing the majority of the data into a very narrow range.

Definition

Independent Feature Scaling

Independent feature scaling is the principle that statistical scaling operations must be applied individually to each feature dimension of a dataset. Formally, for a dataset $D=\{{\mathbf{x}}^{(i)}{\}}_{i=1}^m$ with ${\mathbf{x}}^{(i)}\in {\mathbb{R}}^d$, the scaling parameters ${\theta }_j$ (e.g., ${\min }_j,{\mu }_j$) are computed for each dimension $j\in \{1,\dots ,d\}$ independently:

${\theta }_j=f(\{x_j^{(i)}{\}}_{i=1}^m)$

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