Mean Normalisation

data-analysis

Definition

Mean Normalisation

Mean normalisation is a data preprocessing technique used to rescale numerical features such that they have a zero mean and are bounded by a specific range. Formally, for a feature $j$ with values $x_j$, the transformed value $x_j^{\prime }$ is:

$$x_j^{\prime }=\frac{x_j-{\bar{x}}_j}{\max (x_j)-\min (x_j)}$$

where ${\bar{x}}_j$ is the arithmetic mean of the feature, i.e.:

$$\overline{x_j}=\frac{1}{|X|}\sum _{x\in X}{x}_j$$

Statistical Impact

This transformation centers the distribution around zero while maintaining the relative distances between instances. Similar to min-max scaling, it remains sensitive to outliers due to the use of range-based statistics in the denominator.

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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