Standardisation

data-analysis

Definition

Standardisation

Standardisation (or z-score normalisation) is a data preprocessing technique that transforms numerical features to have a mean of zero and a standard deviation of one. Formally, for a feature $j$ with values $x_j$, the transformed value $x_j^{\prime }$ is:

$$x_j^{\prime }=\frac{x_j-{\bar{x}}_j}{{\sigma }_j}$$

where ${\bar{x}}_j$ is the arithmetic mean and ${\sigma }_j$ is the standard deviation.

Robustness and Assumptions

Unlike min-max scaling, standardisation does not bound values to a fixed range, which makes it more robust to outliers. It is particularly suitable for algorithms that assume a Gaussian distribution of input features, such as linear regression or support vector machines.

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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