Zero-Mean Scaling

Global Algorithm - Multi-Dimensional algorithm

Zero-Mean Scaling algorithm is a algorithm of normalization which uses the mean and standard deviation of the data set to normalize each data point. The basic formulas are stated as follows:

\[\hat{Y}_i = \frac{(Y_i - \mu)}{\sigma}\]

for a one-dimensional data vector,

\[\hat{Y}_{i,j} = \frac{(Y_{i,j} - \mu)}{\sigma}\]

for a two-dimensional data matrix,

\[\hat{Y}_{i,j,k} = \frac{(Y_{i,j,k} - \mu)}{\sigma}\]

for a three-dimensional data matrix, where \(Y\) represents the input data and \(i, j, k\) represent the corresponding indices for the data entry considered. \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the input data.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
\(Y\)	\(Y \in \mathbb R^{N_1}, \mathbb R^{N_1 \times N_2}, \text{ or } \mathbb R^{N_1 \times N_2 \times N_3}, \ldots\)	\(N_1, N_2, N_3 \in \mathbb{N}\)
\(\mu\)	\(\mu \in \mathbb{R}\)		Mean value of \(Y\)
\(\sigma\)	\(\sigma \in \mathbb{R}\)		Standard deviation of \(Y\)

Output Parameters

Parameter	Type	Constraint	Description	Remarks
\(\hat{Y}\)	\(\hat{Y} \in (-1,1)^{N_1}, (-1,1)^{N_1 \times N_2}, \text{ or } (-1,1)^{N_1 \times N_2 \times N_3}, \ldots\)	\(N_1, N_2, N_3 \in \mathbb{N}\)

Single Steps using the Algorithm

• Data Scaling with Zero-Mean Scaling

References

• J. Han, M. Kamber and J. Pei, Data Mining - Concepts and Techniques, 3rd ed., Amsterdam: Morgan Kaufmann Publishers, 2012.