How to Make Data Dimensionless: Key Normalization Techniques Explained

1 Dimensionless Processing of Indicators

Dimensionless processing, also known as indicator normalization, uses mathematical transformations to eliminate the influence of original units and magnitude on raw indicators.

Consequently, indicators have actual values and evaluation values . After dimensionless processing, the resulting values are called indicator evaluation values .

The dimensionless process converts actual indicator values into evaluation values.

Below are several commonly used indicator transformation methods.

2 Standard Sample Transformation Method

Let x_i be the observed value of the i-th indicator. The sample mean \bar{x} and sample standard deviation s are used to compute the standardized observation.

3 Ratio Transformation Method

For extremely large indicators, apply a scaling factor k (e.g., divide by a large constant). For extremely small indicators, apply an inverse scaling factor.

This method preserves proportional relationships before and after transformation, though not all indicators may retain proportionality simultaneously.

4 Vector Normalization Method

For extremely large indicators, divide by the vector norm; for extremely small indicators, apply the inverse operation.

5 Range Transformation Method

For both extremely large and extremely small indicators, map values to a fixed range (e.g., [0,1]), ensuring the transformed maximum and minimum values are consistent.

The drawback is that transformed values are not proportional to the original ones.

6 Efficacy Coefficient Method

This method uses constant parameters: a "translation amount" representing the indicator's baseline value, and a "rotation amount" indicating the scaling factor (increase or decrease).

7 Summary

This article presented several common data dimensionless processing methods (standard transformation, ratio transformation, range transformation, vector normalization, and efficacy coefficient method).

References

ThomsonRen https://github.com/ThomsonRen/mathmodels

司守奎，孙玺菁 Python数学实验与建模