How to Transform Indicators into Dimensionless Scores: Key Normalization Techniques

1. Indicator Dimensionless Processing

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

Consequently, there are actual values and evaluation values . After dimensionless processing, the resulting numbers are referred to as indicator evaluation values . The dimensionless process converts actual values into evaluation values.

2. Standard Sample Transformation Method

Let the sample mean be \(\bar{x}\) and the sample standard deviation be \(s\); the standardized observation is calculated as:

(x - \bar{x}) / s

3. Ratio Transformation Method

For extremely large indicators, apply the transformation:

x / max(x)

For extremely small indicators, apply the transformation:

min(x) / x

This method preserves proportional relationships before and after transformation, though not all indicators may exhibit both properties simultaneously.

4. Vector Normalization Method

For large indicators, use:

x / \sqrt{\sum x^2}

For small indicators, use:

1 / (x / \sqrt{\sum x^2})

5. Range (Extreme Difference) Transformation Method

For large indicators:

(x - min) / (max - min)

For small indicators:

(max - x) / (max - min)

After this transformation, all indicators share the same range, with the best indicator value equal to 1 and the worst equal to 0, but the proportional relationships between indicators are not preserved.

6. Efficacy Coefficient Method

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

7. Summary

The article introduced several common dimensionless processing methods for data—standard transformation, ratio transformation, range transformation, vector normalization, and efficacy coefficient method.

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

司守奎，孙玺菁 Python数学实验与建模