notes on coupling coordination degree (CCD) models

[SpatLyu]

The Coupling Coordination Degree (CCD) model is widely used to quantify the interaction and coordinated development among multiple subsystems or indicators.

At its core, the model consists of three components:

• Coupling degree (C) — measures interaction strength among indicators
• Composite development index (T) — reflects overall system development
• Coupling coordination degree (D) — integrates both interaction and development

This disscussion focuses on three commonly used formulations for computing the coupling degree $C$, followed by the standard procedure for deriving $T$ and $D$.

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Notation

Let a system consist of $n$ indicators:

$$ U = {U_1, U_2, \dots, U_n} $$

For each spatial unit (or observation), $U_i$ denotes the value of the $i$-th indicator.

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Part I. Coupling Degree (C)

1. Standard Coupling Model

$$ C = \left[\frac{\prod\limits_{i=1}^n U_i}{\left(\frac{1}{n}\sum_{i=1}^n U_i\right)^n}\right]^{\frac{1}{n}} $$

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2. Wang et al. Coupling Model

$$ C = \sqrt{\left[1-\frac{\sum\limits_{i>j,j=1}^{n-1} \sqrt{\left(U_i-U_j\right)^2}}{\sum_{m=1}^{n-1}m}\right] \times \left(\prod_{i=1}^n \frac{U_i}{\text{max} U_i}\right)^{\frac{1}{n-1}}} $$

Reference: https://doi.org/10.31497/zrzyxb.20210319

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3. Fan et al. Coupling Model

$$ C = 1-2\sqrt{\frac{n\sum_{i=1}^n U_i^2 - \left(\sum_{i=1}^n U_i\right)^2}{n^2}} $$

Reference: https://doi.org/10.13546/j.cnki.tjyjc.2024.22.007

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Part II. From Coupling to Coordination

Once the coupling degree $C$ is computed using any of the above models, the CCD framework proceeds with the following steps.

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4. Composite Development Index (T)

$$ T = \sum_{i=1}^n w_i \times U_i, \quad \sum_{i=1}^n w_i = 1 $$

Interpretation

• Weighted aggregation of indicators
• Reflects the overall development level
• Weights ( w_i ) are user-defined

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5. Coupling Coordination Degree (D)

$$ D = \sqrt{C \times T} $$

Interpretation

• Integrates interaction (C) and development (T)
• Represents the coordinated development level
• Higher values indicate better coordination

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Summary

Coupling Degree Models

Model	Core Idea	Key Feature
standard	Geometric vs arithmetic mean	Sensitive to imbalance
wang	Difference + normalization	Captures disparity and scale
fan	Variance-based	Stable, inequality-oriented

CCD Workflow

1. Compute C using one of the three models
2. Compute T via weighted aggregation
3. Derive $$D = \sqrt{C \times T}$$

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Notes

• Computation is performed within each observation (row-wise)

• Indicators should be non-negative, and strictly positive for the standard model

• Different formulations emphasize different aspects:

  • Standard → balance
  • Wang → dispersion + relative level
  • Fan → variance / inequality

• The CCD pipeline is inherently two-stage:

  • Stage 1: Coupling measurement (C)
  • Stage 2: Coordination assessment (D via T)

• Choice of coupling model should depend on the research objective and data characteristics