the metacoupling framework in coupling package

[SpatLyu]

The coupling package implements a metacoupling extension of the CCD model, where interactions between spatial units are constructed at the indicator level rather than at the aggregated system level.

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Notation

Let

• $$n$$: number of spatial units
• $$p$$: number of indicators

For unit $$i$$:

$$ \mathbf{U}_i = (U_{i1}, U_{i2}, \dots, U_{ip}) $$

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Intra-System Coupling

$$ C_{\text{intra}}(i) = \mathrm{CCD}(\mathbf{U}_i) $$

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Inter-System Coupling (Indicator-level Mixing)

For each pair $$(i,j)$$, consider all subsets:

$$ S \subset {1,\dots,p}, \quad S \neq \varnothing,; S \neq {1,\dots,p} $$

Each $S$ selects indicators from unit $i$, while the complement is taken from unit $j$.

Define the mixed system:

$$ V_{ij}^{(S)} = \bigl(V_{ij,1}^{(S)}, \dots, V_{ij,p}^{(S)}\bigr) $$

$$ V_{ij,k}^{(S)} = \begin{cases} U_{ik}, & k \in S \\ U_{jk}, & k \notin S \end{cases} $$

This yields all possible indicator-wise recombinations between $i$ and $j$:

$$ \text{combinations} = 2^p - 2 $$

The pairwise coupling is defined as:

$$ C_{ij} = \frac{1}{2^p - 2} \sum_{S} \mathrm{CCD}\bigl(V_{ij}^{(S)}\bigr) $$

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Spatial Aggregation

Given spatial weight matrices:

• $$W_{\text{peri}} = (w^{\text{peri}}_{ij})$$
• $$W_{\text{tele}} = (w^{\text{tele}}_{ij})$$

For each unit $$i$$:

$$ C_{\text{peri}}(i) = \sum_{j=1}^{n} w^{\text{peri}}_{ij} , C_{ij} $$

$$ C_{\text{tele}}(i) = \sum_{j=1}^{n} w^{\text{tele}}_{ij} , C_{ij} $$

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Development Index

Let indicator weights be $$\mathbf{w} = (w_1, \dots, w_p)$$.

$$ T_{\text{intra}}(i) = \sum_{k=1}^{p} w_k , U_{ik} $$

$$ T_{ij} = \frac{1}{2^p - 2} \sum_{S} \left( \sum_{k \in S} w_k U_{ik} + \sum_{k \notin S} w_k U_{jk} \right) $$

$$ T_{\text{peri}}(i) = \sum_{j=1}^{n} w^{\text{peri}}_{ij} , T_{ij} $$

$$ T_{\text{tele}}(i) = \sum_{j=1}^{n} w^{\text{tele}}_{ij} , T_{ij} $$

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Coupling Coordination Degree

$$ D_{\text{intra}}(i) = \sqrt{C_{\text{intra}}(i), T_{\text{intra}}(i)} $$

$$ D_{\text{peri}}(i) = \sqrt{C_{\text{peri}}(i), T_{\text{peri}}(i)} $$

$$ D_{\text{tele}}(i) = \sqrt{C_{\text{tele}}(i), T_{\text{tele}}(i)} $$

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Output

For each spatial unit $$i$$:

$$ \bigl( C_{\text{intra}},; D_{\text{intra}},; C_{\text{peri}},; D_{\text{peri}},; C_{\text{tele}},; D_{\text{tele}} \bigr) $$