4.5 VRM (Value Radiation Model)

4.4.5 Computational Interpretation of Matrix Elements

The computational significance of the Value Correlation Matrix (VCM) lies in the interpretation of its individual matrix elements. Each coefficient,

vij,v_{ij},vij​,

quantifies the degree of value correlation between spatial units uiu_iui​ and uju_juj​, thereby representing the intensity of their interaction within the Urban Land Value Field (ULVF).

Unlike conventional spatial weighting coefficients, the value of vijv_{ij}vij​ does not express geographical distance, neighborhood adjacency, or market price similarity. Instead, it measures the extent to which two spatial units participate in the same urban value field through the combined influence of transportation connectivity, accessibility, socio-economic activity, urban value centers, environmental conditions, land-use functions, and localized spatial characteristics.

Accordingly, each matrix element may be interpreted as the computational strength of value interaction between two locations.

For computational purposes,

0≤vij≤1,0 \leq v_{ij}\leq1,0≤vij​≤1,

where larger values indicate stronger value correlation and smaller values indicate weaker interaction within the Urban Land Value Field.

The interpretation of representative coefficient values is summarized conceptually below.

Correlation coefficient

Computational interpretation

vij=0v_{ij}=0vij​=0

No effective value correlation exists between the two spatial units.

0<vij<0.30<v_{ij}<0.30<vij​<0.3

Weak value interaction. The influence of one location on the other is limited.

0.3≤vij<0.60.3\leq v_{ij}<0.60.3≤vij​<0.6

Moderate value correlation generated by partial similarity of spatial conditions.

0.6≤vij<0.90.6\leq v_{ij}<0.90.6≤vij​<0.9

Strong value interaction resulting from highly compatible urban structures.

vij≈1v_{ij}\approx1vij​≈1

Nearly identical participation in the same Urban Land Value Field.

These numerical ranges are conceptual and may be adjusted according to specific calibration procedures or application contexts.

An important characteristic of the VCM is that high value correlation does not necessarily imply short geographical distance. Two locations connected by an efficient transportation corridor, sharing similar socio-economic functions, or influenced by the same urban value center may exhibit a high correlation coefficient despite being physically distant. Conversely, adjacent locations separated by major physical barriers, incompatible land-use functions, or discontinuities in accessibility may display relatively weak value correlation.

From a computational perspective, the VCM therefore represents a functional interaction network rather than a geometric neighborhood structure.

The diagonal elements,

vii,v_{ii},vii​,

represent the internal reference correlation of each spatial unit with itself. Under the normalized formulation adopted in this study,

vii=1,v_{ii}=1,vii​=1,

indicating perfect self-correlation.

The off-diagonal elements,

vij(i≠j),v_{ij}\quad(i\neq j),vij​(i=j),

describe the pairwise value interactions among different spatial units. Collectively, these coefficients define the complete topology of the Urban Land Value Field and characterize how urban value propagates through the spatial system.

Because every matrix element is derived from the integrated influence of the seven structural mechanisms, changes in transportation infrastructure, accessibility, urban development, land-use configuration, environmental quality, or localized conditions will modify the corresponding correlation coefficients. Consequently, the Value Correlation Matrix evolves continuously with the urban system and provides a dynamic representation of changing spatial relationships.

From a computational viewpoint, the VCM serves as the interaction kernel of the ULVF framework. Subsequent analyses, including the derivation of the Urban Value Field, the construction of the Value Relation Matrix (VRM), and the estimation of spatial value gradients, are all based on the correlation structure represented by its matrix elements.

Therefore, the interpretation of vijv_{ij}vij​ extends beyond a simple numerical coefficient. Each element constitutes a quantitative description of how urban value is transmitted, shared, and accumulated between two spatial units, providing the mathematical basis for modeling the continuous evolution of urban value across space.