4.4.2 Mathematical Definition of the VCM

4.4.2 Mathematical Definition of the Value Correlation Matrix

The Value Correlation Matrix (VCM) provides the mathematical representation of the spatial correlation structure underlying the Urban Land Value Field (ULVF). It quantifies the degree of value interaction between every pair of spatial units within the study area and serves as the computational foundation for subsequent value field analysis.

Assume that the study area consists of nnn spatial units (e.g., land parcels, urban blocks, grid cells, or administrative zones). The Value Correlation Matrix is defined as

VCM=[vij]n×n\mathbf{VCM} = \left[ v_{ij} \right]_{n\times n}VCM=[vij​]n×n​

where

vijv_{ij}vij​

denotes the value correlation coefficient between spatial units iii and jjj.

Unlike conventional spatial weight matrices, whose elements are determined primarily by geographical distance or neighborhood adjacency, the coefficients of the VCM measure the degree to which two spatial units participate in the same Urban Land Value Field. Consequently, the value of vijv_{ij}vij​ reflects the integrated influence of the seven structural mechanisms introduced in the Seven-Variable Structural Model.

The correlation coefficient can therefore be expressed conceptually as

vij=F(S1,S2,S3,S4,S5,S6,S7)ijv_{ij} = F \left( S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}, S_{7} \right)_{ij}vij​=F(S1​,S2​,S3​,S4​,S5​,S6​,S7​)ij​

where

• S1S_{1}S1​ represents the role of road hierarchy and network structure;

• S2S_{2}S2​ denotes transport accessibility;

• S3S_{3}S3​ represents the intensity of socio-economic activities;

• S4S_{4}S4​ describes the influence of urban value centers;

• S5S_{5}S5​ represents environmental and landscape conditions;

• S6S_{6}S6​ denotes land-use function;

• S7S_{7}S7​ represents localized favorable and unfavorable conditions.

The function F(⋅)F(\cdot)F(⋅) integrates the combined effects of these seven structural mechanisms into a single quantitative measure describing the correlation of urban value between spatial units.

The VCM is therefore not constructed from a single explanatory variable but from the collective interaction of multiple spatial processes operating simultaneously within the urban system.

From a mathematical perspective, the matrix satisfies

VCM=[v11v12⋯v1nv21v22⋯v2n⋮⋮⋱⋮vn1vn2⋯vnn]\mathbf{VCM} = \begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1n}\\ v_{21} & v_{22} & \cdots & v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{n1} & v_{n2} & \cdots & v_{nn} \end{bmatrix}VCM=​v11​v21​⋮vn1​​v12​v22​⋮vn2​​⋯⋯⋱⋯​v1n​v2n​⋮vnn​​​

where each matrix element represents the relative correlation intensity between two spatial units.

For practical implementation, the correlation coefficients may be normalized into the interval

0≤vij≤1,0 \leq v_{ij}\leq1,0≤vij​≤1,

where

• vij=0v_{ij}=0vij​=0 indicates that the two spatial units exhibit no effective value correlation;

• 0<vij<10<v_{ij}<10<vij​<1 indicates varying degrees of value interaction;

• vij=1v_{ij}=1vij​=1 represents the maximum attainable value correlation under the adopted analytical framework.

It should be emphasized that these coefficients do not represent market price similarity, Euclidean distance, or topological adjacency. Instead, they quantify the ability of two locations to exchange, transmit, or share urban value through the spatial mechanisms described by the ULVF framework.

Accordingly, two geographically adjacent parcels may exhibit relatively weak value correlation if they are separated by incompatible land-use functions, physical barriers, or discontinuities in the transportation network. Conversely, two distant locations may display a high correlation coefficient when they are efficiently connected by major transportation corridors, belong to the same functional urban center, or participate in similar socio-economic activities.

Mathematically, the Value Correlation Matrix defines the interaction topology of the Urban Land Value Field. Rather than describing isolated spatial attributes, it characterizes the entire network of value relationships existing within the urban system. The VCM therefore constitutes the first quantitative representation of urban value interactions and provides the computational basis for constructing the Urban Value Field and the subsequent Value Relation Matrix (VRM).

Remark

The Value Correlation Matrix should be interpreted as a field-based interaction matrix rather than a conventional spatial weighting matrix. Its elements represent correlations generated by urban value formation mechanisms instead of purely geometric neighborhood relationships. This distinction constitutes one of the principal methodological innovations of the ULVF framework and establishes the mathematical bridge between the conceptual spatial mechanisms and the computational representation of urban value dynamics.