Geary’s c for Multivariate Spatial Data

Abstract

Geary’s c is a prominent measure of spatial autocorrelation in univariate spatial data. It uses a weighted sum of squared differences. This paper develops Geary’s c for multivariate spatial data. It can describe the similarity/discrepancy between vectors of observations at different vertices/spatial units by a weighted sum of the squared Euclidean norm of the vector differences. It is thus a natural extension of the univariate Geary’s c. This paper also develops a local version of it. We then establish their properties.

1. Introduction

Let $y_i$ denote the realization of a single variable y at the vertex/spatial unit $v_i$ for $i=1,\dots ,n$. Spatial autocorrelation is a notion that describes the similarity/discrepancy between observations at different vertices. One way to measure it between $y_i$ and $y_j$ is to use the squared difference:

$$\begin{array}{c}\hfill {(y_i-y_j)}^2.\end{array}$$

(1)

This approach was adopted by Geary [1]. He proposed a spatial autocorrelation measure given by a binary weighted sum of ${(y_i-y_j)}^2$ for $i,j=1,\dots ,n$, which was modified by Cliff and Ord [2,3,4,5]. It is now called Geary’s c and is one of the leading measures of spatial autocorrelation (Geary’s c is a spatial extension of von Neumann [6] ratio. See, e.g., [7]). Subsequently, Anselin [8] developed a local version of Geary’s c, local Geary’s c. To distinguish them, Geary’s c is sometimes referred to as global Geary’s c.

Let ${\mathit{y}}_i={[y_{\left(1\right),i},\dots ,y_{\left(p\right),i}]}^{\top }$ denote the realization of multiple variables $\mathit{y}={[y_{\left(1\right)},\dots ,y_{\left(p\right)}]}^{\top }$ at the vertex/spatial unit $v_i$ for $i=1,\dots ,n$. How can we measure the similarity/discrepancy between ${\mathit{y}}_i$ and ${\mathit{y}}_j$, which are both p-dimensional column vectors? A natural approach is to extend (1). That is, it can be measured with the following squared Euclidean norm of the vector difference:

$$\begin{array}{c}\hfill \parallel {\mathit{y}}_i-{\mathit{y}}_j{\parallel }^2.\end{array}$$

(2)

In this paper, we develop a spatial autocorrelation measure for multivariate spatial data, ${\mathit{y}}_1,\dots ,{\mathit{y}}_n\in {\mathbb{R}}^p$. It uses $\parallel {\mathit{y}}_i-{\mathit{y}}_j{\parallel }^2$. More specifically, as a global Geary’s c, we define it as a weighted sum of $\parallel {\mathit{y}}_i-{\mathit{y}}_j{\parallel }^2$ for $i,j=1,\dots ,n$. This measure allows us to describe the similarity/discrepancy between vectors of observations at different vertices. In other words, we can use it to measure the spatial autocorrelation in multivariate spatial data. In addition, from it, we develop a local version of it, the multivariate local Geary’s c. Subsequently, we establish their properties. We show how our two multivariate autocorrelation measures relate to the two existing univariate autocorrelation measures, (univariate) global and local Geary’s c. We also note that our multivariate global Geary’s c is related to the recently popular graph learning.

Wartenberg [9], Jombart et al. [10], Anselin [11], Lin [12], and Yamada [13,14,15] are relevant to this paper. Among them, (i) Yamada [13,14] presented several results of univariate global Geary’s c using notions of spectral graph theory. This paper depends on them, and it can be regarded as an extension of them. In addition, (ii) Yamada [15] dealt with multivariate extension of Moran’s I, which was developed by Moran [16] and Cliff and Ord [2,3,4,5] and is another prominent measure of spatial autocorrelation (Recent studies on Moran’s I and its extension include [17,18,19,20,21,22,23,24,25]. Thus, this paper can be regarded as a companion paper to it. Finally, (iii) the relevance of [9,10,11,12] to this paper is discussed in the final section. This is mainly because we would like to use our notations below to describe their studies.

This paper is organized as follows: In Section 2, we provide some preliminaries for the following two sections. In Section 3, after defining global and local Geary’s c for multivariate spatial data, we establish their properties. In Section 4, we make additional remarks to the results of the previous section. In the section, we also mention how our multivariate global Geary’s c is related to the recently popular graph learning. Section 5 concludes. In the Appendix A and Appendix B, we provide some of the proofs and two MATLAB/GNU Octave user-defined functions.

2. Preliminaries

Some of the notations. Let ${\mathit{I}}_n$ be the identity matrix of order n and ${\mathit{e}}_{n,i}$ be the i-th column of ${\mathit{I}}_n$, i.e., ${\mathit{I}}_n=[{\mathit{e}}_{n,1},\dots ,{\mathit{e}}_{n,n}]$. Let ${\mathit{\iota }}_n$ be the n-dimensional vector of ones, i.e., ${\mathit{\iota }}_n={[1,\dots ,1]}^{\top }\in {\mathbb{R}}^n$ and ${\mathit{Q}}_{\iota }={\mathit{I}}_n-{\mathit{\iota }}_n{\left({\mathit{\iota }}_n^{\top }{\mathit{\iota }}_n\right)}^{-1}{\mathit{\iota }}_n^{\top }={\mathit{I}}_n-\frac{1}{n}{\mathit{\iota }}_n{\mathit{\iota }}_n^{\top }$. For a matrix $\mathbf{\Gamma }=\left[{\gamma }_{i,j}\right]\in {\mathbb{R}}^{r\times s}$, ${\parallel \mathbf{\Gamma }\parallel }_{\mathrm{F}}$ denotes the Frobenius norm of $\mathbf{\Gamma }$, i.e., ${\parallel \mathbf{\Gamma }\parallel }_{\mathrm{F}}={\left({\sum }_{i=1}^r{\sum }_{j=1}^s{\gamma }_{i,j}^2\right)}^{\frac{1}{2}}$. For a full column rank matrix $\mathbf{\Gamma }\in {\mathbb{R}}^{r\times s}$, denote the column space of $\mathbf{\Gamma }$ and its orthogonal complement by $\mathbb{S}(\mathbf{\Gamma })$ and ${\mathbb{S}}^{\perp }(\mathbf{\Gamma })$, respectively.

Undirected graph. Let $G=(V,E)$ be a simple undirected graph whose vertex set is $V=\{v_1,\dots ,v_n\}$, where $n\ge 2$. In addition, let $\mathit{W}=\left[w_{i,j}\right]\in {\mathbb{R}}^{n\times n}$, where

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{w}_{i,j}>0,\hfill & \{v_i,v_j\}\in E,\hfill \\ w_{i,j}=0,\hfill & \{v_i,v_j\}\notin E.\hfill \end{array}\right.\end{array}$$

Given that $G=(V,E)$ is a simple undirected graph, $\mathit{W}$ is a symmetric matrix such that $w_{i,i}=0$ for $i=1,\dots ,n$. We assume that $\left|E\right|>0$, i.e., $G=(V,E)$ is not an empty graph. Accordingly, $q={\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{i,j}>0$.

Multivariate/multiple spatial data. Let

$$\begin{array}{c}\hfill \mathit{Y}=\left[\begin{array}{ccc}{y}_{\left(1\right),1}& \cdots & y_{\left(p\right),1}\\ ⋮& & ⋮\\ y_{\left(1\right),n}& \cdots & y_{\left(p\right),n}\end{array}\right]=\left[\begin{array}{c}{\mathit{y}}_1^{\top }\\ ⋮\\ {\mathit{y}}_n^{\top }\end{array}\right]=[{\mathit{y}}_{\left(1\right)},\dots ,{\mathit{y}}_{\left(p\right)}]\in {\mathbb{R}}^{n\times p},\end{array}$$

where $y_{\left(h\right),i}$ denotes the realization of variable $y_{\left(h\right)}$ at vertex $v_i$. We refer to ${\mathit{y}}_1,\dots ,{\mathit{y}}_n\in {\mathbb{R}}^p$ as multivariate spatial data and ${\mathit{y}}_{\left(1\right)},\dots ,{\mathit{y}}_{\left(p\right)}\in {\mathbb{R}}^n$ as multiple spatial data. We assume that ${\sum }_{i=1}^n{(y_{\left(h\right),i}-{\overline{y}}_{\left(h\right)})}^2>0$ for $h=1,\dots ,p$, where ${\overline{y}}_{\left(h\right)}=\frac{1}{n}{\sum }_{i=1}^n{y}_{\left(h\right),i}$. That is, by assumption, ${\mathit{y}}_{\left(h\right)}$ does not belong to $\mathbb{S}\left({\mathit{\iota }}_n\right)$.

Standardized multiple/multivariate spatial data. Let $z_{\left(h\right),i}=\frac{y_{\left(h\right),i}-{\overline{y}}_{\left(h\right)}}{s_{\left(h\right)}}$ for $i=1,\dots ,n$ and $h=1,\dots ,p$, where $s_{\left(h\right)}$ is the positive square root of $\frac{1}{n-1}{\sum }_{i=1}^n{(y_{\left(h\right),i}-{\overline{y}}_{\left(h\right)})}^2$. Then, let

$$\begin{array}{c}\hfill \mathit{Z}=\left[\begin{array}{ccc}{z}_{\left(1\right),1}& \cdots & z_{\left(p\right),1}\\ ⋮& & ⋮\\ z_{\left(1\right),n}& \cdots & z_{\left(p\right),n}\end{array}\right]=\left[\begin{array}{c}{\mathit{z}}_1^{\top }\\ ⋮\\ {\mathit{z}}_n^{\top }\end{array}\right]=[{\mathit{z}}_{\left(1\right)},\dots ,{\mathit{z}}_{\left(p\right)}]\in {\mathbb{R}}^{n\times p}.\end{array}$$

By construction, ${\mathit{z}}_{\left(h\right)}$ is related to ${\mathit{y}}_{\left(h\right)}$ as

$$\begin{array}{c}\hfill {\mathit{z}}_{\left(h\right)}=\frac{1}{s_{\left(h\right)}}{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(h\right)}=\frac{1}{s_{\left(h\right)}}({\mathit{y}}_{\left(h\right)}-{\overline{y}}_{\left(h\right)}{\mathit{\iota }}_n),h=1,\dots ,p.\end{array}$$

Then, given that ${\mathit{\iota }}_n^{\top }{\mathit{Q}}_{\iota }=\mathbf{0}$, it follows that ${\mathit{\iota }}_n^{\top }{\mathit{z}}_{\left(h\right)}=0$, which implies that ${\mathit{z}}_{\left(h\right)}$ belongs to ${\mathbb{S}}^{\perp }\left({\mathit{\iota }}_n\right)$.

Geary’s c. Let $c_{\left(h\right)}$ denote the global Geary’s c of ${\mathit{y}}_{\left(h\right)}={[y_{\left(h\right),1},\dots ,y_{\left(h\right),n}]}^{\top }$:

$$\begin{array}{c}\hfill c_{\left(h\right)}=\frac{n-1}{2q}\frac{{\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{i,j}{(y_{\left(h\right),i}-y_{\left(h\right),j})}^2}{{\sum }_{i=1}^n{(y_{\left(h\right),i}-{\overline{y}}_{\left(h\right)})}^2},h=1,\dots ,p.\end{array}$$

(3)

In addition, denote the local Geary’s c of ${\mathit{y}}_{\left(h\right)}$ at vertex $v_i$ by $c_{\left(h\right),i}$:

$$\begin{array}{c}\hfill c_{\left(h\right),i}=\frac{n-1}{2q}\frac{{\sum }_{j=1}^n{w}_{i,j}{(y_{\left(h\right),i}-y_{\left(h\right),j})}^2}{{\sum }_{i=1}^n{(y_{\left(h\right),i}-{\overline{y}}_{\left(h\right)})}^2},i=1,\dots ,n,h=1,\dots ,p.\end{array}$$

(4)

Then, $c_{\left(h\right)}$ and $c_{\left(h\right),i}$ can be represented using $z_{\left(h\right),i}$ as

$$\begin{array}{cc}\hfill & c_{\left(h\right)}=\frac{1}{2q}\sum _{i=1}^n\sum _{j=1}^n{w}_{i,j}{(z_{\left(h\right),i}-z_{\left(h\right),j})}^2,h=1,\dots ,p,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & c_{\left(h\right),i}=\frac{1}{2q}\sum _{j=1}^n{w}_{i,j}{(z_{\left(h\right),i}-z_{\left(h\right),j})}^2,i=1,\dots ,n,h=1,\dots ,p.\hfill \end{array}$$

(6)

By construction, it follows that

$$\begin{array}{c}\hfill c_{\left(h\right)}=\sum _{i=1}^n{c}_{\left(h\right),i}.\end{array}$$

(7)

Graph Laplacian. Let $\mathit{D}=\mathrm{diag}\left(d_1,\dots ,d_n\right)$, where $d_i={\sum }_{j=1}^n{w}_{i,j}$ for $i=1,\dots ,n$. Denote the graph Laplacian corresponding to $\mathit{W}$ by $\mathit{L}$. Then, from, e.g., [26,27],

$$\begin{array}{c}\hfill \mathit{L}=\mathit{D}-\mathit{W}.\end{array}$$

(8)

Global Geary’s c in matrix notation. From [13] (Proposition 1), $c_{\left(h\right)}$ in (5) can be represented using $\mathit{L}$ as

$$\begin{array}{c}\hfill c_{\left(h\right)}=\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }\mathit{L}{\mathit{z}}_{\left(h\right)},h=1,\dots ,p.\end{array}$$

(9)

Spectral decomposition of graph Laplacian. Given that $\mathit{L}$ in (8) is a real symmetric matrix, it can be spectrally decomposed as

$$\begin{array}{c}\hfill \mathit{L}=\mathit{U}\mathbf{\Lambda }{\mathit{U}}^{\top },\end{array}$$

(10)

where $\mathbf{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$ such that ${\lambda }_1\le \cdots \le {\lambda }_n$ and $\mathit{U}=[{\mathit{u}}_1,\dots ,{\mathit{u}}_n]$ is an orthogonal matrix. Given that $\mathit{L}$ is a positive semidefinite matrix such that $\mathit{L}{\mathit{\iota }}_n=\mathbf{0}=0·{\mathit{\iota }}_n$, we can let $({\lambda }_1,{\mathit{u}}_1)=\left(0,\frac{1}{\sqrt{n}}{\mathit{\iota }}_n\right)$. Then, given that $\mathit{U}$ is an orthogonal matrix, it follows that ${\mathit{u}}_1\in \mathbb{S}\left({\mathit{\iota }}_n\right)$ and ${\mathit{u}}_i\in {\mathbb{S}}^{\perp }\left({\mathit{\iota }}_n\right)$ for $i=2,\dots ,n$.

Graph Fourier transform. Let

$$\begin{array}{c}\hfill \mathit{F}={\mathit{U}}^{\top }\mathit{Z}=[{\mathit{U}}^{\top }{\mathit{z}}_{\left(1\right)},\dots ,{\mathit{U}}^{\top }{\mathit{z}}_{\left(p\right)}].\end{array}$$

(11)

${\mathit{U}}^{\top }{\mathit{z}}_{\left(h\right)}$ in (11) is referred to as graph Fourier transform of ${\mathit{z}}_{\left(h\right)}$ [28,29]. Let

$$\begin{array}{c}\hfill \mathit{F}=\left[\begin{array}{ccc}{f}_{\left(1\right),1}& \cdots & f_{\left(p\right),1}\\ ⋮& & ⋮\\ f_{\left(1\right),n}& \cdots & f_{\left(p\right),n}\end{array}\right]=\left[\begin{array}{c}{\mathit{f}}_1^{\top }\\ ⋮\\ {\mathit{f}}_n^{\top }\end{array}\right]=[{\mathit{f}}_{\left(1\right)},\dots ,{\mathit{f}}_{\left(p\right)}]\in {\mathbb{R}}^{n\times p}.\end{array}$$

Then, from (11), it follows that ${\mathit{f}}_{\left(h\right)}={\mathit{U}}^{\top }{\mathit{z}}_{\left(h\right)}$ for $h=1,\dots ,p$. In addition, given that $f_{\left(h\right),i}$ is the i-th entry of ${\mathit{f}}_{\left(h\right)}$, it follows that $f_{\left(h\right),i}={\mathit{u}}_i^{\top }{\mathit{z}}_{\left(h\right)}$ for $i=1,\dots ,n$. Since ${\mathit{u}}_1\in \mathbb{S}\left({\mathit{\iota }}_n\right)$ and ${\mathit{z}}_{\left(h\right)}\in {\mathbb{S}}^{\perp }\left({\mathit{\iota }}_n\right)$, it follows that $f_{\left(h\right),1}={\mathit{u}}_1^{\top }{\mathit{z}}_{\left(h\right)}=0$ for $h=1,\dots ,p$. Accordingly, ${\mathit{f}}_1^{\top }$, which denotes the first row of $\mathit{F}$ in (11), is equal to $\mathbf{0}$.

Weighted average representation of global Geary’s c. Let

$$\begin{array}{c}\hfill c\left({\mathit{u}}_i\right)=\frac{n-1}{q}\frac{{\mathit{u}}_i^{\top }\mathit{L}{\mathit{u}}_i}{{\mathit{u}}_i^{\top }{\mathit{Q}}_{\iota }{\mathit{u}}_i},i=2,\dots ,n,\end{array}$$

(12)

which denotes the global Geary’s c of ${\mathit{u}}_i$ for $i=2,\dots ,n$. Then, given that ${\mathit{u}}_i^{\top }\mathit{L}{\mathit{u}}_i={\mathit{e}}_{n,i}^{\top }\mathbf{\Lambda }{\mathit{e}}_{n,i}={\lambda }_i$ and ${\mathit{u}}_i^{\top }{\mathit{Q}}_{\iota }{\mathit{u}}_i={\mathit{u}}_i^{\top }{\mathit{u}}_i=1$ for $i=2,\dots ,n$, it follows that

$$\begin{array}{c}\hfill c\left({\mathit{u}}_i\right)=\frac{n-1}{q}{\lambda }_i,i=2,\dots ,n.\end{array}$$

(13)

Accordingly, $c_{\left(h\right)}$ can be represented as

$$\begin{array}{cc}\hfill c_{\left(h\right)}& =\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }\mathit{L}{\mathit{z}}_{\left(h\right)}=\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }\mathit{U}\mathbf{\Lambda }{\mathit{U}}^{\top }{\mathit{z}}_{\left(h\right)}=\frac{1}{q}{\mathit{f}}_{\left(h\right)}^{\top }\mathbf{\Lambda }{\mathit{f}}_{\left(h\right)}\hfill \\ \hfill & =\frac{1}{q}\sum _{i=2}^n{\lambda }_i{f}_{\left(h\right),i}^2,h=1,\dots ,p.\hfill \end{array}$$

(14)

Then, substituting (13) into (14) yields

$$\begin{array}{c}\hfill c_{\left(h\right)}=\sum _{i=2}^n{\psi }_{\left(h\right),i}c\left({\mathit{u}}_i\right),\end{array}$$

(15)

where ${\psi }_{\left(h\right),i}=\frac{f_{\left(h\right),i}^2}{n-1}$ for $i=2,\dots ,n$. Here, since ${\psi }_{\left(h\right),i}\ge 0$ and

$$\begin{array}{c}\hfill \sum _{i=2}^n{\psi }_{\left(h\right),i}=1,\end{array}$$

(16)

$c_{\left(h\right)}$ is an weighted average of $c\left({\mathit{u}}_2\right),\dots ,c\left({\mathit{u}}_n\right)$. For a proof of (16), see Appendix A.1. Note that (15) is a restatement of [14] (Proposition 1(a)).

Bounds of global Geary’s c. From [30], it follows that

$$\begin{array}{c}\hfill c_{\left(h\right)}\in \left[\frac{n-1}{q}{\lambda }_2,\frac{n-1}{q}{\lambda }_n\right],h=1,\dots ,p.\end{array}$$

(17)

3. Multivariate Global and Local Geary’s $\mathit{c}$

We define the following measure as the global Geary’s c for multivariate spatial data, ${\mathit{y}}_1,\dots ,{\mathit{y}}_n\in {\mathbb{R}}^p$:

$$\begin{array}{c}\hfill c=\frac{1}{2pq}\sum _{i=1}^n\sum _{j=1}^n{w}_{i,j}{\parallel {\mathit{z}}_i-{\mathit{z}}_j\parallel }^2.\end{array}$$

(18)

We refer to c as multivariate global Geary’s c. Similarly, we define the following measure as the local Geary’s c for multivariate spatial data at vertex $v_i$, ${\mathit{y}}_i\in {\mathbb{R}}^p$:

$$\begin{array}{c}\hfill c_i=\frac{1}{2pq}\sum _{j=1}^n{w}_{i,j}{\parallel {\mathit{z}}_i-{\mathit{z}}_j\parallel }^2,i=1,\dots ,n.\end{array}$$

(19)

which we refer to as multivariate local Geary’s c. Then, by construction, it follows that

$$\begin{array}{c}\hfill \sum _{i=1}^n{c}_i=\sum _{i=1}^n\left(\frac{1}{2pq}\sum _{j=1}^n{w}_{i,j}{\parallel {\mathit{z}}_i-{\mathit{z}}_j\parallel }^2\right)=\frac{1}{2pq}\sum _{i=1}^n\sum _{j=1}^n{w}_{i,j}{\parallel {\mathit{z}}_i-{\mathit{z}}_j\parallel }^2=c.\end{array}$$

(20)

Recall that p denotes the number of variables and ${\mathit{z}}_i^{\top }$ is the i-th row of $\mathit{Z}$, i.e., ${\mathit{z}}_i^{\top }=[z_{\left(1\right),i},\dots ,z_{\left(p\right),i}]$. Accordingly, when $p=1$, given that ${\mathit{z}}_i^{\top }$ reduces to $z_{\left(1\right),i}$ for $i=1,\dots ,n$, it follows that

$$\begin{array}{cc}\hfill & c=\frac{1}{2q}\sum _{i=1}^n\sum _{j=1}^n{w}_{i,j}{(z_{\left(1\right),i}-z_{\left(1\right),j})}^2=c_{\left(1\right)},\hfill \\ \hfill & c_i=\frac{1}{2q}\sum _{j=1}^n{w}_{i,j}{(z_{\left(1\right),i}-z_{\left(1\right),j})}^2=c_{\left(1\right),i}.\hfill \end{array}$$

Thus, the multivariate global (resp. local) Geary’s c is a generalization of the univariate global (resp. local) Geary’s c.

The next two results document how the multivariate local (resp. global) Geary’s c relates to the univariate local (resp. global) Geary’s c’s.

Proposition 1. 

For $i=1,\dots ,n$, $c_i$ in (19) is equal to the simple average of $c_{\left(1\right),i},\dots ,c_{\left(p\right),i}$, i.e.,

$$\begin{array}{c}\hfill c_i=\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right),i}.\end{array}$$

(21)

Proof. 

See Appendix A.2. □

Proposition 2. 

c in (18) is equal to the simple average of $c_{\left(1\right)},\dots ,c_{\left(p\right)}$, i.e.,

$$\begin{array}{c}\hfill c=\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right)}.\end{array}$$

(22)

Proof. 

From (7), (20), and Proposition 1, we have

$$\begin{array}{c}\hfill c=\sum _{i=1}^n{c}_i=\sum _{i=1}^n\left(\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right),i}\right)=\frac{1}{p}\sum _{h=1}^p\left(\sum _{i=1}^n{c}_{\left(h\right),i}\right)=\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right)}.\end{array}$$

□

Remark 1. 

By combining Proposition 2 and (7), c in (18) can be represented by $c_{\left(h\right),i}$ as follows.

$$\begin{array}{c}\hfill c=\frac{1}{p}\sum _{h=1}^p\sum _{i=1}^n{c}_{\left(h\right),i}.\end{array}$$

(23)

Next, we will document matrix form representations of the multivariate Geary’s c’s. In preparation for this, we show a matrix form representation of $c_{\left(h\right),i}$ in (4). Let

$$\begin{array}{c}\hfill {\mathit{L}}_i=\frac{1}{2}{d}_i{\mathit{e}}_{n,i}{\mathit{e}}_{n,i}^{\top }-{\mathit{e}}_{n,i}{\mathit{e}}_{n,i}^{\top }\mathit{W}+\frac{1}{2}\mathrm{diag}\left({\mathit{e}}_{n,i}^{\top }\mathit{W}\right).\end{array}$$

(24)

Proposition 3. 

$c_{\left(h\right),i}$ in (6) can be represented in matrix notation as

$$\begin{array}{c}\hfill c_{\left(h\right),i}=\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }{\mathit{L}}_i{\mathit{z}}_{\left(h\right)}.\end{array}$$

(25)

Proof. 

See Appendix A.3. □

Now, we are ready to document matrix form representations of the multivariate Geary’s c’s.

Proposition 4. 

$c_i$ in (19) can be represented in matrix notation as

$$\begin{array}{c}\hfill c_i=\frac{1}{pq}\mathrm{tr}\left({\mathit{Z}}^{\top }{\mathit{L}}_i\mathit{Z}\right).\end{array}$$

(26)

Proof. 

${\mathit{z}}_{\left(h\right)}^{\top }{\mathit{L}}_i{\mathit{z}}_{\left(h\right)}$ in (25) is the $(h,h)$-th entry of ${\mathit{Z}}^{\top }{\mathit{L}}_i\mathit{Z}\in {\mathbb{R}}^{p\times p}$. In addition, from Proposition 1, it follows that $c_i=\frac{1}{p}{\sum }_{h=1}^p{c}_{\left(h\right),i}$. Combining these results yields

$$\begin{array}{c}\hfill c_i=\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right),i}=\frac{1}{pq}\sum _{h=1}^p{\mathit{z}}_{\left(h\right)}^{\top }{\mathit{L}}_i{\mathit{z}}_{\left(h\right)}=\frac{1}{pq}\mathrm{tr}\left({\mathit{Z}}^{\top }{\mathit{L}}_i\mathit{Z}\right).\end{array}$$

□

Proposition 5. 

c in (18) can be represented in matrix notation as

$$\begin{array}{c}\hfill c=\frac{1}{pq}\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\right).\end{array}$$

(27)

Proof. 

${\mathit{z}}_{\left(h\right)}^{\top }\mathit{L}{\mathit{z}}_{\left(h\right)}$ in (9) is the $(h,h)$-th entry of ${\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\in {\mathbb{R}}^{p\times p}$. In addition, from Proposition 2, it follows that $c=\frac{1}{p}{\sum }_{h=1}^p{c}_{\left(h\right)}$. Combining these results yields

$$\begin{array}{c}\hfill c=\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right)}=\frac{1}{pq}\sum _{h=1}^p{\mathit{z}}_{\left(h\right)}^{\top }\mathit{L}{\mathit{z}}_{\left(h\right)}=\frac{1}{pq}\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\right).\end{array}$$

□

Remark 2. 

We make four remarks on Propositions 4 and 5.

(i) 

From (20), the next equation must follow:

$$\begin{array}{c}\hfill \sum _{i=1}^n{\mathit{L}}_i=\mathit{L}.\end{array}$$

(28)

For a proof of (28), see Appendix A.4.

(ii) 

Given that $\mathrm{tr}\left({\mathit{Z}}^{\top }{\mathit{L}}_i\mathit{Z}\right)=\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }({\mathit{I}}_p\otimes {\mathit{L}}_i)\mathrm{vec}\left(\mathit{Z}\right)$ (e.g., [31], p. 241), from Proposition 4, $c_i$ in (19) can be represented as a quadratic form:

$$\begin{array}{c}\hfill c_i=\frac{1}{pq}\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }({\mathit{I}}_p\otimes {\mathit{L}}_i)\mathrm{vec}\left(\mathit{Z}\right).\end{array}$$

(29)

Similarly, from Proposition 5, c in (18) can be represented as

$$\begin{array}{c}\hfill c=\frac{1}{pq}\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }({\mathit{I}}_p\otimes \mathit{L})\mathrm{vec}\left(\mathit{Z}\right).\end{array}$$

(30)

(iii) 

Given that $\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }=[{\mathit{z}}_{\left(1\right)}^{\top },\dots ,{\mathit{z}}_{\left(p\right)}^{\top }]$, from (29) and (30), it immediately follows that

$$\begin{array}{cc}\hfill & c_i=\frac{1}{pq}\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }({\mathit{I}}_p\otimes {\mathit{L}}_i)\mathrm{vec}\left(\mathit{Z}\right)=\frac{1}{p}\sum _{h=1}^n\left(\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }{\mathit{L}}_i{\mathit{z}}_{\left(h\right)}\right)=\frac{1}{p}\sum _{h=1}^n{c}_{\left(h\right),i},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & c=\frac{1}{pq}\mathrm{vec}{\left(\mathit{Z}\right)}^{\top }({\mathit{I}}_p\otimes \mathit{L})\mathrm{vec}\left(\mathit{Z}\right)=\frac{1}{p}\sum _{h=1}^n\left(\frac{1}{q}{\mathit{z}}_{\left(h\right)}^{\top }\mathit{L}{\mathit{z}}_{\left(h\right)}\right)=\frac{1}{p}\sum _{h=1}^n{c}_{\left(h\right)}.\hfill \end{array}$$

(32)

(31) (resp. (32)) can be regarded as another proof of Proposition 1 (resp. Proposition 2).

(iv)

User-defined MATLAB/GNU Octave function for calculating $c_i$ (resp. c) based on (26) (resp. (27)) is provided in Appendix B.1 (resp. Appendix B.2).

Denote the eigenvalues of $\frac{1}{q}{\mathit{Z}}^{\top }\mathit{L}\mathit{Z}$ by ${\theta }_1,\dots ,{\theta }_p$. Then, since $\mathit{L}$ is a positive semidefinite matrix, $\frac{1}{q}{\mathit{Z}}^{\top }\mathit{L}\mathit{Z}$ is also a positive semidefinite matrix, from which we obtain ${\theta }_i\ge 0$ for $i=1,\dots ,p$. From Proposition 5, c can be represented using these eigenvalues as follows:

Proposition 6. 

c in (18) is equal to the simple average of ${\theta }_1,\dots ,{\theta }_p$, i.e.,

$$\begin{array}{c}\hfill c=\frac{1}{p}\sum _{i=1}^p{\theta }_i.\end{array}$$

(33)

Proof. 

Given that $\frac{1}{q}{\mathit{Z}}^{\top }\mathit{L}\mathit{Z}$ and $\mathrm{diag}({\theta }_1,\dots ,{\theta }_p)$ are similar, it follows that $\frac{1}{q}\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\right)={\sum }_{i=1}^p{\theta }_i$. Then, (33) immediately follows from Proposition 5. □

Again, from Proposition 5, we obtain the following result:

Proposition 7. 

c in (18) can be represented as

$$\begin{array}{c}\hfill c=\frac{1}{pq}{\parallel {\mathit{B}}^{\top }\mathit{Z}\parallel }_{\mathrm{F}}^2,\end{array}$$

(34)

where $\mathit{B}\in {\mathbb{R}}^{n\times \left|E\right|}$ is an incidence matrix corresponding to $G=(V,E)$.

Proof. 

Given that $\mathit{L}=\mathit{B}{\mathit{B}}^{\top }$ (e.g., [27], Proposition 2.3), from Proposition 5, we have $\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\right)=\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{B}{\mathit{B}}^{\top }\mathit{Z}\right)={\parallel {\mathit{B}}^{\top }\mathit{Z}\parallel }_{\mathrm{F}}^2$. □

Example 1. 

Consider the case where $E=\{\{v_1,v_2\},\dots ,\{v_{n-1},v_n\}\}$ in $G=(V,E)$. In the case, $G=(V,E)$ is a path graph, and thus $q=2(n-1)$ and we can let

$$\begin{array}{c}\hfill {\mathit{B}}^{\top }=\left[\begin{array}{ccccc}-1& 1& 0& \cdots & 0\\ 0& -1& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & 0\\ 0& \cdots & 0& -1& 1\end{array}\right]\in {\mathbb{R}}^{(n-1)\times n}.\end{array}$$

(35)

$\mathit{L}$ in the case is explicitly given, e.g., in [32] (p. 68) and [33] (p. 3233). Accordingly, given that ${\mathit{B}}^{\top }\mathit{Z}={[{\mathit{z}}_2-{\mathit{z}}_1,\dots ,{\mathit{z}}_n-{\mathit{z}}_{n-1}]}^{\top }$, from (34), it follows that

$$\begin{array}{c}\hfill c=\frac{1}{2p(n-1)}\sum _{h=1}^p\sum _{i=2}^n{(z_{\left(h\right),i}-z_{\left(h\right),i-1})}^2=\frac{1}{2p(n-1)}\sum _{i=2}^n{\parallel {\mathit{z}}_i-{\mathit{z}}_{i-1}\parallel }^2,\end{array}$$

(36)

which can be regarded as a multivariate von Neumann [6] ratio.

The next result documents that c is also a weighted average of $c\left({\mathit{u}}_2\right),\dots ,c\left({\mathit{u}}_n\right)$.

Proposition 8. 

c in (18) can be represented as a weighted average of $c\left({\mathit{u}}_2\right),\dots ,c\left({\mathit{u}}_n\right)$ as follows:

$$\begin{array}{c}\hfill c=\sum _{i=2}^n{\overline{\psi }}_{i}c\left({\mathit{u}}_i\right),\end{array}$$

(37)

where ${\overline{\psi }}_i=\frac{1}{p}{\sum }_{h=1}^p{\psi }_{\left(h\right),i}$ for $i=2,\dots ,n$.

Proof. 

See Appendix A.5. □

Proposition 9. 

c in (18) belongs to the closed interval given by

$$\begin{array}{c}\hfill \left[\frac{n-1}{q}{\lambda }_2,\frac{n-1}{q}{\lambda }_n\right].\end{array}$$

Proof. 

It immediately follows by combining the results in (17) and Proposition 2. □

4. Additional Remarks

In this section, we make additional remarks to the results obtained in the previous section.

4.1. Local Geary Matrix

Proposition 1, (7), and (23) motivate us to consider the following matrix:

$$\begin{array}{c}\hfill \mathit{C}=\left[\begin{array}{ccc}{c}_{\left(1\right),1}& \cdots & c_{\left(p\right),1}\\ ⋮& & ⋮\\ c_{\left(1\right),n}& \cdots & c_{\left(p\right),n}\end{array}\right]\in {\mathbb{R}}^{n\times p}.\end{array}$$

(38)

Recall that $c_{\left(h\right),i}$ denotes the local Geary’s c of ${\mathit{y}}_{\left(h\right)}$ at vertex $v_i$. We refer to $\mathit{C}$ as local Geary matrix. From (7), Proposition 1, and (23), $c_{\left(h\right)}$, $c_i$, and c can be represented using $\mathit{C}$ as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{Equation}\left(7\right)\text{:}\hfill & c_{\left(h\right)}={\mathit{\iota }}_n^{\top }\mathit{C}{\mathit{e}}_{p,h},h=1,\dots ,p,\hfill \\ \mathrm{Proposition}1\text{:}\hfill & c_i=\frac{1}{p}{\mathit{e}}_{n,i}^{\top }\mathit{C}{\mathit{\iota }}_p,i=1,\dots ,n,\hfill \\ \mathrm{Equation}\left(23\right)\text{:}\hfill & c=\frac{1}{p}{\mathit{\iota }}_n^{\top }\mathit{C}{\mathit{\iota }}_p.\hfill \end{array}\right.\end{array}$$

(39)

4.2. Relationship to Graph Signal Processing

As shown in Proposition 5, our multivariate global Geary’s c in (18) can be represented as $\frac{1}{pq}\mathrm{tr}\left({\mathit{Z}}^{\top }\mathit{L}\mathit{Z}\right)$. We note here that a similar expression has appeared in graph signal processing. Specifically, Dong et al. [34] proposed the following filtering method in our notation:

$$\begin{array}{c}\hfill \underset{\mathit{X}\in {\mathbb{R}}^{n\times p},{\mathit{L}}_{†}\in {\mathbb{R}}^{n\times n}}{min}{\parallel \mathit{Y}-\mathit{X}\parallel }_{\mathrm{F}}^2+\alpha \mathrm{tr}\left({\mathit{X}}^{\top }{\mathit{L}}_{†}\mathit{X}\right)+\beta {\parallel {\mathit{L}}_{†}\parallel }_{\mathrm{F}}^2,\end{array}$$

(40)

where ${\mathit{L}}_{†}$ is a symmetric matrix such that $\mathrm{tr}\left({\mathit{L}}_{†}\right)=n$, ${\mathit{L}}_{†}{\mathit{\iota }}_n=\mathbf{0}$, and its off-diagonal entries are nonpositive. $\alpha$ and $\beta$ are positive regularization parameters. It is very interesting that $\mathrm{tr}\left({\mathit{X}}^{\top }{\mathit{L}}_{†}\mathit{X}\right)$ is included as a penalty term in the above constrained minimization problem. It is notable that $\mathit{X}$ and ${\mathit{L}}_{†}$ are simultaneously estimated by solving (40). See also [35,36].

5. Concluding Remarks

In this paper, we developed global Geary’s c for multivariate spatial data. It describes the similarity/discrepancy between vectors of observations at different vertices/spatial units by a weighted sum of the squared Euclidean norm of the vector differences, and is thus a natural extension of the global Geary’s c for univariate spatial data. In addition, from it, we developed local Geary’s c for multivariate spatial data. Subsequently, we established their properties. The results obtained are summarized in Propositions 1–9. We also mentioned how our multivariate global Geary’s c is related to the recently popular graph learning.

We make a remark. As mentioned in Section 1, Wartenberg [9], Jombart et al. [10], Anselin [11], and Lin [12] are relevant to this paper. Here, we would like to clarify the relationship between those papers and this paper.

First, Anselin [11] defined multivariate local Geary’s c as

$$\begin{array}{c}\hfill \sum _{h=1}^p{c}_{\left(h\right),i}\mathrm{or}\frac{1}{p}\sum _{h=1}^p{c}_{\left(h\right),i}\end{array}$$

(41)

in our notation. As shown in Proposition 1, the latter of (41) is identical to $c_i$ in (19). In this sense, it would be accurate to say that Anselin [11] developed the multivariate local Geary’s c, $c_i$ in (19), and we established its properties. Nevertheless, we emphasize here that we defined the multivariate local Geary’s c by (19) and derived it from the multivariate global Geary’s c in (18).

Second, as an extension of global Moran’s I, Wartenberg [9] developed the following spatial correlation matrix in our notations:

$$\begin{array}{c}\hfill \mathit{M}=\frac{1}{q}\left(\frac{n}{n-1}{\mathit{Z}}^{\top }\mathit{W}\mathit{Z}\right).\end{array}$$

(42)

When $p=1$, given that ${\mathit{z}}_{\left(1\right)}=\frac{1}{s_{\left(1\right)}}{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(1\right)}$ and $s_{\left(1\right)}^2=\frac{1}{n-1}{\mathit{y}}_{\left(1\right)}^{\top }{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(1\right)}$, $\mathit{M}$ reduces to the following scalar:

$$\begin{array}{c}\hfill \frac{1}{q}\left(\frac{n}{n-1}{\mathit{z}}_{\left(1\right)}^{\top }\mathit{W}{\mathit{z}}_{\left(1\right)}\right)=\frac{1}{q}\frac{1}{s_{\left(1\right)}^2}\left(\frac{n}{n-1}{\mathit{y}}_{\left(1\right)}^{\top }{\mathit{Q}}_{\iota }\mathit{W}{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(1\right)}\right)=\frac{n}{q}\frac{{\mathit{y}}_{\left(1\right)}^{\top }{\mathit{Q}}_{\iota }\mathit{W}{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(1\right)}}{{\mathit{y}}_{\left(1\right)}^{\top }{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(1\right)}},\end{array}$$

which is the global Moran’s I for ${\mathit{y}}_{\left(1\right)}$. Moreover, the multivariate global Moran’s I developed by [15] can be represented as

$$\begin{array}{c}\hfill \frac{1}{p}\mathrm{tr}\left(\mathit{M}\right).\end{array}$$

Following [9], Jombart et al. [10] used the spatial correlation matrix given by

$$\begin{array}{c}\hfill \frac{1}{2n}{\mathit{Y}}^{\top }{\mathit{Q}}_{\iota }({\mathit{D}}^{-1}\mathit{W}+\mathit{W}{\mathit{D}}^{-1}){\mathit{Q}}_{\iota }\mathit{Y}.\end{array}$$

(43)

Recalling that ${\mathit{z}}_{\left(h\right)}=\frac{1}{s_{\left(h\right)}}{\mathit{Q}}_{\iota }{\mathit{y}}_{\left(h\right)}$, where ${\mathit{z}}_{\left(h\right)}$ is the h-th column of $\mathit{Z}$, the main difference is the use of the sum of ${\mathit{D}}^{-1}\mathit{W}$ and its transpose instead of $\mathit{W}$. Here, since ${\mathit{D}}^{-1}\mathit{W}{\mathit{\iota }}_n={\mathit{\iota }}_n$, ${\mathit{D}}^{-1}\mathit{W}$ is a row-standardized matrix. In addition, another extension of [9] was recently completed by Lin [12]. He developed the following spatial correlation matrix:

$$\begin{array}{c}\hfill \mathit{G}=\frac{1}{q}{\mathit{Z}}^{\top }\mathit{L}\mathit{Z}.\end{array}$$

(44)

Of these three spatial correlation matrices, $\mathit{G}$ in (44) is the most relevant for this paper. This is because, from Proposition 5, the multivariate global Geary’s c, c in (18), can be represented using $\mathit{G}$ as

$$\begin{array}{c}\hfill c=\frac{1}{p}\mathrm{tr}\left(\mathit{G}\right).\end{array}$$

(45)

In other words, c in (18) is equal to the simple average of the eigenvalues of $\mathit{G}$ (Proposition 6). Thus, our multivariate global Geary’s c can be regarded as a value obtained from the spatial correlation matrix, $\mathit{G}$.