# Understanding Spatial Autocorrelation: An Everyday Metaphor and Additional New Interpretations

## Abstract

**:**

## 1. Introduction

#### 1.1. SA: An Important Geospatial Synoptic Statistic

#### 1.2. SA and Geographic Scale/Resolution

## 2. The Jigsaw Puzzle: An Everyday Object Metaphor

_{i}= $\frac{{\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}}}{{\mathrm{s}}_{\mathrm{Y}}}$ for areal unit i, where $\overline{\mathrm{y}}$ and s

_{y}respectively denote the arithmetic mean and standard deviation of elementary statistics). However, portraying SA rather than a standard bivariate correlation scatterplot replaces X with z

_{i}for the horizontal axis, and, for the vertical axis, Y with the sum of neighboring z

_{i}values (i.e., using algebraic notation, $\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{ij}}{\mathrm{z}}_{\mathrm{j}$, where c

_{ij}= 1 when areal units i and j are, and 0 when they are not, neighbors, perhaps employing the rook adjacency definition). This modified statistical graphic is the Moran scatterplot, whose trend line is proportional to a dataset’s Moran Coefficient (MC; this covariation-based index, arguably the most popular SA quantifier, resembles the Pearson product moment correlation coefficient)—the scaling constant is n divided by the sum of the entries (e.g., 0 s and 1 s) in the attached spatial weights matrix (SWM; which is like an Excel spreadsheet whose row and column labels are the same sequence of areal unit polygons/locations, and whose cell entries are 1 if row and column areal units are neighbors, and 0 otherwise; this tabular data quantifies the topological arrangement of areal units forming a map) indicating which areal units are neighbors. The pattern of the resulting cloud of points, as well as the trend line, reveals the nature and degree of any SA present (see Figure 2). Figure 2 portrays the SA latent in jigsaw puzzles depicted by Figure 3b–d.

## 3. What Is SA? Illustrative SA Jigsaw Puzzle Cases

#### 3.1. A Case of Zero SA

#### The 0/0 Conundrum

**1**. In other words, this neighboring value’s covariation standpoint invents a situation in which the definition of 0/0 implies zero SA; L’Hospital’s rule asymptotically endorses this view. The intuition here is that no arrangement of loose blank regular square tile jigsaw puzzle pieces has observational correlation; any tile can be placed anywhere when completing a puzzle. Meanwhile, the squared paired comparisons index employs the Laplacian SWM version. Copycatting the covariation formula, a rescaling of its eigenvalues delivers its index values. As before, one of these eigenvalues is always zero, with an accompanying eigenvector proportional to the vector

**1**. In other words, this squared paired comparison of neighboring values stance contrives a situation in which the definition of 0/0 implies perfect positive SA; the calculus quotient limit theorem endorses this view. The intuition here is that after organizing a set of loose blank regular square tile jigsaw puzzle pieces into an arrangement with locational tagging, knowing the tile blankness at any particular location in this configuration automatically bestows knowing blankness anywhere else in it. In contrast, L’Hopsital’s rule renders near-zero SA only for typical neighborhood structures, an inconsistency attributable to some of the GR’s weaknesses. Nevertheless, both cases are technically singular, necessitating conceptual instead of computational reasoning for their clarifications.

#### 3.2. A Case of Pure Positive SA

#### 3.3. A Case of Pure Negative SA

#### 3.4. A Positive–Negative SA Mixture Case

#### 3.5. Some Necessary Remarks about SA

## 4. Materials and Methods: Yet More Faces of SA

#### 4.1. Remotely Sensed Data Results: The Case of Strong Positive SA

_{ij}is the row-standardized version of c

_{ij}(the most commonly used spatial weights specification in autoregressive models), and ε denotes a standard independent and identically distributed statistical random error term, produces two notable results: for a binary 0–1 c

_{ij}rook definition of adjacency, a SA parameter estimate, $\hat{\mathsf{\rho}}$, of 0.989 (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$ ≈ 0.005), which is >0.9+, and an approximate residual MC of 0.172 (s

_{MC}≈ 0.024; MC

_{max}= 1.02) and GR of 0.813 (s

_{GR}≈ 0.053)—computed as e

_{i}= ${\hat{\mathsf{\epsilon}}}_{\mathrm{i}}$ = y

_{i}− [$\hat{\mathsf{\rho}}{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{ij}}{\mathrm{y}}_{\mathrm{j}}$ + (1 − $\hat{\mathsf{\rho}}$)$\hat{\mathsf{\mu}}$]—implying the continued presence of more than trace positive SA in the spatial regression residuals, e

_{i}; and, for a c

_{ij}queen definition of adjacency, a SA parameter estimate, $\hat{\mathsf{\rho}}$, of 0.990 (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$ ≈ 0.005), which again is >0.9+, and an approximate residual MC of 0.156 (s

_{MC}≈ 0.017; MC

_{max}= 1.03) and GR of 0.810 (s

_{GR}≈ 0.051), once more implying the continued presence of more than trace positive SA in the spatial regression residuals, e

_{i}. This comparison implies that the remaining residual SA is not a function of the SWM definition.

_{MC}≈ 0.024) and GR of 0.971 (s

_{GR}≈ 0.053), implying the presence of only a trace amount of residual SA for this second specification. These estimates confirm that positive SA is in excess of 0.9 (the average lag-1 spatial correlation is roughly 0.93). Furthermore, the queen adjacency-based parameter estimate becomes $\hat{\mathsf{\rho}}$ ≈ 0.949 (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$ ≈ 0.014), a slight decrease in its magnitude, with an accompanying MA parameter estimate, $\hat{\mathsf{\theta}}$, of −0.684 (${\mathrm{s}}_{\hat{\mathsf{\theta}}}$ ≈ 0.072), and an approximate residual MC of 0.014 (s

_{MC}≈ 0.017) and GR of 0.948 (s

_{GR}≈ 0.051), again implying the presence of only a trace amount of residual SA for this second specification. These estimates also confirm that positive SA exceeds 0.9 (the average lag-1 spatial correlation is roughly 0.92). In other words, for these remotely sensed data, neither a SWM nor a model specification extension uncovers a negative SA component (i.e., no detection of a mixture); rather, these extensions further emphasize that the degree of positive SA latent in remotely sensed images tends to be marked.

^{0}− 1, 2

^{8}− 1] = [0, 255]) for each pixel. Its simple SAR model specification coupled with a rook adjacency definition yields the SA parameter estimate $\hat{\mathsf{\rho}}$ ≈ 0.964 (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$ ≈ 0.014), which, again, slightly decreases to 0.891 by including a companion MA parameter, whose estimate is $\hat{\mathsf{\theta}}$ ≈ −0.654 (${\mathrm{s}}_{\hat{\mathsf{\theta}}}$ ≈ 0.061). The accompanying approximate residual MC decreases from 0.221 (s

_{MC}≈ 0.046) to 0.013, with the corresponding GR increasing from 0.801 (s

_{GR}≈ 0.100) to 1.011, implying nothing more than a trace amount of residual SA being present in this second specification, and again confirming that marked positive SA tends to characterizes remotely sensed data.

_{i}= μ + ESF

_{PSA,i}+ ESF

_{NSA,i}+ ε

_{i}(i = 1, 2, …, n),

_{PSA}and ESF

_{NSA}respectively denote the positive and negative SA ESF (i.e., weighted sums of synthetic SA variates) components, renders residuals containing more than trace SA (its null hypothesis z

_{MC}≈ 5.7 and z

_{GR}≈ −3.0). Replacing the SWM in this probe with one defined by a queen’s adjacency definition results in MC = 0.84 (MC

_{max}= 1.03) and GR = 0.13, converting 53 of the selected eigenvectors to ones representing negative rather than positive SA, although they account for a mere 1.2% of the NDVI geographic variance; an important consequence of this definitional change is the presence of only trace residual SA (its null hypothesis z

_{MC}≈ −0.6 and z

_{GR}≈ −0.5). This MESF finding confirms the existence of a positive–negative SA mixture, with the negative SA component hidden. Figure 7 displays selected akin ESF Tool output for this data analysis; these results require some post-processing to match their SAS counterparts reported in this paragraph (Step 1: save the 338 eigenvectors selected by the “Eigenvector Spatial Filtering Regression” option. Step 2: use the first 267 of these eigenvectors in a “Linear Regression” option, testing the residual SA with the binary 0–1 rook SWM. Step 3: repeat Step 2 testing with the binary 0–1 queen SWM).

#### 4.2. Socio-Economic/Demographic Data Results: The Case of Moderate Positive SA

#### 4.3. Case Studies Discussion

## 5. Summary, Conclusions, and Implications

The north [of England] has wealthy suburbs, like South Wirral, west of Liverpool. They vote Labour. The south has impoverished pockets, like north-east Kent. They vote Conservative. It is as though political opinions derive from the air people breathe.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Web of Science (2012–2018) SA keyword cloud infographics (arbitrary group coloring visually differentiates among perceived SA research communities; node size reflects weighted normalized citation counts, which tend to highlight leading community scholars); compilation and portrayals by Drs. Kai Hu (Jiangnan University) and Qing Luo (Wuhan Institute of Technology). Left (

**a**): authors. Right (

**b**): concepts.

**Figure 2.**Specimen Moran scatterplots portraying the different natures and degrees of SA (n = 25) using z-score axes; gray lines denotes 95% confidence intervals, red lines denote 95% prediction intervals, blue denotes the origin axes, and black denotes trend lines. Left (

**a**): positive (Figure 3b), MC ≈ 0.44, Geary Ratio (GR; a popular paired comparisons (i.e., squared differences in neighboring attribute values) SA index) ≈ 0.52. Middle (

**b**): zero (Figure 3d), MC ≈ −0.03, GR ≈ 0.98. Right (

**c**): negative (Figure 3c), MC ≈ −0.45, GR ≈ 1.42.

**Figure 3.**A specimen aggregated High Peak normalized difference vegetation index (NDVI; from [42]) remotely sensed image as a jigsaw puzzle; the green–yellow–red tertile color palate is directly proportional to the NDVI values (i.e., red denotes H, y

_{H}, yellow denotes M, y

_{M}, and green denotes L, y

_{L}, values). Left (

**a**): the NDVI geographic distribution across a 30-by-30 pixels landscape (rook adjacency criterion; MC ≈ 0.88, GR ≈ 0.10, n = 900) overlaid with a five-by-five jigsaw puzzle dissection (i.e., cutting). Left middle (

**b**): average NDVI values by puzzle piece (mimicking the visualization detected by an adjacent sight cones cluster in a human’s eye). Right middle (

**c**): a Thiessen polygon overlay based upon puzzle piece centroids (puzzle piece physical centers computed by ESRI

^{©}ArcMap) to emphasize the tags and slots. Right (

**d**): a random permutation of the (

**b**) average values.

**Figure 4.**Selected degree-of-difficulty jigsaw puzzle types; red vertical two-way arrows link unassembled puzzle pieces to their solutions. Top left (

**a**): patterned square pieces. Bottom left (

**e**): Figure 4a solution. Top left middle (

**b**): blank square pieces (e.g., https://minifigs.me/products/draw-your-own-personalised-puzzle-various-sizes-custom-lego-jigsaw-puzzle). Bottom left middle (

**f**): a possible (

**b**) solution. Top right middle (

**c**): blank irregular (i.e., random cut) shaped pieces (e.g., https:/www.aliexpress.us/item/2255799894529529.html?gatewayAdapt=glo2usa4itemAdapt) Bottom right middle (

**g**): a possible (

**c**) solution in progress. Top right (

**d**): patterned irregularly shaped pieces. Bottom right (

**h**): (

**d**) solution.

**Figure 5.**Ghostly spatial competition. Left (

**a**): invisible hands creating an all-or-nothing square checkerboard attribute pattern. Middle (

**b**): invisible hands (red) superimposed upon their coordinated physical tabs and slots. Right (

**c**): illustrative invisible hands (red) protruding from the single 1st-level central place; seven (via nesting) 2nd-level central places offer a distinct bundle of goods/services that creates a have/have not hexagonal checkerboard mosaic.

**Figure 6.**The 2010 geographic distribution of Box–Cox transformed percentage of occupied houses across the Dallas–Fort Worth (DFW) Metroplex census tracts. Left (

**a**): 20 (i.e., four rows by five columns) jigsaw puzzle pieces (constructed using BookWidgets: https://www.bookwidgets.com/ widget-library/jigsaw-puzzle (accessed on 23 August 2023)). Right (

**b**): the assembled jigsaw puzzle (MC ≈ 0.63, GR ≈ 0.37; rook adjacency definition).

**Figure 7.**High Peak Box–Cox transformed NDVI SA computation results. Top left (

**a**): a binary (i.e., 0–1) rook SWM Moran scatterplot. Top right (

**b**): output from ESF Tool. Bottom left (

**c**): a binary queen SWM Moran scatterplot. Bottom right (

**d**): output from ESF Tool based upon the first 267 MESF linear regression selected eigenvectors, using binary rook and queen SWMs.

**Figure 8.**2010 Box–Cox transformed population density across the DFW Metroplex census tracts. Left (

**a**): 200 (i.e., 20 rows by 10 columns) jigsaw puzzle pieces (constructed using BookWidgets: https://www.bookwidgets.com/widget-library/jigsaw-puzzle (accessed on 23 August 2023)). Middle (

**b**): the assembled jigsaw puzzle (MC ≈ 0.46, GR ≈ 0.41). Right (

**c**): the rook adjacency definition MESF approximation reproduction of (

**b**).

**Figure 9.**Moran scatterplots, based upon a rook adjacency definition, with superimposed positive and negative SA component trend lines and 95% prediction ellipses (respectively denoted by red and grey). Left (

**a**): Box–Cox transformed High Peak NDVI. Middle (

**b**) Box–Cox transformed DFW population density. Right (

**c**): a US South omitted covariates surrogate.

**Table 1.**Box–Cox transformed 2010 DFW population density (Figure 8b) spatial autoregressive estimation results.

Feature | Rook Adjacency Definition | Queen Adjacency Definition | ||
---|---|---|---|---|

SAR | SARMA ^{‡} | SAR | SARMA ^{‡} | |

$\hat{\mathsf{\rho}}$ (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$) | 0.820 (0.017) | 0.954 (0.014) | 0.844 (0.017) | 0.940 (0.018) |

$\hat{\mathsf{\theta}}$ (${\mathrm{s}}_{\hat{\mathsf{\theta}}}$) | 0 | 0.498 (0.061) | 0 | 0.383 (0.075) |

${\mathrm{r}}_{\hat{\mathsf{\rho}},\hat{\mathsf{\theta}}}$ | 0 | 0.812 | 0 | 0.826 |

Average lag-1 spatial correlation | 0.56 | 0.62 | 0.56 | 0.61 |

pseudo-R^{2} | 0.643 | 0.647 | ||

Residual z_{MC}; residual z_{GR} | −2.0; 1.8 | 1.7; 0.5 | −1.0; 0.9 | 1.5; 0.1 |

^{‡}NOTE: the MA SA parameter sign is the opposite of its nature, in keeping with Box–Jenkins notation (also see [58]).

**Table 2.**Box–Cox transformed 2010 DFW population density (Figure 8b) MESF estimation results.

Feature | Rook Adjacency Definition (SWM Elements Sum = 7074; MC_{max} ≈ 1.175) | Queen Adjacency Definition (SWM Elements Sum = 8494; MC_{max} ≈ 1.125) | ||||||
---|---|---|---|---|---|---|---|---|

Y | PSA | NSA | PSA + NSA | Y | PSA | NSA | PSA + NSA | |

# eigenvectors | 0 | 204 (385) | 66 (400) | 270 (785) | 0 | 186 (365) | 41 (366) | 227 (731) |

MC | 0.64 | 0.89 | −0.42 | 0.81 | 0.59 | 0.84 | −0.38 | 0.78 |

GR | 0.37 | 0.21 | 1.48 | 0.27 | 0.37 | 0.21 | 1.48 | 0.25 |

R^{2} | 0 | 0.750 | 0.052 | 0.802 | 0 | 0.730 | 0.038 | 0.768 |

Residual z_{MC} | 37.8 | −1.0 | 2.2 | 38.6 | 0.8 | 3.2 | ||

Residual z_{GR} | −14.3 | 0.7 | −1.2 | −14.5 | −0.3 | −2.0 |

_{PSA+NSA}≈ 0.93(0.89) + 0.05(−0.42) ≈ 0.81 and GR

_{PSA+NSA}≈ 0.93(0.21) + 0.05(1.48) ≈ 0.27; queen MC

_{PSA+NSA}≈ 0.94(0.84) + 0.03(−0.38) ≈ 0.78 and GR

_{PSA+Nesdf}≈ 0.94(0.21) + 0.03(1.48) ≈ 0.25.

**Table 3.**Spatial autoregressive estimation results for homicide rates across the US South [31].

Feature | Rook Adjacency Definition (SWM Elements Sum = 7700; MC_{max} ≈ 1.111) | Queen Adjacency Definition (SWM Elements Sum = 8096; MC_{max} ≈ 1.152) | ||
---|---|---|---|---|

SAR | SARSM ^{‡} | SAR | SARSM ^{‡} | |

$\hat{\mathsf{\rho}}$ (${\mathrm{s}}_{\hat{\mathsf{\rho}}}$) | 0.585 (0.025) | 0.988 (0.005) | 0.593 (0.025) | 0.988 (0.005) |

$\hat{\mathsf{\theta}}$ (${\mathrm{s}}_{\hat{\mathsf{\theta}}}$) | 0 | 0.880 (0.022) | 0 | 0.877 (0.022) |

${\mathrm{r}}_{\hat{\mathsf{\rho}},\hat{\mathsf{\theta}}}$ | 0 | 0.808 | 0 | 0.805 |

Average lag-1 spatial correlation | 0.31 | 0.38 | 0.31 | 0.39 |

pseudo-R^{2} | 0.326 | 0.326 | ||

Residual z_{MC} | −2.4 | −0.1 | −2.3 | −0.1 |

Residual z_{GR} | −0.2 | −1.1 | −0.4 | −1.3 |

^{‡}NOTE: the MA SA parameter sign is the opposite of its nature, in keeping with Box–Jenkins notation (also see [58]).

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**MDPI and ACS Style**

Griffith, D.A.
Understanding Spatial Autocorrelation: An Everyday Metaphor and Additional New Interpretations. *Geographies* **2023**, *3*, 543-562.
https://doi.org/10.3390/geographies3030028

**AMA Style**

Griffith DA.
Understanding Spatial Autocorrelation: An Everyday Metaphor and Additional New Interpretations. *Geographies*. 2023; 3(3):543-562.
https://doi.org/10.3390/geographies3030028

**Chicago/Turabian Style**

Griffith, Daniel A.
2023. "Understanding Spatial Autocorrelation: An Everyday Metaphor and Additional New Interpretations" *Geographies* 3, no. 3: 543-562.
https://doi.org/10.3390/geographies3030028