# Negative Spatial Autocorrelation: One of the Most Neglected Concepts in Spatial Statistics

## Abstract

**:**

## 1. Introduction

_{i}and y

_{i}, i = 1, 2, …, n. Very explicitly, positive linear correlation means that, relatively speaking, high values of X, x

_{H}, tend to pair with high values of Y, y

_{H}, medium values of X, x

_{M}, tend to pair with medium values of Y, y

_{M}, and low values of X, x

_{L}, tend to pair with low values of Y, y

_{L}. Zero linear correlation means that x

_{H}tends to pair with y

_{H}, y

_{M}, and y

_{L}, x

_{M}tends to pair with y

_{H}, y

_{M}, and y

_{L}, and x

_{L}tends to pair with y

_{H}, y

_{M}, and y

_{L}; no high, medium, or low value pairing preferences emerge. Negative linear correlation means that x

_{H}tends to pair with y

_{L}, x

_{M}tends to pair with y

_{M}, and x

_{L}tends to pair with y

_{H}. This description, well-known to statisticians, is noteworthy in order to use it as a foundation for the ensuing discussion. Figure 1a furnishes a cross-tabulation for n = 254 illustrating a negative correlation (it is roughly −0.5 in this example), the topic of this paper. Its color coding enables the establishment of a parallel for describing correlation associated with Figure 1b, which relates to spatial statistics, a more recent subdiscipline addition to the statistics literature.

_{H}tends to neighbor y

_{H}, y

_{M}tends to neighbor y

_{M}, and y

_{L}tends to neighbor y

_{L}: Neighboring values are relatively similar (see Figure 1b; green counties tend to be neighbors, yellow counties tend to be neighbors, and red counties tend to be neighbors). Zero spatial autocorrelation means that y

_{H}tends to neighbor y

_{H}, y

_{M}, and y

_{L}, y

_{M}tends to neighbor y

_{H}, y

_{M}, and y

_{L}, and y

_{L}tends to neighbor y

_{H}, y

_{M}, and y

_{L}. Negative spatial autocorrelation (NSA) means that y

_{H}tends to neighbor y

_{L}, y

_{M}tends to neighbor y

_{M}, and y

_{L}tends to neighbor y

_{H}: Neighboring values are relatively dissimilar. A perusal of relevant textbooks and internet web pages reveals the presentation of very few real world NSA examples. This concept is one of the most—if not the most—neglected spatial statistics topics, and as such is the motivation for and theme of this article. Griffith [3] and Griffith and Arbia [4] furnish the initial conceptual and foundational discussions about NSA; this paper differs from their work by introducing new traits of NSA (e.g., its relation to jackknifing and model misspecification), beginning with an emphasis on special features of negative correlation in general. Therefore, the primary purpose of this paper is to furnish evidence supporting why statisticians should care about this mostly ignored concept.

#### 1.1. Visualizing Correlation

_{i}, y

_{i}) pairs of points to their z-score equivalents converts the slope of the scatterplot trend line to r. Situations like Figure 2c are the focus of this paper.

**C**, that indicates which locations are neighbors; this matrix articulates the data’s organization. In its simplest form, the entries of this matrix are c

_{ij}= 1 if the matrix row and column label areal unit polygons share a common non-zero length boundary, and c

_{ij}= 0 otherwise; sometimes analysts convert this matrix to its row-standardized version,

**W**[i.e., w

_{ij}= ${\mathrm{c}}_{\mathrm{ij}}/{\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{ij}}$]. Making an analogy with chess, this neighbors designation is known as the rook definition of geographic adjacency; a number of other definitions exist (e.g., the queen, making another analogy with chess, which also includes zero length boundaries (i.e., points), and the nearest neighbors, which may be the set of polygons whose centroids are closest to the centroid of a given polygon). If the trend line representing the scatter of points is linear, then its slope relates to the MC describing the nature and degree of the spatial autocorrelation contained in variable Y. This spatial autocorrelation coefficient may be written as

**C**(see Section 1.3). Now the plotted points may be the z-score pairs (z

_{i}, ${\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{ij}}{\mathrm{z}}_{\mathrm{j}}$). In this context, the slope of the trend line relates to the MC. Situations like the one exemplified by Figure 3c are the focus of this paper. Furthermore, a goal of this paper is to demonstrate that NSA exists in disparate empirical datasets, at least more commonly than believed, and needs to be better understood; its study is more than merely an academic curiosity.

#### 1.2. What Is Special About Negative Correlation?

- Linear regression residuals are negatively correlated;
- Multinomial indicator variables yield a correlation matrix with all negative off-diagonal entries;
- Positively skewed independent variables have a strong tendency to display negative bivariate correlation;
- Negatively correlated replicates can reduce variance in simulation experiments; and,
- Negative bivariate correlation can be relative.

**Y**= a

**1**+

**e**,

**Y**is an n-by-1 vector of variable values,

**1**is an n-by-1 vector of ones, a = $\overline{\mathrm{y}}$ is the ordinary least squares estimate of the regression coefficient, and

**e**is an n-by-1 vector of residuals (

**Y**− a

**1**). The estimate $\overline{\mathrm{y}}$ places a linear constraint on the residuals (i.e., ${\sum}_{\mathrm{i}=1}^{\mathrm{n}-1}{\mathrm{e}}_{\mathrm{i}}$ = −e

_{n}) that results in the expected value E(e

_{i}e

_{j}) = −1/(n − 1), where E(•) denotes the calculus of expectations, even with the standard assumption of independent regression errors.

_{j}denotes the number of one’s indicator variable j contains. For the case of two groups (i.e., a binomial random variable), this off-diagonal formula always reduces to −1. For the 16 census metropolitan regions of Texas (see Figure 1b; the 17th indicator variable would be for the non-metropolitan counties), the correlation matrix is

1.00 | −0.10 | −0.19 | −0.14 | −0.14 | −0.14 | −0.17 | −0.17 | −0.10 | −0.19 | −0.17 | −0.10 | −0.14 | −0.24 | −0.35 | −0.35 | −0.41 |

−0.10 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 0.00 | −0.01 | −0.01 | 0.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 |

−0.19 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.01 | −0.01 | −0.01 | −0.02 | −0.03 | −0.03 | −0.03 |

−0.14 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.02 | −0.02 |

−0.14 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.02 | −0.02 |

−0.14 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.02 | −0.02 |

−0.17 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.03 | −0.03 | −0.03 |

−0.17 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.03 | −0.03 | −0.03 |

−0.10 | 0.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 |

−0.19 | −0.01 | −0.02 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.02 | −0.03 | −0.03 | −0.03 |

−0.17 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.02 | −0.03 | −0.03 | −0.03 |

−0.10 | 0.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.00 | −0.01 | −0.01 | 1.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 |

−0.14 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | −0.01 | 1.00 | −0.01 | −0.02 | −0.02 | −0.02 |

−0.24 | −0.01 | −0.02 | −0.01 | −0.01 | −0.01 | −0.02 | −0.02 | −0.01 | −0.02 | −0.02 | −0.01 | −0.01 | 1.00 | −0.04 | −0.04 | −0.04 |

−0.35 | −0.01 | −0.03 | −0.02 | −0.02 | −0.02 | −0.03 | −0.03 | −0.01 | −0.03 | −0.03 | −0.01 | −0.02 | −0.04 | 1.00 | −0.05 | −0.06 |

−0.35 | −0.01 | −0.03 | −0.02 | −0.02 | −0.02 | −0.03 | −0.03 | −0.01 | −0.03 | −0.03 | −0.01 | −0.02 | −0.04 | −0.05 | 1.00 | −0.06 |

−0.41 | −0.02 | −0.03 | −0.02 | −0.02 | −0.02 | −0.03 | −0.03 | −0.02 | −0.03 | −0.03 | −0.02 | −0.02 | −0.04 | −0.06 | −0.06 | 1.00 |

_{j}ranges from 1 to 17; the rounded correlation coefficients range from −0.004 ≈ 0 to −0.41. Usually constructing a correlation matrix with all negative off−diagonal entries is a challenging feat; encountering such matrices is rare in practice.

_{i}in a given replication has a corresponding –y

_{i}in its complement replication. The feature of interest is a comprehensive covering of the employed probability space: y

_{i}covers part of it, whereas −y

_{i}covers its complement, a different part of the probability space. In other words, the first sequence of random numbers is reused in a simulation experiment. The mathematical statistics property of interest pertains to sums/differences of random variables:

_{1}and its antithetic variable Y

_{2}is a perfect −1, by construction. The benefit of using this antithetic variable replication is variance reduction, as shown by the preceding mathematical statistics theorem (i.e., fewer replications are needed to achieve a smaller standard error of, for example, an arithmetic mean); this variance reduction is between random draws, not the simulated statistic of interest. Another advantage is that statistics of interest are based upon more regularly distributed sample values, allowing better balancing of more extreme simulated pseudo-random numbers.

#### 1.3. What Is Special About NSA?

- NSA manifestations differ between discrete and continuous geographic space;
- NSA links to spatial competition;
- Common spatial autocorrelation indices tend to gauge NSA on a scale shorter than [−1,0];
- Extreme NSA supports the fast calculation of the extreme eigenvalues of certain matrices;
- The boundary between PSA and NSA for the MC is zero rather than some small negative value; and,
- NSA often mixes with PSA, which tends to mask its existence.

**C**expression, which is the matrix version of the non-attribute variable Y portion of the numerator of Equation (1):

**I**−

**11**

^{T}/n)

**C**(

**I**−

**11**

^{T}/n),

**I**denotes the n-by-n identity matrix. Pre- and post-multiplication of the spatial weights matrix

**C**by the projection matrix (

**I**−

**11**

^{T}/n) constitutes the aforementioned specific modification. This matrix operation converts the principal eigenvalue of matrix

**C**to zero, and its corresponding principal eigenvector (this pair of mathematical quantities, one a scalar and the other a vector, is an eigenfunction) to a vector proportional to

**1**; the remaining n − 1 eigenfunctions are approximately unchanged, although all are transformed to have a mean of zero (i.e., each is orthogonal to the vector

**1**). When multiplied by n/

**1**

^{T}

**C1**, the n eigenvalues of matrix expression (2) become MC values, indexing the nature and degree of spatial autocorrelation portrayed by the mapping of their corresponding eigenvectors. For the Texas surface partitioning, the minimum MC is −0.610; Figure 5b portrays its corresponding eigenvector, which depicts the maximum NSA map pattern possible for any set of real numbers combined with the Texas counties spatial weights matrix

**C**based upon the rook adjacency definition. This minimum value indicates that −0.274 effectively is equivalent to −0.449 (≈ −0.274/|−0.610|), a quantity calculated by stretching the endpoint of the MC scale to the more intuitive and appealing limit of −1; in other words, the NSA is moderate, not weak. Regular square tessellation surface partitionings, such as Figure 5a, relate to yardsticks whose extremes are closer to $\pm $ 1; for this 4-by-4 tessellation, the minimum possible MC is −1.079 (see Figure 5c). The minimum for most irregular surface partitioning is closer to −0.5. Furthermore, frequency distributions of eigenvalues, and hence MC values reveal that far more potential NSA than PSA map patterns exist (see Figure 5c). The theoretical NSA upper limit is 75% of the n eigenvalues, which requires spatial weights matrices built using planar graphs comprising only n = 4 completely connected (known as K

_{4}) subgraphs. The practical upper limit is roughly 67%. The apparent empirical upper limit for a reasonable size n (e.g., ≥100) is approximately 60%. All three of these scenarios contain far more distinct NSA than PSA map patterns.

**C**or

**W**, to establish its feasible parameter space. The matrix powering algorithm

**1**

^{T}

**C**

^{τ+1}

**1**/

**1**

^{T}

**C**

^{τ}

**1**almost always converges on the principal eigenvalue of matrix

**C**as τ goes to infinity (its accompanying principal eigenvector is the normalized vector

**E**

_{1}converged upon by a normalized

**C**

^{τ+1}

**1**); the Perron-Frobenius theorem divulges that the principal eigenvalue of matrix

**W**is 1 (its principal eigenvector is proportional to the vector

**1**). The difficulty is determining the most extreme negative eigenvalue of these matrices. Griffith [16] devised an efficient algorithm that exploits NSA to estimate these eigenvalues based upon a Rayleigh quotient: An eigenvector is constructed that maximizes the differences between all neighboring elements, as defined by matrix

**C**or

**W**while being orthogonal to

**E**

_{1}. Griffith et al. [17] show that it is more efficient than many of the standard computer software algorithms for calculating extreme eigenvalues of a matrix.

_{i}e

_{j}). However, if the mean of variable Y is known, then E(MC) = [n/

**1**

^{T}

**C1**] E${\sum}_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}-\mathsf{\mu}){\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{ij}}\left({\mathrm{y}}_{\mathrm{j}}-\mathsf{\mu}\right)$]/E[${\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{j}}-\mathsf{\mu}\right)}^{2}$] = 0. In other words, the expected value of the MC differentiating between PSA and NSA is zero, not −1/(n − 1); this outcome of a negative expected value has nothing to do with spatial autocorrelation, and everything to do with linear regression residuals. This negative quantity operates as a bias adjustment factor, similar to the multiplication of the maximum likelihood sample variance estimator, s

^{2}, by n/(n − 1) to obtain an unbiased estimator. It arises because of the linear constraint placed on the calculation of MC by using the sample mean $\overline{\mathrm{y}}$ when μ is unknown. Although this confusion does not alter the mechanics of linear regression residual spatial autocorrelation evaluations, it does alter the understanding of spatial autocorrelation. A better understanding of the NSA would have avoided this confusion.

## 2. A Brief Overview of Moran Eigenvector Spatial Filtering (MESF)

**R**, that has all ones in its diagonal entries, and all off-diagonal entries that are less than or equal to one in absolute value. Consequently, its eigenvalues are non-negative. A spatial weights matrix

**C**has all zeroes in its diagonal entries, with a sizeable number of zeroes in its off-diagonal entries [planar graph based polygon adjacency definitions result in no more than 6(n − 3) non-zero entries; distance truncations usually result in far less than 50% non-zero entries]. Consequently, its eigenvalues are both positive and negative (see Figure 5c). MESF and PCNM treat a modified version of matrix

**C**(see Equation (2)) like PCA treats matrix

**R**. Eigenfunction calculations are for these specific matrices. PCA uncovers dimensions spanning the variables used to construct matrix

**R**. It computes scores by constructing linear combinations of the original variables, with the coefficients of a given linear combination being the elements of a particular eigenvector; sometimes this synthetic variate is multiplied by the square root of its eigenvector’s corresponding eigenvalue. Matrix

**R**eigenvalues represent variance accounted for by each principal component; this multiplication seeks to preserve the original total variance. These synthetic variates are orthogonal and uncorrelated. Because the square root of negative eigenvalues is a complex number, PCNM initially and erroneously dismissed their eigenfunctions. However, the eigenfunctions of matrix expression (2) pertain to spatial autocorrelation, with negative eigenvalues indexing NSA. The eigenvectors here are also orthogonal and uncorrelated. In contrast to PCA, for MESF and PCNM, the eigenvectors themselves are synthetic variates; they are not used to construct linear combinations. Each element of a given eigenvector links to a particular location, in accordance with the ordering defined by the affiliated spatial weights matrix. Therefore, eigenvectors can be mapped (see Figure 5b), visualizing the nature and degree of spatial autocorrelation designated by their corresponding eigenvalues. In both PCA and MESF/PCNM, the synthetic variates produced can be included as covariates in a regression equation. By doing so, MESF eigenvectors account for residual spatial autocorrelation.

**C**are the eigenfunctions of matrix expression (2), which appears in the numerator of the MC. Eigenvalues, λ, are the n scalar solutions to the nth-order polynomial matrix equation

**I**−

**11**

^{T}/n)

**C**(

**I**−

**11**

^{T}/n) − λ

**I**] = 0;

**E**are the non-trivial vector solutions (i.e.,

**E**≠

**0**) to the equation

**I**−

**11**

^{T}/n)

**C**(

**I**−

**11**

^{T}/n) − λ

**I**]

**E**=

**0**.

**1**, the intercept covariate in a regression model specification; and, (2) this conceptualization relates to the spatially structured random effects component of mixed models [25]. Including eigenvectors as covariates, by selecting relevant ones with a stepwise procedure (see Appendix A), enables spatial autocorrelation to be accounted for in a conventional statistical estimation context, in either a linear or a GLM specification.

## 3. Selected Case Studies Demonstrating the Presence and Importance of NSA

_{0}: zero spatial autocorrelation), which depends upon variables studied. Scientific orthodoxy and societal “big bang for the bucks” viewpoints sometimes stifle, and even eliminate, the types of variables studied, at least until expert opinion governing these paradigms shifts. Over the years, the former has placed far more emphasis on cooperation and expansion, whereas the latter has deemed many static or comparative static investigations fashionable. However, some change in variable subject matter is underway, such as defining the population of areas in terms of the time of day rather than permanent residence a la census reports (i.e., differences between daytime and nighttime population in a location). The final step for concept development is model identification, specification, and estimation (e.g., spatial autoregressive, geostatistical semi-variogram, and MESF equations). These four steps form a natural progression, from concept formation, through the univariate statistical description and sample-to-population inference, to formal model representation.

#### 3.1. Market Area Competition: NSA and Facility Closures

#### 3.2. Journey-To-Work: Shifts In Daytime and Nighttime Populations

^{−1.25}; for simplicity, the inverse itself, 1/Y, also was explored; in either case, because the translation parameter is zero, the variable is equivalent to a nighttime-to-daytime population ratio with an exponent of 1.25).

_{MC}= −3.8), with MC = −0.123. Because its lower limit, in this case, was −0.730, stretching its MC measurement scale endpoint to −1 indicated that this NSA was stronger than it appears (i.e., −0.168 ≈ −0.123/|−0.730|; meanwhile, the positive scale stretched to 1.179, indicating that 0.072 effectively was more like 0.061 ≈ 0.072/1.179, a value closer to the expected value of −1/(1046−1) ≈ −0.001). An MESF analysis revealed that NSA accounted for nearly 20% of the geographic variation in this Box-Cox transformed daytime-to-nighttime ratio.

_{H}or y

_{L}neighboring values were merged (this implementation was inspired by Traun and Loidl [28]). The outcome was a reduction in n from 1046 to 472 (this change of geographic resolution is comparable to using ZIP (zone improvement plan) code tabulation areas rather than census tracts). Adjacent census tracts with relatively similar Box-Cox transformed daytime-to-nighttime population ratios were merged (Figure 10a), which eliminated a major source of PSA when calculating MC values; it essentially constituted the PSA component of the prevailing PSA–NSA mixture. The resulting geographic distribution exhibited starker contrasts (Figure 10b) and yielded a Moran scatterplot (Figure 10c) portraying NSA very similar to that for the PSA-adjusted residuals used to construct Figure 9b. Now MC = −0.123 (z

_{MC}= −4.7). Because the minimum MC value (the original census tract surface partitioning has extreme MC values of −0.730 and 1.179, which is very similar to those for the aggregated census tract surface partitioning of −0.723 and 1.191) here was −0.723, this NSA index value was more like one of −0.185 (≈ −0.123/|−0.723|) when stretching the endpoint of its MC scale to −1. An MESF analysis revealed that PSA accounted for roughly 9% of the geographic variation in this spatially aggregated Box-Cox transformed daytime-to-nighttime ratio, a substantial decrease from about 25%. This MESF analysis also revealed that NSA accounted for 32% of this geographic variation, a substantial increase from nearly 20%. The new weak mixture ESF residuals contained a marginal amount of spatial autocorrelation, producing z

_{MC}= −1.97 (the expected value was 0.011, very close to zero); it was barely statistically but not substantively significant, given its expected value was thus close to zero Therefore, this data experiment confirmed that accounting for PSA through polygon merging removes its masking of NSA. It also confirmed that the role geographic resolution, polygon and variable definitions, and PSA–NSA mixtures played in detecting NSA can be crucial.

#### 3.3. Urban Area Shrinkage

^{ratio}; this transformation increases the Shapiro-Wilk (S–W) normality diagnostic statistic from 0.16 to 0.85 (it remains significant, but is much closer to 1, as well as indicative of better symmetry) (Manly [30] argues for adding e

^{αy}to the family of Box-Cox transformations; here the estimate of α was approximately −1; because this inverse transformation changed the nature of the relationship between Y and other variables, it was subtracted from 1 to preserve these original relationships).

_{H}tending to pair with y

_{H}, in the first quadrant of the scatterplot, and y

_{L}tending to pair with y

_{L}, in the third quadrant of the scatterplot. LISA respectively denotes these pairings as H–H and L–L. In addition, the previous description of NSA trend in a Moran scatterplot is in terms of y

_{L}tending to pair with y

_{H}, in the second quadrant of the scatterplot, and y

_{H}tending to pair with y

_{L}, in the fourth quadrant of the scatterplot. LISA respectively denotes these pairings as L–H and H–L. Figure 11b portrays the LISA statistics map for Detroit, which highlights the prevalence of H–L and L–H population change ratio clusters. Supplementing these pockets of contrast are clusters of similar values, implying the presence of a PSA–NSA mixture geographic distribution. For this case, MC/MC

_{max}= 0.135/1.12493 ≈ 0.120, indicating very weak global PSA.

_{MC}= 0.56 (the expected value was −0.019, very close to zero).

#### 3.4. 1990. Homicide Rates in the US South Revisited

^{2}of 0.333 for the three-year average homicide rates. The analysis summarized here made use of only the 1990 data, and employed a Box-Cox power transformation to better align the empirical homicide rates with a normal distribution. This revised analysis yielded $\widehat{\mathsf{\rho}}$ = 0.259 and a pseudo-R

^{2}of 0.357. Thus, these two sets of results were compatible.

^{2}of 0.311 with quasi-likelihood Poisson regression, and 0.308 with NB regression, which produced a dispersion parameter estimate of 0.106. One expectation was that the spatial autocorrelation latent in these data was a PSA–NSA mixture. A substantive rationale for this contention was the common place features in which many homicides occurred (PSA), on the one hand, and the crime displacement hypothesis (NSA), on the other hand. The analysis summarized here employed MESF because the auto-Poisson model specification was unable to accommodate PSA [13]. Of the 1412 adjusted spatial weights matrix eigenvectors [see matrix expression (2)], the posited candidate set included 352 containing PSA (based on MC

_{j}/MC

_{max}≥0.25), and 482 containing NSA (based on MC

_{j}/MC

_{max}< –0.25). The constructed ESF contained 110 PSA (ESF

_{PSA}) and 102 NSA (ESF

_{NSA}) eigenvectors. The ESF

_{PSA}component (Figure 9a) accounted for roughly 17% of the geographic variation in homicide rates across the US south, and represented moderate PSA (MC = 0.735/1.111 ≈ 0.662), whereas the ESF

_{NSA}component (Figure 13b) accounted for roughly 12%, of this geographic variation, and represented moderate NSA (MC = –0.385/|–0.605| ≈ –0.636). The final pseudo-R

^{2}was 0.596, substantially greater than that for the normal approximation spatial lag specification. In addition, the inclusion of the PSA eigenvectors reduced the NB dispersion parameter from 0.106 to 0.026. Inclusion of the NSA eigenvectors reduced it to 0, returning the specification to a more parsimonious standard Poisson one. Some small amount of residual PSA existed, with MC = 0.018. The exact distribution theory for this residual spatial autocorrelation presently is unknown; hence, a simulation experiment (10,000 replications) was conducted to estimate its mean and variance. The simulation experiment rendered a $\overline{\mathrm{pseudo}-{\mathrm{R}}^{2}}$ of 0.660, slightly better than the observed 0.596. The test statistic was z

_{MC}= 1.72, which was neither statistically (two-tailed test with α = 0.10) nor substantively significant; its expected value of −0.014 was very close to zero.

## 4. Conclusions; Lessons Learned, and Implications

- Developing appropriate quantification modifications that transform NSA index scales to the interval [−1, 0);
- Evaluating the impact of different definitions of spatial weights (e.g., topological adjacency, distance, and nearest neighbors), as well as distance standardization [33], on a resulting NSA value;
- Devising general map pattern descriptions for different degrees of NSA (paralleling the global, regional, and local descriptors for PSA);
- Revisiting various data analytic features that entail a change of studied variables (e.g., denominators of rates and populations at risk);
- Articulating relationships between NSA and both geographic scale and resolution, as suggested by the geostatistical wave-hole semi-variogram model and this paper’s aggregation experimental results for Detroit;
- Seeking an informed answer to the question asking whether or not areal unit polygons should be designed to mask or accentuate NSA;
- Addressing repeatability and replicability of findings by investigating case studies beyond Detroit, the DFW MSA, and the US South with exploratory spatial statistical analysis of other geographic landscapes to see if they, too, exhibit NSA;
- Expanding findings about the full range of geographic flows beyond the DFW MSA journey-to-work analysis presented here;
- Relating MC values in the cross-validation type close-one-store scenario to individual outlet attributes;
- Replacing Thiessen polygons with Huff probabilities in market area competition analyses;
- Establishing the range of PSA–NSA mixtures, and further explicating the notion of hidden NSA;
- Assessing the range of geographic variance accounted for by NSA, specifically to ascertain whether or not 10% is common, and 25% is exceptional;
- Comprehensively evaluating the strategy of separately estimating ESF
_{PSA}and ESF_{NSA}components; - Confirming more cases where ignoring NSA results in specification error;
- Determining the phantom/search degrees of freedom for a given nature and degree of spatial autocorrelation; and,
- Formulating a better understanding of effective geographic sample size as it relates to phantom/search degrees of freedom.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. About PSA-NSA Mixtures and MESF Eigenvector Selection

_{s}, attributable to the combinatorial selection process that considers variables whether or not they are selected (e.g., considerable multiple testing occurs). Spatial autocorrelation represents redundant information in a variable that effectively reduces n to some smaller number of equivalent independent and identically distributed (IID) values [36]; this effective geographic sample size is the corresponding n for IID values void of spatial autocorrelation. Chun et al. [37] exploit this georeferenced data analytic property to establish a PSA cases formula that determines how many eigenvectors should be included in the candidate set from which selection is made by stepwise regression. As the degree of PSA increases, this candidate set increases (i.e., the effective geographic sample size decreases toward its lower limit of one). The quantities p

_{s}and spatial autocorrelation based effective geographic sample size overlap to some degree: As spatial autocorrelation increasingly is accounted for by eigenvectors, its variance inflation in Y decreases as the accompanying effective geographic sample size moves closer and closer to n, with this increase partly accounting for phantom/search degrees of freedom, causing the resulting z/t-scores to increase, changing their zero spatial autocorrelation null hypothesis, H

_{0}, probabilities. For a pure spatial autocorrelation linear regression model specification, the parameter estimates’ variance-covariance matrix becomes

**0**is a 1-by-k vector of zeroes,

**I**is the k-by-k identity matrix, MSE denotes mean squared error, and ESS denotes error sum of squares. ESS contains the variance inflation term TR(

**V**

^{−1})/n, where TR denotes the matrix trace operator, and

**V**is a spatial covariance matrix [e.g., (

**I**− ρ

**W**)

^{T}(

**I**− ρ

**W**) for the simultaneous autoregressive (SAR) model; see Section 3.4]. This variance inflation factor accompanies the adjustment factor TR(

**V**

^{−1})/

**1**

^{T}

**V**

^{−1}

**1**that multiplies n in the matrix to convert it to its effective geographic sample size [36]. As k increases (i.e., a stepwise procedure includes an increasing number of eigenvectors), ρ → 0, TR(

**V**

^{−1})/n → 1, and [TR(

**V**

^{−1})/(

**1**

^{T}

**V**

^{−1}

**1**)]n → n; the MSE decreases (because of k in its denominator, as well as variance accounted for by selected eigenvectors).

_{s})

_{s}. Table A1 reports integer p

_{s}values for the Detroit and DFW MSA normal random variable case studies appearing in this paper. LASSO estimates [35] essentially corroborate these calculations (the LASSO residuals still contain significant spatial autocorrelation; the DFW MSA LASSO results involve only an NSA ESF). Table A2 entries indicate that adjusting this typically underestimated variance [38] fails to change the H

_{0}probabilities enough to alter the sets of selected eigenvectors when a one-tail α = 0.10 test replaces a two-tailed test. Justification for this one-tail replacement of a two-tailed test is that eigenvectors are unique except for a multiplicative factor of −1; in other words, the sign of an eigenvector regression coefficient is arbitrary, converting it to a one-tail test situation. In addition, this is the expected outcome because the residual MC values signify the presence of only trace amounts of spatial autocorrelation. Therefore, the division of α by 2 operates like a Bonferroni type of adjustment. For the Detroit case study, for example, this adjustment retains five PSA eigenvectors that account for only 2.80% of variance, and nine NSA eigenvectors that account for only 4.72% of variance, preventing an increase in the residual z

_{MC}from 0.56 to 2.89, which becomes significant (with an accompanying increase in effective geographic sample size). Because the goal of MESF is to account for spatial autocorrelation, elimination of it in residuals should be the prevailing criterion. The difference between the Detroit R

^{2}and predicted-R

^{2}of 0.502–0.320 is bothersome; however, the associated PRESS (predicted residual sum of squares) statistic falls within the 95% confidence interval of the original MESF PRESS statistic.

_{s}in each estimation situation, being exceptionally ad hoc is a serious weakness of this approach. The topmost question of interest here may be phrased as follows: Especially given that NSA characterizes regression residuals because population parameters are unknown and estimated with sample statistics, is an estimated PSA–NSA mixture ESF portraying heterogeneous spatial autocorrelation (i.e., some individual MC cross-product terms being positive and some being negative, with magnitudes beyond the token ones materializing with genuinely zero spatial autocorrelation) at a given geographic resolution an artifact of the methodology? In other words, does an NSA component emerge strictly to counterbalance part or all of an estimated PSA component, and vice versa? This appendix furnishes evidence to answer this question from a set of simulation experiments, each based upon one of the DFW MSA (aggregated census tracts), Detroit, and US South geographic landscapes studied in this paper in terms of PSA–NSA mixtures. Thus, these experiments have n values of 308, 472, and 1412. In each case, the candidate eigenvector set contains >> 0.5n vectors, implying a need for these simulation experiments [38].

**Table A1.**Summary quantities for determining phantom/search degrees of freedom for constructed ESFs.

Geographic Landscape | Detroit | DFW MSA |
---|---|---|

Standardized response variable variance | 1 | 1 |

SAR residual variance | 0.69799 | 0.86834 |

ESF residual variance | 129.69892/(308 − 1 − 48) | 276.43766/(472 − 1 − 51) |

Estimated search degrees of freedom | 73 | 102 |

Corrected ESF residual variance | 0.69731 | 0.86930 |

LASSO based ESF residual variance | 0.68271 | 0.90344 |

**Table A2.**Frequencies of probabilities under H

_{0}of eigenvectors selected in the empirical analyses.

H_{0} Probability | <0.0001 | 0.0001–0.0050 | 0.0050–0.0100 | 0.0100–0.0500 | 0.0500–0.1000 |
---|---|---|---|---|---|

PSA | |||||

Detroit | 4 (2;1) | 6 (7;6) | 0 (1;2) | 8 (8;3) | 5 (5;6) |

DFW MSA | 0 (0;0) | 0 (0;0) | 0 (0;0) | 10 (9;2) | 6 (7;7) |

US South | 24 | 41 | 10 | 27 | 8 |

NSA | |||||

Detroit | 2 (0;0) | 2 (2;2) | 1 (2;1) | 11 (12;7) | 9 (9;6) |

DFW MSA | 1 (1;1) | 11 (4;1) | 6 (6;3) | 14 (20;17) | 3 (4;9) |

US South | 7 | 43 | 9 | 38 | 5 |

^{2}= 1 for which the probability of the Shapiro-Wilk [i.e., P(S-W)] diagnostic statistic implied a failure to reject the null hypothesis (to avoid contamination by one obvious source of specification error), and linear regression variable selection significance levels of 0.005 and 0.010. These significance levels represented the extremely liberal Bonferroni adjusted probability thresholds that compensated for the phantom/search degrees of freedom, in the presence of zero spatial autocorrelation, of 0.10/308 ≈ 0.00032, 0.10/472 ≈ 0.00021, and 0.10/1471 ≈ 0.00007. The candidate eigenvector sets were determined by MC

_{j}/MC

_{max}> 0.25 for PSA vectors, and MC

_{j}/|MC

_{min}| < −0.25 for NSA vectors. The number of NSA eigenvectors in a candidate set was, respectively, about 70%, 80%, and 90% more than the number of PSA eigenvectors. Table A3 summarizes output from these simulation experiments.

**Table A3.**Summary statistics from the simulation experiments (ranges are in parentheses); a normal random variable with μ = 0 and σ

^{2}= 1, stepwise regression eigenvector selection, and 10,000 replications.

Statistic | $\widehat{\mathsf{\mu}}$ | $\widehat{\mathsf{\sigma}}$ | $\overline{\mathbf{P}\left(\mathbf{S-W}\right)}$ | $\overline{\#\mathbf{v}\mathbf{e}\mathbf{c}\mathbf{t}\mathbf{o}\mathbf{r}\mathbf{s}}$ | Candidate set Size | $\overline{{\mathbf{R}}^{2}}$ | $\overline{\mathbf{P}\mathbf{R}\mathbf{E}\mathbf{S}\mathbf{S}/\mathbf{E}\mathbf{S}\mathbf{S}}$ |
---|---|---|---|---|---|---|---|

α = 0.005 | |||||||

Detroit | −0.000 (−0.223, 0.207) | 0.999 (0.833, 1.156) | 0.548 (0.1001, 0.9998) | 1.228 (0, 10) | 80 + 135 = 215 | 0.037 (0, 0.263) | 1.015 (1.007, 1.075) |

DFW | 0.001 (−0.171, 0.175) | 1.000 (0.872, 1.121) | 0.545 (0.1000, 0.9999) | 1.701 (0, 10) | 105 + 193 = 298 | 0.034 (0, 0.180) | 1.012 (1.004, 1.043) |

US South | 0.000 (−0.097, 0.109) | 0.999 (0.932, 1.071) | 0.543 (0.1002, 0.9995) | 5.903 (0, 17) | 352 + 662 = 1,014 | 0.039 (0.017, 0.107) | 1.010 (1.001, 1.035) |

α = 0.010 | |||||||

Detroit | 0.000 (−0.219, 0.197) | 0.999 (0.844, 1.142 | 0.546 (0.1002, 0.9999) | 2.690 (0, 13) | 80 + 135 = 215 | 0.070 (0, 0.292) | 1.025 (1.007, 1.100) |

DFW | −0.000 (−0.183, 0.176) | 0.999 (0.891, 1.113) | 0.547 (0.1001, 0.9997) | 3.618 (0, 15) | 105 + 193 = 298 | 0.062 (0, 0.242) | 1.020 (1.004, 1.074) |

US South | −0.000 (−0.100, 0.091) | 1.000 (0.932, 1.068) | 0.542 (0.1001, 0.9999) | 12.870 (1, 30) | 352 + 662 = 1,014 | 0.073 (0.005, 0.164) | 1.020 (1.003, 1.047) |

_{0}probabilities were extreme enough that increasing an MSE by subtracting more degrees of freedom in its denominator failed to make them nonsignificant when changing from a two-tailed to a one-tail test using the same α. This table also suggests that mixtures may well tend to have a number of both PSA and NSA eigenvectors.

## Appendix B. The Geographic Distribution of the Spatial Means of the Census Tract Centroids

**Figure A1.**The Thiessen polygon delimited market areas of all 27 Kohl’s stores. Solid red circles denote the store locations. Solid gray triangles denote the spatial means of the census tract centroids, weighted by population.

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**Figure 1.**Correlation illustrations; variable tretile value groupings [red denotes high (H), yellow denotes medium (M), and green denotes low (L) values]. Left (

**a**): A cross-tabulation of paired values bivariate frequencies. Right (

**b**): A choropleth map of 2010 Texas population density by county (demarcated by thin black borders), with thick black borders outlining census metropolitan regions.

**Figure 2.**Specimen scatterplots portraying different natures and degrees of bivariate linear correlation (n = 254) using z-score axes; blue denotes 95% prediction intervals, red denotes 95% confidence intervals, and gray denotes trend lines. Left (

**a**): Positive, r ≈ 0.7. Middle (

**b**): Zero, r ≈ 0. Right (

**c**): r ≈ −0.7.

**Figure 3.**Specimen Moran scatterplots portraying different natures and degrees of spatial autocorrelation (n = 254) using z-score axes; blue denotes 95% prediction intervals, red denotes 95% confidence intervals, and gray denotes trend lines; MC

_{max}= 1.098 and MC

_{min}= −0.635 denote the extreme Moran coefficient (MC) values. Left (

**a**): Positive, MC/MC

_{max}≈ 0.7. Middle (

**b**): Zero, MC ≈ 0. Right (

**c**): MC/|MC

_{min}| ≈ −0.7.

**Figure 4.**Examples involving NSA. Left (

**a**): A wave-hole semi-variogram model plot. Middle (

**b**): Texas environmental quality service regions with their superimposed Thiessen polygon counterparts. Right (

**c**): A z-score axes Moran scatterplot for the polygon-to-region area ratio of Figure 4b; MC = −0.274.

**Figure 5.**NSA benchmark exemplars. Left (

**a**): Selected Texas counties (black boundaries) with superimposed Thiessen polygons (red boundaries); MC = 0.051. Middle (

**b**): The maximum NSA map pattern for the Texas counties surface partitioning; relatively speaking, green denotes y

_{L}, yellow denotes y

_{M}, and red denotes y

_{H}. Right (

**c**): Dot plot frequency distributions of the sets of eigenvalues for Figure 5a (top) and Figure 5b (bottom).

**Figure 6.**Approximate market areas (boundaries denoted by Thiessen polygons) for Kohl’s and Albertsons stores in the Dallas-Forth Worth Metropolitan Statistical Area (DFW MSA), with market area redistributions denoted by red lines, and open stores denoted by gray circles. Left (

**a**): Kohl’s department stores. Right (

**b**): Albertsons grocery stores.

**Figure 7.**z-score axes Moran scatterplots for market area population ratios (after-closure size divided by before-closure size). Left (

**a**): Kohl’s department stores. Right (

**b**): Albertsons grocery stores.

**Figure 8.**The 26 Kohl’s open department stores in the DFW MSA. Left (

**a**): Locations and market areas. Right (

**b**): The 26 NSA Moran scatterplot trend lines using z-score axes, each involving a hypothetical single store closure.

**Figure 9.**A 2000 daytime-to-nighttime population ratio for DFW. Left (

**a**): The geographic distribution of the Box-Cox transformed ratio (relatively speaking, green denotes y

_{L}, yellow denotes y

_{M}, and red denotes y

_{H}). Right (

**b**): A z-score axes Moran scatterplot for this transformed ratio.

**Figure 10.**An aggregated 2000 inverse daytime-to-nighttime population ratio for DFW; relatively speaking, green denotes y

_{L}, yellow denotes y

_{M}, and red denotes y

_{H}. Top (

**a**): An overlay of the aggregated areal unit boundaries on the original census tract resolution geographic distribution. Bottom left (

**b**): The geographic distribution of the geographically aggregated data ratio. Bottom right (

**c**): A z-score axes Moran scatterplot for the aggregated data ratio.

**Figure 11.**Detroit. Left (

**a**): The geographic distribution of the population ratio across census tracts; relatively speaking, green denotes y

_{L}, yellow denotes y

_{M}, and red denotes y

_{H}. Right (

**b**): A Local indicators of spatial association (LISA) statistics map; dark red is high-low (H–L), dark blue is low-high (L–H), light red is high-high (H–H), and light blue is low-low (L–L).

**Figure 12.**z-score axes Moran scatterplots for the 2000–2010 Detroit population change ratio. Left (

**a**): Differentiating individual MC covariation terms according to 95% prediction (dotted line) and confidence limits (blue shaded area). Middle (

**b**): Moran scatterplot with statistically significant LISA statistics identified (blue denotes H–H and L–L; red denotes L–H and H–L), with positive spatial autocorrelation (PSA) (blue), NSA (red), and global (black) trend lines superimposed. Right (

**c**): The residual NSA scatterplot after adjusting for PSA.

**Figure 13.**1990 homicide rate eigenvector spatial filters (ESFs). Left (

**a**): The PSA ESF (ESF

_{PSA}). Right (

**b**): The NSA ESF (ESF

_{NSA}). For the ESF measurement scales, relatively speaking, green denotes y

_{L}, yellow denotes y

_{M}, and red denotes y

_{H}.

**Table 1.**Summary results for a simulation experiment examining the frequency of r <0 for independent and identically distributed variables.

Random Variable | Parameters | Skewness | n | % of r < 0 Batches | Average % of r < 0 |
---|---|---|---|---|---|

exponential | λ = 1 | 2.00 | 100 | 92 | 52.2 |

500 | 77 | 51.2 | |||

1000 | 70 | 50.9 | |||

beta | α = 0.3, β = 25 | 3.39 | 100 | 100 | 56.0 |

gamma | α = 0.1, β = 1 | 4.47 | 100 | 100 | 59.0 |

log-normal | μ =0, σ^{2} = 1 | 6.18 | 100 | 100 | 56.0 |

Weibull | λ = 0.25, κ = 1 | 60.09 | 100 | 100 | 75.0 |

500 | 100 | 73.0 | |||

1000 | 100 | 72.0 |

**Table 2.**MCs for a leave-one-out (i.e., a single store closure) market size competition analysis for Kohl’s department stores in the DFW MSA.

Store ID | MC | MC/|MC_{min} | Prob (H_{0}) | Store ID | MC | MC/|MC_{min}| | Prob (H_{0}) |
---|---|---|---|---|---|---|---|

1 | −0.206 | −0.383 | 0.091 | 14 | −0.181 | −0.336 | 0.129 |

2 | −0.250 | −0.465 | 0.045 | 15 | −0.136 | −0.253 | 0.219 |

3 | −0.280 | −0.520 | 0.027 | 16 | −0.205 | −0.381 | 0.092 |

4 | −0.225 | −0.418 | 0.068 | 17 | −0.311 | −0.578 | 0.014 |

5 | −0.195 | −0.362 | 0.106 | 18 | −0.243 | −0.452 | 0.051 |

6 | −0.213 | −0.396 | 0.082 | 19 | −0.230 | −0.428 | 0.062 |

7 | −0.215 | −0.400 | 0.079 | 21 | −0.250 | −0.465 | 0.045 |

8 | −0.217 | −0.403 | 0.077 | 22 | −0.159 | −0.296 | 0.170 |

9 | −0.236 | −0.439 | 0.057 | 23 | −0.233 | −0.433 | 0.060 |

10 | −0.269 | −0.500 | 0.033 | 24 | −0.202 | −0.375 | 0.095 |

11 | −0.284 | −0.528 | 0.024 | 25 | −0.276 | −0.513 | 0.028 |

12 | −0.200 | −0.372 | 0.098 | 26 | −0.148 | −0.275 | 0.192 |

13 | −0.252 | −0.468 | 0.044 | 27 | −0.183 | −0.340 | 0.125 |

_{max}= 0.855, and MC

_{min}= −0.538 (this value differs from that for Figure 6 because n is different).

Covariate | Model Specification | ||||
---|---|---|---|---|---|

Normal Approximation | Poisson | NB | NB + ESF_{PSA} | Poisson + ESF_{PSA} & ESF_{NSA} | |

Resource deprivation/affluence | 0.4239 * | 0.5250 * | 0.4649 * | 0.4733 * | 0.4937 * |

Population size/density | 0.0806 * | 0.3003 * | 0.2108 * | 0.1825 * | 0.2095 * |

Median age | −0.0035 * | −0.0104 * | −0.0020 * | −0.0005 * | −0.0015 * |

Divorce rate | 0.0453 * | 0.0632 * | 0.0588 * | 0.0840 * | 0.0765 * |

Unemployment rate | −0.0591 * | −0.0731 * | −0.0536 * | −0.0372 * | −0.0463 * |

Deviance statistic | 3.1343 | 1.0926 | 1.1897 | 1.0874 | |

Over-dispersion parameter | 0.1064 | 0.0256 | 0.0000 | ||

(Pseudo)-R^{2} | 0.281 | 0.311 | 0.308 | 0.476 | 0.596 |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Griffith, D.A.
Negative Spatial Autocorrelation: One of the Most Neglected Concepts in Spatial Statistics. *Stats* **2019**, *2*, 388-415.
https://doi.org/10.3390/stats2030027

**AMA Style**

Griffith DA.
Negative Spatial Autocorrelation: One of the Most Neglected Concepts in Spatial Statistics. *Stats*. 2019; 2(3):388-415.
https://doi.org/10.3390/stats2030027

**Chicago/Turabian Style**

Griffith, Daniel A.
2019. "Negative Spatial Autocorrelation: One of the Most Neglected Concepts in Spatial Statistics" *Stats* 2, no. 3: 388-415.
https://doi.org/10.3390/stats2030027