# Chapter 13 Area Data and Spatial Autocorrelation

Areal units of observation are very often used when simultaneous observations are aggregated within non-overlapping boundaries. The boundaries may be those of administrative entities, and may be related to underlying spatial processes, such as commuting flows, but are usually arbitrary. If they do not match the underlying and unobserved spatial processes in one or more variables of interest, proximate areal units will contain parts of the underlying processes, engendering spatial autocorrelation. This is at least in part because the aggregated observations are driven by autocorrelated factors which may or may not themselves have been observed.

With support of data we mean the physical size (lenth, area, volume) associated with an individual observational unit (measurement). It is possible to represent the support of areal data by a point, despite the fact that the data have polygonal support. The centroid of the polygon may be taken as a representative point, or the centroid of the largest polygon in a multi-polygon object. When data with intrinsic point support are treated as areal data, the change of support goes the other way, from the known point to a non-overlapping tesselation such as a Voronoi diagram or Dirichlet tessellation or Thiessen polygons often through a Delaunay triangulation and using a Euclidean plane (projected coordinates). Here, different metrics may also be chosen, or distances measured on a network rather than on the plane. There is also a literature using weighted Voronoi diagrams in local spatial analysis (see for example Boots and Okabe 2007; Okabe et al. 2008; She et al. 2015).

When the intrinsic support of the data is as points, but the underlying process is between proximate observations rather than driven chiefly by distance however measured between observations, the data may be aggregate counts or totals (polling stations, retail turnover) or represent a directly observed characteristic of the observation (opening hours of the polling station). Obviously, the risk of mis-representing the footprint of the underlying spatial processes remains in all of these cases, not least because the observations are taken as encompassing the entirety of the underlying process in the case of tesselation of the whole area of interest. This is distinct from the geostatistical setting in which observations are rather samples taken using some scheme within the area of interest. It is also partly distinct from the practice of taking areal sample plots within the area of interest but covering only a small proportion of the area, typically used in ecological and environmental research.

This chapter then considers a subset of the methods potentially available for exploring spatial autocorrelation in areal data, or data being handled as areal, where the spatial processes are considered as working through proximity understood in the first instance as contiguity, as a graph linking observations taken as neighbours. This graph is typically undirected and unweighted, but may be directed and/or weighted in certain settings, which then leads to further issues with regard to symmetry. In principle, proximity would be expected to operate symmetrically in space, that is that the influence of $$i$$ on $$j$$ and of $$j$$ on $$i$$ based on their relative positions should be equivalent. Edge effects are not considered in standard treatments.

## 13.1 Spatial autocorrelation

When analysing areal data, it has long been recognised that, if present, spatial autocorrelation changes how we may infer, relative to the default position of independent observations. In the presence of spatial autocorrelation, we can predict the values of observation $$i$$ from the values observed at $$j \in N_i$$, the set of its proximate neighbours. Early results (Moran 1948; Geary 1954), entered into research practice gradually, for example the social sciences (Duncan, Cuzzort, and Duncan 1961). These results were then collated and extended to yield a set of basic tools of analysis (Cliff and Ord 1973, 1981).

Cliff and Ord (1973) generalised and extended the expression of the spatial weights matrix representation as part of the framework for establishing the distribution theory for join count, Moran’s $$I$$ and Geary’s $$C$$ statistics. This development of what have become known as global measures, returning a single value of autocorrelation for the total study area, has been supplemented by local measures returning values for each areal unit (Getis and Ord 1992; Anselin 1995).

## 13.2 Spatial weight matrices

Handling spatial autocorrelation using relationships to neighbours on a graph takes the graph as given, chosen by the analyst. This differs from the geostatistical approach in which the analyst chooses the binning of the empirical variogram and function used, and then the way the fitted variogram is fitted. Both involve a priori choices, but represent the underlying correlation in different ways (Wall 2004). In Bavaud (1998) and work citing his contribution, attempts have been made to place graph-based neighbours in a broader context.

One issue arising in the creation of objects representing neighbourhood relationships is that of no-neighbour areal units (Bivand and Portnov 2004). Islands or units separated by rivers may not be recognised as neighbours when the units have areal support and when using topological relationships such as shared boundaries. In some settings, for example mrf (Markov Random Field) terms in mgcv::gam() and similar model fitting functions that require undirected connected graphs, a requirement is violated when there are disconnected subgraphs.

No-neighbour observations can also occur when a distance threshold is used between points, where the threshold is smaller than the maximum nearest neighbour distance. Shared boundary contiguities are not affected by using geographical, unprojected coordinates, but all point-based approaches use distance in one way or another, and need to calculate distances in an appropriate way.

The spdep package provides an nb class for neighbours, a list of length equal to the number of observations, with integer vector components. No-neighbours are encoded as an integer vector with a single element 0L, and observations with neighbours as sorted integer vectors containing values in 1L:n pointing to the neighbouring observations. This is a typical row-oriented sparse representation of neighbours. spdep provides many ways of constructing nb objects, and the representation and construction functions are widely used in other packages.

spdep builds on the nb representation (undirected or directed graphs) with the listw object, a list with three components, an nb object, a matching list of numerical weights, and a single element character vector containing the single letter name of the way in which the weights were calculated. The most frequently used approach in the social sciences is calculating weights by row standardization, so that all the non-zero weights for one observation will be the inverse of the cardinality of its set of neighbours (1/card(nb[[i]])).

We will be using election data from the 2015 Polish Presidential election in this chapter, with 2495 municipalities and Warsaw boroughs (see Figure 13.1 for a tmap map (section 9.5) of the municipality types) , and complete count data from polling stations aggregated to these areal units. The data are an sf sf object:

library(sf)
data(pol_pres15, package="spDataLarge")
head(pol_pres15[, c(1, 4, 6)])
# Simple feature collection with 6 features and 3 fields
# Geometry type: MULTIPOLYGON
# Dimension:     XY
# Bounding box:  xmin: 235000 ymin: 367000 xmax: 281000 ymax: 413000
# Projected CRS: ETRS89 / Poland CS92
#    TERYT                name       types                       geometry
# 1 020101         BOLESŁAWIEC       Urban MULTIPOLYGON (((261089 3855...
# 2 020102         BOLESŁAWIEC       Rural MULTIPOLYGON (((254150 3837...
# 3 020103            GROMADKA       Rural MULTIPOLYGON (((275346 3846...
# 4 020104        NOWOGRODZIEC Urban/rural MULTIPOLYGON (((251770 3770...
# 5 020105          OSIECZNICA       Rural MULTIPOLYGON (((263424 4060...
# 6 020106 WARTA BOLESŁAWIECKA       Rural MULTIPOLYGON (((267031 3870...
library(tmap)
tm_shape(pol_pres15) + tm_fill("types")

Between early 2002 and April 2019, spdep contained functions for constructing and handling neighbour and spatial weights objects, tests for spatial autocorrelation, and model fitting functions. The latter have been split out into spatialreg, and will be discussed in the next chapter. spdep now accommodates objects represented using sf classes and sp classes directly.

library(spdep)
# Loading required package: sp
# Loading required package: spData

### 13.2.1 Contiguous neighbours

The poly2nb() function in spdep takes the boundary points making up the polygon boundaries in the object passed as the pl= argument, and for each observation checks whether at least one (queen=TRUE, default), or at least two (rook, queen=FALSE) points are within snap= distance units of each other. The distances are planar in the raw coordinate units, ignoring geographical projections. Once the required number of sufficiently close points is found, the search is stopped.

args(poly2nb)
# function (pl, row.names = NULL, snap = sqrt(.Machine$double.eps), # queen = TRUE, useC = TRUE, foundInBox = NULL, small_n = 500) # NULL Two other arguments are also worth discussing, foundInBox= and small_n=. The first, foundInBox=, accepted the output of the rgeos gUnarySTRtreeQuery() function to list candidate neighbours, that is polygons whose bounding boxes intersect the bounding boxes of other polygons. From spdep 1.1-7, the GEOS interface of the sf package is used within poly2nb() if foundInBox=NULL and the number of observations is greater than small_n=, to find the candidate neighbours and populate foundInBox internally. In this case, this use of spatial indexing (STRtree queries) in GEOS through sf is the default, as the number of observations is greater than small_n: system.time(nb_q <- poly2nb(pol_pres15, queen=TRUE)) # user system elapsed # 0.859 0.000 0.860 The print method shows the summary structure of the neighbour object: nb_q # Neighbour list object: # Number of regions: 2495 # Number of nonzero links: 14242 # Percentage nonzero weights: 0.229 # Average number of links: 5.71 Raising small_n= above the observation count, we see that the processing time is increased a little; the benefits of indexing are more apparent with larger data sets. system.time(nb_q_legacy <- poly2nb(pol_pres15, queen=TRUE, small_n=2500)) # user system elapsed # 0.998 0.000 0.998 The output objects are identical: all.equal(nb_q, nb_q_legacy, check.attributes=FALSE) # [1] TRUE Much of the work involved in finding contiguous neighbours is spent on finding candidate neighbours with intersecting bounding boxes. Note that nb objects record both symmetric neighbour relationships, because these objects admit asymmetric relationships as well, but these duplications are not needed for object construction. Most of the spdep functions for constructing neighbour objects take a row.names= argument, the value of which is stored as a region.id attribute. If not given, the values are taken from row.names() of the first argument. These can be used to check that the neighbours object is in the same order as data. If nb objects are subsetted, the indices change to continue to be within 1:length(subsetted_nb), but the region.id attribute values point back to the object from which it was constructed. We can also check that this undirected graph is connected using the n.comp.nb() function; while some model estimation techniques do not support graphs that are not connected, it is helpful to be aware of possible problems (Freni-Sterrantino, Ventrucci, and Rue 2018): n.comp.nb(nb_q)$nc
# [1] 1

Neighbour objects may be exported and imported in GAL format for exchange with other software, using write.nb.gal() and read.gal():

tf <- tempfile(fileext=".gal")
write.nb.gal(nb_q, tf)

### 13.2.2 Graph-based neighbours

If areal units are an appropriate representation, but only points have been observed, contiguity relationships may be approximated using graph-based neighbours. In this case, the imputed boundaries tesselate the plane such that points closer to one observation than any other fall within its polygon. The simplest form is by using triangulation, here using the deldir() function in the deldir package. Because the function returns from and to identifiers, it is easy to construct a long representation of a listw object, as used in the S-Plus SpatialStats module and the sn2listw() function internally to construct an nb object (ragged wide representation). Alternatives often fail to return sufficient information to permit the neighbours to be identified.

The soi.graph() function takes triangulated neighbours and prunes off neighbour relationships represented by unusually long edges, especially around the convex hull, but may render the output object asymmetric. Other graph-based approaches include relativeneigh() and gabrielneigh().

The output of these functions is then converted to the nb representation using graph2nb(), with the possible use of the sym= argument to coerce to symmetry. We take the centroids of the largest component polygon for each observation as the point representation; population-weighted centroids might have been a better choice if they were available:

coords <- st_centroid(st_geometry(pol_pres15), of_largest_polygon=TRUE)
suppressMessages(nb_tri <- tri2nb(coords))
nb_tri
# Neighbour list object:
# Number of regions: 2495
# Number of nonzero links: 14930
# Percentage nonzero weights: 0.24
# Average number of links: 5.98

The average number of neighbours is similar to the Queen boundary contiguity case, but if we look at the distribution of edge lengths using nbdists(), we can see that although the upper quartile is about 15 km, the maximum is almost 300 km, an edge along much of one side of the convex hull. The short minimum distance is also of interest, as many centroids of urban municipalities are very close to the centroids of their surrounding rural counterparts.

summary(unlist(nbdists(nb_tri, coords)))
#    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#     247    9847   12151   13485   14994  296974

Triangulated neighbours also yield a connected graph:

n.comp.nb(nb_tri)$nc # [1] 1 The sphere of influence graph trims a neighbour object such as nb_tri to remove edges that seem long in relation to typical neighbours (Avis and Horton 1985). nb_soi <- graph2nb(soi.graph(nb_tri, coords)) nb_soi # Neighbour list object: # Number of regions: 2495 # Number of nonzero links: 12792 # Percentage nonzero weights: 0.205 # Average number of links: 5.13 Unpicking the triangulated neighbours does however remove the connected character of the underlying graph: n_comp <- n.comp.nb(nb_soi) n_comp$nc
# [1] 16

The SoI algorithm has stripped out longer edges leading to urban and rural municipality pairs where their centroids are very close to each other because the rural ones completely surround the urban, giving 15 pairs of neighbours unconnected to the main graph:

table(n_comp$comp.id) # # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # 2465 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 The largest length edges along the convex hull have been removed, but “holes” have appeared where the unconnected pairs of neighbours have appeared. The differences between nb_tri and nb_soi are shown in orange in Figure 13.2. opar <- par(mar=c(0,0,0,0)+0.5) plot(st_geometry(pol_pres15), border="grey", lwd=0.5) plot(nb_soi, coords=st_coordinates(coords), add=TRUE, points=FALSE, lwd=0.5) plot(diffnb(nb_tri, nb_soi), coords=st_coordinates(coords), col="orange", add=TRUE, points=FALSE, lwd=0.5) par(opar) ### 13.2.3 Distance-based neighbours Distance-based neighbours can be constructed using dnearneigh(), with a distance band with lower d1= and upper d2= bounds controlled by the bounds= argument. If spherical coordinates are used and either specified in the coordinates object x or with x as a two column matrix and longlat=TRUE, great circle distances in km will be calculated assuming the WGS84 reference ellipsoid. From spdep 1.1-7, two arguments have been added, to use functionality in the dbscan package for finding neighbours using a kd-tree in two or three dimensions by default, and not to test the symmetry of the output neighbour object. The knearneigh() function for $$k$$-nearest neighbours returns a knn object, converted to an nb object using knn2nb(). It can also use great circle distances, not least because nearest neighbours may differ when uprojected coordinates are treated as planar. k= should be a small number. For projected coordinates, the dbscan package is used to compute nearest neighbours more efficiently. Note that nb objects constructed in this way are most unlikely to be symmetric, hence knn2nb() has a sym= argument to permit the imposition of symmetry, which will mean that all units have at least k= neighbours, not that all units will have exactly k= neighbours. The nbdists() function returns the length of neighbour relationship edges in the units of the coordinates if the coordinates are projected, in km otherwise. In order to set the upper limit for distance bands, one may first find the maximum first nearest neighbour distance, using unlist() to remove the list structure of the returned object. k1 <- knn2nb(knearneigh(coords)) k1dists <- unlist(nbdists(k1, coords)) summary(k1dists) # Min. 1st Qu. Median Mean 3rd Qu. Max. # 247 6663 8538 8275 10124 17979 Here the largest first nearest neighbour distance is just under 18 km, so using this as the upper threshold gives certainty that all units will have at least one neighbour: system.time(nb_d18 <- dnearneigh(coords, 0, 18000)) # user system elapsed # 0.168 0.000 0.168 system.time(nb_d18a <- dnearneigh(coords, 0, 18000, use_kd_tree=FALSE)) # user system elapsed # 0.152 0.000 0.152 all.equal(nb_d18, nb_d18a, check.attributes=FALSE) # [1] TRUE nb_d18 # Neighbour list object: # Number of regions: 2495 # Number of nonzero links: 20358 # Percentage nonzero weights: 0.327 # Average number of links: 8.16 However, even though there are no no-neighbour observations (their presence is reported by the print method for nb objects), the graph is not connected, as a pair of observations are each others’ only neighbours. n_comp <- n.comp.nb(nb_d18) n_comp$nc
# [1] 2
table(n_comp$comp.id) # # 1 2 # 2493 2 Adding 300 m to the threshold gives us a neighbour object with no no-neighbour units, and all units can be reached from all others across the graph. nb_d183 <- dnearneigh(coords, 0, 18300) nb_d183 # Neighbour list object: # Number of regions: 2495 # Number of nonzero links: 21086 # Percentage nonzero weights: 0.339 # Average number of links: 8.45 n_comp <- n.comp.nb(nb_d183) n_comp$nc
# [1] 1

One characteristic of distance-based neighbours is that more densely settled areas, with units which are smaller in terms of area (Warsaw boroughs are much smaller on average, but have almost 30 neighbours). Having many neighbours smooths the neighbour relationship across more neighbours. For use later, we also construct a neighbour object with no-neighbour units, using a threshold of 16 km:

nb_d16 <- dnearneigh(coords, 0, 16000)
nb_d16
# Neighbour list object:
# Number of regions: 2495
# Number of nonzero links: 15850
# Percentage nonzero weights: 0.255
# Average number of links: 6.35
# 7 regions with no links:
# 569 1371 1522 2374 2385 2473 2474

It is possible to control the numbers of neighbours directly using $$k$$-nearest neighbours, either accepting asymmetric neighbours or imposing symmetry:

knn_k6 <- knearneigh(coords, k=6)
knn2nb(knn_k6)
# Neighbour list object:
# Number of regions: 2495
# Number of nonzero links: 14970
# Percentage nonzero weights: 0.24
# Average number of links: 6
# Non-symmetric neighbours list
nb_k6s <- knn2nb(knn_k6, sym=TRUE)
nb_k6s
# Neighbour list object:
# Number of regions: 2495
# Number of nonzero links: 16810
# Percentage nonzero weights: 0.27
# Average number of links: 6.74

Here the size of k= is sufficient to ensure connectedness, although the graph is not planar as edges cross at locations other than nodes, which is not the case for contiguous or graph-based neighbours.

n_comp <- n.comp.nb(nb_k6s)
n_comp$nc # [1] 1 ### 13.2.4 Weights specification Once neighbour objects are available, further choices need to made in specifying the weights objects. The nb2listw() function is used to create a listw weights object with an nb object, a matching list of weights vectors, and a style specification. Because handling no-neighbour observations now begins to matter, the zero.policy= argument is introduced. By default, this is FALSE, indicating that no-neighbour observations will cause an error, as the spatially lagged value for an observation with no neighbours is not available. By convention, zero is substituted for the lagged value, as the cross product of a vector of zero-valued weights and a data vector, hence the name of zero.policy. args(nb2listw) # function (neighbours, glist = NULL, style = "W", zero.policy = NULL) # NULL We will be using the helper function spweights.constants() below to show some consequences of varing style choices. It returns constants for a listw object, $$n$$ is the number of observations, n1 to n3 are $$n-1, \ldots$$, nn is $$n^2$$ and $$S_0$$, $$S_1$$ and $$S_2$$ are constants, $$S_0$$ being the sum of the weights. There is a full discussion of the constants in Bivand and Wong (2018). args(spweights.constants) # function (listw, zero.policy = NULL, adjust.n = TRUE) # NULL The "B" binary style gives a weight of unity to each neighbour relationship, and typically upweights units with no boundaries on the edge of the study area. lw_q_B <- nb2listw(nb_q, style="B") unlist(spweights.constants(lw_q_B)) # n n1 n2 n3 nn S0 S1 S2 # 2495 2494 2493 2492 6225025 14242 28484 357280 The "W" row-standardized style upweights units around the edge of the study area that necessarily have fewer neighbours. This style first gives a weight of unity to each neighbour relationship, then divides these weights by the per unit sums of weights. Naturally this leads to division by zero where there are no neighbours, a not-a-number result, unless the chosen policy is to permit no-neighbour observations. We can see that $$S_0$$ is now equal to $$n$$. lw_q_W <- nb2listw(nb_q, style="W") unlist(spweights.constants(lw_q_W))[c(1,6:8)] # n S0 S1 S2 # 2495 2495 958 10406 Inverse distance weights are used in a number of scientific fields. Some use dense inverse distance matrices, but many of the inverse distances are close to zero, so have little practical contribution, especially as the spatial process matrix is itself dense. Inverse distance weights may be constructed by taking the lengths of edges, changing units to avoid most weights being too large or small (here from m to km), taking the inverse, and passing through the glist= argument to nb2listw(): gwts <- lapply(nbdists(nb_d183, coords), function(x) 1/(x/1000)) lw_d183_idw_B <- nb2listw(nb_d183, glist=gwts, style="B") unlist(spweights.constants(lw_d183_idw_B))[c(1,6:8)] # n S0 S1 S2 # 2495 1841 534 7265 No-neighbour handling is by default to prevent the construction of a weights object, making the analyst take a position on how to proceed. try(lw_d16_B <- nb2listw(nb_d16, style="B")) # Error in nb2listw(nb_d16, style = "B") : Empty neighbour sets found Use can be made of the zero.policy= argument to many functions used with nb and listw objects. lw_d16_B <- nb2listw(nb_d16, style="B", zero.policy=TRUE) unlist(spweights.constants(lw_d16_B, zero.policy=TRUE))[c(1,6:8)] # n S0 S1 S2 # 2488 15850 31700 506480 Note that by default the adjust.n= argument to spweights.constants() is set by default to TRUE, subtracting the count of no-neighbour observations from the observation count, so $$n$$ is smaller with possible consequences for inference. The complete count can be retrieved by changing the argument. ## 13.3 Measures of spatial autocorrelation Measures of spatial autocorrelation unfortunately pick up other mis-specifications in the way that we model data (Schabenberger and Gotway 2005; McMillen 2003). For reference, Moran’s $$I$$ is given as (Cliff and Ord 1981, 17): $I = \frac{n \sum_{(2)} w_{ij} z_i z_j}{S_0 \sum_{i=1}^{n} z_i^2}$ where $$x_i, i=1, \ldots, n$$ are $$n$$ observations on the numeric variable of interest, $$z_i = x_i - \bar{x}$$, $$\bar{x} = \sum_{i=1}^{n} x_i / n$$, $$\sum_{(2)} = \stackrel{\sum_{i=1}^{n} \sum_{j=1}^{n}}{i \neq j}$$, $$w_{ij}$$ are the spatial weights, and $$S_0 = \sum_{(2)} w_{ij}$$. First we test a random variable using the Moran test, here under the normality assumption (argument randomisation=FALSE, default TRUE): set.seed(1) x <- rnorm(nrow(pol_pres15)) mt <- moran.test(x, lw_q_B, randomisation=FALSE, alternative="two.sided") Inference is made on the statistic $$Z(I) = \frac{I - E(I)}{\sqrt{\mathrm{Var}(I)}}$$, the z-value compared with the Normal distribution for $$E(I)$$ and $$\mathrm{Var}(I)$$ for the chosen assumptions; this x does not show spatial autocorrelation with these spatial weights: glance_htest <- function(ht) c(ht$estimate, "Std deviate"=unname(ht$statistic), "p.value"=unname(ht$p.value))
glance_htest(mt)
# Moran I statistic       Expectation          Variance       Std deviate
#         -0.004772         -0.000401          0.000140         -0.369320
#           p.value
#          0.711889

The test however fails to detect a missing trend in the data as a missing variable problem, finding spatial autocorrelation instead:

beta <- 0.15e-02
t <- st_coordinates(coords)[,1]/1000
x_t <- x + beta*t
mt <- moran.test(x_t, lw_q_B, randomisation=FALSE, alternative="two.sided")
glance_htest(mt)
# Moran I statistic       Expectation          Variance       Std deviate
#          0.043403         -0.000401          0.000140          3.701491
#           p.value
#          0.000214

If we test the residuals of a linear model including the trend, the apparent spatial autocorrelation disappears:

lmt <- lm.morantest(lm(x_t ~ t), lw_q_B, alternative="two.sided")
glance_htest(lmt)
# Observed Moran I      Expectation         Variance      Std deviate
#        -0.004777        -0.000789         0.000140        -0.337306
#          p.value
#         0.735886

A comparison of implementations of measures of spatial autocorrelation shows that a wide range of measures is available in R in a number of packages, chiefly in the spdep package, and that differences from other implementations can be attributed to design decisions (Bivand and Wong 2018). The spdep package also includes the only implementations of exact and Saddlepoint approximations to global and local Moran’s I for regression residuals (Tiefelsdorf 2002; Bivand, Müller, and Reder 2009).

### 13.3.1 Global measures

We will begin by examining join count statistics, where joincount.test() takes a factor vector of values fx= and a listw object, and returns a list of htest (hypothesis test) objects defined in the stats package, one htest object for each level of the fx= argument. The observed counts are of neighbours with the same factor levels, known as same-colour joins.

args(joincount.test)
# function (fx, listw, zero.policy = NULL, alternative = "greater",
#     sampling = "nonfree", spChk = NULL, adjust.n = TRUE)
# NULL

The function takes an alternative= argument for hypothesis testing, a sampling= argument showing the basis for the construction of the variance of the measure, where the default "nonfree" choice corresponds to analytical permutation; the spChk= argument is retained for backward compatibility. For reference, the counts of factor levels for the type of municipality or Warsaw borough are:

table(pol_pres15$types) # # Rural Urban Urban/rural Warsaw Borough # 1563 303 611 18 Since there are four levels, we re-arrange the list of htest objects to give a matrix of estimated results. The observed same-colour join counts are tabulated with their expectations based on the counts of levels of the input factor, so that few joins would be expected between for example Warsaw boroughs, because there are very few of them. The variance calculation uses the underlying constants of the chosen listw object and the counts of levels of the input factor. The z-value is obtained in the usual way by dividing the difference between the observed and expected join counts by the square root of the variance. The join count test was subsequently adapted for multi-colour join counts (Upton and Fingleton 1985). The implementation as joincount.mult() in spdep returns a table based on nonfree sampling, and does not report p-values. joincount.multi(pol_pres15$types, listw=lw_q_B)
#                               Joincount Expected Variance z-value
# Rural:Rural                    3087.000 2793.920 1126.534    8.73
# Urban:Urban                     110.000  104.719   93.299    0.55
# Urban/rural:Urban/rural         656.000  426.526  331.759   12.60
# Warsaw Borough:Warsaw Borough    41.000    0.350    0.347   68.96
# Urban:Rural                     668.000 1083.941  708.209  -15.63
# Urban/rural:Rural              2359.000 2185.769 1267.131    4.87
# Urban/rural:Urban               171.000  423.729  352.190  -13.47
# Warsaw Borough:Rural             12.000   64.393   46.460   -7.69
# Warsaw Borough:Urban              9.000   12.483   11.758   -1.02
# Warsaw Borough:Urban/rural        8.000   25.172   22.354   -3.63
# Jtot                           3227.000 3795.486 1496.398  -14.70

So far, we have used binary weights, so the sum of join counts multiplied by the weight on that join remains integer. If we change to row standardised weights, where the weights are not unity in all cases, the counts, expectations and variances change, but there are few major changes in the z-values.

Using an inverse distance based listw object does, however, change the z-values markedly, because closer centroids are upweighted relatively strongly:

joincount.multi(pol_pres15$types, listw=lw_d183_idw_B) # Joincount Expected Variance z-value # Rural:Rural 3.46e+02 3.61e+02 4.93e+01 -2.10 # Urban:Urban 2.90e+01 1.35e+01 2.23e+00 10.39 # Urban/rural:Urban/rural 4.65e+01 5.51e+01 9.61e+00 -2.79 # Warsaw Borough:Warsaw Borough 1.68e+01 4.53e-02 6.61e-03 206.38 # Urban:Rural 2.02e+02 1.40e+02 2.36e+01 12.73 # Urban/rural:Rural 2.25e+02 2.83e+02 3.59e+01 -9.59 # Urban/rural:Urban 3.65e+01 5.48e+01 8.86e+00 -6.14 # Warsaw Borough:Rural 5.65e+00 8.33e+00 1.73e+00 -2.04 # Warsaw Borough:Urban 9.18e+00 1.61e+00 2.54e-01 15.01 # Warsaw Borough:Urban/rural 3.27e+00 3.25e+00 5.52e-01 0.02 # Jtot 4.82e+02 4.91e+02 4.16e+01 -1.38 The implementation of Moran’s $$I$$ in spdep in the moran.test() function has similar arguments to those of joincount.test(), but sampling= is replaced by randomisation= to indicate the underlying analytical approach used for calculating the variance of the measure. It is also possible to use ranks rather than numerical values (Cliff and Ord 1981, 46). The drop.EI2= agrument may be used to reproduce results where the final component of the variance term is omitted as found in some legacy software implementations. args(moran.test) # function (x, listw, randomisation = TRUE, zero.policy = NULL, # alternative = "greater", rank = FALSE, na.action = na.fail, # spChk = NULL, adjust.n = TRUE, drop.EI2 = FALSE) # NULL The default for the randomisation= argument is TRUE, but here we will simply show that the test under normality is the same as a test of least squares residuals with only the intercept used in the mean model. The spelling of randomisation is that of Cliff and Ord (1973). mt <- moran.test(pol_pres15$I_turnout, listw=lw_q_B, randomisation=FALSE)
glance_htest(mt)
# Moran I statistic       Expectation          Variance       Std deviate
#          0.691434         -0.000401          0.000140         58.461349
#           p.value
#          0.000000

The lm.morantest() function also takes a resfun= argument to set the function used to extract the residuals used for testing, and clearly lets us model other salient features of the response variable (Cliff and Ord 1981, 203). To compare with the standard test, we are only using the intercept here, and as can be seen, the results are the same.

ols <- lm(I_turnout ~ 1, pol_pres15)
lmt <- lm.morantest(ols, listw=lw_q_B)
glance_htest(lmt)
# Observed Moran I      Expectation         Variance      Std deviate
#         0.691434        -0.000401         0.000140        58.461349
#          p.value
#         0.000000

The only difference between tests under normality and randomisation is that an extra term is added if the kurtosis of the variable of interest indicates a flatter or more peaked distribution, where the measure used is the classical measure of kurtosis. Under the default randomisation assumption of analytical randomisation, the results are largely unchanged.

mtr <- moran.test(pol_pres15$I_turnout, listw=lw_q_B) glance_htest(mtr) # Moran I statistic Expectation Variance Std deviate # 0.691434 -0.000401 0.000140 58.459835 # p.value # 0.000000 Of course, from the very beginning, interest was shown in Monte Carlo testing, also known as a Hope-type test and as a permutation bootstrap. By default, moran.mc() returns a "htest" object, but may simply use boot::boot() internally and return a "boot" object when return_boot=TRUE. In addition the number of simulations of the variable of interest by permutation, that is shuffling the values across the observations at random, needs to be given as nsim=. set.seed(1) mmc <- moran.mc(pol_pres15$I_turnout, listw=lw_q_B, nsim=999, return_boot = TRUE)

The bootstrap permutation retains the outcomes of each of the random permutations, reporting the observed value of the statistic, here Moran’s $$I$$, the difference between this value and the mean of the simulations under randomisation (equivalent to $$E(I)$$), and the standard deviation of the simulations under randomisation.

If we compare the Monte Carlo and analytical variances of $$I$$ under randomisation, we typically see few differences, arguably rendering Monte Carlo testing unnecessary.

c("Permutation bootstrap"=var(mmc$t), "Analytical randomisation"=unname(mtr$estimate[3]))
#    Permutation bootstrap Analytical randomisation
#                 0.000144                 0.000140

Geary’s global $$C$$ is implemented in geary.test() largely following the same argument structure as moran.test(). The Getis-Ord $$G$$ test includes extra arguments to accommodate differences between implementations, as Bivand and Wong (2018) found multiple divergences from the original definitions, often to omit no-neighbour observations generated when using distance band neighbours. It is given by (Getis and Ord 1992, 194). For $$G_*$$, the $$\sum_{(2)}$$ constraint is relaxed by including $$i$$ as a neighbour of itself (thereby also removing the no-neighbour problem, because all observations have at least one neighbour).

Finally, the empirical Bayes Moran’s $$I$$ takes account of the denominator in assessing spatial autocorrelation in rates data (Assunção and Reis 1999). Until now, we have considered the proportion of valid votes cast in relation to the numbers entitled to vote by spatial entity, but using EBImoran.mc() we can try to accommodate uncertainty in extreme rates in entities with small numbers entitled to vote. There is, however, little impact on the outcome in this case.

Global measures of spatial autocorrelation using spatial weights objects based on graphs of neighbours are, as we have seen, rather blunt tools, which for interpretation depend critically on a reasoned mean model of the variable in question. If the mean model is just the intercept, the global measures will respond to all kinds of mis-specification, not only spatial autocorrelation. A key source of mis-specification will typically also include the choice of entities for aggregation of data.

### 13.3.2 Local measures

Building on insights from the weaknesses of global measures, local indicators of spatial association began to appear in the first half of the 1990s (Anselin 1995; Getis and Ord 1992, 1996). In addition, the Moran plot was introduced, plotting the values of the variable of interest against their spatially lagged values, typically using row-standardised weights to make the axes more directly comparable (Anselin 1996). The moran.plot() function also returns an influence measures object used to label observations exerting more than propotional influence on the slope of the line representing global Moran’s $$I$$. In Figure 13.3, we can see that there are many spatial entities exerting such influence. These pairs of observed and lagged observed values make up in aggregate the global measure, but can also be explored in detail. The quadrants of the Moran plot also show low-low pairs in the lower left quadrant, high-high in the upper right quadrant, and fewer low-high and high-low pairs in the upper left and lower right quadrants.

infl_W <- moran.plot(pol_pres15$I_turnout, listw=lw_q_W, labels=pol_pres15$TERYT, cex=1, pch=".", xlab="I round turnout", ylab="lagged turnout")

If we extract the hat value influence measure from the returned object, Figure 13.4 suggests that some edge entities exert more than proportional influence (perhaps because of row standardisation), as do entities in or near larger urban areas.

pol_pres15$hat_value <- infl_W$hat
tm_shape(pol_pres15) + tm_fill("hat_value")

Bivand and Wong (2018) discuss issues impacting the use of local indicators, such as local Moran’s $$I$$ and local Getis-Ord $$G$$. Some issues affect the calculation of the local indicators, others inference from their values. Because $$n$$ statistics may be being calculated from the same number of observations, there are multiple comparison problems that need to be addressed. Although the apparent detection of hotspots from values of local indicators has been quite widely adopted, it remains fraught with difficulty because adjustment of the inferential basis to accommodate multiple comparisons is not often chosen, and as in the global case, mis-specification also remains a source of confusion. Further, interpreting local spatial autocorrelation in the presence of global spatial autocorrelation is challenging (Ord and Getis 2001; Tiefelsdorf 2002; Bivand, Müller, and Reder 2009). The mlvar= and adjust.x= arguments to localmoran() are discussed in Bivand and Wong (2018), and permit matching with other implementations. The p.adjust.method= argument uses an untested speculation that adjustment should only take into account the cardinality of the neighbour set of each observation when adjusting for multiple comparisons; using stats::p.adjust() is preferable.

Taking "two.sided" p-values because these local indicators when summed and divided by the sum of the spatial weights, and thus positive and negative local spatial autocorrelation may be present, we obtain:

locm <- localmoran(pol_pres15$I_turnout, listw=lw_q_W, alternative="two.sided") all.equal(sum(locm[,1])/Szero(lw_q_W), unname(moran.test(pol_pres15$I_turnout, lw_q_W)$estimate[1])) # [1] TRUE Using stats::p.adjust() to adjust for multiple comparisons, we see that almost 29% of the local measures have p-values < 0.05 if no adjustment is applied, but only 12% using Bonferroni adjustment, with two other choices also shown: pvs <- cbind("none"=locm[,5], "bonferroni"=p.adjust(locm[,5], "bonferroni"), "fdr"=p.adjust(locm[,5],"fdr"), "BY"=p.adjust(locm[,5], "BY")) apply(pvs, 2, function(x) sum(x < 0.05)) # none bonferroni fdr BY # 715 297 576 424 In the global measure case, bootstrap permutations could be used as an alternative to analytical methods for possible inference. In the local case, conditional permutation may be used, retaining the value at observation $$i$$ and randomly sampling from the remaining $$n-1$$ values to find randomised values at neighbours, and is provided as localmoran_perm(), which will use multiple nodes to sample in parallel if provided, and permits the setting of a seed for the random number generator across the compute nodes: library(parallel) set.coresOption(ifelse(detectCores() == 1, 1, detectCores()-1L)) # NULL system.time(locm_p <- localmoran_perm(pol_pres15$I_turnout, listw=lw_q_W, nsim=499, alternative="two.sided", iseed=1))
#    user  system elapsed
#   0.599   1.506   0.767

The outcome is that almost 32% of observations have two sided p-values < 0.05 without multiple comparison adjustment, and under 3% with Bonferroni adjustment.

pv <- locm_p[,5]
apply(pvsp, 2, function(x) sum(x <= 0.05))
#       none bonferroni        fdr         BY
#        797         76        463        161

We can see what is happening by tabulating counts of the standard deviate of local Moran’s $$I$$, where the two-sided $$\alpha=0.05$$ bounds would be $$0.025$$ and $$0.975$$, but Bonferroni adjustment is close to $$0.00001$$ and $$0.99999$$. Without adjustment, almost 800 observations are significant, with Bonferroni adjustment, only 68 in the conditional permutation case:

brks <- qnorm(c(0, 0.00001, 0.0001, 0.001, 0.01, 0.025, 0.5, 0.975, 0.99, 0.999, 0.9999, 0.99999, 1))
(tab <- table(cut(locm_p[,4], brks)))
#
#  (-Inf,-4.26] (-4.26,-3.72] (-3.72,-3.09] (-3.09,-2.33] (-2.33,-1.96]
#             0             0             1             4             5
#     (-1.96,0]      (0,1.96]   (1.96,2.33]   (2.33,3.09]   (3.09,3.72]
#           459          1239           195           316           145
#   (3.72,4.26]   (4.26, Inf]
#            55            76
sum(tab[c(1:5, 8:12)])
# [1] 797
sum(tab[c(1, 12)])
# [1] 76
pol_pres15$locm_Z <- locm[,4] pol_pres15$locm_p_Z <- locm_p[,4]
tm_shape(pol_pres15) + tm_fill(c("locm_Z", "locm_p_Z"), breaks=brks, midpoint=0, title="Standard deviates of\nLocal Moran's I") + tm_facets(free.scales=FALSE) + tm_layout(panel.labels=c("Analytical", "Conditional permutation"))

Figure 13.5 shows that conditional permutation scales back the proportion of standard deviate values taking extreme values, especially positive values. As we will see below, the analytical standard deviates of local Moran’s $$I$$ should probably not be used if alternatives are available.

In presenting local Moran’s $$I$$, use is often made of “hotspot” maps. Because $$I_i$$ takes high values both for strong positive autocorrelation of low and high values of the input variable, it is hard to show where “clusters” of similar neighbours with low or high values of the input variable occur. The quadrants of the Moran plot are used, by creating a categorical quadrant variable interacting the input variable and its spatial lag split at their means. The quadrant categories are then set to NA if, for the chosen standard deviate and adjustment, $$I_i$$ would be considered insignificant. Here, for the conditional permutation standard deviates, Bonferroni adjusted, 14 observations belong to “Low-Low clusters”, and 54 to “High-High clusters”:

quadr <- interaction(cut(infl_W$x, c(-Inf, mean(infl_W$x), Inf), labels=c("Low X", "High X")), cut(infl_W$wx, c(-Inf, mean(infl_W$wx), Inf), labels=c("Low WX", "High WX")), sep=" : ")
pol_pres15$hs_an_q <- quadr is.na(pol_pres15$hs_an_q) <- !(pol_pres15$locm_Z < brks[6] | pol_pres15$locm_Z > brks[8])
b <- table(pol_pres15$hs_an_q) pol_pres15$hs_cp_q <- quadr
is.na(pol_pres15$hs_cp_q) <- !(pol_pres15$locm_p_Z < brks[2] | pol_pres15$locm_p_Z > brks[12]) c <- table(pol_pres15$hs_cp_q)
t(rbind("Moran plot quadrants"=a, "Unadjusted analytical"=b, "Bonferroni cond. perm."=c))
#                  Moran plot quadrants Unadjusted analytical
# Low X : Low WX                   1040                   370
# High X : Low WX                   264                     3
# Low X : High WX                   213                     0
# High X : High WX                  978                   342
#                  Bonferroni cond. perm.
# Low X : Low WX                       18
# High X : Low WX                       0
# Low X : High WX                       0
# High X : High WX                     58
tm_shape(pol_pres15) + tm_fill(c("hs_an_q", "hs_cp_q"), colorNA="grey95", textNA="Not significant", title="Turnout hotspot status\nLocal Moran's I") + tm_facets(free.scales=FALSE) + tm_layout(panel.labels=c("Unadjusted analytical", "Cond. perm. with Bonferroni"))

Figure 13.6 shows the impact of using analytical or conditional permutation standard deviates, and no or Bonferroni adjustment, reducing the counts of observations in “Low-Low clusters” from 370 to 14, and “High-High clusters” from 342 to 54; the “High-High clusters” are metropolitan areas.

The local Getis-Ord $$G$$ measure is reported as a standard deviate, and may also take the $$G^*$$ form where self-neighbours are inserted into the neighbour object using include.self(). The observed and expected values of local $$G$$ with their analytical variances may also be returned if return_internals=TRUE.

system.time(locG <- localG(pol_pres15$I_turnout, lw_q_W)) # user system elapsed # 0.014 0.000 0.013 system.time(locG_p <- localG_perm(pol_pres15$I_turnout, lw_q_W, nsim=499, iseed=1))
#    user  system elapsed
#   0.123   0.762   0.788

Once again we face the problem of multiple comparisons, with the count of areal unit p-values < 0.05 being reduced by an order of magnitude when employing Bonferroni correction:

pv <- 2 * pnorm(abs(c(locG)), lower.tail = FALSE)
colnames(pvsp) <- c("none", "bonferroni", "fdr", "BY")
apply(pvsp, 2, function(x) sum(x < 0.05))
#       none bonferroni        fdr         BY
#        789         69        468        156

In the $$G_i$$ case, however, there is no systematic differencce between the analytical and conditional permutation standard deviates.

library(ggplot2)
p1 <- ggplot(data.frame(Zi=locm[,4], Zi_perm=locm_p[,4])) + geom_point(aes(x=Zi, y=Zi_perm), alpha=0.2) + xlab("Analytical") + ylab("Conditional permutation") + coord_fixed() + ggtitle("Local Moran's I")
p2 <- ggplot(data.frame(Zi=c(locG), Zi_perm=c(locG_p))) + geom_point(aes(x=Zi, y=Zi_perm), alpha=0.2) + xlab("Analytical") + ylab("Conditional permutation") + coord_fixed() + ggtitle("Local G")
gridExtra::grid.arrange(p1, p2, nrow=1)

Figure 13.7 shows that, keeping fixed aspect in both panels, conditional permutation changes the range and distribution of the standard deviate values for $$I_i$$, but that for $$G_i$$, the two sets of standard deviates are equivalent.

pol_pres15$locG_Z <- c(locG) pol_pres15$hs_G <- cut(c(locG), c(-Inf, brks[2], brks[12], Inf), labels=c("Low", "Not significant", "High"))
table(pol_pres15$hs_G) # # Low Not significant High # 14 2426 55 m1 <- tm_shape(pol_pres15) + tm_fill(c("locG_Z"), midpoint=0, title="Standard\ndeviate") m2 <- tm_shape(pol_pres15) + tm_fill(c("hs_G"), title="Bonferroni\nhotspot status") tmap_arrange(m1, m2, nrow=1) As can be seen from Figure 13.7, we do not need to contrast the two estimation methods, and showing the mapped standard deviate is as informative as the “hotspot” status for the chosen adjustment (Figure 13.8). In the case of $$G_i$$, the values taken by the measure reflect the values of the input variable, so a “High cluster” is found for observations with high values of the input variable, here high turnout in metropolitan areas. Very recently, Geoda has been wrapped for R as rgeoda (Li and Anselin 2021), and will provide very similar functionalities for the exploration of spatial autocorrelation in areal data as spdep. The active objects are kept as pointers to a compiled code workspace; using compiled code for all operations (as in Geoda itself) makes rgeoda perform fast, but leaves less flexible when modifications or enhancements are desired. The contiguity neighbours it constructs are the same as those found by poly2nb(), as almost are the $$I_i$$ measures. The difference is as established by Bivand and Wong (2018), that localmoran() calculates the input variable variance divinding by $$n$$, but Geoda uses $$(n-1)$$, as can be reproduced by setting mlvar=FALSE: library(rgeoda) # Loading required package: digest # # Attaching package: 'rgeoda' # The following object is masked from 'package:spdep': # # skater system.time(Geoda_w <- queen_weights(pol_pres15)) # here # user system elapsed # 0.111 0.004 0.114 summary(Geoda_w) # name value # 1 number of observations: 2495 # 2 is symmetric: TRUE # 3 sparsity: 0.00228786229774178 # 4 # min neighbors: 1 # 5 # max neighbors: 13 # 6 # mean neighbors: 5.70821643286573 # 7 # median neighbors: 6 # 8 has isolates: FALSE system.time(lisa <- local_moran(Geoda_w, pol_pres15["I_turnout"], cpu_threads=ifelse(parallel::detectCores() == 1, 1, parallel::detectCores()-1L), permutations=499, seed=1)) # user system elapsed # 0.288 0.009 0.079 all.equal(card(nb_q), lisa_num_nbrs(lisa), check.attributes=FALSE) # [1] TRUE all.equal(lisa_values(lisa), localmoran(pol_pres15$I_turnout, listw=lw_q_W, mlvar=FALSE)[,1], check.attributes=FALSE)
# [1] TRUE

### References

Anselin, L. 1995. “Local indicators of spatial association - LISA.” Geographical Analysis 27 (2): 93–115.

Anselin, L. 1996. “The Moran Scatterplot as an ESDA Tool to Assess Local Instability in Spatial Association.” In Spatial Analytical Perspectives on GIS, edited by M. M. Fischer, H. J. Scholten, and D. Unwin, 111–25. London: Taylor & Francis.

Assunção, R.M., and E. A. Reis. 1999. “A New Proposal to Adjust Moran’s $$I$$ for Population Density.” Statistics in Medicine 18: 2147–62.

Avis, D., and J. Horton. 1985. “Remarks on the Sphere of Influence Graph.” In Discrete Geometry and Convexity, edited by J. E. Goodman, 323–27. New York: New York Academy of Sciences, New York.

Bavaud, F. 1998. “Models for Spatial Weights: A Systematic Look.” Geographical Analysis 30: 153–71. https://doi.org/10.1111/j.1538-4632.1998.tb00394.x.

Bivand, Roger S., and David W. S. Wong. 2018. “Comparing Implementations of Global and Local Indicators of Spatial Association.” TEST 27 (3): 716–48. https://doi.org/10.1007/s11749-018-0599-x.

Bivand, R. S., W. Müller, and M. Reder. 2009. “Power Calculations for Global and Local Moran’s $$I$$.” Computational Statistics and Data Analysis 53: 2859–72.

Bivand, R. S., and B. A. Portnov. 2004. “Exploring Spatial Data Analysis Techniques Using R: The Case of Observations with No Neighbours.” In Advances in Spatial Econometrics: Methodology, Tools, Applications, edited by L. Anselin, R. J. G. M. Florax, and S. J. Rey, 121–42. Berlin: Springer.

Boots, B., and A. Okabe. 2007. “Local Statistical Spatial Analysis: Inventory and Prospect.” International Journal of Geographical Information Science 21 (4): 355–75. https://doi.org/10.1080/13658810601034267.

Cliff, A. D., and J. K. Ord. 1973. Spatial Autocorrelation. London: Pion.

Cliff, A. 1981. Spatial Processes. London: Pion.

Duncan, O. D., R. P. Cuzzort, and B. Duncan. 1961. Statistical Geography: Problems in Analyzing Areal Data. Glencoe, IL: Free Press.

Freni-Sterrantino, Anna, Massimo Ventrucci, and Håvard Rue. 2018. “A Note on Intrinsic Conditional Autoregressive Models for Disconnected Graphs.” Spatial and Spatio-Temporal Epidemiology 26: 25–34. https://doi.org/https://doi.org/10.1016/j.sste.2018.04.002.

Geary, R. C. 1954. “The Contiguity Ratio and Statistical Mapping.” The Incorporated Statistician 5: 115–45.

Getis, A., and J. K. Ord. 1992. “The Analysis of Spatial Association by the Use of Distance Statistics.” Geographical Analysis 24 (2): 189–206.

Getis, A., and J. K. Ord. 1992. “The Analysis of Spatial Association by the Use of Distance Statistics.” Geographical Analysis 24 (2): 189–206.

1996. “Local Spatial Statistics: An Overview.” In Spatial Analysis: Modelling in a Gis Environment, edited by P. Longley and M Batty, 261–77. Cambridge: GeoInformation International.

Li, Xun, and Luc Anselin. 2021. Rgeoda: R Library for Spatial Data Analysis. https://CRAN.R-project.org/package=rgeoda.

McMillen, Daniel P. 2003. “Spatial Autocorrelation or Model Misspecification?” International Regional Science Review 26: 208–17.

Moran, P. A. P. 1948. “The Interpretation of Statistical Maps.” Journal of the Royal Statistical Society, Series B (Methodological) 10 (2): 243–51.

Okabe, A., T. Satoh, T. Furuta, A. Suzuki, and K. Okano. 2008. “Generalized Network Voronoi Diagrams: Concepts, Computational Methods, and Applications.” International Journal of Geographical Information Science 22 (9): 965–94. https://doi.org/10.1080/13658810701587891.

Ord, J. K., and A. Getis. 2001. “Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation.” Journal of Regional Science 41 (3): 411–32.

Schabenberger, O., and C. A. Gotway. 2005. Statistical Methods for Spatial Data Analysis. Boca Raton/London: Chapman & Hall/CRC.

She, Bing, Xinyan Zhu, Xinyue Ye, Wei Guo, Kehua Su, and Jay Lee. 2015. “Weighted Network Voronoi Diagrams for Local Spatial Analysis.” Computers, Environment and Urban Systems 52: 70–80. https://doi.org/https://doi.org/10.1016/j.compenvurbsys.2015.03.005.

Tiefelsdorf, M. 2002. “The Saddlepoint Approximation of Moran’s I and Local Moran’s $${I}_i$$ Reference Distributions and Their Numerical Evaluation.” Geographical Analysis 34: 187–206.

Upton, G., and B. Fingleton. 1985. Spatial Data Analysis by Example: Point Pattern and Qualitative Data. New York: Wiley.

Wall, M. M. 2004. “A Close Look at the Spatial Structure Implied by the CAR and SAR Models.” Journal of Statistical Planning and Inference 121: 311–24.