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Nearest neighbor methods

Nearest neighbor (NN) methods include at least sixdifferent groups of statistical methods. All have incommon the idea that some aspect of the similaritybetween a point and its NN can be used to make use-ful inferences. In some cases, the similarity is thedistance between the point and its NN; in others,the appropriate similarity is based on other identi-fying characteristics of the points. I will discuss indetail NN methods for spatial point processes andfield experiments because these are commonly usedin biology and environmetrics. I will very brieflydiscuss NN designs for field experiments, in whicheach pair of treatments occurs as neighbors equallyfrequently. I will not discuss NN estimates of proba-bility density functions (pdfs) [23], NN methods fordiscrimination orclassification [59], or NN linkage(i.e. simple linkage) in hierarchical clustering [32,pp. 57–60]. Although these last three methods havebeen applied to environmetric data, they are muchmore general.

Nearest Neighbor Methods for SpatialPoint Processes

Spatial point process data describe the locations of‘interesting’ events (see Point processes, spatial)and (possibly) some information about each event.Some examples include locations of tree trunks [52],locations of bird nests [11], locations of potteryshards, and locations of cancer cases [20]. I willfocus on the most common case where the locationis recorded in two dimensions (x, y). Similar tech-niques can be used for three-dimensional data (e.g.locations of galaxies in space) or one-dimensionaldata (e.g. nesting sites along a coastline or along ariverbank). Usually, the locations of all events in adefined area are observed (completely mapped data),but occasionally only a subset of locations is observed(sparsely sampled data). Univariate point process datainclude only the locations of the events; marked pointprocess data include additional information about theevent at each location [65]. For example, the speciesmay be recorded for each tree, some cultural identi-fication may be recorded for each pottery shard, andnest success or nest failure may be recorded for eachbird nest.

Location or marked location data can be used toanswer many different sorts of questions. The scien-tific context for a question depends on the area ofapplication, but the questions can be grouped intogeneral categories. One very common category ofquestion concerns the spatial pattern of the obser-vations. Are the locations spatially clustered? Dothey tend to be regularly distributed, or are theyrandom (i.e a realization of a homogeneousPois-son process)? A second common set of questionsconcerns the relationships between different types ofevents in a marked point process. Do two differentspecies of tree tend to occur together? Are locationsof cancer cases more clustered than a random sub-set of a control group (see Disease mapping)? Athird set of questions deals with the density (num-ber of events per unit area). What is the averagedensity of trees in an area? What does a map ofdensity look like? Methods to answer each of thesetypes of question are discussed in the followingsections.

Theoretical treatments of NN methods for spa-tial point patterns can be found in [18], [25], [58]and [65]. Applications of NN methods can be foundin many articles and books, including [40], [53], [64]and [66].

Describing and Testing Spatial PatternsUsing Completely Mapped Data

Describing and testing spatial patterns of locationshas a long history. Historically, the primary con-cern was with the question of randomness [1, 2, 15,48]. Are locations randomly distributed throughoutthe study area (i.e. are the locations a realization ofa Poisson process with homogeneous intensity), ordo the locations indicate some structure (i.e. clus-tering or repulsion between locations)? Because ofthe many connotations of randomness and the impor-tance of a homogeneous Poisson process as a bench-mark, it is commonly calledcomplete spatial ran-domness(CSR).

In this section I will describe NN tests basedon completely mapped data. Locations of all eventsare recorded in an arbitrary study region. Often,the study region is square or rectangular, but thisis not a requirement. Tests for the less commoncase of sampled data are described in the nextsection.

Encyclopedia of Environmetrics, Online © 2006 John Wiley & Sons, Ltd.This article is © 2006 John Wiley & Sons, Ltd.This article was published in Encyclopedia of Environmetrics in 2002 by John Wiley & Sons, Ltd.DOI: 10.1002/9780470057339.van007

2 Nearest neighbor methods

Tests Based on Mean Nearest Neighbor Distance

The distances between NNs provide informationabout the pattern of points. DefineW as the dis-tance from a randomly chosen event to the nearestother event in a homogeneous Poisson process withintensity (expected number of points per unit area;seePoisson intensity) of �. The pdf and cumulativedistribution function (cdf) of W are

g�w D 2�w exp��w2 �1

G�w D 1 � exp��w2 �2

so the mean and the variance ofW are

E�W D 1

2�1/2 �3

andvar�W D 4 �

4��4

Based on these moments, Clark and Evans [15]proposed a test of CSR. The conditional moments,E�Wj O� and var�Wj O�, are calculated by substitut-ing the observed density,O� (total number of pointsdivided by the total area of the study region), into(3) and (4). The observed mean NN distance,w,is computed by identifying the NN of each point,finding the distance between NNs, then averag-ing. Clark and Evans proposed that the standard-ized mean

ZCE D w � E[Wj O�]

�N�1var[Wj O�]1/2�5

has a standard normal distribution if the process is ahomogeneous Poisson process [15].

TheZCE statistic and the many users of it ignoretwo problems: nonindependence of some NN dis-tances, andedge effects. In a completely mappedarea, many of the distances between NNs are cor-related. The problem is most severe with reflex-ive NNs. Two points, A and B, are reflexive NNswhen B is the NN of A, and A is the NN ofB [16]. Other authors have called these ‘isolatedNNs’ [51] or ‘mutual NNs’ [61]. When A andB are reflexive NNs, each point has the value ofW, which inflates the variance of the mean NNdistance. This problem is not restricted to a fewpoints. When points exhibit CSR in two dimensions,approximately 62.15% of the points are reflexiveNNs [16].

Edge effects arise because the distribution ofW(2) assumes an unbounded area, but the observedNN distances are calculated from points in a definedstudy area. When a point is near the edge of thestudy area, it is possible that the true NN is a pointjust outside the study area, not a more distant pointthat happens to be in the study area. Edge effectslead to overestimation (positive bias) of the meanNN distance. Edge effects can be practically impor-tant; neglecting them can alter conclusions about thespatial pattern [10].

Edge effects may be minimized by including abuffer area that surrounds the primary study area [15].NN distances are calculated only for points in the pri-mary study area, but locations in the buffer area areavailable as potential NNs. With a sufficiently largebuffer area, this approach can eliminate edge effects,but it is wasteful since an appropriately large bufferarea may contain many locations.

Using simulations, Donnelly [31] derived edge-corrected approximations to E�Wj O� and var�wj O�when the study region is rectangular:

E�Wj O� ³ 0.5(A

N

)1/2

C 0.0514P

NC 0.041

P

N

3/2�6

var�wj O� ³ 0.0703A

N2 C 0.037PA1/2

N5/2 �7

whereN is the observed number of points,A is thearea of the study region, andP is the total perime-ter of the study region. These approximations can beused to test CSR by substituting them into (5), thencomparing the resultingz -statistic to a standard nor-mal distribution. This test has reasonable power todetect departures from CSR [57].

One difficulty with tests based on the mean NNdistance is that the mean is just a single summary ofthe pattern. Two point patterns may have the samemean NN distance, but one exhibits CSR and theother does not. One such pattern would have a fewpatches of clustered points and an appropriate num-ber of widely scattered individuals. The points inthe clusters have small NN distances, but the widelyscattered individuals have large NN distances. Withthe appropriate mix of clustered and scattered points,the mean NN distance could be exactly that givenby (3).


Nearest neighbor methods 3

Distribution of Nearest Neighbor Distances

An alternative is to consider the entire distribu-tion, G�w, of NN distance [24]. CSR can be testedby comparing the observed distribution functionof NN distances, G�w, to the theoretical cdf(2). A variety of test statistics have been sug-gested, including Kolmogorov–Smirnov type statis-tics: supw jG�w � G�wj, Cramer–von Mises typestatistics:

∫w[G�w � G�w]2, or Anderson–Darling

type statistics:∫w[G�w � G�w]2/[G�w�1 � G�w].

The Kolmogorov–Smirnov statistic seems to be themost commonly used. The usual critical values forthe one-sample Kolmogorov–Smirnov test are notappropriate here because of nonindependence ofNN distances, especially for reflexive NNs, as dis-cussed above. Instead, a Monte Carlo test must beused [28].

The Monte Carlo approach computes the˛-levelcritical value or theP value by simulation (seeSim-ulation and Monte Carlo methods). The number ofpoints,N, is fixed at the observed number.N ran-dom locations are simulated as a realization of CSRin the study area, and the Kolmogorov–Smirnov (orother) test statistic is computed. This is repeatedRtimes to giveR values from the sampling distributionof the test statistic under the null hypothesis of CSR.The one-sided -level critical value is the �R C 1thlargest value from the sampling distribution. ThePvalue is computed as (1C number of random valueslarger than the observed value)/�1 C R [5].

Edge effects complicate the estimation ofG�w.One solution is to include a border strip, as dis-cussed above, but this may ignore a considerableamount of information. A variety of edge-correctedestimators ofG�w has been proposed; four of themare summarized in [18, pp. 613–614, 637–638].The edge-corrected cdf can also be estimated by aKaplan–Meier estimator [3]. Edge corrections reducethe bias in the estimator, but they increase thesampling variance [36].

Although edge-corrected estimators are needed ifthe observed distribution function is to be comparedwith the theoretical distribution function under CSR(2) they may not be needed for a Monte Carlotest of CSR. A more powerful test of CSR is touse a nonedge-corrected estimator (2) and comparethe biased estimate ofG�w to the biased mean,G�w, under CSR [36]. The biased meanG�wand pointwise prediction intervals are computed by

simulation. Values ofG�w above the simulatedmean indicate clustering of points (an excess ofshort distances to NNs). Values ofG�w below thesimulated mean indicate regularity (few to no pointswith short distances to NNs).

Graphical diagnostics based on the empirical dis-tribution of NN distances provide additional infor-mation about the spatial process. The most commongraphical diagnostic is a quantile–quantile plot ofeitherG�w or G�w on thex-axis andG�w on they-axis. G�w would be used when the object is toevaluate the fit of the theoretical NN distribution;G�w would be used when the theoretical NN dis-tribution is unknown and the object is to evaluate thefit to a process that can be simulated.

Monte Carlo simulation of the spatial processprovides both an estimate ofG�w and the samplingdistribution ofG�w conditional on a specific spatialprocess.Quantiles of the sampling distribution ofG�w can be calculated for interesting distances,w. If the spatial process is simulated many times(e.g. 199 or 999), the quantiles can be estimatedrelatively precisely. The 0.05 and 0.95 quantiles canbe approximated by the minimum and maximum ofR D 19 simulations. The 0.01 and 0.99 quantiles canbe approximated by the minimum and maximum ofR D 99 simulations.

The simulated mean and quantiles ofG�w areplotted against the observedG�w. If the observedcurve of G�w falls entirely between the lower andupper bounds, then there is no evidence againstthe null hypothesis (e.g. that the locations are arealization of CSR). An excursion outside the boundsindicates a departure from CSR. If the observedG�w falls below the lower bound at short distances,there are too few NNs at short distances, which isconsistent with a regular pattern, or one where there isinhibition of nearby points. If the observedG�w liesabove the upper bound at short distances, there aretoo many NNs at short distances, which is consistentwith a clustered process.

The Monte Carlo approach is not limited to testingCSR. It can be used to evaluate the fit of any processthat can be simulated. For example, a set of locationsmight be compared with aPoisson cluster processor a Strauss process [25, 31].

NN methods have been extended in a variety ofways. Tests can be based on other functions of theNN distances (e.g. squared [12] or smallest [62] NNdistances), but such tests have not been widely used.



Distances to second NN, or the third NN, or perhapsan even further neighbor have been suggested as away to look at patterns on a larger scale. The set ofcdfs of distance to the NN, distance to the secondNN, . . . , distance tonth NN is related toRipley’s Kfunction, another commonly used method to analyzespatial patterns [56; 65, p. 267]. Finally, the distancebetween a randomly chosen point and the nearestevent also provides information about the spatialpattern [25, 66].

Point–Event Distances

The point–event distribution,F�x, considers thedistance between a randomly chosen location (not thelocation of an event) and the nearest event. This canbe estimated by choosingm locations in the studyarea and computing the distance from each locationto its NN. As with G�w, edge effects complicateestimation of the cdf. An edge-corrected estimator is

OFR�x D number of�xi � x, di > x

number of�di > x�8

where xi is the distance between a point and itsneighboring event, anddi is the distance between apoint and its nearest boundary. When the events area realization of CSR,X, the point–event distance,andW, the NN distance, have the same distribution,soF�x D 1 � exp��x2. However, the effects ofdeviations from randomness onF(x) are opposite tothose onG�w. Values of F�x above the expectedvalue indicate regularity. Values below the expectedvalue indicate clustering.

The NN distance distribution, point–event dis-tance distribution, and Ripley’sK function providedifferent insights into the spatial pattern. The NN dis-tribution function, G�w, is slightly more powerfulat detecting departures from CSR in the direction ofregularity [24]. The point–event distribution functionprovides information about the empty space betweenpoints. It appears to be the more powerful methodfor detecting departures in the direction of cluster-ing [25, 66]. Ripley’s K function simultaneouslyexamines the spatial pattern at many distance scalesand is now the most popular approach for completelymapped data. However, it is possible to constructpoint processes with the sameG�w but a differ-ent Ripley’sK�t function [65, p. 267]. Conversely,Baddeley and Silverman [4] illustrate two processeswith the same Ripley’sK function but very different

NN distance distributions and point–event distancedistributions.

Are Events Points or Circles?

All the distributional theory of the previous sectionsassumes that events occupy no space. Treating eventsas points assumes that it is possible for two events(e.g. locations of tree trunks or bird nests) to be aninfinitesimally small distance from each other. Thepoint assumption is reasonable when the area of theevents is small relative to the spacing between theevents. The assumption is likely to be appropriate forbird nests (generally small) or tree trunks (generallylow density), but not for ant nests (large size relativeto the density of nests in an area). If events areincorrectly assumed to be points, the analysis ofthe spatial pattern indicates a tendency to regularitybecause two events do not occur within a smalldistance (the physical size of the event) of each other.

An approximation to the mean event–event dis-tance,W, for nonoverlapping circles under CSR is

W ³ d C exp��d2[1 � �2�d1/2]p�

�9

whered and� are, respectively, the diameter of thecircles and the number of circles per unit area [63].In (9) a low intensity,�, and a small and constantdiameter,d, are assumed. The distribution,G�w,of NN distances for nonoverlapping circles can beestimated by Monte Carlo simulation by using asequential inhibition algorithm [24]. The distributionof event–event distances can be complicated whenthe circles are large or the density is high. Forexample, it may not be possible to fit the requirednumber of large circles into the study area.

Algorithms and Computing

The simplest way to compute NN distances is bydirect enumeration; that is, computing the distancesbetween all pairs of points, then reporting the smallestdistance for each point as the NN distance. For alarge number of points, this becomes impractical,and more efficient algorithms have been developedto approach the problem. Possible approaches includesubdividing the region into smaller subregions [47],computing the Direchlettessellation (also known asthe Voronoi tessellation, Thiessen tessellation, andother less common names [66, p. 96]) and using that



to identify NNs, or computing the quadtree, a sortedmatrix of locations that simplifies the search for NNs,and using that to identify NNs [35]. Murtagh [47]reviews the properties of these algorithms.

Functions or procedures for NN spatial analy-sis are included in few statistical programs, butdirect enumeration is very simple to program whenneeded. Packages of functions for spatial point patternanalysis usually include functions for point–pointand point–event analyses. Many of these are S orS-PLUS libraries, for example Splancs [60], spa-tial [67], andS + SPATIALSTATS [45].

Example: Trees in a Swamp Hardwood Forest

Figure 1 shows the locations of all 630 trees (stems>11.5 cm diameter at breast height) and the locationsof 91 cypress trees in a 1 ha plot of swamp hardwoodforest in South Carolina, USA. There are 13 differenttree species in this plot, but over 75% of thestems are one of three species: black gum (Nyssasylvatica), water tupelo (Nyssa aquatica), or baldcypress (Taxodium distichum). Visually, trees seemto be scattered randomly throughout the plot, butcypress trees seem to be clustered in three bands.NN statistics provide a way to test the hypothesis thatstems are randomly distributed throughout the plot. Iwill illustrate tests based on mean NN distance,G�w,andF�x using the locations of all 630 trees and thelocations of the 91 cypress trees.

For all 630 tree locations, the mean NN distanceis 1.99 m. If 630 points were randomly distributed ina 200 mð 50 m rectangle, the expected NN distanceis 2.034 m, with astandard error (SE) of 0.044 m,using Donnelly’s approximations, (6) and (7). Thereis no evidence of departure from CSR�z D �0.973with a two-sidedP value of 0.33). The effect of theedge corrections is minimal probably because the plotis large and the NN distance is small. The uncorrectedexpected NN distance is 1.992 m, with an SE of0.042 m, using (3) and (4).

Conclusions using the distribution of NNdistances,G�w, are similar. G�w was estimatedwithout edge corrections, soG�w must be comparedwith simulated values, not the theoretical expectation(2). The observed cdf, the theoretical expectedvalue (2), and the average simulated cdf are verysimilar (Figure 2a), although the observedG�wis slightly larger than the expected value at shortdistances. The differences can be seen more clearly

x coordinate50100 30

0

50

100

150

200

y co

ordi

nate

Figure 1 Marked plot of tree locations in a 50 mð 200 mplot of hardwood swamp in South Carolina, USA. Circlesare locations of cypress trees, squares are locations of blackgum trees, and dots are locations of any other species

if 1 � exp��w2 (2) is subtracted from all curves(Figure 2b). AlthoughG�w is larger than both thetheoretical expected value and the average simulatedvalue, it lies within the pointwise 95% confidencelimits. None of the three summary statistics(Kolmogorov–Smirnov, Cramer–von Mises, or



Distance (m)

cdf

1 2 3 40

2

4

6

8

(a) Distance (m)

cdf

1 2 3 4

–4

–2

0

4

2

(b)

Distance (m)5 10 15

0

2

4

6

8

10

(c)

cdf

5 10 15

Distance (m)

5

0

10

–10

–5

15

–15

(d)

cdf

Figure 2 Cdf plots of NN distances for all trees and for cypress trees only, in a 50 mð 200 m plot (see Figure 1).(a) All trees: comparison of observed cdf (solid line), theoretical cdf (dotted line) and average simulated cdf under CSR(dashed line). (b) All trees: comparison of observed cdf (solid line) and average simulated cdf (dashed line). The dottedlines represent the pointwise 0.025 and 0.975 quantiles of the simulated cdf under CSR. For clarity, the theoretical cdf issubtracted from all plotted cdfs. (c) Cypress trees only: comparison of observed cdf (solid line), theoretical cdf (dotted line)and average simulated cdf under CSR (dashed line). (d) Cypress trees only: comparison of observed cdf (solid line) andaverage simulated cdf (dashed line). The dotted lines represent the pointwise 0.025 and 0.975 quantiles of the simulatedcdf under CSR. For clarity, the theoretical cdf is subtracted from all plotted cdfs

Anderson–Darling) is significant at D 0.05. Forexample, the observed Kolmogorov–Smirnov teststatistic of 0.044 is less than the simulated 90thpercentile, 0.052. The estimatedP values forthe Kolmogorov–Smirnov, Cramer–von Mises, andAnderson–Darling test statistics are 0.19, 0.31, and0.40, respectively.

In contrast, the distribution of point–event dis-tances suggests there is some clustering of tree loca-tions. The observedF�x is below the theoreticaland average simulated curves (Figure 3a, b) andoutside the pointwise 95% confidence bounds atlarge distances (Figure 3b). BecauseF�x falls belowthe expected values, distances from randomly cho-sen points are stochastically greater than expectedif events exhibited CSR. The greater than expectedabundance of large empty spaces provides evi-dence of clustering of the events. TheP valuesfor the Kolmogorov–Smirnov, Cramer–von Mises,and Anderson–Darling summary statistics range from

0.004 to 0.009. The conclusion of some evidence forclustering of all tree locations matches the conclusionusing Ripley’sK function.

For the 91 cypress trees, the mean NN distanceis 5.08 m, which is slightly smaller than the edge-corrected expected distance of 5.55 m, with an SE of0.33 m. Using the NN distance, there is no evidenceof a nonrandom distribution; thez-statistic is�1.41,with a two-sidedP value of 0.16. The effect of theedge corrections is larger when the density of pointsis smaller. The uncorrected expected NN distance forthe 91 cypress trees is 5.24 m, with an SE of 0.29 m.

However, the distributions ofG�w and F�x forthe 91 cypress trees provide evidence of clusteringof cypress trees. There is an unusually large numberof NN distances between 3 m and 7 m (Figures 2cand 2d). This excess is significant;G�w is ator above the pointwise 0.975 quantiles of simu-lated values (Figure 2d). This excess is consis-tent with clustering of cypress trees. There are



Distance (m)1 2 3 4

0

2

4

6

8

(a)

cdf

–4

0

–2

2

4

Distance (m)1 2 3 4

(b)

cdf

(c) Distance (m)

0

2

4

6

8

10

5 10 15

cdf

Distance (m)(d)5 10 15

–2

–1

0

1

2

cdf

Figure 3 Cdf plots of point–event distances for all trees and for cypress trees only, in a 50 mð 200 m plot (see Figure 1).(a) All trees: comparison of observed cdf (solid line), theoretical cdf (dotted line) and average simulated cdf under CSR(dashed line). (b) All trees: comparison of observed cdf (solid line) and average simulated cdf (dashed line). The dottedlines represent the pointwise 0.025 and 0.975 quantiles of the simulated cdf under CSR. For clarity, the theoretical cdf issubtracted from all plotted cdfs. (c) Cypress trees only: comparison of the observed cdf (solid line), theoretical cdf (dottedline) and average simulated cdf under CSR (dashed line). (d) Cypress trees only: comparison of observed cdf (solid line)and average simulated cdf (dashed line). The dotted lines represent the pointwise 0.025 and 0.975 quantiles of the simulatedcdf under CSR. For clarity, the theoretical cdf is subtracted from all plotted cdfs

also significant fewer (at least pointwise) NN dis-tances at 13 m. All three summary statistics aresignificant (Kolmogorov–SmirnovP valueD 0.034,Cramer–von MisesP valueD 0.007, Anderson–Dar-ling P valueD 0.011). The point–event distancesare stochastically greater than expected under CSR(Figures 3c and 3d). The observed distribution,F�x,lies outside the pointwise 95% confidence bandsfor many distances. All three summary statistics arehighly significant�P D 0.001. The conclusion thatcypress trees are strongly clustered matches that usingRipley’s K function.

Directed Tests

The tests in the previous section are general tests ofCSR against an unspecified alternative. Other testsmay be more powerful when the alternative is morespecific (e.g. events are associated with specific sites,

or the density of events increases from east to west).Association between point events and a nonpointstochastic process can be tested using the NN distancefrom each event to the nearest part of the secondprocess [7]. However, most directed tests [43, 55,68] use features other than NN distance.

Describing and Testing Spatial Patternswith Use of a Sample of Nearest NeighborDistances

Although mapped data are often easy to collect, astatistician might view this as wasteful because of thehigh degree of correlation between NN distances (e.g.the perfect correlation for pairs of reflexive NNs). Analternative is to measure NN distances only on a ran-dom sample of individuals. Because the NN distancesare calculated from a simple random sample of points,



the distributional theory for both the mean NN dis-tance and the distribution function is much simpler.Many different tests of CSR have been developedfor use with a random sample of point–event, NN,or point–event–event distances. These are summa-rized and evaluated in [66, pp. 59–64] and [18,pp. 602–614].

The most straightforward way to select a randomsample of NN distances is to enumerate all indi-viduals in the statistical population, select a simplerandom sample of events, and measure the distancesfrom the selected events to their NNs. This can betime-consuming and is usually impractical [66]. Enu-meration can be avoided by clever use of subregions(described by Byth and Ripley [14]), or by randomlyselecting points (not events). The distance from therandomly selected point to the nearest event is arandom sample from the distribution of point–eventdistances,F�x, but the distance from that event toits closest event is not a random sample fromG�wbecause the point–event and event–event distancesare correlated. The distributions of all quantities inthe point–event–event sample when events exhibitCSR have been derived [17].

An alternative that is easy to implement in thefield is theT-square sample [8], illustrated in [18]and [66], a modified point–event–event sample. Apoint, A, is randomly chosen, and the NN, B, isfound. Then, the study area is divided into two halfplanes by a line through B and perpendicular to AB(hence the name,T-square). Attention is restricted tothe half plane that does not contain point A. Thedistance to the NN,Z, of B in that half plane ismeasured. When points exhibit CSR,Z and X (thedistance from point A to NN B) are independent,and the distribution ofZ/

p2 is the same as the

distribution of NN distances,G�w [8].

Estimating Density

A random sample of point to NN distances can beused to estimate�, the average density of eventsin the study area. When events exhibit CSR, themaximum likelihood estimate is

O� D n

(

n∑iD1

X2i

)�1

�10

where Xi is the distance from a randomly chosenpoint i to its NN [46]. An unbiased estimate is [54]:

O�P D �n � 1

(

n∑iD1

X2i

)�1

�11

Both estimators are very dependent on the CSRassumption and can be biased if locations are clus-tered or regularly distributed.

Many other estimators have been proposed. Uptonand Fingleton [66, pp. 118–133] summarize and pro-vide examples of calculations for many of these.Byth [13] evaluated the robustness of many estima-tors to deviations from CSR. She recommended anestimator,O�T, based on two quantities fromT-squaresampling:X, the distance from point to nearest event,andZ, the distance to the NN in a half plane:

O�T D n2

[2p

2

(n∑

iD1

Xi

)(n∑

iD1

Zi

)]�1

�12

Nearest Neighbor Methods to ExamineSpatial Patterns of More Than One Typeof Point

Additional information about an event is often avail-able. For example, tree locations might be markedwith the species of tree (a mark with discrete lev-els) or the size of the tree (a mark with continuouslevels). The methods in the previous sections canbe used to analyze patterns in all events (ignor-ing the marks) or subgroups of events (e.g. justspecies A or just trees larger than 50 cm). How-ever, other interesting questions could be asked aboutthe relationship between the two (or more) sets oflocations.

Multivariate spatial point patterns are those whereevents can be classified into different types, that is,the marks are discrete [18, p. 707]. Usually, thenumber of different types is small; bivariate pat-terns, with two types of marks, are the most com-mon. Some questions that could be asked aboutpoint processes with discrete marks are as fol-lows:

1. Are the processes that generate locations withdifferent marks independent?

2. Are marks randomly assigned to locations? Con-ditional on the observed locations of superposi-tion of the two marked processes, are the marksindependent?



3. Are marks segregated? Are locations with onetype of mark surrounded by locations with thesame mark?

These questions about the relationships between pro-cesses make no assumptions about the marginal pat-tern of each process. In particular, either process (orthe superposition of processes) may be independent,clustered, or regularly distributed. The two generalmethods to answer these questions are the compar-ison of distribution functions [39, 44] and the NNcontingency table [30, 52]. Other approaches includeRipley’s K functions or parametric point processmodels.

Define the following multitype extensions of thepoint–event and NN distances.Xi is the distancefrom a randomly chosen point to the nearest eventwith mark i, with cdf Fi�x. Wij is the distancefrom an event with marki to the nearest eventwith mark j, with cdf Gij�w. If the process withmark i is independent of the process with markj, then

Fi�x D Gji�x �13

Fj�x D Gij�x �14

and Xi and Xj are independent [27, 39]. Note thatproperty (13) does not imply property (14), so twotests are needed [39].

For sparsely sampled data, Goodall [39] suggestsa t test of Xi D Xj. Diggle and Cox [27] considernonparametric versions of thet test, tests of equalityof distribution, and tests of the correlation betweenXi and Xj. Details and a comparison of the testsare given in [27]. Analyses of completely mappeddata tend to focus on the comparison of distributionfunctions in (13) and (14). Monte Carlo tests are usedbecause of the nonindependence of point–event andevent–event distances.

Two different simulation methods could be usedin the Monte Carlo test. The choice depends on thenull hypothesis. If the null hypothesis is independencebetween marks (question 1, above), then toroidalshifts or some parametric model should be used togenerate the randomization distribution. If the nullhypothesis is random assignment of marks condi-tional on the set of events, then random labeling ofevents should be used to generate the randomizationdistribution. In general, these two hypotheses are notequivalent and the sampling distributions are not thesame.

Nearest Neighbor Contingency Tables

The NN contingency table focuses on the ecolog-ically important question of segregation [30, 52].This table describes marks of events and their NNs,not the distance between them (Table 1). In sparselysampled data, the counts (NAA , NAB, NBA, andNBB)are independent Poisson random variables or condi-tionally independent given the row marginal totals(NA and NB) under the null hypothesis of randomlabeling [52]. The hypothesis can be tested with atraditional 1 degree of freedom (df)!2 test of inde-pendence [52] (seeCategorical data).

In completely mapped data, the sampling distri-bution of the counts is different [16]. If events arerandomly labeled, the expected values of the countsdepend only on the number of each type of event(NA and NB) and the total number of events,N(Table 2) [30]. The variances and covariances dependon the number of events of each type, the number ofreflexive NNs, and the number of shared NNs; fora derivation of the formula, see [30]. The first twomoments of the cell counts can be used to test forsegregation of type A events [NAA > E�NAA ], testfor segregation of type B events [NBB > E�NBB],

Table 1 Cell counts in an NN contingency table

Mark of neighbor

Mark of point A B Total

A NAA NAB NAB NBA NBB NBTotal MA MB N

N is the number of points in the spatial pattern.NA andNB arethe number of type A and type B points.NAA is the numberof type A points with type A NNs.NAB is the number of typeA points with type B NNs.NBA andNBB are the number oftype B points with type A or type B NNs, respectively.MA

andMB are the column sums, i.e. the total number of pointswith type A neighbors or type B neighbors, respectively

Table 2 Expected cell counts for NN contin-gency table for completely mapped data

To:

From: A B

ANA�NA � 1

N � 1

NANB

N � 1

BNBNA

N � 1

NB�NB � 1

N � 1

See Table 1 for definitions ofM, NA andNB



or construct an omnibus 2 df!2 test of randomlabeling. If the numbers of points are large, thenthe distributions of test statistics can be adequatelyapproximated by asymptotic normal and!2 distri-butions. If the number of points is small, then thedistributions should be determined by Monte Carlosimulation.

Patterns withk marks�k > 2 can be analyzed byconsidering all pairs of marks two at a time (usingdistance methods or 2ð 2 NN contingency tables),or by considering thek ð k contingency table. Theexpected counts and their variances under randomlabeling follow the same form as those for a 2ð 2contingency table, but there are more possible formsfor the covariance between two counts. Details aregiven in [29].

Other approaches that have been suggested forthe analysis of multitype point processes include thecomparison of bivariate Ripley’sK functions [26,44], empty space methods [44] (comparisons ofpoint–event distance distributions), and mark corre-lation functions [65].

Example, Part 2

Cypress and black gum are two of the three abundantspecies in the 1-ha plot of swamp forest consid-ered in Example 1. An interesting ecological questionis whether these two species are spatially segre-gated; that is, do cypress trees tend to be found nearother cypress trees and do black gum trees tend tobe found near other black gum trees? The markedplot of locations (Figure 1) suggests that cypresstrees and black gum occur in different clusters. Con-firming this requires an analysis of the bivariatespatial pattern. The three tests that will be illus-trated are the comparisons of cdfs (13–14) [27], theindependence of distances [27], and the NN contin-gency table [30]. The ecological background suggeststhat random labeling is the more appropriate nullhypothesis.

The cdf of distances from randomly chosen pointsto the nearest black gum tree,FG�x, and the cdfof point–event distances to the nearest cypress tree,FC�x, were estimated without edge corrections byusing a randomly located grid of points [14]. Thecdfs of distances from black gum trees to the nearestcypress tree,GGC�x, and from cypress trees to thenearest black gum tree,GCG�x, were also estimatedwithout edge corrections. Both species show the same

5 10 15 20

0

0.2

0.4

0.6

0.8

1.0

Distance (m)(a)

cdf

5 10 15 20

0

0.2

0.4

0.6

0.8

1.0

Distance (m)(b)

cdf

0

0

5

5

10

10

15

15

20

20

Distance to black gum (m)(c)

Dis

tanc

e to

cypr

ess

(m)

Figure 4 Cdf of point–event distances for all treesand for cypress trees only. (a) Comparison of cdfs ofpoint to cypress tree distances [FC�x, dotted line] andblack gum tree to cypress tree distances [GGC�x, solidline]. (b) Comparison of cdfs of point to black gumtree distances [FG�x, dotted line] and cypress tree to blackgum tree distances [GCG�x, solid line]. (c) Relationshipbetween distances from a randomly chosen point to thenearest cypress tree and nearest black gum tree

pattern. Cypress trees are stochastically further fromblack gum trees than from randomly chosen points(Figure 4a). Also, black gum trees are stochasticallyfarther from cypress trees than from randomly chosenpoints (Figure 4b).



The observed differences can be comparedwith those found under random labeling by usingthe Kolmogorov–Smirnov two-sample statistics,maxjFG�x � GCG�xj and maxjFC�x � GGC�xj,as the test statistics. The observed maximumdifferences are not unusually large (P valueD 0.109for black gum and 0.083 for cypress). The distancefrom a randomly chosen point to the nearest blackgum tree, XG, is negatively correlated with thedistance from the same point to the nearest cypresstree, XC (Figure 4c; Kendall’s O$ D �0.12, with aone-sidedP value, by randomization, of 0.001). Thisresult is consistent with the spatial segregation of thetwo species. The different results from the three testsare consistent with Diggle and Cox’s [27] observationthat the correlation test is more powerful than theKolmogorov–Smirnov test for the sparsely sampledspatial patterns they studied.

The NN contingency table indicates that bothspecies have an excess of NNs of the same species(Table 3). The variances of the cell counts are38.88 for black gum–black gum and 25.55 forcypress–cypress. TheP values can be computed byMonte Carlo randomization or by a normal approx-imation [30]. In either case, the one-sidedP valuesare small (0.001 or less) for both species.

Nearest Neighbor Methods for FieldExperiments

A different set of NN methods can be used to ana-lyze spatially structured field experiments. One of themost common applications is in agronomic varietytrials, where many treatments are compared by takingsmall plots arranged in a rectangular lattice. Tradi-tional methods of controlling for between-plot hetero-geneity, such as using a randomized complete block(RCB) design, may not be very effective because the

Table 3 Observed counts�NO, expected counts�NE),and z scores for the cypress tree and black gum tree NNcontingency table

Species of neighbor

Species ofBlack gum tree Cypress tree

point NO NE z NO NE z Total

Black gum 149 121.1 4.47 33 60.9�4.47 182Cypress 43 60.9�3.54 48 30.1 3.54 91Total 192 81 273

large number of treatments forces the blocks to belarge. NN methods use information from adjacentplots to adjust for within-block heterogeneity and soprovide more precise estimates of treatment meansand differences. If there is within-plot heterogeneityon a spatial scale that is larger than a single plotand smaller than the entire block, then yields fromadjacent plots will be positively correlated. Informa-tion from neighboring plots can be used to reduce orremove the unwanted effect of the spatial heterogene-ity and hence improve the estimate of the treatmenteffect. Data from neighboring plots can also be usedto reduce the influence of competition between adja-cent plots. Each of these approaches will be brieflydiscussed below.

Papadakis [49, 50] proposed an analysis of covari-ance to reduce the effects of small-scale spatial het-erogeneity in yields. The value of the covariate foreach plot is obtained by averagingresiduals from theneighboring plots. The choice of neighboring plotsdepends on the crop, the plot size and shape, and thespacing between plots. In many row crops, the neigh-boring plots are defined as the two adjacent plots inthe row, except that plots at an end of a row have onlyone neighbor. In other situations, it may be appropri-ate to consider four neighbors, which include the twobetween-row neighbors. If the spatial heterogeneity issuch that the effects of within-row and between-rowneighbors are quite different, then one could computeseparate covariates for within-row and between-rowneighbors. Once the covariates are computed, treat-ment effects are re-estimated using an analysis ofcovariance. For adjustment in one dimension (e.g.along crop rows), the model would be

Yi D & C Xi$i C ˇRi C εi �15

whereYi is the observed yield on theith plot, Riis the mean residual on neighbors of theith plot,ˇ is the spatial dependence parameter,& and $iare the parameters in the model for the treatmenteffects,Xi is the row of the design matrix for theith plot, andεi is the residual variability in yields,which are assumed to be uncorrelated. Whenˇ is0, observations on adjacent plots are independent;larger positive values of �ˇ < 1 correspond toincreasing spatial correlation between neighboringplots. The observed value of depends on plotsize, plot shape, plot spacing, and the scale of thespatial heterogeneity. Values are often close to 1 whenplots are small. In the absence of treatments, and



ignoring edge effects, the Papadakis model impliesthat correlations between plot yields have a first-orderautoregressive structure, with corr�Yi, Yj D *ji�jj,whereˇ D 2*/�1 C *2. Values ofˇ close to 1 implythat * is also close to 1.

The autoregressive correlation structure impliedby the ad hoc Papadakis model is one exam-ple of the random field approach to a spatiallydesigned experiment [6, 18, 73]. Many other mod-els, including the iterated Papadakis method, theWilkinson NN [71] model, the Besag and Kemp-ton [9] first-order difference models, the Williamsmodel [69] and the Gleeson–Cullis autoregressiveintegrated moving average (ARIMA) models [19, 38](seeTime series) correspond to different specifica-tions of a spatial correlation matrix. Computationsare handled either by a general purpose residualmaximum likelihood (REML) algorithm for linearmixed effects models [e.g. PROC MIXED inSAS,lme( ) in S-PLUS], or by specialized software fora particular model (e.g. TwoD for two-dimensionalGleeson–Cullis models [37]).

The properties of these methods have been exten-sively discussed over the past 20 years. Dagnelie [21,22] provides a relatively recent review and his-torical summary of the Papadakis model. Wu andDutilleul [72] use uniformity trial data to compareautoregressive models, difference models, and tra-ditional RCB analyses. The efficiency of a spatialanalysis, relative to an RCB design, is usually greaterthan 1.2 and can be as high as 2 [22]. However, itcan give biased estimates of treatment effects [71].The Papadakis method appears to work best whenthere are at least three replicates per treatment, manytreatments (greater than 10), and strong but patchyspatial heterogeneity [22]. When there is an under-lying trend, first-order difference models appear towork well.

Medium-scale spatial heterogeneity usually causesa positive correlation between adjacent plots. Whenthere is competition between plots, neighbors canhave a negative effect on the response in adjacentplots [41]. The Papadakis model (15) can be extendedto estimate treatment-specific competitive effects.The choice of covariate should be influenced bybiological mechanisms. If competition for sunlightis important, a reasonable covariate could be thedifference between the mean height of plants in theplot and the mean height on neighboring plots. Ifdisease spread is important, a reasonable covariate

could be the mean disease severity on neighboringplots [42]. The coefficient for the covariate [ˇ in (15)]estimates the strength of the competitive relationship.

Experimental design for a study that will use someform of neighbor-adjusted analysis usually focuses onneighbor balance; that is, ensuring that all pairs oftreatments occur in adjacent plots equally frequently.Adjacent plots can be defined as only those withinthe same row (one-dimensional neighbor-balanceddesigns [70]) or as including both those in the samerow and those in the same column (two-dimensionalneighbor-balanced designs [34]). The choice willdepend on the size, shape, and spacing of the plotsand on the biological and physical mechanisms influ-encing the correlation between plots. Methods forconstruction of one-dimensional or two-dimensionaldesigns can be found in [33], [34], and [70].

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(See alsoPoint processes, spatial–temporal; Spa-tial analysis in ecology; Spatial design, optimal)

PHILIP M. DIXON


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