spatial statistics point patterns. spatial statistics increasing sophistication of gis allows...

Spatial Statistics

Point Patterns

Spatial Statistics

• Increasing sophistication of GIS allows archaeologists to apply a variety of spatial statistics to their data– Predictive Modeling– Intra-site Spatial Analysis

Predictive Modeling 1

• Goal is to predict where sites will be located

• Usually involves two samples: known sites, surveyed areas where sites have not been found

• For each group, we collect data: slope, aspect, distance to water, soil, vegetation zone, etc

Predictive Modeling 2

• Must convert nominal scales to dichotomies or an interval scale of some kind

• Analysis by logistic regression or discriminant functions

• For new areas we compute the probability that a site will be found

Predictive Modeling 3

• Problems:– Usually purely inductive– Goal is management not

anthropology– Independent variables are those

gathered for other reasons

Intra-Site Spatial Analyses

• Nearest Neighbor– Can use on any point plot data (sites,

houses, artifacts)– Find distance to nearest neighbor for

each item– Mean nearest neighbor compared to

expected value (random distribution)

Nearest Neighbor

• If observed mean distance is significantly less than expected, the points are clustered

• If the mean distance is significantly more than expected, the points are evenly spread

• But problems with borders

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Mean Distance = 1.04Expected Dist = 0.81Probability = 0.008 *R = Mean/Exp = 1.29

Mean Distance = .74Expected Dist = 0.63Probability = 0.091R = Mean/Exp = 1.18

Mean Distance = 0.52Expected Dist = 0.78Probability = 0.002 *R = Mean/Exp = 0.67

Clustered Random Regular

Point Patterns in R

• Package spatstat• Create a ppp object (point process)• Plotting and analytical tools are

extensive

# Fix duplicate coordinates by adding 1-2 mm to eachload("C:/Users/Carlson/Documents/Courses/Anth642/R/Data/BTF3a.RData")Win3a <- data.frame(x=c(982,982,983,983,984.5,985,985,987,987,986.2, 985,985,984.5,984,983.5,983,982.7,982.5), y=c(1015.5,1021,1021,1022,1022,1021.3,1018,1018,1017.6,1017, 1017,1016.9,1016.8,1016.6,1016.3,1016,1015.6,1015.5))# coordinates must be counterclockwiseWin3a <- Win3a[order(as.numeric(rownames(Win3a)), decreasing=TRUE),]library(spatstat)BTF3a.p <- ppp(BTF3a$East, BTF3a$North, window=owin(poly=Win3a, unitname=c("meter", "meters")), marks=BTF3a$Type)summary(BTF3a.p)plot(BTF3a.p, main="Bifacial Thinning Flakes", cex=.75, chars=16, cols=c("red", "blue", "green"))legend("topright", c("BTF", "CBT", "NCBT"), pch=16, col=c("red", "blue", "green"))plot(split(BTF3a.p), main="Bifacial Thinning Flakes") mtext("BTF = Fragments CBT = Cortex NCBT = No Cortex", side=1)

Marked planar point pattern: 230 pointsAverage intensity 12.8 points per square meter Multitype: frequency proportion intensityBTF 87 0.378 4.86CBT 49 0.213 2.74NCBT 94 0.409 5.25

Window: polygonal boundarysingle connected closed polygon with 18 verticesenclosing rectangle: [982, 987]x[1015.5, 1022]metersWindow area = 17.91 square meters Unit of length: 1 meter

plot(982, 982, xlim=c(982, 987), ylim=c(1015, 1022), main="Bifacial Thinning Flakes", xlab="", ylab="", axes=FALSE, asp=1, type="n")contour(density(BTF3a.p, adjust=.5), add=TRUE)polygon(Win3a)points(BTF3a.p, pch=20, cex=.75)

plot(density(BTF3a.p, adjust=.5), main="Bifacial Thinning Flakes")polygon(Win3a, lwd=2)points(BTF3a.p, pch=20, cex=.75)

plot(density(BTF3a.p, adjust=.5), main="Bifacial Thinning Flakes")polygon(Win3a, lwd=2)points(BTF3a.p, pch=20, cex=.75)

windows(10, 5)V <- split(BTF3a.p)A <- lapply(V, density, adjust=.5)plot(as.listof(A), main="Bifacial Thinning Flakes")

Tab <- quadratcount(BTF3a.p, xbreaks=982:987, ybreaks=1015:1022)quadrat.test(Tab)

# Warning: Some expected counts are small; chi^2 approximation may# be inaccurate

# Chi-squared test of CSR using quadrat counts

# data: # X-squared = 274.8859, df = 19, p-value < 2.2e-16

# Quadrats: 20 tiles (levels of a pixel image)

E <- kstest(BTF3a.p, "x")plot(E)N <- kstest(BTF3a.p, "y")plot(N)EN

Spatial Kolmogorov-Smirnov test of CSR

data: covariate 'x' evaluated at points of 'BTF3a.p' and transformed to uniform distribution under CSRI D = 0.1101, p-value = 0.007611alternative hypothesis: two-sided

Spatial Kolmogorov-Smirnov test of CSR

data: covariate 'y' evaluated at points of 'BTF3a.p' and transformed to uniform distribution under CSRI D = 0.2891, p-value < 2.2e-16alternative hypothesis: two-sided

Gest(BTF3a.p)plot(Gest(BTF3a.p), main="Nearest Neighbor Function G")# Above poisson is clustered, below is regularplot(envelope(BTF3a.p, Gest, nsim=39, rank=1), main="Nearest Neighbor Envelope")# The test is constructed by choosing a fixed value of r, and rejecting the null# hypothesis if the observed function value lies outside the envelope at this# value of r. This test has exact significance level alpha = 2 * nrank/(1 + nsim). plot(envelope(BTF3a.p, Gest, nsim=19, rank=1, global=TRUE), main="Global Nearest Neighbor Envelope")# The estimated K function for the data transgresses these limits if and only if# the D-value for the data exceeds Dmax. Under H0 this occurs with probability# 1/(M + 1). Thus, a test of size 5% is obtained by taking M = 19.plot(alltypes(BTF3a.p, "G"))plot(alltypes(BTF3a.p, "Gdot"))