Point Pattern Analysis

11.3.1 Average Nearest Neighbor (ANN)

• The average distance from each point to its nearest point
  
• An extension: plot the ANN values for different order neighbors (x-orders & y-ANN values for a specific order)

• Different point patterns return different ANN vs. neighbor order plots.

11.3.2.1 Ripley’s K function

• Summarizes the distance between points for all distances.
  
• First, count the number of points at different distance bands for each point.
  
• Second, compute the mean of the counts for each distance bands.
  
• Third, divide the mean by the area event density (n/a).

• Plot K and compare it to a plot under an CSR (complete spatial random) process (i.e., K expected).
  
• K values above K expected : clustering of points at a given distance band
  
• K values below K expected : dispersion of points at a given distance band (i.e., inhibition)

11.3.2.1 Besag’s L function

• Helps determine small differences between K and K expected.
  
• Transform values for K and K expected _ so that the latter is always zero.
• L values above zero : clustering of points at a given distance band
  
• L values below zero : dispersion of points at a given distance band (i.e., inhibition)

11.3.3 The Pair Correlation Function g

• Note that K/L functions are cumulative.
  
• Modify K function so that, instead of counting all points within a distance r, count only those points falling within a narrow distance band around r.
  
• This modified function is called g function, which presents more accurate trends for spatial dispersion/clustering.

1. Compute the average distance to the nearst neighbors (ANN) for Starbucks in Massachusetts

# load the data 
load(url("http://github.com/mgimond/Spatial/raw/master/Data/ppa.RData"))

# rescale from meter to kilometer 
marks(starbucks)  <- NULL
Window(starbucks) <- ma # redefine the window of starbuck ppp 
starbucks.km <- rescale(starbucks, 1000, "km")

# nndist function: distance to the kth nearest neighbor from each point 
nndist(starbucks.km, k=1) 
nndist(starbucks.km, k=2) 
nndist(starbucks.km, k=1:2)

mean(nndist(starbucks.km, k=1))
mean(nndist(starbucks.km, k=2))

# compute the average nndist for up to the 100th nearest neighbor from each point 
# base R way 
ANN <- apply(nndist(starbucks.km, k=1:100),2,FUN=mean) # MARGIN = 2 (over column)

# tidyverse way 
ANN <-
  map(1:100, ~nndist(starbucks.km, k=.)) %>% # returns a list of lists 
  map_dbl(~mean(.))                          # returns a nemeric vector 

# plot 
# https://www.rdocumentation.org/packages/graphics/versions/3.5.3/topics/plot
plot(ANN ~ eval(1:100), type="b", main=NULL, las=1)

2. Plot Ripley’s K function for Starbucks in Massachusetts

# Estimates Ripley's reduced second moment function K(r)
# ?spatstat::Kest 

# create a K object 
K <- Kest(starbucks.km)
plot(K, main=NULL, las=1, legendargs=list(cex=0.8, xpd=TRUE)) 

# three K(r) values under different edge-corrections 
# Kpois(r) presents the K expected under the CSR process (Poisson process)   

3. Plot Besag’s L function for Starbucks in Massachusetts

# Calculates an estimate of the L-function (Besag's transformation of Ripley's K-function)
# ?spatstat::Lest 

# create an L object 
L <- Lest(starbucks.km, main=NULL)

# plot diagonal/horizontal L 
plot(L, main=NULL, las=1, legendargs=list(cex=0.8, xpd=TRUE))

plot(L, . -r ~ r, main=NULL, las=1, legendargs=list(cex=0.8, xpd=TRUE))

4. Plot the pair correlation fuction

# Estimate the pair correlation function.
# ?spatstat::pcf

# create a pcf object 
g  <- pcf(starbucks.km)
plot(g, main=NULL, las=1, legendargs=list(cex=0.8, xpd=TRUE))