Project 3, Part B: Kernel Density Estimation

Kernel Density Analysis

When we plot points on a map, sometimes the points are plotted on top of one another, making it difficult to see where the greatest number of events are taking place within a study area. To solve this problem, kernel density analysis is often used.

Kernel density visualization is performed in spatstat using the density() function, which we have already seen in action in Lesson 3A. The second (optional) parameter in the density function is the bandwidth (sigma). R's definition of bandwidth requires some care in its use. Because it is the standard deviation of a Gaussian (i.e., normal) kernel function, it is actually only around 1/2 of the radius across which events will be 'spread' by the kernel function. Remember that the spatial units we are using here are feet.

It's probably best to add contours to a plot by storing the result of the density analysis, and you can add the points too.

R provides a function that will suggest an optimal bandwidth to use. Once the optimal bandwidth value has been calculated, you can use this in the density function. You may not feel that the optimal value is optimal. Or you may find it useful to consider what is 'optimal' about this setting.

#KDE Analysis

#Density
K1 <-density(crimesppp) # Using the default bandwidth
plot(K1, main=NULL, las=1)
contour(K1, add=TRUE, col = "white")
#add a label to the top of the plot
mtext("default bandwidth") 

#Density–changing bandwidth
K2 <-density(crimesppp, sigma=500) # set different bandwidths. This data is in projected feet.
plot(K2, main=NULL)
contour(K2, add=TRUE, col = "white")
mtext("500 feet") 

#Density–optimal bandwidth

#R provides a function that will suggest an optimal bandwidth to use:
#You can replicate this code to find out the other bandwidth values. when you examine 
#individual crime types later in the lesson.
KDE_opt <-bw.diggle(crimesppp)

K3 <-density(crimesppp, sigma=KDE_opt) # Using the diggle bandwidth
plot(K3, main=NULL)
contour(K3, add=TRUE, col = "white")

# Convert the bandwidth vector to a character string with 5 character places 
diggle_bandwidth <-toString(KDE_opt, width = 5) 
# Add the bandwidth value to the plot title 
mtext(diggle_bandwidth) 

#Now that you are familiar with how KDE works, lets create a KDE for one particular crime type. 

#To select an individual crime specify the crimetype as follows:
xhomicide <-crimesppp[crimes_shp_prj$crimet2=="homicide"]

KHom<-density(xhomicide) # Using the default bandwidth
plot(KHom, main=NULL, las=1)
contour(KHom, add=TRUE, col = "white")
plot(xhomicide, pch=19, col="white", use.marks=FALSE, cex=0.75, add=TRUE) #the pch=19 sets the symbol type 
#to a solid filled circle, the col="white" sets the fill color for the symbols,
#we set use.marks=FALSE so that we aren't plotting a unique symbol (ie square, circle, etc) for each observation, 
#cex=0.75 sets the size of the symbol to be at 75% of the default size, 
#and add=TRUE draws the crime points on top of the existing density surface.
mtext("homicide default bandwidth") 
 

Note that you can experiment with the plot symbol, color, and size if you would like in the plot() function. Here is a link to a quick overview page discussing plot() commands.

Remember to select the specific crime of the overall data set before continuing. 

#To select an individual crime specify the crimetype as follows:
xhomicide <-crimesppp[crimes_shp_prj$crimet2=="homicide"]

For the remainder of this project: in addition to homicide, also examine two additional crime types using KDE. Review the plots created during Lesson 2 to select your crimes to analyze.