81  Bivariate Kernel Density Plot

81.1 Definition

The Bivariate Kernel Density Plot is an alternative for the Scatter Plot. It is obtained by applying Kernel Density Estimation for a bivariate data set (\(x, y\)) and drawing contours of equal density on top of the Scatter Plot.

81.1.1 Horizontal axis

The horizontal axis represents the values for the \(x\) variable.

81.1.2 Vertical axis

The vertical axis represents the values for the \(y\) variable.

81.1.3 Third axis

The third axis represents the density of scatter points and is not drawn. Instead, the density is represented by contour lines which connect all points of equal density.

81.2 R Module

81.2.1 Public website

The Bivariate Kernel Density Plot can be found on the public website:

To investigate a trivariate dataset, one can also use the Trivariate Scatterplot module that can be found on the public website:

81.2.2 RFC

The Bivariate Kernel Density Plot is also available in RFC under the menu item “Descriptive / Multivariate Descriptive Statistics”.

To compute the Bivariate Kernel Density Plot on your local machine, the following script can be used in the R console:

library(ks)
df = data.frame(y = c(20,16,19.8,18.4,17.1,15.5,14.7,17.1,15.4,16.2,15,17.2,16,17,14.4), 
                x = c(88.6,71.6,93.3,84.3,80.6,75.2,69.7,82,69.4,83.3,79.6,82.6,80.6,83.5,76.3))

fhat <- kde(df)
cor(fhat$x)
plot(fhat, display = "image", drawpoints = F)

          y         x
y 1.0000000 0.8351438
x 0.8351438 1.0000000

To compute the Bivariate Kernel Density Plot, the R code uses the kde function from the ks library.

81.3 Purpose

The Bivariate Kernel Density Plot is often used as an exploratory tool -- i.e. to visualize the relationship between two quantitative variables.

Compared with Chapter 70 and Chapter 80, this plot combines both ideas: it keeps the bivariate view of a scatterplot but adds density contours to reveal where points are concentrated.

81.4 Pros & Cons

81.4.1 Pros

The Bivariate Kernel Density Plot has the following advantages:

  • it is a much better tool than the Scatter Plot because it provides more information
  • it is easily understood by most readers
  • it allows the researcher to identify the shape of the relationship between both variables (e.g. linear versus non-linear)

81.4.2 Cons

The Bivariate Kernel Density Plot has the following disadvantages:

  • there are not many software packages which are able to compute the Bivariate Kernel Density Plot
  • it does not always display the true nature of the relationship between both variables

81.5 Example

A useful application is to inspect whether a visually linear cloud in a scatterplot is actually concentrated in one dense core or split into multiple dense regions. Multiple high-density regions may indicate subgroups, non-linearity, or outliers.

81.6 Task

Compare the bivariate density plot with the ordinary scatterplot for the same variables. Do both plots support the same interpretation?