Visualising rate of change: application to age-specific fertility
  Visualisation methods help in the discovery of characteristics that might not have been apparent using mathematical models and summary statistics. However, visualisation methods have not received much attention in demography, with the exceptions of scatter plot and Lexis surface. We utilise a phase-plane plot to visualise the rate of change, obtained from derivatives of a continuous function. The phase-plane plot bears a resemblance to hysteresis loops, isogrowth curves, and solutions to differential equations. Using Australian and Chilean fertility, we present phase-plane plots. Similarly to the scatter plot and Lexis surface, the phase-plane plot identifies the age with maximum fertility rate and displays skewness of fertility distribution. Unlike the scatter plot and Lexis surface, the phase-plane plot identifies the age with maximum positive or negative velocity (i.e., trend), can compare the magnitude of the rate of change between any two years based on the size of the radius of circles. The phase-plane plot allows the visualisation of dynamic changes in fertility for a given age over the years and is potentially useful for visualising dynamic changes in birth-cohort fertility. Via the animate package in LaTeX, a dynamic phase-plane plot is also proposed to visualise changes in fertility over age or year. 
Han Lin Shang
stat.AP