Visualizing rate of change

an application to age-specific fertility rates

Han Lin Shang*

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Citations (Scopus)


Visualization methods help in the discovery of characteristics that might not have been apparent by using mathematical models and summary statistics. However, visualization methods have not received much attention in demography, with the exceptions of scatter plots and Lexis surfaces. We utilize a phase plane plot to visualize 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 the fertility distribution. Unlike the scatter plot and Lexis surface, the phase plane plot identifies the age with maximum positive or negative velocity (i.e. the trend) and can compare the magnitude of the rate of change between any two years on the basis of the size of the radius of circles. The phase plane plot enables visualization of dynamic changes in fertility for a given age over the years and is potentially useful for visualizing dynamic changes in birth cohort fertility. Via the animate package in LATEX, a dynamic phase plane plot is also proposed to visualize changes in fertility over age or year.
Original languageEnglish
Pages (from-to)249-262
Number of pages14
JournalJournal of the Royal Statistical Society. Series A: Statistics in Society
Issue number1
Publication statusPublished - Jan 2019
Externally publishedYes


  • Derivatives of functions
  • Hysteresis loop
  • Isogrowth curve
  • Non-linear systems

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