Cyclical biological events should be analysed using circular statistics

Journal of Ecology recently published a new review by Staggemeier et al. “The circular nature of recurrent life cycle events: a test comparing tropical and temperate phenology.”

Here authors Vanessa Graziele Staggemeier, Maria Gabriela Gutierrez & Patrícia Morellato discuss their study in more detail and highlight the importance of circular statistics.


Since von Humboldt, many naturalists have been fascinated and intrigued by the contrasts between temperate and tropical zones. They have been impressed by the diversity of life forms and reproductive strategies of plants living in habitats where environmental factors (such as strong seasonality, cold and high latitudes) do not limit biological activity.

Temperate ecosystems feature a marked annual resting season, which defines the start of the growing and reproductive season for the entire plant community. Whereas tropical communities have an indeterminate starting point – since the mild climate, even when seasonal, allows species to reproduce throughout the year (Fig. 1).

Figure 1. Tropical evergreen forest at Santa Virginia Station, São Paulo, Brazil. In non-seasonal forests, where temperature and rainfall are not liming factors, species are observed flowering year-round due to the absence of a restrictive season.
Source: http://www.recod.ic.unicamp.br/ephenology/client/index.html


These differences must be considered when analysing recurrent events. Many biological events have a cyclical nature and can be called phenological events; examples include leafing, flowering and fruiting, bird migration and insect emergence. We cannot use linear statistics to analyse patterns among biological events, when these events have no starting point i.e. when there is no true zero. In these instances, we need to account for the circular nature of biological events. Circular statistics is an elegant and accurate solution to calculate basic statistics such as mean, standard deviation and variance for directional or temporal data with a circular distribution (Fig. 2). In our article, we explore and illustrate the consequences of using linear statistics for data that are distributed year-round.

Figure 2. Linear (A, B) & circular (C, D) histograms representing the distribution of simulated FFD (first flowering date) for 1000 temperate communities affected by a resting season, during which species cannot reproduce (A, C), and for 1000 non-resting tropical communities, where species can reproduce throughout the year (B, D). Absolute differences in the mean dates of FFD per community, estimated by linear (yellow arrows) and circular (blue arrows) statistics, were negligible for resting communities (E). For non-resting communities (F), however, absolute differences were larger, demonstrating the error of using a linear scale for circular measures of recurrent events. The linear FFD mean date of non-resting systems differed from the circular mean date and fell outside the phenological activity peak (Sep–Dec).


Circular statistics emerged as a field of research in the 70s and are now being widely used to study a variety of questions. Examples of such questions include: flooding dynamics, the direction of animals flight, the arrangement of proteins, neuronal activity, cognitive psychology, respiratory disease dynamics associated with variation on airborne pollen, volcano activity, crime patterns, feeding time or foraging activity of animals, effects of power lines on flight behaviour, plant colonisation and directional growth responses, plant niche preferences, root distribution and, of course, phenology. In circular statistics, instead of analysing the raw data, such as dates of a year, we convert dates into angles (unit is degrees). Sine and cosine values for each angle represent the cartesian coordinates, and all further calculations are based on trigonometric moments. Unfortunately, although this field of statistics is important to ecology, it has not yet been included in Ecology textbooks.

In the first section of our paper, we compare linear and circular approaches to analyse phenological patterns in resting and non-resting systems. We then briefly review the use of first flowering date (FFD) in phenological studies and investigate the implications of applying linear analyses to inherently circular data. We use simulated datasets with known phenological and phylogenetic structures, both with and without a rest season. Specifically, we look at the comparative methods applied to estimate phylogenetic signals using phenological metrics derived from linear and circular statistics. We show that standard analyses of linear ordinal dates may yield misleading descriptive statistics (see Fig. 2) and lead to incorrect inferences of phylogenetic signal.  Finally, we provide Rscript on Dryad Digital Repository (Staggemeier et al., 2019). This will the analyses conducted in this theoretical study to be reproduced and can be used in future studies on circular data. The advancement of this field of research urgently requires the development of comparative phylogenetic methods to properly deal with the circular nature of phenological data.


Vanessa Graziele Staggemeier, Maria Gabriela Gutierrez & Patrícia Morellato, Universidade Estadual Paulista (UNESP), Rio Claro, Brasil


Read the full paper onlineThe circular nature of recurrent life cycle events: a test comparing tropical and temperate phenology

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