Seasonality Index
A novel seasonality index (S) was developed for this dataset to assess how strongly seasonal the vocal activity was for each species. This index defines seasonal behavior as behavior concentrated during a particular time of year, occurring at the same time across years. The index is made up of two components: signal strength (R) and directional agreement (D) (Formula 1).
Signal strength (R) is a measure of how concentrated vocal activity is within a single time of year. To calculate this, the number of vocally active hours per day is averaged by month and each monthly value is treated as a vector, where the direction of the vector corresponds with the month of the year, placed radially on a circle (Figure 2). The magnitude of the sum of all 12 vectors is divided by the total magnitude of all vectors to get signal strength (R) (Formula 2, Figure 2iii). Subscript i indexes the month. Signal strength (R) has a maximum value of 1 when all activity is concentrated within a single month, and a minimum value of 0 when activity is equally distributed throughout the year.
Directional agreement (D) is a measure of the similarity between years. To calculate directional agreement (D), all 12 monthly vectors are added together for each year to find the direction of the resultant vector θ. Directional agreement is the inverse of the difference between θyear 1 and θ year 2 (Formula 3, Figure 2ii). Directional agreement (D) has a maximum value of 1 when peak vocal activity occurs at the same time in both years, and a minimum value of 0 when peak activity happens at opposite times of year between the two years; for example, if peak activity occurred on January 1st of the first year and July 1stof the second, then D=0.
The seasonality index (S) will have a maximum value of 1 when vocal activity is concentrated entirely in the same single month in both years. S will have a minimum value of zero either if activity is evenly distributed throughout the year, or if peak activity happens at opposite times of year in the two years. R code used to calculate the seasonality index is available in the Supporting Information.
Formula 1:
\begin{equation} S=R*D=Seasonality\ index\nonumber \\ \end{equation}
Formula 2:
Formula 3:
\begin{equation} \theta_{i}=direction\ of\ the\ sum\ of\ all\ vectors\ [\overrightarrow{v_{1}\ }:\overrightarrow{v_{12}}]\ from\ \text{year}_{i}\nonumber \\ \end{equation}\begin{equation} dd=\ {|\theta}_{1}-\theta_{2}|=directional\ disagreement\nonumber \\ \end{equation}\begin{equation} \text{if\ }\left(\left|\theta_{1}-\theta_{2}\right|>\pi\right),\ \ then\ dd=(2\pi-\left|\theta_{1}-\theta_{2}\right|)\nonumber \\ \end{equation}