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}