Statistical analysis
With the goal of determining which parts of the time series corresponded
to the activity period or the hibernation period, we ran an algorithm
for calculating the Iterated Cumulative Sums of Squares (ICSS), which
detects retrospective changes of variance for identifying breaking
points.
We applied several metric-based and model-based approaches as leading
indicators of early warning signals (EWS) of critical transitions for
changes in body temperatures during hibernation
(Th ) and estivation (euthermic temperature,Te ). We tested most indicators reviewed in (Dakoset al. 2012) to assess the limitations in their application and
interpretation. First, we used the ‘earlywarnings’ package in R for
calculating metric-based indicators: BDS tests, conditional
heteroskedasticity (CH), nonparametric Drift-Diffusion-Jump (DDJ)
models, and generic EWS (temporal autocorrelation at lag-1, standard
deviation SD, and skewness) (details on how each indicator was applied
are in Appendix S1, Table S1). We performed sensitivity analyses to
assess the reliability of generic EWS depending on choices for data
transformation, detrending and filtering (Dakos et al. 2012).
Second, we ran model-based indicators on standardized data. We began by
running a potential analysis for assessing the existence of both
flickering and the occurrence of two stable states in the body
temperature time series. We then fitted threshold AR(p ) models to
identify transitions between alternative states due to flickering in the
time series. By using the Kalman filter and AIC values, we assessed
which model with different orders (p = 1, 2, 3) best fit the
data. The models also estimated the threshold value c and the
variance of the process error and were as follows:
\(T\left(t\right)=\varnothing_{0}+\sum_{i=1}^{p}{\varnothing_{i}\left(T\left(t-i\right)-\varnothing_{0}\right)+\varepsilon(t)}\),
where T(t) represents the changes of body temperature over timet ; parameters \(\varnothing_{i}\ \)had two sets of values
depending on T (t -1) being lower or higher than the
threshold value c ; ε (t ) was a white noise process
representing environmental variability. We also calculated the Kendall
τ, which indicates the strength of the trend in the indicators for body
temperatures. We also fitted time-varying AR(p ) models and
compared their fit to those obtained from threshold models to confirm
than the latter better described the flickering features of the body
temperature time series. All AR(p ) models were fitted using the
package ‘setar’ in R.
We also assessed the influence of air temperature on the dynamics of
body temperature in our studied dormice. To simplify the analysis (i.e.
avoiding including seasonality), we partitioned the time series between
three periods: activity prior to hibernation, hibernation, and activity
after hibernation, as indicated by the breaking point analysis (Appendix
S1, Table S2). The models added the air temperature covariate (A )
into the AR(p ) models and were as follows:
\(T\left(t\right)={\beta A_{i}+\varnothing}_{0}+\sum_{i=1}^{p}{\varnothing_{i}\left(T\left(t-i\right)-\varnothing_{0}\right)+\varepsilon(t)}\),
Here, we added the slope \(\beta\) of the effect of air temperatureAi on body temperature T . We then used AIC
values of each model (with and without air temperature as explanatory
variables) to select the best model. Fitting of models was carried out
using the package ‘TSA’ in R.