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.