## 2 Methods
### Algorithm  
The regular ENMO, as its name indicates, is the Euclidean norm of the accelerometer data minus gravitational acceleration. The latter equals 1 g for signals measured in \(g\) unit. For reliability purposes, accelerometers must first be calibrated to avoid an inaccurate internal approximation for the gravitational acceleration in the devices. In our case, the off-the-shelf smartwatches turned out to be imprecisely calibrated. To compensate for this, the gravitational acceleration was instead approximated by (i) low-pass filtering the Euclidean norm (Chebyshev type I IIR filter, third order, \(f_{c}=0.01\) Hz) and (ii) computing the median of the filtered signal values which were between 0.9 and 1.1 as the smartwatch approximation of the gravitational acceleration value is expected to be close to 1. Figure 1 illustrates this method. The low-pass filtering step of this method was inspired by multiple previous studies, which showed notably good results [(20; 21)]. The ENMEG, \(E[k]\), was then computed on the raw accelerometer data by calculating the Euclidean norm and by subtracting the previously estimated constant:  
\[E[k]=\sqrt{\gamma_{x}[k]^{2}+\gamma_{y}[k]^{2}+\gamma_{z}[k]^{2}}-g_{est} \tag{1}\]  
where \(\gamma_{x}[k]\), \(\gamma_{y}[k]\) and \(\gamma_{z}[k]\) denote the \(k^{th}\) values of the acceleration vector along axis x, y and z and \(g_{est}\) the estimated gravitational acceleration. Negative values of \(E[k]\) were set to 0. Figure 1: The gravitational acceleration estimation \(g_{est}\) is computed as the median of the low-pass filtered (in blue) ENMEG (\(E[k]\), in black) considered between the boundary \([0.9,1.1]\) (in grey). It was observed that its dynamic was too fast for our application. To be meaningful, we defined that a physical activity of interest should be evaluated over the course of a timespan. Therefore, for smoothing purposes, a moving average was computed on a one-minute sliding window as:  
\[\overline{E}[k]=\frac{\sum_{i=k-\frac{n}{8}}^{k+\frac{n}{8}-1}E[i]}{n} \tag{2}\]  
with \(n=1800\) in our case, which corresponds to 1 min sampled at 30 Hz. The sliding window size was decreased at the edges, where there were not enough samples to fill it. The sliding window length was empirically selected by considering that 1 min was enough to smooth brief and intense motions which should not be considered as intense activity (e.g., picking up a distant object, stretching while working, etc. ), but also to preserve high frequency content. Finally, three thresholds were applied to discriminate between the timestamps spent in four different categories of activity, as summarized in Table 1.  
It is foreseen that the thresholds values are different for each individual. To personalize the method to each subject's abilities, the thresholds were individually estimated by computing the histogram of \(\overline{E}\) and by looking for the minimum value within certain ranges of magnitude, as illustrated in Figure 2.  
These ranges for individual threshold estimation were defined by the spread of valleys between each peak of activity type on the aggregated histogram of all time series across subjects. These ranges are summarized in Table 2.  
The activity categories and their interpretations can be developed as follows: \begin{table}
\begin{tabular}{c c} \hline \hline
**Activity Intensity** & **Thresholds** \\ \hline Rest & \([0,\tau_{1}]\) \\ Low & \(]\tau_{1},\tau_{2}]\) \\ Moderate & \(]\tau_{2},\tau_{3}]\) \\ High & \(]\tau_{3},+\infty[\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Activity intensity and their corresponding ranges. \begin{table}
\begin{tabular}{c c} \hline \hline
**Thresholds** & **Ranges (\(10^{-3}\) g)** \\ \hline \(\tau_{1}\) & \(R_{1}=[0,5]\) \\ \(\tau_{2}\) & \(R_{2}=[5,100]\) \\ \(\tau_{3}\) & \(R_{3}=[100,400]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thresholds and their corresponding identified ranges for estimation.