## 1 Introduction
### State of the Art
#### 1.1.2 Physical Activity Quantification  
[PERSON] et al. introduced a simple and efficient accelerometry-based methodology for the assessment activity of both arms [14]. A sensor was placed on each upper limb of subjects and accelerometry was recorded while they were preforming specific tasks. After data collection, two variables of interest were computed: the Magnitude Ratio and the Bilateral Magnitude. The first one was a representation of the acceleration ratio between the two arms and the latter reflected the overall intensity of the activity performed with both arms. The same team applied this concept in a study involving non-disabled patients and stroke patients [15].  
Similar solutions to our method were proposed in past studies. Notably in [16], the authors present a smartwatch-based system for real-time and online mobility monitoring. The tri-axial accelerometer data was used to compute a vector magnitude, the angle between the forearm and the horizontal line and some other simple spectral parameters. These parameters were then used to monitor the min spent in light, moderate or vigorous activity or even performing sedentary activities.  
Introduced by [PERSON] et al. in 2013, the ENMO metric was recently used by [PERSON] et al. for a comparison to the ActiGraph GT3X to monitor physical activity. The device is one of the major references in the field of physical activity monitoring [17; 18]. The ENMO is a simple and straightforward tool to analyse physical activity via accelerometry without any filtering or complex computation. The study from [PERSON] et al. emphasized that this metric, computed on hip and wrist accelerometers, could discriminate between sedentary behaviours and light activity.  
As our final application will involve elderly and potentially impaired post stroke patients, it is reasonable to consider that any machine learning algorithm trained on healthy subjects may provide poor results. Furthermore, to the best of our knowledge, the large data sets, in the \"Big Data\" sense, specialized on disabled subjects and needed to train machine learning algorithms are seldom and difficult to obtain in the framework of feasibility studies. Here, a simpler approach is preferable which can be easily adapted to the use case of stroke patients and effortlessly implemented on a mobile system, such as smartwatches or smartphones.  
However, the main flaw of the ENMO is its sensitivity to imprecisely calibrated sensors. Such sensors usually lead to biased measurements, which can be problematic with method requiring the subtraction of a fixed offset, as with ENMO and its hypothesis of a 1 g constant gravitational acceleration. To tackle this problem, [PERSON] et al. proposed a calibration method to be applied after data collection but before the ENMO computation [19]. In fact, it was also used by [PERSON] et al. in [18]. The proposed method considers the acceleration signals when the sensors are close to immobility to estimate the error bias and gain on each axis. It also takes into consideration the local temperature during measurement to compensate for thermally induced errors. The efficiency of the algorithm is demonstrated in the article, however strong limitations remain, namely: (i) the necessity to have a certain amount of data (a 72 h long signal seems to be an important boundary in the approach) and (ii) the method reaches its full potential when temperature measurements are provided, which are not always available.  
In that regard, we decided to base our method on a modified version of the ENMO metric, which we refer to as the Euclidean norm minus the estimated gravity (ENMEG) and which integrate a straightforward internal compensation for imprecise sensors.