# Unsupervised Classification of Human Activity with Hidden Semi-Markov Models
## 1 Introduction  
Current national guidelines suggest that people engage in at least 150 min of moderate activity per week (or 75 min of vigorous activity). These guidelines are supported by years of research showing that active individuals have reduced risks of cardiovascular disease, cancer and mortality [1, 2]. Recently, some countries such as Australia and the UK have introduced guidelines to reduce sedentary behaviour, but sufficient evidence is lacking for specific time-based recommendations. Likewise, the current research evidence is not enough to advise on the type and intensity of activity one should engage in during nonsedentary time [3]. More research is needed to collect further evidence to inform such guidelines. Accelerometers are vital in collecting data to understand the effects of physical behaviours and support guideline development. Data collected through accelerometers can be processed to classify the type and intensity of activities through various signal processing techniques. The most common method to classify human activity is the cut-off or thresholding approach. The method requires the development of a calibration study to measure the acceleration magnitude while engaging in activities of varying intensities. The acceleration magnitude is then regressed against the metabolic equivalent (MET) values for the recordedactivities to obtain device-specific cut-off values. Conventionally, two cutoffs of 1.5 and 3 METs are used to split activity into sedentary, light intensity and moderate-to-vigorous intensity [4,5]. Cut-off points are developed using studies carried out in laboratory settings, which are not reflective of free-living conditions. Calibration studies are required practically for every study, as applying established cutoffs could create biased results due to them being established on different populations, devices and activity types. As Ref. [6] found, even when using the same accelerometer device, the cut-off points ranged from 191 to 2743 counts-per-minute (CPM) for moderate-intensity activity and from 4945 to 7526 CPM for vigorous-intensity activity, depending on whether the calibration was done on laboratory or free-living activities, reflecting the poor generalisability of the cut-off approach. Supervised machine learning can be applied to accelerometer data to recognise activities. Such methods require a large number of labelled data to train the models, which can be hard to obtain in free-living settings. For example, it could require the participants to complete a very detailed activity log, which does not guarantee the accuracy of the labelled set as it is subject to recall bias. [PERSON] et al. [7] achieved a high accuracy (\(\geq\)90%) with random forests paired with a hidden Markov model, but used a ground-truth set of labelled data acquired with a camera to train their model, which is extremely resource ineffective and would require participants to wear a camera for several days, on top of their accelerometer. Alternative solutions include training the model on laboratory-acquired acceleration data, but the prediction accuracy of the classification model can be reduced by 20-30% when cross-validated on free-living acceleration data [8]. Furthermore, using existing data sets to train a model would not be feasible, given the heterogeneity in the population, device type and placement. A further problem related to conventional supervised methods is that they do not capture the time dependence of the data and assume the data to be independent [9]. Unsupervised learning methods offer a viable alternative to the methods outlined above without requiring labelled data. Clustering methods are at the heart of unsupervised learning, and standard techniques such as K-means [10,11] and Gaussian mixture models [11] have been applied to human activity recognition. However, simple clustering has some disadvantages when it comes to accelerometer data, mainly because the data are assumed to be independent and, consequently, the time dependency is lost.