Most expert projections indicate that in 2030, there will be over one billion people aged 60 or over. The vast majority of them prefer to spend their last years at home, and almost a third of them live alone. This creates a growing need for technology-based solutions capable of helping older people to live independently in their places. Despite the wealth of solutions proposed for this general problem, there are very few support systems that can be reproduced on a larger scale. In this study, we propose a method to monitor the activity of the elderly living alone and detect deviations from the previous activity patterns based on the idea that the residential living environment can be modeled as a collection of behaviorally significant places located arbitrarily in a generic space. Then we use virtual pheromones—a concept defined in our previous work—to create images of the pheromone distribution maps, which describe the spatiotemporal evolution of the interactions between the user and the environment. We propose a method to detect deviations from the activity routines based on a simple statistical analysis of the resulting images. By applying this method on two public activity recognition datasets, we found that the system is capable of detecting both singular deviations and slow-deviating trends from the previous activity routine of the monitored persons.

The weighted independent set problem on $P_5$ -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of $P_5$ -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of $P_5$ -free graphs. The size of a minimum independent feedback vertex set that belongs to a $P_5$ -free graph with n vertices can be computed in $O(n^{16})$ time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the $P_5$ -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an $O(n(n+m))$ recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in $O(n)$ time, the size of a minimum independent feedback vertex set; and determine in $O(n+m)$ time the number of stability, the dominating number and the minimum clique cover.