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Top1. Introduction
According to the ubiquitous computing paradigm, the term introduced by Mark Weiser (1991), many networked and closely cooperating sensors and computing devices which are implanted, worn, or carried by the user synthesize a dynamic computing system. Such a system (Friedemann, 2001) is required to present the characteristics of transparency, adaptability, availability, awareness, reliability, scalability, connectability, and reusability. The end-user is supported unobtrusively, the way someone is using eyeglasses, available everywhere and at any time. The supporting software is always aware of the evolving situational context, the feature of context-awareness.
In the ubiquitous computing environment, independently operating sensors measure physical magnitudes. The values of the sensors measuring the same magnitudes must be fused in order to obtain the dominating value for each measured magnitude. The operation of the measuring sensors can be cooperative, competitive, or complementary. Moreover, the sensors can be classified as either physical, directly measuring devices, or virtual, resulting from the physical sensors’ data correlation without succeeding, so far, to find a generalized fusion model (Zhang, Cao, Zhou, & Guo, 2009) for home healthcare.
The disseminated physical sensors at home are characterized by practical limitations caused by user’s mobility or energy availability and thus introducing uncertainty. There are two types of uncertainty, the objective uncertainty, caused by random events and situations at home, and the subjective uncertainty, caused by missed information and by the holding state of knowledge. Uncertainty develops when sensors contribute incomplete, imprecise, or conflicting measurements. The determination of uncertainty requires precise hypotheses probability values from mutually exclusive sensors which is hard to obtain in real-time conditions. Instead, intervals of probabilities of sensors’ readings can be considered, expressing the confidence about the sensors’ reliability applying the Dempster-Shafer evidence theory -DSET- (Zhang et al, 2009) (a generalization of the Bayesian probability theory) referring to discrete and finite readings from single or multiple sensors applied on sets of events. The DSET method presents problems related with the emblematic Zadeh’s paradox (1986) and the excessive computational overhead which is partially reduced applying DSET’s extensions such as those proposed by Yger (1987), Boujelben et al. (2009), and Zhang et al. (2009).
Home healthcare is an alternative paradigm provided that the necessary infrastructure is in place supporting the inhabitants’ health status. The provision of health services at home presents advantages over the experienced conventional ways with respect to financial savings, the quality level of the provisional health services, and the avoidance of hospital infections. Healthcare faces three categories of patients, first, the healthy population, second, the people experiencing a health crisis, and third, the chronic patients. The provision of healthcare services at home can be based on a ubiquitous computing system which is capable of supporting the dynamic development of consecutive contexts on which the decision making relies on. However, the available analytic mathematics has limited practical expressive power to describe dynamic abstractions of context in home healthcare. The developing context at home evolves in time and it is tedious or impractical to analyze and examine with analytical mathematics due to the unavoidable large number of details.
The complexity developed by the interrelationships among the sensing and the computing devices supporting healthcare at home obliges to look for alternative mathematical tools. Denotational mathematics propose an alternative way of handling complex data structures beyond the conventional use of sets and calculus functions which find difficulties in representing relationships, behaviors, concepts, and knowledge. The complexity of the analytic mathematics for the formal description of medical contexts at home leads to the adoption of denotational mathematics for the mathematical manipulation of complex abstractions representing the contextual entities, their relationships, and their behaviors.