A Bound for the Accuracy of Sensors Acquiring Compositional Data

Among the many challenges that the Internet of Things poses, the accuracy of the sensor network and relative data flow is of the foremost importance: sensors monitor the surrounding environment of an object and give information on its position, situation or context, and an error in the acquired data can lead to inappropriate decisions and uncontrolled consequences. Given a sensor network that gathers relative data - that is data for which ratios of parts are more important than absolute values - acquired data have a compositional nature and all values need to be scaled. To analyze these data a common practice is to map bijectively compositions into the ordinary euclidean space through a suitable transformation, so that standard multivariate analysis techniques can be used. In this paper an error bound on the commonly used asymmetric log-ratio transformation is found in the Simplex. The purpose is to highlight areas of the Simplex where the transformation is ill conditioned and to isolate values for which the additive log-ratio transform cannot be accurately computed. Results show that the conditioning of the transformation is strongly affected by the closeness of the transformed values and that not negligible distortions can be generated due to the unbounded propagation of the errors. An explicit formula for the accuracy of the sensors given the maximum allowed tolerance has been derived, and the critical values in the Simplex where the transformation is component-wise ill conditioned have been isolated.