Error probabilities for local extrema in gene expression data

Current approaches for the prediction of functional relations from gene expression data often do not have a clear methodology for extracting features and are not accompanied by a clear characterisation of their performance in terms of the inherent noise present in such data sets. Without such a characterisation it is unclear how to focus on the most probable functional relations present. In this article, we start from the fundamental theory of scale-space for obtaining features (i.e., local extrema) from gene expression profiles. We show that under the assumption of Gaussian distributed noise, repeatedly measuring a local extrema behaves like a bivariate Gaussian distribution. Furthermore, the error of not re-observing local extrema is phrased in terms of the integral over the tails of this bivariate Gaussian distribution. Using integration techniques developed in the 1950s, we demonstrate how to compute these error probabilities exactly.