Stochastic monotony signature and biomedical applications

We introduce a new concept, the stochastic monotony signature of a function, made of the sequence of the signs that indicate if the function is increasing or constant (sign +), or decreasing (sign -). If the function results from the averaging of successive observations with errors, the monotony sign is a random binary variable, whose density is studied under two hypotheses for the distribution of errors: uniform and Gaussian. Then, we describe a simple statistical test allowing the comparison between the monotony signatures of two functions (e.g., one observed and the other as reference) and we apply the test to four biomedical examples, coming from genetics, psychology, gerontology, and morphogenesis.