We consider the problem of experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. We further develop the method introduced by Rosenblum and Pikovsky [Phys. Rev. E 64, 045202 (2001)], suggesting an alternative approach. Next, we consider another framework for identification of directionality, based on the idea of mutual predictability. Our algorithms provide directionality index that shows whether the coupling between the oscillators is unidirectional or bidirectional, and quantifies the asymmetry of bidirectional coupling. We demonstrate the efficiency of three different algorithms in determination of directionality index from short and noisy data. These techniques are then applied to analysis of cardiorespiratory interaction in healthy infants. The results reveal that the direction of coupling between cardiovascular and respiratory systems varies with the age within the first 6 months of life. We find a tendency to change from nearly symmetric bidirectional interaction to nearly unidirectional one (from respiration to the cardiovascular system).