Side-channel attacks of maximal efficiency require an accurate knowledge of the leakage function. Template attacks have been introduced by Chari et al. at CHES 2002 to estimate the leakage function using available training data. Schindler et al. noticed at CHES 2005 that the complexity of profiling could be alleviated if the evaluator has some prior knowledge on the leakage function. The initial idea of Schindler is that an engineer can model the leakage from the structure of the circuit. However, for some thin CMOS technologies or some advanced countermeasures, the engineer intuition might not be sufficient. Therefore, inferring the leakage function based on profiling is still important. In the state-of-the-art, though, the profiling stage is conducted based on a linear regression in a non-orthonormal basis. This does not allow for an easy interpretation because the components are not independent. In this paper, we present a method to characterize the leakage based on a Walsh-Hadamard orthonormal basis with staggered degrees, which allows for direct interpretations in terms of bits interactions. A straightforward application is the characterization of a class of devices in order to understand their leakage structure. Such information is precious for designers and also for evaluators, who can devise attack bases relevantly.