Dimensionality, granularity, and differential residual weighted entropy

While Shannon's differential entropy adequately quantifies a dimensioned random variable's information deficit under a given measurement system, the same cannot be said of differential weighted entropy in its existing formulation. We develop weighted and residual weighted entropies of a dimensioned quantity from their discrete summation origins, exploring the relationship between their absolute and differential forms, and thus derive a ‟differentialized” absolute entropy based on a chosen ‟working granularity” consistent with Buckingham's ╬á-theorem. We apply this formulation to three common continuous distributions: exponential, Gaussian, and gamma and consider policies for optimizing the working granularity.