  In this paper we present a theoretical foundation for a representation of a
data set as a measure in a very large hierarchically parametrized family of
positive measures, whose parameters can be computed explicitly (rather than
estimated by optimization), and illustrate its applicability to a wide range of
data types. The pre-processing step then consists of representing data sets as
simple measures. The theoretical foundation consists of a dyadic product
formula representation lemma, a visualization theorem. We also define an
additive multiscale noise model which can be used to sample from dyadic
measures and a more general multiplicative multiscale noise model which can be
used to perturb continuous functions, Borel measures, and dyadic measures. The
first two results are based on theorems. The representation uses the very
simple concept of a dyadic tree, and hence is widely applicable, easily
understood, and easily computed. Since the data sample is represented as a
measure, subsequent analysis can exploit statistical and measure theoretic
concepts and theories. Because the representation uses the very simple concept
of a dyadic tree defined on the universe of a data set and the parameters are
simply and explicitly computable and easily interpretable and visualizable, we
hope that this approach will be broadly useful to mathematicians,
statisticians, and computer scientists who are intrigued by or involved in data
science including its mathematical foundations.
