Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise

Abstract

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 preprocessing step then consists of representing data sets as simple measures. The theoretical foundation consists of a dyadic product formula representation lemma, and a visualization theorem. We also define an additive multiscale noise model that can be used to sample from dyadic measures and a more general multiplicative multiscale noise model that can be used to perturb continuous functions, Borel measures, and dyadic measures. The first two results are based on theorems in [15, 3, 1]. 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.

Inference of a Dyadic Measure and Its Simplicial Geometry from Binary Feature Data and Application to Data Quality

“…In [18] Kolaczyk and Nowak developed a systematic approach to multiscale probability models. They showed that multiscale factorizations, similar to Lemma 3.20 in [12] and the refomulation of it for dyadic measures on binary tree structured spaces, Lemma 2.1 in [2], arise when conditions for a "multiresolutionanalysis (MRA)" of likelihoods are satisfied and shown that these conditions characterize the Gaussian, Poisson and multinomial models. They also quantified the risk behavior of certain nonparametric, complexity penalized likelihood estimators based on their factorizations.…”

“…The product coefficient parameters uniquely determine the measure µ by the Dyadic Product Formula Representation ( Lemma 2.1) [2], even when the binary tree is infinite. The basic observation is that µ(S(n)) equals µ(D) multipled by the product of the factors from the root to a node n divided by 2 −scale(n) .…”

“…The two sets of product coefficients are related by a Bayes formula, so the new parameters are related by rational algebraic formulas to the original set of parameters. One quantitative formulation of this Bayes-type rule is given in Appendix 2 of version 2 of [2].…”

“…The restricted set of dyadic measures that they used is very far from the case of general measures and in particular the L 2 condition they studied does not hold for multifractal measures which typically arise in for many real data sets. In [2] the author and collaborators realized the result could be re-formulated for dyadic measures on sets with binary tree structures (not just the unit interval) and used to provide an algorithmizable theoretically-based method for representing finite data samples from universes with a binary tree structure as vectors of product coefficient parameters of measures. They exploited the fact that the weak star limit result holds for the much more general class of non-negative dyadic measures.…”

“…In this paper we will describe a new method for representing a data set for which binary feature have been defined. The method algorithmically infers two related representations: a representation of the data set as a non-parametric multi-scale dyadic measure using the dyadic product formula representation [2] [12] [20] and a summary of the simplicial geometry of the support of the measure in terms of the betti numbers of a variant of a nerve simplicial complex determined by the dyadic measure. Both of these representations are based on mathematical theory.…”