Additive Functional Regression for Densities as Responses

Abstract

Define K-disjoint partitions (I k : 1 ≤ k ≤ K) of an index set {1, . . . , n} such thatThen (i) we set a baseline bandwidth vectord ) for each sub-sample X n,(−k) , and (ii) compute the prediction performance of the trained model on the test set k) ) are the back transformed density estimators (2.5) from the smooth backfitting log-quantile density estimators ĝ(•, X i ; X n,(−k) , αh (k) ) based on the sub-sample X n,(−k) with bandwidth αh (k) . Also, fi are the marginal density estimators of f i defined as in (2.9), based on Y i , respectively.

Additive regression for predictors of various natures and possibly incomplete Hilbertian responses

“…For such compositional data with zero entries, one may apply the parametric approach in [38], for example. For density-valued responses with continuous predictors, [11] considered an additive model but on the transformed conditional Fréchet mean via log-quantile and log-hazard transformations studied in [33]. It is different from our additive model that assumes additivity directly on the conditional mean based on the 'Aitchison' geometry given in Section 2.1.…”

“…It is different from our additive model that assumes additivity directly on the conditional mean based on the 'Aitchison' geometry given in Section 2.1. The target in [11] is to estimate the conditional Fréchet mean minimizing the expected Wasserstein distance from Y, while our target is to estimate the conditional mean minimizing E( Y • 2 ) with the Aitchison norm • .…”

Object oriented spatial analysis of natural concentration levels of chemical species in regional-scale aquifers

“…Note that the choice of the space within which the analysis is embedded (also termed as the feature space) is critical in Object Oriented Spatial Statistics (O2S2) -and, more generally, in Object Oriented Data Analysis -because it has a key impact on the results and on their interpretation. We also note that the Bayes space approach we consider shares some similarities with the approach advocated by Petersen and Müller (2016); Han et al (2019), who propose a strategy for the analysis of distributional datasets consisting of (a) mapping the data in L 2 through an isometric isomorphism, (b) performing an unconstrained analysis in L 2 , and (c) mapping back the results to the space of densities. We model observations X s1 , ...X sn upon considering the mathematical framework of O2S2 and assuming they are a partial observation of a random field {X s , s ∈ D} valued in B 2 , whose elements are random NBL densities.…”

Glucodensities: A new representation of glucose profiles using distributional data analysis

“…In the case of regression models with a density function as response, the literature is not very extensive to the current date. 43–47 In this article, we use the model proposed in Petersen and Müller 45 which allows us to incorporate the desired metric dW2 and is a direct generalization of classical linear regression. The primary rationale for the use of this model is that, unlike the other approaches cited above, there is a methodology developed to perform inferential procedures such as confidence bands and hypothesis testing in order to establish the significance of the input variables in the model.…”