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, 96 (1), 149-162

Bayesian Nonparametric Functional Data Analysis Through Density Estimation

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Bayesian Nonparametric Functional Data Analysis Through Density Estimation

Abel Rodríguez et al. Biometrika.

Abstract

In many modern experimental settings, observations are obtained in the form of functions, and interest focuses on inferences on a collection of such functions. We propose a hierarchical model that allows us to simultaneously estimate multiple curves nonparametrically by using dependent Dirichlet Process mixtures of Gaussians to characterize the joint distribution of predictors and outcomes. Function estimates are then induced through the conditional distribution of the outcome given the predictors. The resulting approach allows for flexible estimation and clustering, while borrowing information across curves. We also show that the function estimates we obtain are consistent on the space of integrable functions. As an illustration, we consider an application to the analysis of Conductivity and Temperature at Depth data in the north Atlantic.

Figures

Figure 1
Figure 1
Oceanography study. Heatmap with the probabilities of pairwise joint classification. Pixel (i, j) represents the posterior probability of locations i and j being clustered together. The axes correspond to the longitude/latitude where the data were collected. Four tight groups are clear from the plot, roughly agreeing with spatial location.
Figure 2
Figure 2
Oceanography study. Fitted curves (solid line), along with pointwise credible bands (dashed lines) for each of the 4 groups of curves. Credible bands are tight even in groups with very few curves. Panel (d) is an exception, and the additional uncertainty is due to very sparse data.
Figure 3
Figure 3
Oceanography study. Density estimates for the depth of the isothermal layer at 12°C for representative curves on each cluster. These are preferable to more straightforward measures like the conditional mean when multi-modality is expected, like in panel (b). Results are in line with the functional estimates shown above.

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