  We develop a new method for multivariate scalar on multidimensional
distribution regression. Traditional approaches typically analyze isolated
univariate scalar outcomes or consider unidimensional distributional
representations as predictors. However, these approaches are sub-optimal
because: i) they fail to utilize the dependence between the distributional
predictors: ii) neglect the correlation structure of the response. To overcome
these limitations, we propose a multivariate distributional analysis framework
that harnesses the power of multivariate density functions and multitask
learning. We develop a computationally efficient semiparametric estimation
method for modelling the effect of the latent joint density on multivariate
response of interest. Additionally, we introduce a new conformal algorithm for
quantifying the uncertainty of regression models with multivariate responses
and distributional predictors, providing valuable insights into the conditional
distribution of the response. We have validated the effectiveness of our
proposed method through comprehensive numerical simulations, clearly
demonstrating its superior performance compared to traditional methods. The
application of the proposed method is demonstrated on tri-axial accelerometer
data from the National Health and Nutrition Examination Survey (NHANES)
2011-2014 for modelling the association between cognitive scores across various
domains and distributional representation of physical activity among older
adult population. Our results highlight the advantages of the proposed
approach, emphasizing the significance of incorporating complete spatial
information derived from the accelerometer device.
