  Aggregate data often appear in various fields such as socio-economics and
public security. The aggregate data are associated not with points but with
supports (e.g., spatial regions in a city). Since the supports may have various
granularities depending on attributes (e.g., poverty rate and crime rate),
modeling such data is not straightforward. This article offers a multi-output
Gaussian process (MoGP) model that infers functions for attributes using
multiple aggregate datasets of respective granularities. In the proposed model,
the function for each attribute is assumed to be a dependent GP modeled as a
linear mixing of independent latent GPs. We design an observation model with an
aggregation process for each attribute; the process is an integral of the GP
over the corresponding support. We also introduce a prior distribution of the
mixing weights, which allows a knowledge transfer across domains (e.g., cities)
by sharing the prior. This is advantageous in such a situation where the
spatially aggregated dataset in a city is too coarse to interpolate; the
proposed model can still make accurate predictions of attributes by utilizing
aggregate datasets in other cities. The inference of the proposed model is
based on variational Bayes, which enables one to learn the model parameters
using the aggregate datasets from multiple domains. The experiments demonstrate
that the proposed model outperforms in the task of refining coarse-grained
aggregate data on real-world datasets: Time series of air pollutants in Beijing
and various kinds of spatial datasets from New York City and Chicago.
