  Count data occur widely in many bio-surveillance and healthcare applications,
e.g., the numbers of new patients of different types of infectious diseases
from different cities/counties/states repeatedly over time, say,
daily/weekly/monthly. For this type of count data, one important task is the
quick detection and localization of hot-spots in terms of unusual infectious
rates so that we can respond appropriately. In this paper, we develop a method
called Poisson assisted Smooth Sparse Tensor Decomposition (PoSSTenD), which
not only detects when hot-spots occur but also localizes where hot-spots occur.
The main idea of our proposed PoSSTenD method is articulated as follows. First,
we represent the observed count data as a three-dimensional tensor including
(1) a spatial dimension for location patterns, e.g., different
cities/countries/states; (2) a temporal domain for time patterns, e.g.,
daily/weekly/monthly; (3) a categorical dimension for different types of data
sources, e.g., different types of diseases. Second, we fit this tensor into a
Poisson regression model, and then we further decompose the infectious rate
into two components: smooth global trend and local hot-spots. Third, we detect
when hot-spots occur by building a cumulative sum (CUSUM) control chart and
localize where hot-spots occur by their LASSO-type sparse estimation. The
usefulness of our proposed methodology is validated through numerical
simulation studies and a real-world dataset, which records the annual number of
10 different infectious diseases from 1993 to 2018 for 49 mainland states in
the United States.
