  Spatial classification with limited feature observations has been a
challenging problem in machine learning. The problem exists in applications
where only a subset of sensors are deployed at certain spots or partial
responses are collected in field surveys. Existing research mostly focuses on
addressing incomplete or missing data, e.g., data cleaning and imputation,
classification models that allow for missing feature values or model missing
features as hidden variables in the EM algorithm. These methods, however,
assume that incomplete feature observations only happen on a small subset of
samples, and thus cannot solve problems where the vast majority of samples have
missing feature observations. To address this issue, we recently proposed a new
approach that incorporates physics-aware structural constraint into the model
representation. Our approach assumes that a spatial contextual feature is
observed for all sample locations and establishes spatial structural constraint
from the underlying spatial contextual feature map. We design efficient
algorithms for model parameter learning and class inference. This paper extends
our recent approach by allowing feature values of samples in each class to
follow a multi-modal distribution. We propose learning algorithms for the
extended model with multi-modal distribution. Evaluations on real-world
hydrological applications show that our approach significantly outperforms
baseline methods in classification accuracy, and the multi-modal extension is
more robust than our early single-modal version especially when feature
distribution in training samples is multi-modal. Computational experiments show
that the proposed solution is computationally efficient on large datasets.
