Stochastic Local Interaction (SLI) Model: Interfacing Machine Learning and Geostatistics

Machine learning and geostatistics are two powerful approaches used to model spatial data. Each approach has advantages and disadvantages, but they both suffer from poor scaling of the computational load with data size due to the inversion of large covariance matrices. We present a new method for the modeling of spatial data that combines ideas from statistical physics, computational geometry, and machine learning. The proposed Stochastic Local Interaction (SLI) model is based on an explicit precision (inverse covariance) matrix and can be applied to data in $d$-dimensional spaces. The SLI model is defined by means of a Gaussian joint probability density function, which is expressed in terms of energy functionals that involve local interaction constraints implemented by means of kernel functions. The variability of the sampling density is accounted by means of local kernel bandwidths. The SLI model leads to a semi-analytical expression for interpolation (prediction), which is valid in any number of dimensions and does not require the inversion of a covariance matrix.