  Identifying spatial heterogeneous patterns has attracted a surge of research
interest in recent years, due to its important applications in various
scientific and engineering fields. In practice the spatially heterogeneous
components are often mixed with components which are spatially smooth, making
the task of identifying the heterogeneous regions more challenging. In this
paper, we develop an efficient clustering approach to identify the model
heterogeneity of the spatial additive partial linear model. Specifically, we
aim to detect the spatially contiguous clusters based on the regression
coefficients while introducing a spatially varying intercept to deal with the
smooth spatial effect. On the one hand, to approximate the spatial varying
intercept, we use the method of bivariate spline over triangulation, which can
effectively handle the data from a complex domain. On the other hand, a novel
fusion penalty termed the forest lasso is proposed to reveal the spatial
clustering pattern. Our proposed fusion penalty has advantages in both the
estimation and computation efficiencies when dealing with large spatial data.
Theoretically properties of our estimator are established, and simulation
results show that our approach can achieve more accurate estimation with a
limited computation cost compared with the existing approaches. To illustrate
its practical use, we apply our approach to analyze the spatial pattern of the
relationship between land surface temperature measured by satellites and air
temperature measured by ground stations in the United States.
