  The problem of large-scale spatial multiple testing is often encountered in
various scientific research fields, where the signals are usually enriched on
some regions while sparse on others. To integrate spatial structure information
from nearby locations, we propose a novel approach, called {\bf STR}ucture-{\bf
A}daptive {\bf W}eighting (STRAW) procedure, for large-scale spatial multiple
testing. The STRAW procedure is capable of handling a broad range of spatial
settings by leveraging a class of weighted p-values and is fully data-driven.
Theoretical results show that the proposed method controls the false discovery
rate (FDR) at the pre-specified level under some mild conditions. In practice,
the local sparsity level, defined as the probability of the null hypothesis
being not true, is commonly unknown. To address this issue, we develop a new
method for estimating the local sparsity level by employing the kernel-smooth
local false discovery rate (Lfdr) statistic. The superior numerical performance
of the STRAW procedure is demonstrated by performing extensive simulation
studies and a real data analysis.
