Geographically weighted regression model-assisted (GWRMA) estimation to improve precision of estimates combining remote sensing and ground plot data

• Incorporating spatial heterogeneity into model-assisted estimator improves precision. • Geographically Weighted Regression Model Assisted (GWRMA) estimator is proposed. • GWRMA precision gain over conventional model-assisted estimators can be substantial. • GWRMA advantage increases with larger sample size and greater spatial heterogeneity. • Typical model-assisted variance estimator severely underestimates GWRMA variance. Remotely sensed data play a vital role providing auxiliary variables that can improve precision of sample-based estimates without incurring the substantial cost of increasing sample size. In the setting of design-based inference, model-assisted estimators using generalized regression models provide a practical option to exploit such auxiliary variables to obtain estimates with smaller uncertainties than the Horvitz-Thompson (HT) estimator. Spatial heterogeneity is one of the main characteristics of spatial geographical populations and spatial nonstationarity in regression model parameters can greatly impact model predictions. However, model-assisted estimators typically do not incorporate spatial heterogeneity in the assisting models, and this may result in failing to gain the full precision improvement available from model-assisted estimation. Geographically Weighted Regression (GWR) offers a practical way to incorporate spatial heterogeneity of the data into the model. In this study, we substantiate the benefit of employing a GWR model-assisted estimator (GWRMA) in which GWR is used as a linking model between the response and auxiliary variables. The illustrative example we use to demonstrate the precision enhancing capacity of GWRMA is estimating tree volume in forest inventory monitoring. The performance of GWRMA was compared with the HT estimator and a linear regression model-assisted (LRMA) estimator when both the response and auxiliary variables are continuous. Several factors potentially impacting the estimators and their precision were investigated using Monte-Carlo simulation applied to populations constructed to represent different levels of spatial heterogeneity and correlation between the response and auxiliary variables. Because both LRMA and GWRMA estimators are asymptotically unbiased, variance is the key criterion for comparing the estimators. For the constructed populations with spatial heterogeneity, the GWRMA estimator achieved standard errors smaller than the HT and LRMA estimators, and the improvement in precision increased with increasing sample size. The precision advantage of the GWRMA estimator relative to the LRMA estimator was attributable to the capacity of the GWRMA estimator to adapt to spatial variation in the regression model leading to better local prediction of the target variable. The conventional model-assisted variance estimator substantially underestimated the variance of the GWRMA estimator. An alternative variance estimator based on averaging the GWRMA and LRMA variance estimates avoided the severe underestimation of the conventional model-assisted variance estimator. This average variance estimator yielded confidence intervals that exceeded the nominal 90% coverage but still had shorter length than the 90 % confidence intervals from LRMA estimator. Further exploration of variance estimators for the GWRMA estimator is a necessary next step to improve utility of the GWRMA estimator.