Unit-level and area-level small area estimation under heteroscedasticity using digital aerial photogrammetry data

Abstract

In many applications, estimates are required for small sub-populations with so few (or no) sample plots that direct estimators that do not utilize auxiliary variables (e.g. remotely sensed data) are not applicable or result in low precision. This problem is overcome in small area estimation (SAE) by linking the variable of interest to auxiliary variables using a model. Two types of models can be distinguished based on the scale on which they operate: i) Unit-level models are applied in the well-known area-based approach (ABA) and are commonly used in forest inventories supported by ne-resolution 3D remote sensing data such as airborne laser scanning (ALS) or digital aerial photogrammetry (AP); ii) Area-level models, where the response is a direct estimate based on a sample within the domain and the explanatory variables are aggregated auxiliary variables, are less frequently applied. Estimators associated with these two model types can make use of sample plots within domains if available and reduce to so-called synthetic estimators in domains where no sample plots are available. We used both model types and their associated model-based estimators in the same study area with AP data as auxiliary variables. Heteroscedasticity, i.e. for continuous dependent variables typically an increasing dispersion of residuals with increasing predictions, is often observed in models linking eld-and remotely sensed data. This violates the model assumption that the distribution of the residual errors is constant. Complying with model assumptions is required for model-based methods to result in reliable estimates. Addressing heteroscedasticity in models had considerable impacts on standard errors. When complying

Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR

“…While at large scales both LiDAR-based and field-based estimates were very similar and had equivalent accuracies, at the stand-level, LiDAR based estimates, clearly had smaller errors than their field-based counterparts do. Qualitatively, this result for the change in V, BA and B is similar to the results obtained in [15,17] for the structural variables themselves and shows that the LiDAR auxiliary information allows for gains in efficiency when estimating changes in AOIs with small sample sizes. However, due to the low correlation of LiDAR and structural changes, values of CV δ and CV y were oftentimes larger than 50%.…”

“…The predictor most correlated with the variable of interest (i.e., the predictor used to model the error variance) was the same for 2009 and 2015. For V and B, the variance of model errors was a function of the square of the mean LiDAR elevation (Elev_mean 2 ), and the exponents of the error variance function were very close to those obtained in [4,17,27] for V, and in [27] for B. For BA, variance of model errors was a function of the percentage of first returns above two meters (PcFstAbv2).…”

“…However, due to the low correlation of LiDAR and structural changes, values of CV δ and CV y were oftentimes larger than 50%. These values of CV are larger than those observed for structural variables in similar AOIs in previous studies [4,14,17]. While differences were not of large magnitude RMSE y , tended to be smaller than RMSE δ .…”

“…Small area estimation (SAE) techniques, especially empirical best linear unbiased predictors (EBLUPs) in combination with the ABA approach have been used to obtain estimates, and their corresponding measures of uncertainty for subpopulations such as municipalities [14], groups or management units [15] and stands [4,14,16,17]. SAE techniques allow correcting the potential bias problems of synthetic predictions (i.e., predictions developed assuming that a general model developed at the population level holds for all subpopulations) and also permit reducing the large variance problems of direct estimators when AOIs sample sizes are small [13].…”

Precision of exogenous post-stratification in small-area estimation based on a continuous national forest inventory

“…As a result, sample sizes in small areas (or domains) such as municipalities are often too small for sufficiently precise estimation from field data only, that is, for direct domain estimation (Rao 2003, p. 1). Model-based approaches relying on auxiliary data and a regression model fitted to the training data from a larger region can overcome this problem by borrowing strength from outside the area of interest; therefore, these approaches are commonly used for small-area estimation (Lappi 2001;Breidenbach and Astrup 2012;Breidenbach et al 2016Breidenbach et al , 2018McRoberts et al 2017). Rao (2003) describes a wide range of small-area estimation methods and regards methods using sample data outside the small area of interest and linking models as indirect domain estimation.…”