Population and Stand-Level Inference in Forest Inventory with Penalized Splines

Abstract

Penalized splines have potential to decrease estimates of variance in forest inventories with a design-based population-level inference, and a model-based domain-level inference by decreasing the likelihood of a model misspecification. We provide examples with second-order (B2) B-splines and radial basis (RB) functions as extensions to a linear working model (WM). Bias was not prominent, yet greater with B2 and in particular with RB than with WM, and decreased with sample size. Important reductions in the variance of a population mean were achieved with both B2 and RB, but at the domain-level only with RB. The proposed regression estimator of variance generated estimates of variance being slightly smaller than the observed variance. A consistent and larger underestimation was seen with the popular difference estimator of variance. Study Implications: Forest inventories supported by light detection and range (LiDAR) data require—in the estimation phase—a model for linking LiDAR metrics to attributes of interest. Formulating a parametric model can be a challenge and unsatisfactory if the goodness of fit varies across the range of the attribute of interest. A semiparametric model provides more flexibility and lessens the chance of a model misspecification, albeit with the potential of overfitting. A penalty directed at reducing overfitting is required. A flexible semiparametric model is potentially also better suited for applications to small areas like stands than a parametric model. We demonstrate that important reductions in variance are indeed possible, but also that they depend on the form of the nonparametric part of the chosen model and the level of inference (population versus domains). With regard to practical application, reliable estimates of forest attributes at stand-level are of special interest within the scope of forest-management planning, as silvicultural treatments are always stand-oriented, at least with small-scale forestry under Central European conditions, and stand-related volume (basal area, tree density) belongs to the set of relevant parameters for management decisions regarding harvest and regeneration measures.