Artificial neural networks as an alternative method to nonlinear mixed-effects models for tree height predictions

Highlights

• At the plot level, mixed-effects models provided the most accurate tree height predictions.

• By grouping similar plots the ANN predictions improved.

• The ANNs are more competitive if enough tree height measurements are available.

• The BAL tree competition variable increased the accuracy of ANN models.

Abstract

Tree heights are one of the most important aspects of forest mensuration, but data are often unavailable due to costly and time-consuming field measurements. Therefore, various types of models have been developed for the imputation of tree heights for unmeasured trees, with mixed-effects models being one of the most commonly applied approaches. The disadvantage here is the need of sufficient sample size per tree species for each plot, which is often not met, especially in mixed forests. To avoid this limitation, we used principal component analysis (PCA) for the grouping of similar plots based on the most relevant site descriptors. Next, we compared mixed-effects models with height-diameter models based on artificial neural networks (ANN). In terms of root mean square error (RMSE), mixed-effects models provided the most accurate tree height predictions at the plot level, especially for tree species with a smaller number of tree height measurements. When plots were grouped using the PCA and the number of observations per category increased, ANN predictions improved and became more accurate than those provided by mixed-effects models. The performance of ANN also increased when the competition index was included as an additional explanatory variable. Our results show that in the pursuit of the most accurate modelling approach for tree height predictions, ANN should be seriously considered, especially when the number of tree measurements and their distribution is sufficient.

Introduction

Forest inventories are one of the primary sources for national and international reporting schemes (e.g. FAO, 2020), with wood volume, forest biomass and carbon stock among the most important attributes to be reported (Vidal et al., 2016). Methods for wood volume estimates are usually country-specific relying on diverse volume model types, the most common being the use of different volume functions, taper-curves, and breast-height form factor functions (Gschwantner et al., 2019). Directly and indirectly all of them require diameter at breast height (DBH, or D) and tree height (H) estimates. In addition to volume and biomass estimates, tree heights are of key importance for assessment of forest components such as productivity, site indexes and forest development in general. Consequently almost all growth and yield models require information on tree height to predict forest dynamics (Barreiro and Tomé, 2017), where tree heights could be needed at the tree (e.g. Pretzsch et al., 2002, Buchacher and Ledermann, 2020), plot or stand level (e.g. Härkönen et al., 2019). Therefore, highly accurate estimates of tree heights are crucially important in several forestry subdisciplines, as well in related ecological and environmental disciplines.

Field measurements of tree heights are time-consuming and therefore often measured only for a subsample of trees, with unmeasured heights predicted using height-diameter (H-D) models (Soares and Tomé, 2002, Mehtätalo et al., 2015). There are two types of H-D model; the first requires only DBH to predict tree height (H-D function), while the second incorporates stand-level predictors in addition to DBH. The former are called simple and the latter generalised models (Mehtätalo et al., 2015). In the literature two- and three- parameter H-D functions exists (Kindermann, 2016). Simple models are particularly useful in even aged stands with a small number of species in homogenous stand and site conditions. However, the tendency in European forests and beyond is to promote uneven aged and mixed species stand structures (Bravo-Oviedo et al., 2014, Pach et al., 2018), which require more complex modelling approaches that rely on DBH combined with additional stand and tree characteristics (Temesgen and Von Gadow, 2004).

In more complex forest communities, tree heights are often predicted on the basis of information at the plot level, with the plot index entering the model as a random effect of a mixed-effects model (Zuur et al., 2009, Bronisz and Mehtätalo, 2020). Therefore, we assume the same species-specific H-D curve within each plot. Here the fixed part of the model describes the predicted H-D curve for a typical plot in the used database (fixed-effect prediction) and the random effect provides a calibrated prediction (random-effect prediction), which together describe the plot specific H-D relationship. Consequently, the use of mixed-effects models also enable prediction on new plots (Mehtätalo et al., 2015).

The usefulness of including site or plot effects in H-D relationships has been reported for many species, e.g. Pinus Sylvestris (Lappi, 1991), Quercus pagoda (Lynch et al., 2005), Pinus taeda (Trincado et al., 2007), Pseudotsuga menziesii (Temesgen et al., 2008), and Betula pendula (Bronisz and Mehtätalo, 2020). VanderSchaaf (2014) fitted mixed-effects models for ten different conifer species in the Northwest USA, while Mehtätalo et al. (2015) tested modelled H-D relationships using a dataset representing a wide range of tree species.

The advantage of such approaches is the consideration of local site conditions that importantly affect H-D relationships, while the problems with convergence of model could arise if the number of height measurements per plot is low (Harrison et al., 2018), which is often the case in uneven aged mixed forests, with greater diversity of tree species and diameter distributions.

In recent years, machine learning (ML) has seen increased application in various sciences, including forestry. Besides the decision-tree learning and support vector machine, one commonly used method is the artificial neural network (ANN) (Liu et al., 2018). ANNs have the ability to acquire and maintain information based knowledge and can be defined as a set of processing units, represented by artificial neurons, interlinked by a multitude of interconnections (artificial synapses), implemented by vectors and matrices of synaptic weights (da Silva et al., 2017). The ANN model can be applied to various kinds of problems, from classification, clustering and optimisation to function approximation, and has already been applied in various forestry disciplines, such as forest fire prediction (Safi and Bouroumi, 2013), prediction of insect outbreaks (Park and Chung, 2006), and species distribution models (Scrinzi et al., 2007). Apart from these, the ANN has also been tested in tree height modelling for eucalyptus trees (Vieira et al., 2018), common beech (Fagus sylvatica) from northwestern Spain (Castaño-Santamaría et al., 2013) and Crimean juniper (Juniperus excels) (Özçelik et al., 2013).

The primary aim of our study was to explore the application of mixed-effects H-D models for height predictions within predominantly uneven aged mixed forests using forest inventory plot data. Slovenia has a long tradition of close to nature forest management with strong emphasis on natural regeneration, and consequently, a high proportion of uneven aged and mixed stands (Diaci, 2006). Secondly, focused on less representative tree species, we explore the application of grouping plots based on site factors derived from principal component analysis (PCA) (Jolliffe and Cadima, 2016), which are later used as (nested) random effects and categorical independent variables. Finally, we test and compare a new methodological approach based on artificial neural networks (ANN) for tree height predictions for variety of different tree species specific to central Europe. Inclusion of additional explanatory variables, namely competition, was tested for the ANN. Competition is often among the most effective in explaining forest stand dynamics (Jevšenak and Skudnik, 2021, Vospernik, 2021) and studies have reported a significant effect of competition on the modelling of height (Temesgen and Von Gadow, 2004) and height growth (Sharma and Brunner, 2017).