Modelling tree survival for investigating climate change effects Nicole Augustin School of Mathematics University of Edinburgh UK

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Modelling tree survival for investigating climate

change effects

Nicole Augustin, School of Mathematics, University of Edinburgh, UK

Axel Albrecht , Forest Research Institute Baden-W¨

urttemberg, Freiburg, Germany

Karim Anaya-Izquierdo, Department of Mathematical Sciences, University of Bath, UK

Alice Davis, Mayden, The Old Dairy, Melcombe Road, UK

Stefan Meining, Ofﬁce of Environmental Monitoring, Freiburg, Germany

Heike Puhlmann, Forest Research Institute Baden-W¨

urttemberg, Freiburg, Germany

and Simon Wood, School of Mathematics, University of Edinburgh, UK

August 21, 2024

Abstract

Using German forest health monitoring data we investigate the main drivers leading to

tree mortality and the association between defoliation and mortality; in particular (a) whether

defoliation is a proxy for other covariates (climate, soil, water budget); (b) whether defolia-

tion is a tree response that mitigates the effects of climate change and (c) whether there is a

threshold of defoliation which could be used as an early warning sign for irreversible dam-

age. Results show that environmental drivers leading to tree mortality differ by species, but

arXiv:2210.02247v2 [stat.AP] 20 Aug 2024

some are always required in the model. The defoliation effect on mortality differs by species

but it is always strong and monotonic. There is some evidence that a defoliation threshold

exists for spruce, ﬁr and beech.

We model tree survival with a smooth additive Cox model allowing for random effects

taking care of dependence between neighbouring trees and non-linear functions of spatial

time varying and functional predictors on defoliation, climate, soil and hydrology character-

istics. Due to the large sample size and large number of parameters, we use parallel com-

puting combined with marginal discretization of covariates. We propose a ’boost forward

penalise backward’ scheme based on combining component-wise gradient boosting with in-

tegrated backward selection.

Keywords: component-wise gradient boosting, forest health, integrated backward selection,

smooth additive Cox model, spatial frailty model.

1 Introduction

Forest health is monitored in Europe by The International Co-operative Programme on Assess-

ment and Monitoring of Air Pollution Effects on Forests (ICP Forests, Eichhorn et al. (2017)) in

cooperation with the European Union. Recently climate change has contributed to the decline

in forest health and these data are increasingly being used to investigate the effects of climate

change on forests in order to decide on forest management strategies for mitigation. Forests in

Germany have been badly affected and climate change now appears to be the major cause of

defoliation (Eickenscheidt et al., 2019; Augustin et al., 2009).

Here we focus on two main questions to investigate climate change effects on tree mortal-

ity. These are, what are the main drivers leading to tree mortality; and, what is the association

between defoliation and mortality? Regarding the second question, we explore (a) whether de-

foliation is a proxy for other covariates (climate, soil, water budget); (b) is defoliation a tree

response that mitigates the effects of climate change and (c) whether there is a threshold of de-

foliation which could be used as an early warning sign for irreversible damage. If this threshold

is found, it has practical relevance for forest management. Trees with defoliation greater than

this threshold value but not dead yet could be harvested in an anticipatory manner or stabilised.

We focus in our analysis on the main species: Norway spruce, Silver ﬁr, Scots pine, oak (Pe-

donculate oak and Sessile oak) and common beech. To answer these questions we use extensive

yearly data on tree mortality and crown defoliation, an indicator of tree health, from a monitor-

ing survey carried out in Baden-W¨

urttemberg, Germany since 1983, which includes a part of the

ICP transnational grid. On a spatial grid, defoliation, mortality and other tree and site speciﬁc

variables are recorded. In some cases the grid locations are no longer observed which leads to

censored data. Also recruitment of trees happens throughout when new grid points are added or

trees are replaced.

There are a number of statistical challenges with regard to modelling tree survival for in-

vestigating climate change effects. The large number of correlated predictor variables seriously

impacts the complexity of model selection. Some of these variables are time varying and many

variables have non-linear effects. For example, there is an optimal range of Julian day of budburst

implying a non-linear effect of this variable. At the survey grid locations, several trees, each a

maximum of 50m apart, are observed over time and we need to account for this short range spa-

tial correlation. An appropriate model for this type of short range correlation would include a

spatial frailty term, i.e. a random effect for grid location. With more than one thousand locations,

this results in some very large models.

Several approaches have been applied to modelling tree survival. Neuner et al. (2015) used

parametric Weibull accelerated failure time models to analyse the survival of Norway spruce

and European beech in Baden-W¨

urttemberg, Bavaria and Rhineland-Palatinate. In this study the

survival time was age of tree at death. Nothdurft (2013) looked at age-dependent survival of

Norway spruce, Silver ﬁr, Scots pine and common beech in Baden-W¨

urttemberg using a Cox

model with a frailty term, to account for correlation between individuals in survey data which

overlaps with ours. Li et al. (2015) and Thapa et al. (2016) model the survival of the Loblolly

Pine in the Piedmont, Atlantic Coastal Plain and Gulf Coastal Plain regions of the US, also using

Cox frailty models with special attention to the parametric form of the spatial correlation (Li

et al., 2015). In Thapa et al. (2016) the survival time is time in years since plot establishment.

Lee et al. (2011) use a survival model for interval censored data in a joint model of tree growth and

mortality. Zhou and Hanson (2018) ﬁt a Bayesian semiparametric model to arbitrarily censored

spatially referenced survival data.

We model tree hazard at calendar time using a Cox model as a function of the predictor

variables on climate, soil characteristics and water budget. This approach assumes the baseline

hazard changes with calendar time and means the baseline hazard can explain some temporal

trends common to all trees driven by hidden explanatory variables which change with calendar

time. In contrast, others (Neuner et al., 2015; Nothdurft, 2013) use tree age as the survival

time and in that case the baseline hazard is a function of age and can explain trends common

to all trees changing with age, but this would not be so suited to investigating the relationship

between mortality and environmental variables. Maringer et al. (2021) model tree mortality from

experimental research plots in Baden-W¨

urttemberg, with stand age as the survival time, using an

accelerated failure time model with linear effects.

We use a smooth additive Cox model for the hazard, which allows for a spatial frailty term

taking care of dependence between neighbouring trees and non-linear smooth functions of spatial,

time varying and functional predictors modelling any spatial trend. For parameter estimation,

including smoothness parameters, we use a penalised version of the partial likelihood of the Cox

model.

In order to account for time varying covariates we use a Poisson generalised additive mixed

model for pseudodata to ﬁt the Cox model (Whitehead, 1980). This is possible because the

Poisson likelihood for the pseudodata is identical to the partial likelihood up to a constant of pro-

portionality. This approach also accounts automatically for the varying risk set size induced by

the sampling protocol, causing left truncation, and this is similar to the adjustment for left trun-

cation in epidemiological cohort studies (Thi´

ebaut and B´

enichou, 2004; Pencina et al., 2007). As

outlined in Wood (2017), Section 7.8.1 and Bender et al. (2018) the use of a Poisson generalised

additive mixed model allows the estimation of time-varying effects in these smooth additive Cox

models. The computational cost due to data expansion in the GAMM approach is not worse

than in the traditional partial likelihood based approach as soon as time varying predictors and

non-linear effects are involved. In addition using the GAMM approach gives access to the entire

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ModellingtreesurvivalforinvestigatingclimatechangeeffectsNicoleAugustin,SchoolofMathematics,UniversityofEdinburgh,UKAxelAlbrecht,ForestResearchInstituteBaden-W¨urttemberg,Freiburg,GermanyKarimAnaya-Izquierdo,DepartmentofMathematicalSciences,UniversityofBath,UKAliceDavis,Mayden,TheOldDairy,MelcombeRo...

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Modelling tree survival for investigating climate change effects Nicole Augustin School of Mathematics University of Edinburgh UK

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