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Swiss Federal Institute of Technology Zurich Seminar forStatistics
Department of Mathematics
Master Thesis Winter 2015
Jurgen Zell
Analyzing growth and mortality
of Picea Abies for a growth simulator
in Switzerland
Submission Date: February 10th 2016
Adviser: Dr. Martin MachlerMain Reader: Dr. Martin Machler
iii
For my family: Katja, Karl and Alva
iv Preface
Preface
The intention of the work was the development of a single tree growth simulator, whichuses the modern way of statistics with very old data from the experimental forest researchplots.
This was achieved with the help of Martin Machler, who has an enormous background andknowledge in the field of modern statistics. He always had time when I needed it and thepatients to think about the problems at hand. Unambiguously he has a fast and preciseway of thinking from which I profited throughout this work.
The analysis was only possible with the help of generations of forest researcher and as-sistants working and measuring on this long term plots. Andreas Zingg, who is currentlyresponsible for this data, had always time and good ideas when I was stuck in some dataproblems or interpretation difficulties.
v
vi Abstract
Abstract
The thesis is about modeling growth and mortality of Picea Abies. The data are complexand stem from experimental forest management trials all over Switzerland. In the firstpart growth was modeled. 65% of the total variation can be explained by many differentexplanatory variables. The second part is about mortality and contains a logistic regressionmodel, which is compared to a Survival analysis approach.
vii
viii CONTENTS
Contents
Notation xi
1 Background 1
2 Growth 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Tree height, crown base, crown width . . . . . . . . . . . . . . . . . 32.1.2 Crown surface and cross sectional areas . . . . . . . . . . . . . . . . 4
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Model building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.4 Examining the full model . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Mortality 153.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Modeling probability to die using logistic regression . . . . . . . . . . . . . . 153.3 Discussion logistic regression model . . . . . . . . . . . . . . . . . . . . . . . 183.4 Introduction survival analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Survival Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5.1 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Modeling times to event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Discussion: Prediction and comparison . . . . . . . . . . . . . . . . . . . . . 23
4 Summary 254.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Bibliography 27
A Complementary information 31A.1 Growth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 Logistic regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.3 Survival model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
LIST OF FIGURES ix
List of Figures
2.1 Crown forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Calculation of crown competition index . . . . . . . . . . . . . . . . . . . . 52.3 Overview increment and interval lengths . . . . . . . . . . . . . . . . . . . . 62.4 Basal area increment over diameter . . . . . . . . . . . . . . . . . . . . . . . 72.5 Residuals of a BWinPro-model, with data from Switzerland . . . . . . . . . 82.6 Residuals of different transformations . . . . . . . . . . . . . . . . . . . . . 92.7 Residuals growth model against plots . . . . . . . . . . . . . . . . . . . . . . 112.8 Residuals growth model against period length and the year of the survey . . 112.9 Residuals sum of squares of different transformations . . . . . . . . . . . . . 122.10 Exemplary effects of the final growth model . . . . . . . . . . . . . . . . . . 13
3.1 Maximum stand density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Smoothed explanatory variables estimated by a generalized additive model 183.3 Exemplary predicted probabilities to die, logit model . . . . . . . . . . . . . 193.4 Kaplan-Meier estimates of the trees, depending on the type of thinning . . 203.5 Residuals of a automatically reduced Weibull survival model . . . . . . . . . 213.6 Taxonomy of different hazard functions. . . . . . . . . . . . . . . . . . . . . 223.7 Predicted survival curves and hazard functions . . . . . . . . . . . . . . . . 233.8 Cross validation: comparing logistic regression model with Survival model. . 24
x LIST OF TABLES
List of Tables
2.1 Residual sum of squares for different transformations. . . . . . . . . . . . . 8
Notation
• d: tree diameter at breast height in cm
• dq: root mean square diameter at breast height in cm
• h: measured tree height, h estimated tree height in m and rh relative estimated treeheight: h/max(h)
• hq90: 90%-quantile of tree heights in a stand
• h: mean height in a stand
• v: tree volume, v(d) estimated tree volume (solid volume) based on measured diam-eter in m3
• A: total area of a plot in hectare (104 m2)
• a: age of a tree in years
• ba: basal area of a tree (at breast height) in m2: π40000d
2
• bat: total basal area for the plot in m2/ha: π4A
∑ni=1 d
2i
• ti: i.th time of measurement in years
• dt: period length as difference to 5 years: (ti − ti−1)− 5
• bai: basal area increment normalized to 5-year period: 5π4(d2ti−d2ti−1
)
ti−ti−1
• rbai: relative basal area increment: bai/ba
• cba: cumulative basal area: cbaj = 1bat
∑ni=1
(1di≤djbai
)• bal: basal area of larger trees: balj = 1
A
∑ni=1
(1di>djbai
)• rbal: relative basal area of larger trees: bal/bat
• bah: percent of basal area previously harvested
• nt: total number of trees per hectare in a plot
• nm: number of trees recently dead in previous interval (mortality)
• nh: percent of stems previously harvested
• csa: crown surface area in m2 (definition: 2.1.2.1)
• cb: crown base (height were the living crown starts), in m
xi
xii Notation
• cca: cross sectional area of a crown in m2
• sdi: stand density index nt(dq/25)1.605
• cc: crown competition index (definition: 2.1.2.2)
• ccc: change in crown competition index (definition: 2.1.2.2)
• nphy.∗: yearly precipitation sum as mean over different period lengths. “phy” standsfor physiological year, meaning that the sum was taken from October until Septem-ber, in contrast to calender years (Rohner, Weber, and Thurig, 2016). “*” takes thevalues “1, 2, 5, 10, per” and stands for previous time spans in years and “per” standsfor “period”, e.g. the exact time intervals as given by the re-measurement of thetrees in mm
• tphy.∗: mean physiological year temperature, over intervals like nphy.∗, in ◦C.
• mphy.∗: Is the aridity index by deMartonne (Pretzsch, 2010, p. 520) A = nphy.∗tphy.∗+10 .
Since this index means smaller values have higher aridity, its inverse interpretationis more intuitive and I labeled it “moisture-index” (m∗).
The following abbreviations are (geological) site variables and are represented by factors.They are widely used in forestry and agriculture1.
• wat.hold.cap: water holding capacity
• wat.perm: water permeability
1http://www.blw.admin.ch/dienstleistungen/00334/00337/index.html?lang=de
Chapter 1
Background
In the field of forest science there is a need to predict future states of our forests, since thechanges in our environment are fast compared to the rotation length or the lifespan of treesin middle Europe. Hence practitioners need to decide today with a long-lasting effect onthe future of our forests. The thesis is aimed to use modern statistical tools to analyze anddescribe two prominent processes in forests: growth and mortality. They should includethe effects of climate and human interventions. These can be used in single tree growthsimulator, hopefully helping to find a good bases for the decisions in the forests.
Therefore the overall aim is the prediction of future forest stands1, when human interven-tion and climate are changing beyond the range of the given observations. It focuses ontwo processes and on the most prominent tree species in Switzerland, Picea Abies. Hencethe models should not be simply be best in prediction accuracy, but must also predict inphysiological reasonable manner. This balance was found by building the models forwardby hand and stepwise comparison.
A growth simulator on a stand level is a prediction mechanism, which aims to describethe development of single trees in a stand. There are also mechanistic biological drivenmodels, which describe the fundamental laws of biology. They rely on few parameters andmany laws. For example the successional development of a forest may be in the focus (socalled “gap models” (Bugmann, 2001)). A growth simulator usually relies on many data,hence many parameters and less fundamental laws. Its simplest form are yield tables2.Growth simulators can also be used on larger scales, based on national forest inventories.In this case usually the knowledge of the current stand is measured with high variance oris unknown, hence they aim to predict on a large scale, normally not using explicit standinformation (e.g. MASSIMO in Switzerland (Thurig and Kaufmann, 2010). More detailscan be found here3, on page 162 pp.).
The data are measurements form the experimental forest management (EFM) plots inSwitzerland. They are unique, in the sense that we have remeasured trees since 1880. Thatis the time, when the first systematic growth and yield research plots were established.The data are measured and maintained by a research group around Andreas Zingg at WSL(Birmensdorf, Switzerland). Several weeks of the work were needed to systematically bringthe data together and to prepare them for analysis. This is not straight forward since many
1stand: uniform treatment over a set of trees under comparable site conditions2http://www.efi.int/portal/virtual_library/databases/efiscen/yield_tables/3https://www.lfi.ch/publikationen/publ/LFI2_Methoden.pdf
1
2 Background
generations of forest researchers were working on different research questions. Hence, thereis no guarantee for consistency the data; just at the moment WSL has started to bringthe data into a modern data base system.
The analysis is embedded in a project at WSL to develop a single tree growth simulator forSwitzerland, which is part of larger Research Programme on Forests and Climate Change4.
4http://www.wsl.ch/fe/waldressourcen/projekte/ra_param_klimasensitiv_waldmgmt/index_DE
Chapter 2
Growth
First some background models are briefly presented. They predict values, which are as-sumed to have a potential explanatory value. Only the expectation of the models areused. Multiple imputation was not used, for two reasons: the amount of data was largeand the exemplary nature of the procedure: they are tested against other variables, basedon measurements; if a similar model may be achieved based on measured values, thenthere is no further need for multiple imputation.
2.1 Introduction
Biging and Dobbertin (1995) have shown that basal area increment can well be explainedby crown measures like crown surface area or indexes based on crown sectional areas ina stand. These measures are physiological well interpretable. Hence those explanatoryvariables are used in growth models e.g. BWinPro (Hansen and Nagel, 2014)) or Silva(Pretzsch, Biber, and Dursky, 2002) (both Germany) or MOSES, which is based on crownmodels developed in Hasenauer and Monserud (1996). Describing the tree crown forms asmathematical models further allows to show a three dimensional representation of trees(and growth) on the computer screen.
In the long term experimental forest management plots there are measures for the totaltree height, crown base and crown width, but not for crown surface area or crown sectionalareas on different heights of a tree. Still, based on the measured values it is possible toconstruct models, which allow the calculation of tree height, crown surface area and crosssectional areas through the crowns. The latter two serve only as potential explanatoryvariables in the basal area increment model, which itself is based on the full data set.Therefore the models are briefly described.
2.1.1 Tree height, crown base, crown width
Tree height is measured from 5.19% of all trees. The models only rely on diameter atbreast height, but are tree species and survey specific. An often used model is the so-called “Michajlow” function (Michajlow, 1952) producing stable predictions, since it hasan interpretable and meaningful form:
3
4 Growth
h = 1.3 + β0 exp(−β1/d)
with h the tree height, measured in m, β0 + 1.3 the asymptote of the function and β1represents the slope. The inflection point of this function is β1.
Crown base (cb, in meters) is measured from 3.81% of all observations. Models are treespecies specific and use d, h/d, mixture of deciduous trees (in percent of basal area), cbaof individual trees and bat as explanatory variables. A mixed model was estimated withrandom effects on plots and trees nested within a plot.
Crown width (in 3.3% of the observations) is measured in four radii per tree and accordingto Rohle and Huber (1985) the mean of those were used as radius for the crown width.Comparable models to those of crown base were estimated.
2.1.2 Crown surface and cross sectional areas
The above models allow an estimate of crown surface area csa using a paraboloid (r =estimated radius of the crown, l estimated crown length (l = h− cb), h = estimated totaltree height (see function f.csurf in A.4):
csa = πr
6l2
((4l2 + r2)3/2 − r3
)(2.1.2.1)
Further for the crown sectional areas assumptions on the crown forms are needed. Herethe plausible forms as given in (Pretzsch, 2010, p. 307) were used and are shown in thefollowing figure 2.1. The code to produce the crown forms and the graph is given in A.4.They are species specific and differentiate between an upper sun lighted and the shadedpart of the crown, the latter usually in form of a cone.
0 5 10 15 20
0.0
1.0
2.0
3.0
tree height
crow
n ra
dius
sprucebeechfir
Figure 2.1: Crown forms of different tree species, with the same tree height, maximumcrown width and crown base.
Now crown sectional areas at any height and corresponding indexes can be calculated. Afamous one is the cc, a crown competition index. It relates the sum of cross sectional areas
2.2 Data 5
(cca) at the height of maximum crown width of a current tree j to total area of the stand(A). The function f.csah (A.4) was used to calculate it.
cc(h)j =1
A
n∑i=1
i 6=j
cca(h)i (2.1.2.2)
ccc is the change of that index, calculated after harvesting has taken place. The followingfigure (2.2) illustrates the calculation of cc.
Figure 2.2: Calculation of crown competition index. Summing of cross sectional crownareas at a given height. Figure taken from Pretzsch (2010).
2.2 Data
Basic entity here is a tree, standing on a research plot, which is usually remeasured overdifferent interval lengths. The only variable, which is measured for each tree at any timeis its diameter at breast height (d). Instead of using the increment of that variable, it iscommon practice in forest science (e.g. Monserud and Sterba (1996)) to use the basal areaincrement. To make that variable comparable with respect to the different period lengths,the following normalization was used:
baiti = 5π4 (d2ti − d
2ti−1
)
ti − ti−1
As an overview the left panel in figure 2.3 shows the total amount of re-measured d andhence bai observations by tree species. The thesis only focuses on Picea Abies, the mostpromiment species in this data set, but also in Switzerland1. On the right panel in figure2.3, the distribution of time intervals is plotted. Unfortunately in the 1980’s the intervallength were prolonged for some of the research plots. Still a standard interval is five years.
1https://www.lfi.ch/
6 Growth
Pic
ea a
bies
Fagu
s sy
lvat
ica
Abi
es a
lba
Que
rcus
spp
.
oc
Larix
spp
.
Pin
us s
ylve
stris
Pse
udot
suga
m.
olb
Ace
r sp
p.
osb
Increment and diameter obs.
0e+00
1e+05
2e+05
3e+05
4e+05
Time intervals
periode length
Fre
quen
cy
0 5 10 15 20 25 30
050
000
1000
0015
0000
2000
0025
0000
3000
00
Figure 2.3: Left: Overview of increment data by tree species. “olb”: other long-livingbroadleaved species, “osb”: other short-living broadleved species, “oc”: other coniferspecies. Right: Distribution of length of time intervals.
Also note that sampling usually takes place in the dormant season and that directly aftermeasurement the planed felling takes place. For simplicity the previous increment isassociated to the end of an interval (only for the growth data). The explanatory variablese.g. climate are calculated over the previous period (or some other previous time span).
2.3 Model
2.3.1 Nonlinearity
The basal area increment shows an asymptotic behavior over diameter at breast height.Simply plotting all observed basal area increments for Picea Abies (on a yearly basis) overd is given in the following figure 2.4. The simple nonlinear regression explains around55% of the total variation and follows a formulation given in Quicke, Meldahl, and Kush(1994).
bai = exp(β0 + β1(1− exp(β2d))) (2.3.1.1)
2.3.2 Transformations
Here I start with refitting the BWinPro-model and use it as an example of a log-trans-formation of bai data. Next some other transformations are tested and their resultingresiduals and residual sum of squares are reported.
In total 2.11% of bai measures are negative, even after a small trimming of 0.1%. Sincethese values are physiologically less plausible, one might ignore them, resulting in an
2.3 Model 7
0 50 100 150
−50
050
100
150
200
250
300
dbh [cm]
incr
emen
t qcm
/y
loess f=1/250asympt. regression
Figure 2.4: Basal area increment over diameter at breast height (d), shown with a 2-dimensional kernel density plot, together with a simple nonlinear regression and a sym-metric local polynomial regression fit.
overestimation of the expected growth. Based only on the positive observations it ispossible to reconstruct the growth functions as it is given in BWinPro (Hansen and Nagel,2014). They use a (natural) logarithmic transformation of bai within the following model:
log bai = β0 + β1 log csa+ β2 log a+ β3cc+ β4ccc+ β5 log dt (2.3.2.1)
Where csa is the crown surface area of a tree in m2, a age of a tree in years, cc and cccthe crown competition indexes (see above), dt is the difference of re-measurement yearscompared to five year standard.
Using the above sketched process of estimating height, crown base and crown radius, it ispossible to reconstruct this model with (only the positive) increment data in Switzerland.The resulting graphs 2.5 are Tukey-Anscombe plots both on the transformed and originalscale of bai. The figure in the middle is only for comparison with Albrecht, Kohnle, andNagel (2010).
The left panel in figure 2.5 shows that the logarithmic transformation shrinks larger ob-servations too strong compared to small observations. Further the small values seem tobe overestimated. The curvature of the smoother curve indicates the inclusion of further(quadratic) explanatory variables. Especially on the original scale the asymmetry of theresiduals is evident, resulting in the overestimation in smaller increments. Albrecht et al.(2010) fitted the same model to data from Baden-Wurttemberg (SW-Germany) and foundthe same overestimation problem.
Since bai shows an asymptotic behavior over d, it is reasonable to use models on the basisof equation 2.3.1.1. Models based on this equation are not transformable to a linear model.Several transformations were tested on this simple model. With a small trimming of 0.1%,the minimum of 5-year bai is −19.4. Adding a value of α = 20 allows a log transformationwithout loss of observations. The following transformations were tested:
8 Growth
−1 0 1 2 3 4 5 6
−4
−2
02
4TA−plot, transformed
fitted
resi
dual
s
−2 0 2 4 6
−2
02
46
fitted over observed, transformed
observedfit
ted
0 100 200 300
−20
00
200
400
600
TA−plot, backtransformed
fitted
resi
dual
s
Figure 2.5: Residuals of a BWinPro-model, with data from Switzerland. Left panel:transformed Tukey-Anscombe plot, middle panel: fitted over observed, right panel: back-transformed Tukey-Anscombe plot.
• y′ = sign (y) |y|1/n, with n = 2, 3, 4
• y′ = log (y + α), with α > min(y)
The square-root transformation (n=2) as indicated by figure 2.6 shows a good represen-tation of the residuals, they a are nearly symmetric over the predicted values and thereis no obvious heteroscedasticity visible. With increasing strength of the transformation(increasing n), the residuals plots get worse.
A log transformation using an α close to the minimum of y is better compared to thelog transformation only on positive observations. Still the residuals are asymmetric. Amodel based on log transformation may result in negative expectation values on the back-transformed scale for small increments, which is the price for a fixed constant α.
The resulting residuals sum of squares, given in table 2.1 show that the graphical interpre-tation is in accordance with the sum of squares. Hence for the following model buildingthe square-root transformation was used. With the full model the same analysis withchanging α will be repeated, since the residuals will be changed.
n=2 n=3 n=4 log, α=20
1885.82 1977.64 2054.96 1911.66
Table 2.1: Residual sum of squares in million, for different back-transformed models.
2.3.3 Model building
After trimming (0.1%), the distribution of bai still contains observations which might beseen as outliers. There are at least two sources of uncertainty in the measurements. Thecaliper is used to measure d, which is sensitive to the form of tree. This is an measurementerror. Further trees might be confused with each other, resulting in extreme values for bai.In the data these two errors are not distinguishable. Because of the large amount of data,an outlier detection by hand seems unfeasible. Hence for stable parameter estimation therobust estimation method with the function nlrob from robustbase (Rousseeuw, Croux,Todorov, Ruckstuhl, Salibian-Barrera, Verbeke, Koller, and Maechler, 2015) was used.
2.3 Model 9
2 4 6 8 10 12 14 16
−20
−10
010
20
n=2
fitted
resi
dual
s
1 2 3 4 5 6−
8−
6−
4−
20
24
6
n=3
fitted
resi
dual
s
1.0 1.5 2.0 2.5 3.0 3.5
−6
−4
−2
02
n=4
fitted
resi
dual
s
3.0 3.5 4.0 4.5 5.0 5.5 6.0
−4
−2
02
log trans with alpha=20
fitted
resi
dual
s
Figure 2.6: Residuals of different transformations. The strength of transformation in-creases from top-left to bottom-right.
The residuals of the basis model 2.3.1.1 were analyzed with a generalized additive model(package mgcv, function gam (Wood, 2011)), to find the most relevant explanatory variablesand their (nonlinear) relationship on the basis model.
Then a forward selection process started. Therefore the potential explanatory variableswere grouped into physiological reasonable and interpretable themes. The aim of thesimulator is predictions outside the observed range of explanatory variables, hence thisprocedure makes sense against a “best” model fit.
In the following list the physiological groups are defined and sequentially variables areadded until a full model was found:
i.) Between tree competition: Four variables were used: Basal area of larger trees(bal), relative basal area of larger trees (rbal), cumulative basal area (cba) and crowncompetition (cc, 2.1.2.2). bal was nearly linear in the gam-graph and clearly outper-
10 Growth
formed cba, but also cc.
ii.) A measure of decrement of growth, which is assumed to be a tree size relatedeffect. Test variables were: a (age), crown surface area (csa, see equation 2.1.2.1),d2h and an estimated volume v(d). a performed best and was further used.
iii.) Site: Available explanatory variables are: nitrogen deposition (in kg N/ha/a, ndep),water holding capacity (wat.hold.cap), water permeability (wat.perm), nutrient avail-ability (nut.av), slope, aspect and relief. Nitrogen deposition was not linear inits effect on growth, hence a quadratic term was used. Simple contrasts betweensouth/south-east, north/north-west and the remaining rest were used to representaspect. Slope (in %) was transformed by arcsin(
√x). Nitrogen deposition had the
strongest effect, followed by aspect, water permeability and slope.
iv.) Stand density: was measured by a stand density index sdi (Reineke, 1933), totalbasal area (bat) and logarithmic number of stems (per hectare) log10(nt). Totalbasal area performed best.
v.) Thinning effects: Thinning effects were measured by an indicator variable treat,which holds 1, if the stand was ever thinned and 0 if not, basal area of (previously)harvested trees (bah), number of trees per hectare harvested (nh) and the changeof crown competition ccc see 2.1.2.2. bah in a nonlinear form and ”Treatment”performed best.
vi.) Mixture with other species: An effect measured by shares of basal area ofbroadleaved trees (mixB) is in the full model and if the leading species (in basalarea) is a coniferous species (leadcon).
vii.) Climatic effects: these are complicated effects, since I found many interactionsamong them and they are non-linear. I used temperature, precipitation and mois-ture and tested the effects over different previous period lengths. Longer periodlengths showed better results (10, 5 and per); for comparison I choose the periodlength according to the given re-measurement intervals (∗.per). The final modelincluded a quadratic term in moisture (mphy.per) and interactions between pre-cipitation (nphy.per) and temperature (tphy.per) as well as between moisture andprecipitation. All climate variables were used in scaled form.
2.3.4 Examining the full model
The full model can be found in the Appendix A.1. Since we have repeated measures oftrees on plots, a mixed model may also be useful. An examination of the residuals overplots shows some systematic deviation (see figure: 2.7). But a mixed model is, due to thecomplex structure, not easy to fit.
Further the residuals were plotted against the period length, to see if the linear interpo-lation has a systematic effect (see figure 2.8). Obviously that is not the case. The rightpanel in the same figure shows that historical effects remain in the model, which can notbe further explained. Obviously in the last years the trees grow a bit faster then the modelpredicts. It can not be attributed to a climate signal, since I also tested changes over timeof the climate variables.
In the last step the transformations from above were repeated, to see if they are still valid
2.4 Discussion 11
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2044
2045
3010
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−4
−2
0
2
4
6
Figure 2.7: Residuals of the final growth model over the first 50 plots.
1 2 3 4 5 6 7 8 9 11 13 15 22
−6
−4
−2
02
46
interval length
1920 1940 1960 1980 2000
−6
−4
−2
02
46
year of survey
year of survey
resi
dual
s
Figure 2.8: Residuals growth model over period length and the year of the survey.
under a full model. Further a grid search for α was performed (see figure 2.9). Still thesquare-root transformation is best in terms of back-transformed residual sum of squares.
2.4 Discussion
A comparison with basal area increment models as presented in (pp. 174, Ch. 3.2.2.5Brassel and Lischke, 2001), reveals, that the presented full model in A.1 for incrementhas much more flexibility in terms of explanatory variables. Kaufmann (in Brassel andLischke (2001)) reports R2 in the range 40-60% for the major tree species, whereas the fullmodel (A.1) has an R2 of 65% (all simple in-sample measures). Kaufmann’s model relieson national forest inventory data, which means, that a measure of the stand structure isnearly impossible or has extremely large variance (due to the small sample size).
Another comparison is possible by fitting a “full” (log-linearized, with fixed α = 20) linearmodel (ignoring the second exponent in 2.3.1.1) and automatically reducing it by the step-function, according to the BIC-criterion. This model has a R2 of 72% (transformed scale)
12 Growth
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8.05
transformations
α
RS
S
ln(y + α)
sign(y)*|y|^(1/2)
sign(y)*|y|^(1/3)
sign(y)*|y|^(1/4)
Figure 2.9: Residuals sum of squares for different transformations of the full model (inmillion), for spruce, back-transformed.
and 69% on the back-transformed scale. This means that we loose about 4% explanatorypower (in-sample) for a better interpretable model. A cross-validation of this comparisonmay show an opposite effect.
The way I choose to select the models (forward) and the thoughts and assumption at anyof the steps contains some subjectivity in model building. Although I have tried to bereproducible and transparent, different persons, with the same data would certainly cometo different models. After analyzing many of them I can suppose that the overall modelstructure, complexity and predictability of the models would not be greatly changed.
The complicated structure of the climate effects, due to the interactions and non-linearityis probably easier to understand in a graphical representation. Therefore in the middlepanel of figure 2.10 all three climate effects were simultaneously altered. Since climatevariables were scaled before entering the model, they are comparable to each other. Henceturning the values from −2 to 2 means from low temperatures, low precipitation andlow moisture up to high levels. This means that Picea Abies growth well in cold anddry conditions. Increasing temperature, precipitation and moisture means that growthdeclines. Under warm and wet conditions it is changing to better growth. The sameinterpretation is possible for the right panel in figure 2.10, which reports the combinedeffect of temperature and precipitation (and hence moisture) on growth for fixed d=30 cm,the other variables were fixed at 0. Best conditions can be found in the lower left corner(cold and dry) and upper right corner (warm and wet).
The total effect sizes of climate are small compared to the between tree competitionmeasure by bal (maximum basal area can reach 100 m2/ha). The other effects seemreasonable with the interpretation of their sign (e.g. strong.water.perm was estimated to
2.4 Discussion 13
0 20 40 60 80 100
010
020
030
040
050
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d in cm
pred
icte
d ba
i [qc
m in
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ears
]
BAL in qm
0204060
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050
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pred
icte
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i [qc
m in
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ears
]
change of clim. var
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reci
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tion
210
215
220
225
230
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235
235
240
240
245 250
255
−2 −1 0 1 2
−2
−1
01
2
Figure 2.10: Exemplary effects of the final growth model. Left: Effect of basal areaof larger trees, which measures between tree competition. Middle: Simultaneous effectof climate (scaled variables) on the predicted bai. Right: Simultaneous effect of scaledtemperature, precipitation (and hence moisture) on bai, d fixed at 30 cm.
be −0.27, hence a we have a decreasing effect on growth, if the permeability of the soil isstrong, or: bat is estimated to be negative, meaning that increasing total basal area, as ameasure of stocking density on a stand level, reduces growth).
Under the model building procedure I found, that the crown related measures like cc orccc or csa on a direct model comparison with measures based directly d are less important.Although the literature states the opposite (e.g. Biging and Dobbertin (1995)), the valuesI have calculated and modeled come with many assumptions and are “smooth”, becauseonly the expectation values of the models were used; hence we loose (compared to a fullmeasure) information. At this point one may want to use multiple imputation as techniqueto get more reliable estimates on the unknown patterns in the model matrix.
An interesting method to encompass with the problems of the data at hand may be the useof a Bayesian framework. Eitzel, Battles, York, Knape, de Valpine, and de Valpine (2013)have successfully shown that a methodology for long-term monitored forests is applicable.Their method was able to account for the observation error, missing data, uneven timeintervals and auto-correlation through the repeated measures. A full mixed model shouldcontain the effect on plot level and trees within a plot. Especially on the last problema successful fitting with standard likelihood methods (nlme (Pinheiro, Bates, DebRoy,Sarkar, and R Core Team, 2015) or glmer in lme4 (Bates, Machler, Bolker, and Walker,2015)) was not possible. Hence the reported standard errors in A.1 should carefully beinterpreted.
14 Growth
Chapter 3
Mortality
In order to describe mortality, the first modeling choice was logistic regression analysis.After some discussion with my mentor, we came to the conclusion, that the process ofdying my better be understood with a time to event analysis (Survival analysis). In thelast section 3.7 of this chapter both methods will be compared.
3.1 Introduction
Although only 3% of all remeasured trees were classified as being recently dead, thesetrees have a large importance. Mortality gains importance since forest reserves are beingestablished and by stopping human intervention, mortality becomes a dominant processin the forests (Wunder, Reineking, Bigler, and Bugmann, 2007). Often the simulators areused to predict growth without treatment, such that mortality is the driving process. Inthe EFM data (and hence the model) there is no distinction for the reason of dying.
On a stand level the number of trees and their size will reach a limit. One limit is themaximum stand density found by Reineke (1933). On a log-log scale (number of treesper hectare over the quadratic mean diameter) we observe a threshold. Figure 3.1 showsthe surveys in Switzerland (divided in tree species groups) together with the assumedmaximum stand density.
The limit (the black line) was plotted by using a fixed negative slope of -1.605 and fixingthe line at the highest observed intercept in the data. The full single tree growth sim-ulator can be validated against this threshold. A deeper analysis (linearity, estimationof the boundary and dependence) of the threshold is not useful for the development ofthe simulator, since it should be based on single tree observations, i.e. using most of thepossible information.
3.2 Modeling probability to die using logistic regression
In contrast to the growth models, recently dead trees are binary observations. Y = {0, 1}is a Bernoulli variable and is 1 if the tree died and 0 otherwise. One might use the sumof dead trees from each survey (nm) as a realization of a binomial distributed randomvariable (nm ∼ B(n, π)).
15
16 Mortality
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34
56
78
910
European beech
log(dq)
log(
NH
A)
max(inter.) =
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Douglas fir
log(dq)lo
g(N
HA
)
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910
Oak spp
log(dq)
log(
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A)
max(inter.) =
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1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
34
56
78
910
Norway spruce
log(dq)
log(
NH
A)
max(inter.) =
12.7
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34
56
78
910
Scots pine
log(dq)
log(
NH
A)
max(inter.) =
12.43
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34
56
78
910
Larch spp
log(dq)
log(
NH
A)
max(inter.) =
12.27
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34
56
78
910
other conifers
log(dq)lo
g(N
HA
)
max(inter.) =
12.47
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1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
34
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78
910
Silver fir
log(dq)
log(
NH
A)
max(inter.) =
12.46
Figure 3.1: Observed realizations of the maximum stand density relationship by treespecies. Dots are surveys of plots. Line is the maximum stand density given by a definitionin Reineke (1933).
To model Y many flexible classification methods may well be used like linear discriminantanalysis (Crow and Hicks, 1990), neuronal networks (Hasenauer, Merkl, and Weingartner,2001), classification trees (Dobbertin and Biging, 1998), Bayesian analysis (Wyckoff andClark, 2000) or Survival analysis (Woodall, Grambsch, and Thomas, 2005). Further thelinear logistic regression (Bigler and Bugmann, 2003, 2004) can also be extended to a mixedeffects logistic regression (e.g. Rohner, Bigler, Wunder, Brang, and Bugmann, 2012).
One problem are the different time spans for the observed nm: one might consider tonormalize (e.g. to a yearly mortality probability) π = 1− (Nt/N0)
1/t like in Rohner et al.(2012)). This contains the assumption that π is constant over the normalization period.
However, it is also possible to use the Bernoulli observations and to stay on the singletree level. This has the advantage of using most of the information present in the data;that is the approach which was used in this analysis. To encompass different time spans Iintroduced dt = (ti − ti−1)− 5 as the difference to the 5-year standard period and used itas explanatory variable in the model.
The probability of being dead
π(x) = P [Y = 1|X = x] (3.2.0.1)
can be modeled by the logistic linear regression with the logit transformation:
log
(π(x)
1− π(x)
)= g(x) (3.2.0.2)
On the transformed scale (log of odds-ratio) it is a linear model and hence all the advan-tages in linear model building can be used.
3.2 Modeling probability to die using logistic regression 17
To get an overview first a full model with automated model selection was fitted. Since thephysiological reasonable interpretation and the extrapolation into unobserved regions ofthe data is of interest, a balance between statistical performance and interpretation needsto be found. Hence a forward selection process with variable selection by hand makessense, since collinearity can also well be handled. Preferably only one of several competingvariables is added as a representative of a interpretable effect at a step. All continuousvariables were scaled:
i.) relative basal area increment rbai as a measure of current vitality (or stress, orstress memory) of the tree showed excellent classification properties compared tosolely bai.
ii.) To measure size and age related effects d, a, v(d) and crown surface area csa wereused. a performed best. Although only 87% of the trees have a single tree age, itwas further used (second choice would be d).
iii.) On a stand level density related effects were estimated by nt, total basal area batand the stand density index sdi (Reineke, 1933). sdi performed best.
iv.) As a measure of between tree competition bal, rbal, cba, the estimated treeheight (h), and the relative tree height (rh) were tested. bal performed best and wasfurther used.
v.) Thinning effects were measured by percent of basal area being harvested bah, per-centage of trees being harvested nh and the change in crown competition ccc; furthera variable treat, containing the information if the stand was ever thinned, was used.
vi.) Mixture with broadleaved trees (measured in shares of basal area) and the leadingtree species (lead.sp, the species which holds the maximum share on basal area) weretested. lead.sp, a factor with eight levels was further used.
vii.) Climatic effects: like in the growth model, interactions and non-linearity arepresent. I tested different combinations: a maximal version, maximal without pre-cipitation and maximal without temperature. Maximal means all three main effectsplus interactions between precipitation and moisture, between temperature and mois-ture and non-linearity in temperature and precipitation (by quadratic terms). Theremaining model includes the maximal version without temperature and withoutsquared precipitation. Age was not significant anymore.
viii.) Site related effects are: water holding capacity, water permeability, relief, nutrientavailability, exposition, slope and soil depth. Water permeability improved the modelsignificantly, but with small effect sizes. It was further used in the model.
ix.) The variable dt is a measure of different period length, compared to the standard5-year period. It was a necessary variable, since different period length were used.It was included as a linear and quadratic term.
The final model was fitted with the following call to R. Its full summary can be found inthe appendix A.2.
1 m.final.logit ← glm(tot ∼ rbai + sdi + bal + bah + I(bah^2) + treat +
2 lead.sp + nphy.per + tphy.per + mphy.per + wat.perm + dt +
3 I(dt^2) + nphy.per:mphy.per , family = binomial , data = subset(fi.s,
4 !is.na(nphy.per) & !is.na(wat.perm)))
18 Mortality
−2 0 2 4 6 8 10
−0.
050.
050.
150.
25
bah
s(ba
h,1.
98)
−4 −3 −2 −1 0 1 2 3
−0.
050.
050.
150.
25
nphy.per
s(np
hy.p
er,1
.74)
−2 −1 0 1 2
−0.
050.
050.
150.
25
tphy.per
s(tp
hy.p
er,1
.75)
−1 0 1 2 3
−0.
050.
050.
150.
25
mphy.per
s(m
phy.
per,1
.9)
−2 0 2 4 6
−0.
050.
050.
150.
25
dt
s(dt
,2.8
5)
Figure 3.2: Smoothed explanatory variables estimated by a generalized additive model
3.3 Discussion logistic regression model
Although the logistic regression approach used here is not correct in the sense that it usesrepeated measures, it will briefly be discussed.
In order to include the random effects, two levels will be needed. The first one is the effecton plot level (FNUM) and within each plot the effect of the repeated measured trees. Hencea call to glmer (in package lme4), with BID the identifier of individual trees would looklike this:
1 m.glmer ← glmer( tot ∼ bai.rel + sdi + bal + ba.h + I(ba.h^2) + treat + lead.sp +
2 nphy.per + I(nphy.per^2) + tphy.per + mphy.per + wat.perm +
3 delta.t5 + I(delta.t5^2) + nphy.per:mphy.per + (1| FNUM/BID),
4 data=fi.s, family=binomial)
On my computer it took 2 hours to be executed and the model failed to converge after10.000 evaluations. Beside the long computation time, the process of dying can better beexplained by Survival analysis, so mixed logistic regression was not further used.
However, to check the linearity of the explanatory variables generalized additive models(function gam in mgcv (Wood, 2011)) were used and graphically interpreted. As an examplethe comparable full gam model (including the interaction on precipitation and moisture andrbai, bal, sdi as linear effects) shows the strong curvature of bah (basal are harvested) infigure 3.2. This effect is interesting, because the increase in logs odds ratio means a higherprobability to die. The interpretation is, that the stands get vulnerable to storm, whenharvesting increases. The simultaneous inclusion of the variable treat shows a decreasingeffect on the probability to die, if stands were treated. Both effects are graphically shown
3.4 Introduction survival analysis 19
−2 0 2 4 6
0.00
0.01
0.02
0.03
0.04
basal area harvested, scaled
π
Treatment
YesNo
π
scaled temperaturesc
aled
pre
cipi
tatio
n
0.01
0.02
0.02
0.03
0.04
0.05
0.06
−2 −1 0 1 2−
2−
10
12
Figure 3.3: Exemplary predicted probabilities (holding the other scaled variables at 0).Left: Effects of thinning. Right: Effects of climate (temperature, precipitation and mois-ture).
in figure 3.3. Further the right panel shows the effect of climate. The dry and warm areain the bottom right corner shows a clearly higher probability to die.
3.4 Introduction survival analysis
Survival analysis intends to model the time until an event occurs (Kleinbaum and Klein,2012). Here an event is the death of a tree. The time until the tree dies is a random vari-able. Since all the potential explanatory variables are time dependent, each observationis spitted into a series of left truncated observations. I assumed that the time depen-dent variables are constant within the interval. The time dependent variables were linearinterpolated to the midpoint of an interval and are fully comparable to the explanatoryvariables previously used in the logistic regression. Since many of the trees over a fullrotation period are being harvested before they die, nearly all observations are also rightcensored. When a tree was dying in an interval it is also a left censored observation, sincethe true age of death is unknown, but lies inside the interval.
Survival analysis can encompass censoring of observations by using the distribution F (t) =1− S(t) (with S(t) the survival function) and density function f(t) together in the likeli-hood. Therefore an indicator variable ci is used. It is 1, if the observation is uncensoredand 0 otherwise. Further the left truncation (si) is accounted for by using the condition,that the tree was already alive until it was observed.
More formally the likelihood of left and right censored observations for a series of left
20 Mortality
truncated observations becomes (Jackson, 2015):
l(θ|tmin, tmax, c, s) =
∏i: ci=1
fi(ti)∏
i: ci=0
(Si(t
mini )− Si(tmaxi )
) /∏i
Si(si) (3.4.0.3)
Another important function is the hazard function: h(t) = −dS(t)/dtS(t) which is the instan-
taneous death rate. It can be interpreted like a density, hence it needs to be integrated tomake it comparable to a probability.
3.5 Survival Data
A common and intuitive way to describe survival data is presenting a Kaplan-Meier esti-mate. This is a nonparametric survivor curve possibly with a confidence interval. In figure3.4 the survivor curves are presented depending on the type of thinning. Within each typethe strength of thinning is controlled and is changing, depending on the research question.This variable is assumed to be an potential explanatory variable for the regression. Asindicated by the left panel, the difference on the survivor curve for the type of thinningseems not be important, hence in the further analysis only treated and untreated standswere differentiated, as shown in the right panel. A noticeable change in the form of thecurve is visible at around 100 years and later at 160 years.
0 50 100 150 200 250
0.0
0.2
0.4
0.6
0.8
1.0
Kaplan−Meier estimate
age in years
S(t
)
Thinning regimes
Thinning from aboveNoneThinning from below
0 50 100 150 200 250
0.0
0.2
0.4
0.6
0.8
1.0
Kaplan−Meier estimate
age in years
S(t
)
Treated
YesNo
Figure 3.4: Kaplan-Meier estimates of the trees, depending on the type of thinning.
3.5.1 Distribution
The formulation of the above likelihood is general in terms of the distribution function.Several common distribution functions can be used in survival analysis. For location-scale
3.5 Survival Data 21
families a common approach is to model the location parameter by explanatory variables,whilst the scale parameter is fixed, given the data.
I started to search for a distribution by estimating large survival models (using survreg
in package survival (Therneau and Grambsch, 2000)), which were automatically reducedto a reasonable model by stepAIC in package MASS (Venables and Ripley, 2002). As anexample the residuals of a Weibull distribution of such a reduced model are presented inthe left panel of figure 3.5. The obvious deviation was also found under different otherlocation-scale families. Yet the two parameter distributions seemed not to be flexibleenough to deal with the full dataset.
0.5 1.0 1.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
qq−Plot Weibull distribution
theoretical quantiles
quan
tiles
of t
he d
eath
obs
erva
tions
0 50 100 150 200 250
01
23
4
cumulative hazard
age in years
cum
ulat
ive
haza
rd
conv. Weibullflex. Weibull
Figure 3.5: Left: Residuals of a automatically reduced Weibull survival model, Right:cumulative hazard of similiar groups within the data: for each a Kaplan-Meier estimate,overlaid by a conventional and flexible Weibull model fit.
Possibly the different sites from which the data were sampled have an influence on thesurvivor curve. This means that probably some regional effects are present in the data.Hence another approach was tested to find groups of similar explanatory variables withrespect to the classification of dying. I used a decision tree (function rpart in the similarnamed package (Therneau, Atkinson, and Ripley, 2015)) to separate common observations.rpart found five similar groups. Then a simple survival model with the group as onlyexplanatory variable was fitted. Beside the conventional Weibull model W (β′X,σ) a moreflexible approach W (β′X,σ′X) (using flexsurvreg in the package flexsurv (Jackson,2015)) was also tested and the results are graphically shown in figure 3.5 in the rightpanel. Although the more flexible Weibull regression approach fits much better to thecumulative hazard, it still deviates clearly from the Kaplan-Meier curves.
For that reason a three parametric distribution was further used. The Generalized Gamma(GG) distribution in the following parametrization (equation 3.5.1.1) is a stable versionand it is implemented in flexsurvreg for survival analysis (Jackson, 2015). In this formβ is the location, σ the scale and λ the shape parameter.
f(t;β, σ, λ) =|λ|
σtΓ(λ−2)
[λ−2(e−βt)λ/σ
]λ−2
exp[−λ−2(e−βt)λ/σ
](3.5.1.1)
22 Mortality
Cox, Chu, Schneider, and Munoz (2007) discusses the flexibility and the possible forms ofthe resulting hazard functions. E.g. the Weibull distribution is a simplification of the GGdistribution by fixing the shape parameter to 1 (W (µ, σ) = GG(µ, σ, 1), see also figure3.6).
3.6 Modeling times to event
In the model building a fully comparable approach to the logistic regression 3.2 was used.The forward selection process was implemented by hand, with the difference that I startedwith the variable treat in the most simple model. This is due to the differences observedin the Kaplan-Meier estimate from above (figure 3.4).
I started conventionally, meaning that I only regressed the location parameter. In a secondrun I tested to regress also the scale and shape parameter. Since the amount of potentialexplanatory variables is large (>20) and the estimation of a larger model takes on mycomputer easily more than a few hours, I had to restrict which parameters needs to bemodeled and how flexible it should be. Figure 3.6 shows the possible forms of the hazardfunctions, depending on the scale (σ) and shape (λ) parameter.
Figure 3.6: Taxonomy of different hazard functions within the GG distribution. Figureaccording to Cox et al. (2007). The four regions, indicated by the arrows, representincreasing and decreasing hazard functions and bathtub and arc-shaped forms.
After some tests on larger models I found that a flexible σ is mostly better than flexibleλ model (differences in AIC). As explanatory variables in σ I tested treat, bah (per centof basal are harvested, a measure of the thinning intensity) and age.a.100.
From the logistic regression model it is already known, that bah has a strong negativeinfluence on the survival, which can be interpreted as a destabilization effect on the stand.If we assume, that under large bah the stands gets more vulnerable to storm effects, thismight also change the following survival curve (and hence the hazard function). FurtherI implemented an indicator variable age.a.100 which is 1 if the stand is above 100 yearsand 0 else. This was an idea based on the Kaplan-Meier curves from above. At this ageusually the thinning from above is changing to final harvesting and hence the form of thesurvival function might be changed as well.
3.7 Discussion: Prediction and comparison 23
The resulting best model found by comparing AIC in a forward selection process has thefollowing form; its corresponding estimated parameters and standard errors are reportedin A.3.
1 ini ← c(4.59 ,0.16 ,0.42 ,0.34 ,0.15 , -0.22 ,0.05 , -0.07 ,0.01 ,0.15 , -0.26 ,0.24 ,0.29 ,
2 0.15 , -0.02 ,0.15 ,0.03 ,0 ,0.05 ,0.06 ,0.01 ,0.02 , -0.25 ,0 ,0.63 , -0.04 ,0.04 ,0.12 ,0.56)
3 m.final.surv ← flexsurvreg(formula = Surv(t1, t2, tot , type = "interval") ∼4 rbai + d + nt + rbal + bah + I(bah^2) + lead.sp + nphy.per +
5 tphy.per + I(tphy.per^2) + mphy.per + relief + age.a.100 +
6 nphy.per:mphy.per + tphy.per:mphy.per ,
7 anc = list(sigma = ∼ bah + age.a.100),
8 data = fi.s, dist = "gengamma", inits = ini)
3.7 Discussion: Prediction and comparison
Prediction in Survival analysis means, that a Survival curve S(t,X = x) is fully described.This is - compared to logistic regression - not a single number, hence it is so flexible and candescribe processes. In left panel of figure 3.7 an example is given with an plotted hazardfunction. It is not continuous at t=100, since I have introduced an indicator variableage.a.100; it shows the change in total instantaneous death rate and the change in formof the curve, due to the changes in σ. The right panel shows predicted death probabilitiescalculated as a the difference of S(t = 80, X = x) − S(t = 90, X = x), which is theprobability to die between 80 and 90 years, under simultaneous changes in temperatureand precipitation (and hence also moisture) and holding the other variables constant atzero. The probability to die is higher in warm and dry regions, comparable to figure 3.3,but shows higher gradients.
0 50 100 150
0.00
0.02
0.04
0.06
0.08
0.10
time to event
haza
rd
age < 100age>100
scaled temperature
scal
ed p
reci
pita
tion
0.05
0.1
0.1
0.15
0.15
0.2
0.2
0.25
0.25
−2 −1 0 1 2
−2
−1
01
2
Figure 3.7: Predicted survival curves and hazard functions. Left: Hazard function overtime, right: probability to die between 80 and 90 years under different climates.
Comparing Survival analysis with the predicted probabilities form the logistic regressionneeds an estimate of a probability to die in the survival analysis. I used the difference ofS(ti−1, X = x)− S(ti, X = x) as it was defined by the interval length of the observations;this is the probability to die inside the interval length. In order to estimate the prediction
24 Mortality
accuracy of the two methods, a 10-fold cross validation was calculated. The loss functionwas l(π, Y ) =
∑(π − Y )2. The result in figure 3.8 shows, that the logistic regression model
is better in predicting. This is contradicting, since the logistic regression model containsremeasured trees, hence the model is “wrong” and may be over-fitted, which should bevisible in the cross-validation. Whereas in the Survival analysis the likelihood 3.4.0.3 onlyuses the censored part of the observations (and hence the correct amount of information)from the data.
I was not able to find a satisfying answer to that problem, but made the following obser-vations:
• Survival analysis was better (Weibull and GG), if small models were used (few ex-planatory variables).
• When the predictive power of the explanatory variables was high, the logistic modelprediction were usually better.
• The difference between models based on Weibull or GG-distribution was not large,also the for the flexible modeling of scale and shape parameters.
Maybe a simulation method could help to better understand the contradicting mechanism.
●
logit GG Weibull
400
420
440
460
480
500
520
10−
fold
pre
dict
ed lo
ss
logit logit, autom. GG Weibull
360
380
400
420
440
460
480
10−
fold
pre
dict
ed lo
ss
Figure 3.8: Left: Loss of the 10-fold cross validated logistic regression model, comparedto predicted probability to die of the survival models. Right: the same for a subset ofdata, including an stepwise reduced logistic regression model. GG=Generalized GammaDistribution.
Chapter 4
Summary
The thesis is about two fundamental processes of Picea Abies in the forests of Switzerland.It is based on the broad experimental forest management data, established in 1880 inSwitzerland.
The first part deals with the analysis of growth and shows the importance of a usefultransformation of the basal area increment model. In a preliminary part crown forms andcrown competition indexes on a single tree level were calculated. It was assumed, thatcrown related indexes should well explain the increment (Biging and Dobbertin, 1995).Comparing the estimated crown indexes against simple other measures based on fullyobserved diameter at breast height, shows that the latter should be preferred. At least inthis context, with the amount of data and simple imputation technique applied.
The model was found by a forward selection process, introducing groups of reasonablephysiological interpretation. Basal area increment was explained by a nonlinear functionof diameter at breast height and a multiplicative part containing:
• basal area of larger trees, as a measure of between species competition
• age of the trees (a decremental part of the growth process)
• site related effects: nitrogen deposition, exposition and water holding capacity
• total basal area, measuring stand density
• thinning effects were estimated by basal area harvested and an indicator variableholding the information if the stand was ever thinned
• mixture effects were estimated by relative shares of broadleaved trees and if theleading tree species is a coniferous species
• climate effects estimated by precipitation, temperature and moisture
Especially the climatic effects contain non-linearity and interactions. The resulting effectson growth are graphically shown. Better growth for Picea Abies can be reported for coldand drier sites.
The second part of the thesis is about mortality. A logistic regression model was applied topredict the probability to die on a single tree level. A comparable forward selection processcompared to the growth model was established. The logistic regression model containsrepeated measures of trees and hence my be wrong in the reported standard errors and
25
26 Summary
the model may be over-fitted. Hence the promising Survival analysis was further used todescribe the process of dying. It is promising since the description of the process must bea function over time, which is estimated by this method. Further the censoring of the dataand the truncation can well be handled in the likelihood. Although theoretically better,the resulting predicted probabilities were less precise than the predictions based on the“wrong” logistic regression model.
4.1 Future Work
After analyzing the data I have recognized that the amount of data, the changing intervals,the missing values in the data and the repeated measured trees on different plots arechallenging to be modeled. An alternative may be a Bayesian framework, which hasshown to work well under such conditions (Eitzel et al., 2013). Alternatively the mixedmodels could further be used, if the problems reported in the thesis, can be solved.
A second interesting work could be to explain or analyze in depth the contradictory resultof the predictions of a“wrong”logistic regression model against a theoretical better foundedSurvival analysis approach.
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30 BIBLIOGRAPHY
Appendix A
Complementary information
A.1 Growth model
The final growth model has the following form.
Call:
nlrob(formula = I(sign(bai) * sqrt(abs(bai))) ~ b0 + b1 * (1 -
exp(b2 * d)) + b3 * bal + b4 * a + b5 * ndep + b6 * ndep^2 +
b7 * NNW + b8 * SSE + b9 * strong.water.perm + b10 * bat +
b11 * Treatment + b12 * bah^2 + b13 * mixB + b14 * nphy.per +
b15 * tphy.per + b16 * mphy.per + b17 * nphy.per * mphy.per +
b18 * tphy.per * nphy.per + b19 * tphy.per * mphy.per + b20 *
mphy.per^2 + b21 * leadcon, data = fi6, start = c(b0 = 0.08,
b1 = 7, b2 = -0.03, b3 = -0.04, b4 = -0.003, b5 = 1e-04,
b6 = 0, b7 = -0.001, b8 = 0.001, b9 = -0.001, b10 = -0.001,
b11 = 0.001, b12 = 0.001, b13 = -0.001, b14 = 0.001, b15 = 0,
b16 = 0, b17 = 0, b18 = 0, b19 = 0, b20 = 0, b21 = 0))
Residuals:
Min 1Q Median 3Q Max
-15.31435 -1.22400 0.03149 1.22021 14.58023
Parameters:
Estimate Std. Error t value Pr(>|t|)
b0 4.018e+00 6.438e-02 62.417 <2e-16 ***
b1 1.844e+01 1.211e-01 152.268 <2e-16 ***
b2 -2.884e-02 3.445e-04 -83.714 <2e-16 ***
b3 -7.504e-02 6.447e-04 -116.386 <2e-16 ***
b4 -1.601e-02 3.340e-04 -47.948 <2e-16 ***
b5 -1.462e-01 3.922e-03 -37.276 <2e-16 ***
b6 2.927e-03 7.224e-05 40.517 <2e-16 ***
b7 -3.581e-01 1.350e-02 -26.537 <2e-16 ***
b8 2.491e-01 2.177e-02 11.440 <2e-16 ***
b9 -2.680e-01 2.028e-02 -13.215 <2e-16 ***
b10 -3.028e-02 9.252e-04 -32.723 <2e-16 ***
31
32 Complementary information
b11 3.502e-01 1.737e-02 20.161 <2e-16 ***
b12 9.199e+00 3.041e-01 30.252 <2e-16 ***
b13 -1.413e+00 4.135e-02 -34.182 <2e-16 ***
b14 1.544e-01 1.115e-02 13.839 <2e-16 ***
b15 -2.433e-01 1.356e-02 -17.947 <2e-16 ***
b16 -2.330e-01 1.610e-02 -14.474 <2e-16 ***
b17 2.400e-01 1.294e-02 18.550 <2e-16 ***
b18 3.059e-01 1.431e-02 21.380 <2e-16 ***
b19 -2.399e-01 1.700e-02 -14.109 <2e-16 ***
b20 -1.163e-01 1.301e-02 -8.934 <2e-16 ***
b21 9.785e-01 2.113e-02 46.301 <2e-16 ***
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
Robust residual standard error: 1.812
Convergence in 9 IRWLS iterations
Robustness weights:
149462 weights are ~= 1. The remaining 37910 ones are summarized as
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.1592 0.5795 0.7359 0.7139 0.8681 0.9990
A.2 Logistic regression model
The output of the final logistic regression model (3.2) with standard errors.
## summary(m.final.logit)
## Call:
## glm(formula = tot ~ rbai + sdi + bal + bah + I(bah^2) + treat +
## lead.sp + nphy.per + tphy.per + mphy.per + wat.perm + dt +
## I(dt^2) + nphy.per:mphy.per, family = binomial, data = subset(fi.s,
## !is.na(nphy.per) & !is.na(wat.perm)))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9159 -0.2382 -0.0959 -0.0318 5.7792
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -6.273780 0.062574 -100.261 < 2e-16 ***
## rbai -3.445999 0.052155 -66.072 < 2e-16 ***
## sdi -0.168557 0.021869 -7.707 1.28e-14 ***
## bal 0.293124 0.022717 12.903 < 2e-16 ***
## bah 0.702263 0.024334 28.859 < 2e-16 ***
## I(bah^2) -0.053028 0.004905 -10.811 < 2e-16 ***
## treatNo 0.821258 0.093291 8.803 < 2e-16 ***
## lead.spbeech -0.248918 0.092111 -2.702 0.00688 **
## lead.spdouglas 0.702260 0.312280 2.249 0.02452 *
A.3 Survival model 33
## lead.spoak -0.120147 0.119494 -1.005 0.31468
## lead.sppine 0.280548 0.164460 1.706 0.08803 .
## lead.splarch 0.092319 0.105114 0.878 0.37980
## lead.spoc -0.044106 0.103230 -0.427 0.66919
## lead.spfir -0.361902 0.092734 -3.903 9.52e-05 ***
## nphy.per -0.031291 0.031704 -0.987 0.32364
## tphy.per 0.062155 0.031763 1.957 0.05037 .
## mphy.per -0.279647 0.045686 -6.121 9.30e-10 ***
## wat.perminhibited -0.177741 0.088540 -2.007 0.04470 *
## wat.permweakly inhibited -0.182622 0.034891 -5.234 1.66e-07 ***
## wat.permstrongly inhibited -3.234202 1.009270 -3.204 0.00135 **
## wat.permexcessiv -0.593792 0.064730 -9.173 < 2e-16 ***
## dt 0.587229 0.025083 23.412 < 2e-16 ***
## I(dt^2) -0.098903 0.010584 -9.344 < 2e-16 ***
## nphy.per:mphy.per 0.327342 0.016934 19.331 < 2e-16 ***
## ---
## Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 45812 on 128251 degrees of freedom
## Residual deviance: 31081 on 128228 degrees of freedom
## AIC: 31129
##
## Number of Fisher Scoring iterations: 8
A.3 Survival model
Printed output of from the final flexsurvreg model.
## m.final.surv <- flexsurvreg(Surv(t1, t2, tot, type = "interval") ~
## rbai + d + nt + rbal + bah + I(bah^2) + lead.sp + nphy.per +
## tphy.per + I(tphy.per^2) + mphy.per + relief + age.a.100 +
## nphy.per:mphy.per + tphy.per:mphy.per,
## anc = list(sigma = ~ bah + age.a.100),
## data = fi.s, dist = "gengamma", inits = ini)
## est L95% U95% se
## mu 4.59102 4.5698 4.6122 0.0108
## sigma 0.15802 0.1524 0.1639 0.0029
## Q 0.41978 0.3655 0.4741 0.0277
## rabl 0.33783 0.3240 0.3517 0.0071
## d 0.14869 0.1400 0.1574 0.0045
## nt -0.22332 -0.2343 -0.2123 0.0056
## rbal 0.05112 0.0428 0.0595 0.0043
## bah -0.07300 -0.0804 -0.0656 0.0038
## I(bah^2) 0.00841 0.0064 0.0105 0.0010
## lead.spbeech 0.14897 0.1291 0.1688 0.0101
34 Complementary information
## lead.spdouglas -0.25886 -0.3495 -0.1683 0.0462
## lead.spoak 0.23672 0.2032 0.2702 0.0171
## lead.sppine 0.29265 0.2549 0.3304 0.0193
## lead.splarch 0.14810 0.1095 0.1867 0.0197
## lead.spoc -0.02148 -0.0515 0.0086 0.0153
## lead.spfir 0.15303 0.1287 0.1773 0.0124
## nphy.per 0.02903 0.0221 0.0360 0.0036
## tphy.per 0.00206 -0.0091 0.0132 0.0057
## I(tphy.per^2) 0.04696 0.0391 0.0548 0.0040
## mphy.per 0.06160 0.0512 0.0720 0.0053
## reliefflat 0.01089 0.0022 0.0196 0.0045
## reliefslope toe 0.02427 -0.0128 0.0614 0.0189
## reliefsummit -0.24769 -0.2842 -0.2112 0.0186
## reliefdepression 0.00066 -0.1484 0.1497 0.0760
## age.a.100 0.62505 0.5853 0.6648 0.0203
## nphy.per:mphy.per -0.03539 -0.0400 -0.0308 0.0023
## tphy.per:mphy.per 0.03759 0.0281 0.0471 0.0049
## sigma(bah) 0.11903 0.1045 0.1335 0.0074
## sigma(age.a.100) 0.55980 0.4991 0.6205 0.0310
A.4 Functions
The following function can calculate the crown radius at any height in the tree. It wasused to produce figure 2.1.
1 f.crh ← function(h, H, hcb , cr , BA){
2 # calc crown radius at any height ’h’ in between height of crown base ’hcb ’
3 # and total height ’H’. cr: crown radius (at maximum)
4 # h is scalar , but H, hcb , cr , BA may be vectors of the same length
5 stopifnot(
6 BA %in% c("Fichte","Laerche","Douglasie",
7 "Tanne","Buche","Eiche" , "Kiefer")
8 )
9
10 # form of the sunlight part
11 form ← ifelse(BA %in% c("Fichte","Laerche","Douglasie"), 1,
12 ifelse(BA == "Tanne", 1/2, 1/3))
13 # fraction ’f’ of radius max for the lower part in the frustum
14 f.max ← ifelse(BA %in% c("Fichte","Tanne","Laerche","Douglasie"), 0.5,
15 ifelse(BA == "Kiefer", 2/3,
16 ifelse(BA %in% c("Buche","Eiche"), 1/3, 1)))
17 # fraction of upper crown part (’o’: oben)
18 f.o ← ifelse(BA %in% c("Fichte","Laerche","Douglasie"), 2/3,
19 ifelse(BA == "Kiefer" ,0.68,
20 ifelse(BA == "Tanne", 0.5,
21 ifelse(BA %in% c("Buche","Eiche"), 0.4, 0.35))))
22
23 cr.at.cb ← f.max*cr # crown radius at crown base
24 l ← H-hcb # ’l’ length of crown
25 lo ← l*f.o # lenght "oben" (sunlight)
26 lu ← l*(1 - f.o) # length "unten" (shaded)
27 h.at.max.r ← lu + hcb # height at max crown radius
28 dist ← H - h # distance to the tip
29
30 # for the linear part (the shaded part of the crown):
31 d ← (cr.at.cb - cr)/(l-lo)
32 c ← cr - d*lo
A.4 Functions 35
33 # for the upper part (sunlight)
34 a ← cr/lo^(form)
35
36 cr.at.h ← ifelse( (h < hcb) | (h>=H), 0, # h below crown base or above tree
height
37 ifelse( h < h.at.max.r,
38 c + d*dist , # linear shaded case
39 a*dist^form # sunlight part
40 )
41 )
42 cr.at.h
43 }
Here is the call to the function f.crh, producing the crown radius over the whole treeheight (see figure: 2.1.)
1 curve( f.crh(h=x, H= 20, hcb= 8, cr=3, BA="Fichte" ), 0, 20,
2 xlab="tree height", ylab="crown radius")
3 curve( f.crh(h=x, H= 20, hcb= 8, cr=3, BA="Buche" ), add=T, col=2)
4 curve( f.crh(h=x, H= 20, hcb= 8, cr=3, BA="Tanne" ), add=T, col=3)
5 legend("topleft", legend=c("spruce","beech","fir"), col=1:3, lty=1, bty="n")
Given the function f.crh it is possible to calculate the cross sectional areas of stand, usingf.csah, which is needed to calculate crown indeces like cc (2.1.2.2).
1 f.csah ← function(h, data){
2 # calc. cross sectional area (csa) in a height ’h’.
3 # if tree height of a competitor is below h, competitor is ignored
4 # if height of maxim crown width is above h, full crown area is used
5 # else the cross sectional area is calculated
6 # h: scalar; data: dataframe cont. H.hat , cb , hcb , BA
7
8 # tree heights must be larger then h, if below ignore:
9 data ← data[data$H.hat > h, ]
10
11 # if height of max crown width is above (or equal) relevent crown height
12 # use total crown area
13 radii.above ← data$cd[data$hmc >= h]/2
14 # from the rest calc cross sectional area
15 data ← data[data$hmc < h,]
16 radii.sect ← f.crh(h=h, H=data$H.hat , hcb=data$cb, data$cd/2, data$BA.cform)
17 # sum up: csa: cross sectional area at ’h’
18 pi*(sum(radii.sect ^2) + sum(radii.above ^2))
19 }
Here the simple function f.csurf, calculating crown surface area of a tree:
1 f.csurf ← function(cd, cb , h){
2 # calc crown surface given crown diameter ’cd ’, height of crown base ’cb ’ and
3 # ’h’ height of tree , assuming a paraboloid
4 r ← cd/2 # crownradius
5 cl ← h - cb # crownlength
6 pi*r/(6*cl^2)*( (4*cl^2+r^2)^(3/2) - r^3 )
7 }
36 Complementary information
![Phenotype features in juvenile populations of Picea abies and … · Norway spruce (Picea abies [L.] Karst.) mature trees has been used for a long time. It is based mainly on the](https://img.pdfslide.us/doc/110x75/60a18f23d0ce166df4619aeb/phenotype-features-in-juvenile-populations-of-picea-abies-and-norway-spruce-picea.jpg)
