how to quantify nutrient export: additive biomass functions for … · 2015-10-15 ·...
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Introduction data methods Results Discussion Literatur
How to quantify nutrient export:Additive Biomass functions for spruce fit with
Nonlinear Seemingly Unrelated RegressionIBS-DR Biometry Workshop, Wurzburg
Christian Vonderach
Forest Research Institute Baden-Wurttemberg
Wurzburg, 07.10.2015
Introduction data methods Results Discussion Literatur
1 Introduction
2 data
3 methods appliedNonlinear Seemingly Unrelated Regression
4 resultsNSUR fitcomparison
5 Discussion
Introduction data methods Results Discussion Literatur
what it is about
EnNa: Energywood and sustainability (funded by FNR)
havesting removes wood (i. e. C) and also nutrients (Ca, K,Mg, P, . . . )
→ sustainability required regaring C and also Ca, K, Mg, P, . . .
nutrient balance:
NB = VW + DP − SI︸ ︷︷ ︸soil
−HV
HV =∑
trees =∑
compartments
nutrient concentration differs within different compartments
→ compartment-specific biomass functions required
Introduction data methods Results Discussion Literatur
collected data
spruce (Piceaabies)
6 datacompilations(incl. Wirthet al., 2004)
homogenisation
stump/Bcoarse wood/Bsmall woodneedles
≈ 1200 trees
(only referenced shown; +CH|DK|B)
Introduction data methods Results Discussion Literatur
Overview of collected data
age dbh height
0
50
100
0 50 100 150 200 0 25 50 75 0 10 20 30 40
num
ber
of d
ata
sets
0
10
20
30
40
0 20 40 60 80dbh [cm]
heig
ht [m
]
0
1000
2000
3000
0 20 40 60 80dbh [cm]
abov
egro
und
biom
asse
[kg]
Introduction data methods Results Discussion Literatur
general methodological design
wanted: biomass functions for all compartments, and the totalmass
maintain additivity (see Parresol, 2001)1 BMtotal =
∑BMcomp
with var(ytotal) =∑c
i=1 var(yi) + 2∑∑
i<j cov(yi , yj )2 Nonlinear Seemingly Unrelated Regression (NSUR)
NSUR requires rectangulardata set (i. e. no NA’s)
but some of the studiescontain NA’s
complete caseimputation
Introduction data methods Results Discussion Literatur
general methodological design
wanted: biomass functions for all compartments, and the totalmass
maintain additivity (see Parresol, 2001)1 BMtotal =
∑BMcomp
with var(ytotal) =∑c
i=1 var(yi) + 2∑∑
i<j cov(yi , yj )2 Nonlinear Seemingly Unrelated Regression (NSUR)
NSUR requires rectangulardata set (i. e. no NA’s)
but some of the studiescontain NA’s
complete caseimputation
Introduction data methods Results Discussion Literatur
general methodological design
wanted: biomass functions for all compartments, and the totalmass
maintain additivity (see Parresol, 2001)1 BMtotal =
∑BMcomp
with var(ytotal) =∑c
i=1 var(yi) + 2∑∑
i<j cov(yi , yj )2 Nonlinear Seemingly Unrelated Regression (NSUR)
NSUR requires rectangulardata set (i. e. no NA’s)
but some of the studiescontain NA’s
complete caseimputation
Introduction data methods Results Discussion Literatur
SUR: seemingly-unrelated regression I
linear SUR-Regression, see Zellner (1962):
ysur = Xβ + ε with ε ∼ N (0,Σc ⊗ IN ) (1)
with the stacked column vectors ysur = [y ′1y′2 · · ·y ′m ]′,
β = [β′1β′2 · · ·β′m ]′ and error term ε = [ε′1ε
′2 · · · ε′m ]′.
The design matrix X now is blockdiagonal:
X =
X1 0 · · · 00 X2 · · · 0...
......
0 0 · · · XM
where N=number of Observation, M=number of equations
Introduction data methods Results Discussion Literatur
SUR: seemingly-unrelated regression II
the variance-covariance matrix of the errors is:
Σ = Σc ⊗ IN =
σ11 σ12 · · · σ1Mσ21 σ22 · · · σ2M
......
...σM1 σM2 · · · σMM
⊗ IN (2)
Zellner (1962, S. 350) and Rossi et al. (2005, S. 66):
”In a formal sense, we regard (1) as a single-equation regression
model [. . . ]“.”Given Σ, we can transform (1) into a system with
uncorrelated errors“ [. . . ]”by a matrix H , so that
E (H εε′H ′) = HΣH ′ = I .“”This means that, if we premultiply
both sides of (1) by [H ], the transformed system has uncorrelatederrors“.
Introduction data methods Results Discussion Literatur
SUR: seemingly-unrelated regression III
the resulting model fulfills the LS-assumptions and theLS-estimator is (Zellner, 1962):
βsur = (X ′H ′HX )−1X ′H ′Hy = (X ′Σ−1X )−1X ′Σ−1y (3)
where the covariance matrix of the estimator is:
Var(β) = (X ′Σ−1X )−1 (4)
whereΣ−1 = Σ−1
c ⊗ I (5)
BUT: Σ is not known and must be estimated from the data.
Introduction data methods Results Discussion Literatur
weighted nonlinear seemingly unrelated regression I
in the non-linear case, the model is (see Parresol, 2001):
ynsur = f (X , β) + ε mit ε ∼ N (0,Σ⊗ IN ) (6)
with the stacked column vectors ynsur = [y ′1y′2 · · ·y ′m ]′,
f = [f ′1f′2 · · · f ′m ]′ and error term ε = [ε′1ε
′2 · · · ε′m ]′.
if a weighted regression is needed (as in this case):
Ψ(θ) =
Ψ1(θ1) 0 · · · 0
0 Ψ2(θ2) · · · 0...
......
0 0 · · · ΨM (θM )
(7)
Introduction data methods Results Discussion Literatur
weighted nonlinear seemingly unrelated regression II
Considering a univariate gnls, the estimated parameter vectorminimises the (weighted) sum of squares of the residuals
S (β) = ε′Ψ−1ε = [Y − f (X ,β)]′Ψ−1[Y − f (X ,β)] (8)
with weights-matix Ψ.In the NSUR-model, this term is updated to:
S (β) = ε′∆′(Σ−1 ⊗ I )∆ε
= [Y − f (X ,β)]′∆′(Σ−1 ⊗ I )∆[Y − f (X ,β)](9)
where ∆ =√
Ψ−1 and Σ (still) not known.Parresol (2001) estimates Σ from the residuals of an univariategnls-fit (i , j ):
σij =1√
N −Ki
√N −Kj
εi∆′i∆j εj (10)
Introduction data methods Results Discussion Literatur
weighted nonlinear seemingly unrelated regression III
to estimate β, we can use the Gauss-Newton-Minimisation method(Parresol, 2001):
βn+1 = βn + ln ·[F(βn)′∆′(Σ−1 ⊗ I )∆F(βn)]−1
F(βn)′∆′(Σ−1 ⊗ I )∆[y − f (X , βn)](11)
where F(βn) is the jacobian.the covariance-matrix of the parameter estimates is:
Σb = [F(βn)′∆′(Σ−1 ⊗ I )∆F(βn)]−1 (12)
and the NSUR-system-variance is:
σ2NSUR =S (b)
MN −K(13)
Introduction data methods Results Discussion Literatur
wait, what about study-effects?
gnls cannot model random effects
and hence, the NSUR-code can’t as well
but we are not interested in these anyway. . .
ycorr = yobs − ( f (Aβ + Bb, ν)︸ ︷︷ ︸fixed+random effects
− f (Aβ, ν)︸ ︷︷ ︸fixed effects
)
5 10 15 20
510
1520
raw data
x
y
5 10 15 20
510
1520
nlme−fit
x
y
5 10 15 20
510
1520
gnls−fit
x
y
Introduction data methods Results Discussion Literatur
wait, what about study-effects?
gnls cannot model random effects
and hence, the NSUR-code can’t as well
but we are not interested in these anyway. . .
ycorr = yobs − ( f (Aβ + Bb, ν)︸ ︷︷ ︸fixed+random effects
− f (Aβ, ν)︸ ︷︷ ︸fixed effects
)
5 10 15 20
510
1520
raw data
x
y
5 10 15 20
510
1520
nlme−fit
x
y
5 10 15 20
510
1520
gnls−fit
x
y
Introduction data methods Results Discussion Literatur
wait, what about study-effects?
gnls cannot model random effects
and hence, the NSUR-code can’t as well
but we are not interested in these anyway. . .
ycorr = yobs − ( f (Aβ + Bb, ν)︸ ︷︷ ︸fixed+random effects
− f (Aβ, ν)︸ ︷︷ ︸fixed effects
)
5 10 15 20
510
1520
raw data
x
y
5 10 15 20
510
1520
nlme−fit
x
y
5 10 15 20
510
1520
gnls−fit
x
y
Introduction data methods Results Discussion Literatur
effect on biomass data
20 40 60 80
050
010
0015
0020
0025
00
Comparison of gnls−, nlme− and gnls/nlme−full−model for coarse wood mass
BHD
coar
se w
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obsgnlsnlmegnls−nlme
Introduction data methods Results Discussion Literatur
NSUR step-by-step
NSUR procedure
1 fit nlme-model with Study as grouping variable
2 remove difference between fixed-effects and random-effects
3 fit an univariate, unweighted nls-model
4 deduce weights for the summary compartment
5 fit weighted gnls-model
6 estimate Σ from weighted residuals
7 fit NSUR-model using Σ and weights from univariate fits
Introduction data methods Results Discussion Literatur
results for spruce
how the model looks like
stump a11 · dbha12 · stumpha13 · agea14 · hsla15stumpB a21 + a22 · dbha23 · stumpha24 · heighta25
cw a31 · dbha32 · heighta33 ·D03a34 · agea35cwB a41 · dbha42 · heighta43 ·D03a44 · agea45 · hsla46
sw a51 + a52 · dbha53 · heighta54 ·D03a55 · cla56needles a61 + a62 · dbha63 · heighta64 ·D03a65 · agea66 · hsla67 · cla68
totalBM stump + stumpB + cw + cwB + sw + needles
eqn stump stumpB cw cwB sw needles totalBM
r2 0.919 0.874 0.976 0.948 0.866 0.802 0.977
Introduction data methods Results Discussion Literatur
results for spruce
how the model looks like
stump a11 · dbha12 · stumpha13 · agea14 · hsla15stumpB a21 + a22 · dbha23 · stumpha24 · heighta25
cw a31 · dbha32 · heighta33 ·D03a34 · agea35cwB a41 · dbha42 · heighta43 ·D03a44 · agea45 · hsla46
sw a51 + a52 · dbha53 · heighta54 ·D03a55 · cla56needles a61 + a62 · dbha63 · heighta64 ·D03a65 · agea66 · hsla67 · cla68
totalBM stump + stumpB + cw + cwB + sw + needles
eqn stump stumpB cw cwB sw needles totalBM
r2 0.919 0.874 0.976 0.948 0.866 0.802 0.977
Introduction data methods Results Discussion Literatur
observed and fitted
20 40 60 80
050
100
200
Stock
BHD
Sto
ck
20 40 60 80
02
46
810
Stockrinde
BHD
Sto
ckrin
de
20 40 60 80
050
015
0025
00
DerbGesamt
BHD
Der
bGes
amt
20 40 60 80
050
100
150
200
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BHD
Der
bGes
amtR
inde
20 40 60 80
010
030
050
0NichtDerbmitRinde
BHD
Nic
htD
erbm
itRin
de
20 40 60 80
050
100
150
Nadeln
BHD
Nad
eln
20 40 60 80
010
0020
0030
00
oiBT
BHD
oiB
T
Introduction data methods Results Discussion Literatur
effect of random-effects-correction
20 40 60 80
050
150
250
stump
dbh
stum
p
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010
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dbh
coar
se w
ood
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050
150
250
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dbh
coar
se w
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bark
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020
040
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0
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dbh
smal
l woo
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dbh
tota
l woo
d
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Introduction data methods Results Discussion Literatur
confidence intervals
20 40 60 80
050
150
250
stump
dbh
| CI: mfull−NSUR
20 40 60 80
05
1015
20
stump bark
dbhst
ump
bark
| CI: mfull−NSUR
20 40 60 80
010
0030
00
coarse wood
dbh
coar
se w
ood
| CI: mfull−NSUR
20 40 60 80
050
150
250
coarse wood bark
dbh
| CI: mfull−NSUR
20 40 60 80
020
040
060
0
small wood
dbh
smal
l woo
d
| CI: mfull−NSUR
20 40 60 80
050
100
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needles
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need
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| CI: mfull−NSUR
20 40 60 80
010
0030
0050
00
total wood
| CI: mfull−NSUR
Introduction data methods Results Discussion Literatur
comparison to NFI3
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comparison to Wirth et al. 2004
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Introduction data methods Results Discussion Literatur
NSUR-Method
additivity maintained
study effect included
seem to be comparable to NFI-results
comparability to Wirth et al. (2004) limited
prediction intervals not yet set up
confidence & prediction intervals for univariate functions
differences to Wirth still to be evaluated
Introduction data methods Results Discussion Literatur
NSUR-Method
additivity maintained
study effect included
seem to be comparable to NFI-results
comparability to Wirth et al. (2004) limited
prediction intervals not yet set up
confidence & prediction intervals for univariate functions
differences to Wirth still to be evaluated
Introduction data methods Results Discussion Literatur
THANK YOU!
? mixed effects correction OK
? NSUR-method sensible
? any other suggestions
Introduction data methods Results Discussion Literatur
THANK YOU!
? mixed effects correction OK
? NSUR-method sensible
? any other suggestions
Introduction data methods Results Discussion Literatur
Joosten, R., Schumacher, J., Wirth, C., Schulte, A., 2004. Evaluating tree carbonpredictions for beech (Fagus sylvatica L.) in western Germany. Forest Ecology andManagement 189 (1-3), 87–96.
Kandler, G., Bosch, B., 2013. Methodenentwicklung fur die 3. Bundeswaldinventur:Modul 3 Uberprufung und Neukonzeption einer Biomassefunktion -Abschlussbericht. Tech. rep., FVA-BW.
Little, R. J. A., Rubin, D. B., 1987. Statistical analysis with missing data. Wiley seriesin probability and mathematical statistics : Applied probability and statistics. Wiley,New York [u.a.].
Parresol, B. R., 2001. Additivity of nonlinear biomass equations. Canadian Journal ofForest Research-Revue Canadienne De Recherche Forestiere 31 (5), 865–878.
Rossi, P., Allenby, G., McCulloch, R., 2005. Bayesian Statistics and Marketing. Wiley.
van Buuren, S., Groothuis-Oudshoorn, K., 2011. mice: Multivariate Imputation byChained Equations in R. Journal of Statistical Software 45 (3), 1–67.
Wirth, C., Schumacher, J., Schulze, E.-D., 2004. Generic biomass functions forNorway spruce in Central Europe - a meta-analysis approach toward prediction anduncertainty estimation. Tree Physiology 24 (2), 121–139.
Zellner, A., 1962. An Efficient Method of Estimating Seemingly Unrelated Regressionsand Tests for Aggregation Bias. Journal of the American Statistical Association57 (298), 348–368.