how to quantify nutrient export: additive biomass ...€¦ · aboveground biomasse [kg]...

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


1 Introduction

2 data

3 methods appliedNonlinear Seemingly Unrelated Regression

4 resultsNSUR fitcomparison

5 Discussion


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


collected data

spruce (Piceaabies)

6 datacompilations(incl. Wirthet al., 2004)

homogenisation

stump/Bcoarse wood/Bsmall woodneedles

≈ 1200 trees

(only referenced shown; +CH|DK|B)


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]


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


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

croever

image removed


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


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“.


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.


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)


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)


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)


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

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y

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nlme−fit

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510

1520

gnls−fit

x

y


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

ood

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obsgnlsnlmegnls−nlme


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


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


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

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BHD

Der

bGes

amt

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050

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bGes

amtR

inde

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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


effect of random-effects-correction

20 40 60 80

050

150

250

stump

dbh

stum

p

●

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mfullfull

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20

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dbh

stum

p ba

rk

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010

0030

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dbh

coar

se w

ood

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050

150

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dbh

coar

se w

ood

bark

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l woo

d●

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confidence intervals

20 40 60 80

050

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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

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040

060

0

small wood

dbh

smal

l woo

d

| CI: mfull−NSUR

20 40 60 80

050

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150

needles

dbh

need

les

| CI: mfull−NSUR

20 40 60 80

010

0030

0050

00

total wood

| CI: mfull−NSUR


comparison to NFI3

20 40 60 80

010

0020

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l woo

d

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comparison to Wirth et al. 2004

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coarse wood + B

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se w

ood

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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


THANK YOU!

? mixed effects correction OK

? NSUR-method sensible

? any other suggestions


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.

how to quantify nutrient export: additive biomass ...€¦ · aboveground biomasse [kg]...

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