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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 Regression

IBS-DR Biometry Workshop, W¨ urzburg Christian Vonderach

Forest Research Institute Baden-W¨ urttemberg

W¨ urzburg, 07.10.2015

Introduction data methods Results Discussion Literatur

1

Introduction

2

data

3

methods applied Nonlinear Seemingly Unrelated Regression

4

results NSUR fit comparison

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

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collected data

spruce (Picea abies) 6 data compilations (incl. Wirth et al., 2004) homogenisation stump/B coarse wood/B small wood needles ≈ 1200 trees

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

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Overview of collected data

age dbh height 50 100 50 100 150 200 25 50 75 10 20 30 40

number of data sets

10 20 30 40 20 40 60 80

dbh [cm] height [m]

1000 2000 3000 20 40 60 80

dbh [cm] aboveground biomasse [kg]

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general methodological design

wanted: biomass functions for all compartments, and the total mass maintain additivity (see Parresol, 2001)

1 BMtotal = BMcomp

with var(ˆ ytotal) = c

i=1 var(ˆ

yi) + 2

i<j cov(ˆ

yi, ˆ yj)

2 Nonlinear Seemingly Unrelated Regression (NSUR)

Introduction data methods Results Discussion Literatur

general methodological design

wanted: biomass functions for all compartments, and the total mass maintain additivity (see Parresol, 2001)

1 BMtotal = BMcomp

with var(ˆ ytotal) = c

i=1 var(ˆ

yi) + 2

i<j cov(ˆ

yi, ˆ yj)

2 Nonlinear Seemingly Unrelated Regression (NSUR)

NSUR requires rectangular data set (i. e. no NA’s) but some of the studies contain NA’s

complete case imputation

Introduction data methods Results Discussion Literatur

general methodological design

wanted: biomass functions for all compartments, and the total mass maintain additivity (see Parresol, 2001)

1 BMtotal = BMcomp

with var(ˆ ytotal) = c

i=1 var(ˆ

yi) + 2

i<j cov(ˆ

yi, ˆ yj)

2 Nonlinear Seemingly Unrelated Regression (NSUR)

NSUR requires rectangular data set (i. e. no NA’s) but some of the studies contain NA’s

complete case imputation

[image removed]

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SUR: seemingly-unrelated regression I

linear SUR-Regression, see Zellner (1962): ysur = X β + ǫ with ǫ ∼ 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 ǫ = [ǫ′ 1ǫ′ 2 · · · ǫ′ m]′.

The design matrix X now is blockdiagonal: X =      X1 · · · X2 · · · . . . . . . . . . · · · XM      where N=number of Observation, M=number of equations

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SUR: seemingly-unrelated regression II

the variance-covariance matrix of the errors is: Σ = Σc ⊗ IN =      σ11 σ12 · · · σ1M σ21 σ22 · · · σ2M . . . . . . . . . σM 1 σM 2 · · · 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 ǫǫ′H ′) = H ΣH ′ = I . “ ” This means that, if we premultiply both sides of (1) by [H ], the transformed system has uncorrelated errors “.

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SUR: seemingly-unrelated regression III

the resulting model fulfills the LS-assumptions and the LS-estimator is (Zellner, 1962): ˆ 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) where Σ−1 = Σ−1

c

⊗ I (5) BUT: Σ is not known and must be estimated from the data.

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weighted nonlinear seemingly unrelated regression I

in the non-linear case, the model is (see Parresol, 2001): ynsur = f (X , β) + ǫ mit ǫ ∼ 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 ǫ = [ǫ′ 1ǫ′ 2 · · · ǫ′ m]′.

if a weighted regression is needed (as in this case): Ψ(θ) =      Ψ1(θ1) · · · Ψ2(θ2) · · · . . . . . . . . . · · · ΨM (θM )      (7)

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weighted nonlinear seemingly unrelated regression II

Considering a univariate gnls, the estimated parameter vector minimises the (weighted) sum of squares of the residuals S(β) = ǫ′Ψ−1ǫ = [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) = ǫ′∆′(Σ−1 ⊗ I )∆ǫ = [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 univariate gnls-fit (i, j): σij = 1 √N − Ki N − Kj ǫi ˆ ∆′

i ˆ

∆j ǫj (10)

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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)′ ˆ ∆′(ˆ Σ−1 ⊗ I ) ˆ ∆F(βn)]−1 F(βn)′ ˆ ∆′(ˆ Σ−1 ⊗ I ) ˆ ∆[y − f (X , βn)] (11) where F(βn) is the jacobian. the covariance-matrix of the parameter estimates is: ˆ Σb = [F(βn)′ ˆ ∆′(ˆ Σ−1 ⊗ I ) ˆ ∆F(βn)]−1 (12) and the NSUR-system-variance is: ˆ σ2

NSUR =

S(b) MN − K (13)

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wait, what about study-effects?

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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. . .

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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 5 10 15 20

raw data

x y 5 10 15 20 5 10 15 20

nlme−fit

x y 5 10 15 20 5 10 15 20

gnls−fit

x y

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effect on biomass data

20 40 60 80 500 1000 1500 2000 2500

Comparison of gnls−, nlme− and gnls/nlme−full−model for coarse wood mass

BHD coarse wood

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• bs

gnls nlme gnls−nlme

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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

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results for spruce

how the model looks like

stump a11 · dbha12 · stumpha13 · agea14 · hsla15 stumpB a21 + a22 · dbha23 · stumpha24 · heighta25 cw a31 · dbha32 · heighta33 · D03a34 · agea35 cwB a41 · dbha42 · heighta43 · D03a44 · agea45 · hsla46 sw a51 + a52 · dbha53 · heighta54 · D03a55 · cla56 needles a61 + a62 · dbha63 · heighta64 · D03a65 · agea66 · hsla67 · cla68 totalBM stump + stumpB + cw + cwB + sw + needles

Introduction data methods Results Discussion Literatur

results for spruce

how the model looks like

stump a11 · dbha12 · stumpha13 · agea14 · hsla15 stumpB a21 + a22 · dbha23 · stumpha24 · heighta25 cw a31 · dbha32 · heighta33 · D03a34 · agea35 cwB a41 · dbha42 · heighta43 · D03a44 · agea45 · hsla46 sw a51 + a52 · dbha53 · heighta54 · D03a55 · cla56 needles 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

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• bserved and fitted

20 40 60 80 50 100 200

Stock

BHD Stock 20 40 60 80 2 4 6 8 10

Stockrinde

BHD Stockrinde 20 40 60 80 500 1500 2500

DerbGesamt

BHD DerbGesamt 20 40 60 80 50 100 150 200

DerbGesamtRinde

BHD DerbGesamtRinde 20 40 60 80 100 300 500

NichtDerbmitRinde

BHD NichtDerbmitRinde 20 40 60 80 50 100 150

Nadeln

BHD Nadeln 20 40 60 80 1000 2000 3000

• iBT

BHD

• iBT

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effect of random-effects-correction

20 40 60 80 50 150 250

stump

dbh stump

• mfull

full

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40 60 80 5 10 15 20

stump bark

dbh stump bark

• mfull

full

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40 60 80 1000 3000

coarse wood

dbh coarse wood

• mfull

full

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40 60 80 50 150 250

coarse wood bark

dbh coarse wood bark

• mfull

full

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40 60 80 200 400 600

small wood

dbh small wood

• mfull

full

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40 60 80 50 100 150

needles

dbh needles

• mfull

full

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40 60 80 2000 4000

total wood

dbh total wood

• mfull

full

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confidence intervals

20 40 60 80 50 150 250

stump

| CI: mfull−NSUR 20 40 60 80 5 10 15 20

stump bark

| CI: mfull−NSUR 20 40 60 80 1000 3000

coarse wood

| CI: mfull−NSUR 20 40 60 80 50 150 250

coarse wood bark

| CI: mfull−NSUR 20 40 60 80 200 400 600

small wood

| CI: mfull−NSUR 20 40 60 80 50 100 150

needles

| CI: mfull−NSUR 20 40 60 80 1000 3000 5000

total wood

| CI: mfull−NSUR

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comparison to NFI3

20 40 60 80 1000 2000 3000 4000

total wood

dbh total wood

• mfull

NFI3

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comparison to Wirth et al. 2004

20 40 60 80 500 1000 1500 2000 2500 3000

coarse wood + B

dbh coarse wood + B

• mfull

Wirth et al. 2004

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40 60 80 200 400 600 800

small wood

dbh small wood

• mfull

Wirth et al. 2004

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40 60 80 50 100 150 200 250

needles

dbh needles

• mfull

Wirth et al. 2004

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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

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

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THANK YOU!

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 carbon predictions for beech (Fagus sylvatica L.) in western Germany. Forest Ecology and Management 189 (1-3), 87–96. K¨ andler, G., B¨

• sch, B., 2013. Methodenentwicklung f¨

ur die 3. Bundeswaldinventur: Modul 3 ¨ Uberpr¨ ufung und Neukonzeption einer Biomassefunktion -

• Abschlussbericht. Tech. rep., FVA-BW.

Little, R. J. A., Rubin, D. B., 1987. Statistical analysis with missing data. Wiley series in probability and mathematical statistics : Applied probability and statistics. Wiley, New York [u.a.]. Parresol, B. R., 2001. Additivity of nonlinear biomass equations. Canadian Journal of Forest 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 by Chained Equations in R. Journal of Statistical Software 45 (3), 1–67. Wirth, C., Schumacher, J., Schulze, E.-D., 2004. Generic biomass functions for Norway spruce in Central Europe - a meta-analysis approach toward prediction and uncertainty estimation. Tree Physiology 24 (2), 121–139. Zellner, A., 1962. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of the American Statistical Association 57 (298), 348–368.