Modelling of tree height growth Comparison of different es 09.11.2012 1/20
Model demands h = f (t) + ε One inflection point Asymptote parallel to the abscissa Function through the origin (0, 0) Good numerical properties (1) 2/20
set 23502 measurements 6339 trees 181 plots Norway spruce Lower saxony 3/20
s 4/20 Basic model (h = f(t), Sloboda 1971) with h k =f (ψ, t k ) + ɛ k h k =ψ (1) ( ψ (4) ψ (1) ) exp (ψ (3) ψ (2) (ψ (3) 1)t k (2) 1) ψ (ψ (3) ( ψ (3) 1) 1)t 0 + ɛ k ψ = (ψ (1),..., ψ (4) ) T (3) Plot i Tree j on plot i Measurement k on tree j on plot i (2) (Nothdurft 2007, 2011)
Modelling of the parameters ψ = A β + B b (4) ψ (1) = ι (1) ψ (4) = ι (4) + r (4) i + r (4) ψ (1) ψ (4) free parameter modelled using fixed and random effects of covariates (fixed effects): h k = f (ψ, t k, Temp i ) + ɛ k ψ (4) = β Temp Temp i + ι (4) + r (4) i + r (4) (5) (6) 5/20
Basic model (h = f(t), Korf function): h = A exp( Bt C ) + ε (7) Linearised (if C fixed heuristically, Lappi 1997): ln(h) = ln(a) Bt C + ε (8) As mixed model: ln(h k ) = ln(a ) B t C k + ε k (9) Modelling of covariates (not tested so far, D fixed heuristically): ln(h k ) = ln(a ) B t C k + β Temp Temp D i + ε k (10) 6/20
s Polynomial spline h = Zβ + ε (11) with 1 t 1... t1 L Z =. 1 t n... tn L { (t 1 k 1 ) L, k 1 t 1 0, else { (t n k 1 ) L, k 1 t n 0, else...... { (t m k 1 ) L, k m t 1 0, else {. (t n k m) L, k m t n 0, else (12) Estimate β (least squares) ˆβ = (Z T Z) 1 Z T y (13) 7/20
GAM Model demands One inflection point (using less nodes) Asymptote parallel to abscissa (2 artificial heights at high ages based on Korf asymptote) Function through the origin (removing first column) t 1... t1 L Z =. t n... tn L { (t 1 k 1 ) L, k 1 t 1 0, else { (t n k 1 ) L, k 1 t n 0, else...... { (t m k 1 ) L, k m t 1 0, else {. (t n k m) L, k m t n 0, else (14) Good numeric qualities (least squares) 8/20
All curves (nonlinear model) Red: all random effects set to zero Blue: random effects on plot level Green: random effects on plot and tree level 9/20
Curves for one plot (nonlinear model) 10/20
Curve for one tree (nonlinear model) 11/20
Residuals (nonlinear model) Prob 0.0 0.2 0.4 0.6 0.8 2 1 0 1 2 Residuals 12/20
All curves (linearisation ) C = 0.757 (heuristically) 13/20
Curves for one plot (linearisation ) 14/20
Curve for one tree (linearisation ) 15/20
Residuals (linearisation, height scale) Prob 0.0 0.2 0.4 0.6 0.8 2 1 0 1 2 Residuals 16/20
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Thank you for your attention! 18/20
Modelling of covariates Temp influences A: ln(h k ) = ι + r i + r + β Temp Temp D i Temp influences B: B t C k + ε k (15) ln(h k ) = ln(a k ) (ι + r i + r + β Temp Temp i )t C k + ε k (16) ln(h k ) = ln(a k ) (ι+r i +r )t C k β TempTemp i t C k +ε k (17) 19/20
Lappi, J. (1997). A longitudinal analysis of height/diameter curves. Forest Science, 43, 555 570..II, 300. Nothdurft, A. (2007). Ein nichtlineares, hierarchisches und gemischtes Modell für das Baum-Höhenwachstum der Fichte (Picea abies (L.) Karst.) in Baden-Württemberg. PhD thesis, Forstwissenschaftlicher Fachbereich der Georg-August-Universität Göttingen., S. (2011). Ein raumbezogenes, klimasensitives, nichtlineares, hierarchisches gemischtes Modell für das Baum-Höhenwachstum der Fichte (Picea abies (L.) Kast.) in Reinbeständen im niedersächsischen Staatswald. Master s thesis, Georg-August-Universität Göttingen. Sloboda, B. (1971). Zur Darstellung von Wachstumsprozessen mit Hilfe von Differentialgleichungen erster Ordnung. Mitteilungen der Forstlichen Versuchs- und Forschungsanstalt Baden-Württemberg, 1971, 32. II, 900. 20/20