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Incorporating shape constraints in generalized additive modelling of the height-diameter relationship for Norway spruce
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Measurements of tree heights and diameters are essential in forest assessment and modelling. Tree heights are used for estimating timber volume, site index and other important variables related to forest growth and yield, succession and carbon budget models. However, the diameter at breast height (dbh) can be more accurately obtained and at lower cost, than total tree height. Hence, generalized height-diameter (h-d) models that predict tree height from dbh, age and other covariates are needed. For a more flexible but biologically plausible estimation of covariate effects we use shape constrained generalized additive models as an extension of existing h-d model approaches. We use causal site parameters such as index of aridity to enhance the generality and causality of the models and to enable predictions under projected changeable climatic conditions.
We develop unconstrained generalized additive models (GAM) and shape constrained generalized additive models (SCAM) for investigating the possible effects of tree-specific parameters such as tree age, relative diameter at breast height, and site-specific parameters such as index of aridity and sum of daily mean temperature during vegetation period, on the h-d relationship of forests in Lower Saxony, Germany.
Some of the derived effects, e.g. effects of age, index of aridity and sum of daily mean temperature have significantly non-linear pattern. The need for using SCAM results from the fact that some of the model effects show partially implausible patterns especially at the boundaries of data ranges. The derived model predicts monotonically increasing levels of tree height with increasing age and temperature sum and decreasing aridity and social rank of a tree within a stand. The definition of constraints leads only to marginal or minor decline in the model statistics like AIC. An observed structured spatial trend in tree height is modelled via 2-dimensional surface fitting.
We demonstrate that the SCAM approach allows optimal regression modelling flexibility similar to the standard GAM but with the additional possibility of defining specific constraints for the model effects. The longitudinal character of the model allows for tree height imputation for the current status of forests but also for future tree height prediction.
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Incorporating shape constraints in generalized additive modelling of the height-diameter relationship for Norway spruce
),
Abstract
Measurements of tree heights and diameters are essential in forest assessment and modelling. Tree heights are used for estimating timber volume, site index and other important variables related to forest growth and yield, succession and carbon budget models. However, the diameter at breast height (dbh) can be more accurately obtained and at lower cost, than total tree height. Hence, generalized height-diameter (h-d) models that predict tree height from dbh, age and other covariates are needed. For a more flexible but biologically plausible estimation of covariate effects we use shape constrained generalized additive models as an extension of existing h-d model approaches. We use causal site parameters such as index of aridity to enhance the generality and causality of the models and to enable predictions under projected changeable climatic conditions.
We develop unconstrained generalized additive models (GAM) and shape constrained generalized additive models (SCAM) for investigating the possible effects of tree-specific parameters such as tree age, relative diameter at breast height, and site-specific parameters such as index of aridity and sum of daily mean temperature during vegetation period, on the h-d relationship of forests in Lower Saxony, Germany.
Some of the derived effects, e.g. effects of age, index of aridity and sum of daily mean temperature have significantly non-linear pattern. The need for using SCAM results from the fact that some of the model effects show partially implausible patterns especially at the boundaries of data ranges. The derived model predicts monotonically increasing levels of tree height with increasing age and temperature sum and decreasing aridity and social rank of a tree within a stand. The definition of constraints leads only to marginal or minor decline in the model statistics like AIC. An observed structured spatial trend in tree height is modelled via 2-dimensional surface fitting.
We demonstrate that the SCAM approach allows optimal regression modelling flexibility similar to the standard GAM but with the additional possibility of defining specific constraints for the model effects. The longitudinal character of the model allows for tree height imputation for the current status of forests but also for future tree height prediction.
References(36)
Brezger, A, Lang S (2006) Generalized structured additive regression based on Bayesian P-splines. Comput Stat Data Anal 50(4): 967-991.
Calama, R, Montero G (2004) Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. Can J For Res 34: 150-163.
Castedo-Dorado, F, Diéguez-Aranda U, Barrio Anta M, Sánchez Rodríguez M, von Gadow K (2006) A generalized height-diameter model including random components for radiata pine plantations in northwestern Spain. For Ecol Manag 229(1-3): 202-213.
Curtis, RO (1967) Height-diameter and height-diameter-age equations for second-growth douglas-fir. For Res 13(4): 365-375.
De Martonne, E (1926) Une nouvelle fonction climatologique: l'indice d'aridité. La Météorologie (1942)21: 449-458.
Eerikäinen, K (2003) Predicting the height-diameter pattern of planted Pinus kesiya stands in Zambia and Zimbabwe. For Ecol Manag 175: 355-366.
Eilers, PH, Marx BD (1996) Flexible smoothing with B-splines and penalties. Stat Sci 11: 89-121.
Hastie, T, Tibshirani R (1993) Varying-coefficient models. J R Stat Soc Ser B 55(4): 757-796.
Hofner, B, Müller J, Hothorn T (2011) Monotonicity-constrained species distribution models. Ecology 92(10): 1895-1901.
Hökkä, H (1997) Height-diameter curves with random intercepts and slopes for trees growing on drained peatlands. For Ecol Manag 97: 63-72.
Huang, S, Price D, Titus SJ (2000) Development of ecoregion-based height-diameter models for white spruce in boreal forests. For Ecol Manag 129: 125-141.
Huang, S, Titus SJ, Wiens DP (1992) Comparison of nonlinear height-diameter functions for major Alberta tree species. Can J For Res 22: 1297-1304.
Jayaraman, K, Lappi J (2001) Estimation of height-diameter curves through multilevel models with special reference to even-aged teak stands. For Ecol Manag 142: 155-162.
Kangas, A, Maltamo M (2002) Anticipating the variance of predicted stand volume and timber assortments with respect to stand characteristics and field measurements. Silva Fennica 36(4): 799-811.
Lappi, J (1997) A longitudinal analysis of height/diameter curves. For Sci 43(4): 555-570.
Lin, X, Zhang D (1999) Inference in generalized additive mixed models by using smoothing splines. J R Stat Soc Ser B 61: 381-400.
Mehtätalo, L (2005) Height-diameter models for Scots pine and birch in Finland. Silva Fennica 39(1): 55-66.
Mehtätalo, L (2004) A longitudinal height diameter model for norway spruce in finland. Can J For Res 34(1): 131-140.
Nanos, N, Calama R, Montero G, Gil L (2004) Geostatistical prediction of height/diameter models. For Ecol Manag 195(1-2): 221-235.
Nothdurft, A, Wolf T, Ringeler A, Böhner J, Saborowski J (2012) Spatio-temporal prediction of site index based on forest inventories and climate change scenarios. For Ecol Manag 279: 97-111.
Pya, N, Wood SN (2015) Shape constrained additive models. Stat Comput 25(3): 543-559.
Schmidt, M, Kiviste A, Gadow K (2011) A spatially explicit height-diameter model for Scots pine in Estonia. Eur J For Res 130: 303-315.
Scott, R, Mitchell S (2005) Empirical modelling of windthrow risk in partially harvested stands using tree neighbourhood and stand attributes. For Ecol Manag 218: 193-209.
Sharma, M, Parton J (2007) Height-diameter equations for boreal tree species in Ontario using a mixed-effects modeling approach. For Ecol Manag 249: 187-198.
Silverman, BW (1985) Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J R Stat Soc Ser B 47: 1-52.
Thornthwaite, CW (1931) The climates of North America: according to a new classification. Geogr Rev 21(4): 633-655.
Wahba, G (1983) Bayesian confidence intervals for the cross validated smoothing spline. J R Stat Soc Ser B 45: 133-150.
Wood, SN (2000) Modelling and smoothing parameter estimation with multiple quadratic penalties. J R Stat Soc Ser B 62: 413-428.
Wood, SN (2006b) On confidence intervals for generalized additive models based on penalized regression splines. Aust N Z J Stat 48(4): 445-464.
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Acknowledgements
The forest data were provided by the Lower Saxony forest planning agency. NP has been partly funded by the EPSRC grant EP/K005251/1.
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