The Further Development of Stem Taper and Volume Models Defined by Stochastic Differential Equations

Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 229)

Abstract

Stem taper process measured repeatedly among a series of individual trees is standardly analyzed by fixed and mixed regression models. This stem taper process can be adequately modeled by parametric stochastic differential equations (SDEs). We focus on the segmented stem taper model defined by the Gompertz, geometric Brownian motion and Ornstein-Uhlenbeck stochastic processes. This class of models enables the representation of randomness in the taper dynamics. The parameter estimators are evaluated by maximum likelihood procedure. The SDEs stem taper models were fitted to a data set of Scots pine trees collected across the entire Lithuanian territory. Comparison of the predicted stem taper and stem volume with those obtained using regression based models showed a predictive power to the SDEs models.

Keywords

Diameter Geometric Brownian motion Gompertz process Ornstein-Uhlenbeck process Taper Transition probability density Stochastic differential equation Volume 

References

  1. 1.
    Jana D, Chakraborty S, Bairagi N (2012) Stability, nonlinear oscillations and bifurcation in a delay-induced predator-prey system with harvesting. Eng Lett 20(3):238–246Google Scholar
  2. 2.
    Boughamoura W, Trabelsi F (2011) Variance reduction with control variate for pricing asian options in a geometric L’evy model. IAENG Int J Appl Math 41(4):320–329MathSciNetGoogle Scholar
  3. 3.
    Bartkevičius E, Petrauskas E, Rupšys P, Russetti G (2012) Evaluation of stochastic differential equations approach for predicting individual tree taper and volume, Lecture notes in engineering and computer science. In: Proceedings of the world congress on engineering, WCE 2012, 4–6 July 2012, vol 1. London, pp 611–615Google Scholar
  4. 4.
    Suzuki T (1971) Forest transition as a stochastic process. Mit Forstl Bundesversuchsanstalt Wein 91:69–86Google Scholar
  5. 5.
    Tanaka K (1986) A stochastic model of diameter growth in an even-aged pure forest stand. J Jpn For Soc 68:226–236Google Scholar
  6. 6.
    Rupšys P, Petrauskas E, Mažeika J, Deltuvas R (2007) The gompertz type stochastic growth law and a tree diameter distribution. Baltic For 13:197–206Google Scholar
  7. 7.
    Rupšys P, Petrauskas E (2009) Forest harvesting problem in the light of the information measures. Trends Appl Sci Res 4:25–36CrossRefGoogle Scholar
  8. 8.
    Rupšys P, Petrauskas E (2010) The bivariate Gompertz diffusion model for tree diameter and height distribution. For Sci 56:271–280Google Scholar
  9. 9.
    Rupšys P, Petrauskas E (2010) Quantifying tree diameter distributions with one-dimensional diffusion processes. J Biol Syst 18:205–221CrossRefGoogle Scholar
  10. 10.
    Rupšys P, Bartkevičius E, Petrauskas E (2011) A univariate stochastic gompertz model for tree diameter modeling. Trends Appl Sci Res 6:134–153CrossRefGoogle Scholar
  11. 11.
    Rupšys P, Petrauskas E (2012) Analysis of height curves by stochastic differential equations. Int J Biomath 5(5):1250045Google Scholar
  12. 12.
    Rupšys P, Petrauskas E, Bartkevičius E, Memgaudas R (2011) Re-examination of the taper models by stochastic differential equations. Recent advances in signal processing, computational geometry and systems theory pp 43–47Google Scholar
  13. 13.
    Kozak A, Munro DD, Smith JG (1969) Taper functions and their application in forest inventory. For Chron 45:278–283Google Scholar
  14. 14.
    Max TA, Burkhart HE (1976) Segmented polynomial regression applied to taper equations. For Sci 22:283–289Google Scholar
  15. 15.
    Kozak A (2004) My last words on taper equations. For Chron 80:507–515Google Scholar
  16. 16.
    Trincado J, Burkhart HE (2006) A generalized approach for modeling and localizing profile curves. For Sci 52:670–682Google Scholar
  17. 17.
    Yang Y, Huang S, Meng SX (2009) Development of a tree-specific stem profile model for White spruce: a nonlinear mixed model approach with a generalized covariance structure. Forestry 82:541–555CrossRefGoogle Scholar
  18. 18.
    Rupšys P, Petrauskas E (2010) Development of q-exponential models for tree height, volume and stem profile. Int J Phys Sci 5:2369–2378Google Scholar
  19. 19.
    Westfall JA, Scott CT (2010) Taper models for commercial tree species in the Northeastern United States. For Sci 56:515–528Google Scholar
  20. 20.
    Petrauskas E, Rupšys P, Memgaudas R (2011) Q-exponential variable form of a stem taper and volume models for Scots pine Pinus Sylvestris) in Lithuania. Baltic For 17(1):118–127Google Scholar
  21. 21.
    Itô K (1936) On stochastic processes. Jpn J Math 18:261–301Google Scholar
  22. 22.
    Oksendal BK (2002) Stochastic differential equations, an introduction with applications. Springer, New York, p 236Google Scholar
  23. 23.
    Uhlenbeck GE, Ornstein LS (1930) On the theory of Brownian motion. Physical Rev 36:823–841MATHCrossRefGoogle Scholar
  24. 24.
    Max TA, Burkhart HE (1976) Segmented polynomial regression applied to taper equations. For Sci 22:283–289Google Scholar
  25. 25.
    Fisher RA (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc A-Math Phys Eng Sci 222:309–368MATHCrossRefGoogle Scholar
  26. 26.
    Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Forest ManagementAleksandras Stulginskis UniversityKaunasLithuania

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