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 ŠUMARSKI LIST 9-10/2019 str. 34     <-- 34 -->        PDF Single and Double Entry Volume Equations – Jednoulazne i dvoulazne volumne jednadžbe To develop single and double entry volume equations predicting the relationships between individual tree volume and dbh and total height, some nonlinear regression models were used in this study. These single and double entry volume equations were developed by using the nonlinear regression analysis, including the total tree volume as the dependent variable and dbh for single entry equation and dbh and total height for the double entry equation as the independent variables. This linear regression analysis was performed by using PROC MODEL procedure of the SAS/ETS V9 software. In regression analyses, these nonlinear models that provided better predictive volume values than other equations were used to obtain the individual tree volume predictions: The single-entry volume equation: V = b0 . dbhb1          (4) The double-entry volume equation: V = b0 . (dbh2 . h) (5) Taper-based volume predictions – Volumna predviđanja temeljena na konusu (taper-based) From various taper equations, Fang et al. (2000)’s equation presented better predictive results for modeling individual tree taper and compatible volume in many studies (Diéguez Aranda et al. 2006; Corral-Rivas et al. 2007; Crecente-Campo et al. 2009; Pompa et al. 2009; Li and Weiskittel 2010; Tang et al. 2016). In this study, the taper equation of Fang et al. (2000) was fitted to obtain the individual taper-based tree volume predictions and compared with the predictions obtained from the volume equations and ANN models. Fang et al. (2000) presented the segmented taper equation that assumes three sections with a variable-form factor (Corral-Rivas et al. 2007). Fang et al. (2000)’s stem taper equation is as follows:                 d = c1[H(k–b1)/b1(1 – g)(k–b)/ba1l1+l2a2l2]0.5 (6) Where k =  g = h/H  p1 = h1/H p2 = h2/H  b = b11–(l1+l2)b2l1b3l2 a1 = (1–p1)(b2–b1)k/b1b2  a2 = (1–p2)(b3–b2)k/b2b3  r0 = ((1 – 0.3)/H)k/b1 r1 = (1 – p1)k/b1  r2 = (1 – p2)k/b2   Where, d: stem diameter over bark (cm) at a height h (m), D: diameter at breast height over bark (cm), h: measuring the height (m), H: total height (m), p1 and p2 are relative heights from ground level in the two inﬂection points, a1, a2, a3, b1, b2, b3, c1 are the equation parameters to be predicted by nonlinear regression. Fang et al. (2000) integrated its segmented taper equation to obtain compatible total volume equation. This compatible volume equation is as follows:                 V = a0dbha1ha2      (7) The prediction of the parameters of this segmented taper equation of Fang et al. (2000) was obtained by using the PROC MODEL. Artificial Neural Network Models – Modeli umjetne neuronske mreže Within the scope of this study, the predictions of individual tree volumes were obtained by using Artificial Neural Network (ANN) models. ANN is a mathematical modeling method inspired by biological neural systems, such as the human brain, and the estimations of ANN is created using computer software, developed according to the physiology of the human brain (Gurney 1999; Demuth and Beale 2001). The artificial neural network model comprises layers with structurally connected nerves. Essentially, these layers are classified as the input layer, the hidden layer and the output layer (Kurup and Dudani 2002). These layers cover artificial neurons called process elements. As a result of their structure, composed of multiple non-linear artificial neurons that can be organized as several layers and work coordinately, they are very successful in solving non-linear complex problems and creating estimations (Nasr et al. 2003). For ANN, miscellaneous neural structures, input, and output (target) variables in the system to be estimated are defined; then, using these definitions, ANN analyses the data and presents weight values to provide the best possible estimations with minimum error (Fausett 1994). In ANN literature, this process is called network training. ANN is used to provide several different weight values and using these values the output estimations are obtained through the addition and activation functions. Using the estimation values obtained through ANN, the change of errors calculated are analyzed, based on the observation values of the output variable defined in the first place, and the process is complete when the errors reach a minimum level and the error-related changes have reached a fixed point (Fausett 1994; Demuth and Beale 2001).