Developing a Model for Curve-Fitting a Tree Stem’s Cross-Sectional Shape and Sapwood–Heartwood Transition in a Polar Diagram System Using Nonlinear Regression

Abstract

A function from the domain (x-set) to the codomain (y-set) connects each x element to precisely one y element. Since each x-point originating from the domain corresponds to two y-points on the graph of a closed curve (i.e., circle, ellipse, superellipse, or ovoid) in a rectangular (Cartesian) diagram, it does not fulfil the function’s requirements. This non-function phenomenon obstructs the nonlinear regression application for fitting observed data resembling a closed curve; thus, it requires transforming the rectangular coordinate system into a polar coordinate system. This study discusses nonlinear regression to fit the circumference of a tree stem’s cross-section and its sapwood–heartwood transition by transforming rectangular coordinates (x, y) of the observed data points’ positions into polar coordinates (r, θ). Following a polar coordinate model, circular curve fitting fits a log’s cross-sectional shape and sapwood–heartwood transition. Ellipse models result in better goodness of fit than circular ones, while the rotated ellipse is the best-fit one. Deviation from the circular shape indicates environmental effects on vascular cambium differentiation. Foresters have good choices: (1) continuing using the circular model as the simplest one or (2) changing to the rotated ellipse model because it gives the best fit to estimate a tree stem’s cross-sectional shape; therefore, it is more reliable to determine basal area, tree volume, and tree trunk biomass. Computer modelling transforms the best-fit model’s formulas of the rotated ellipse using Python scripts provided by Wolfram engine libraries.

Annual Tree-Ring Curve-Fitting for Graphing the Growth Curve and Determining the Increment and Cutting Cycle Period of Sungkai (Peronema canescens)

“…Use of Webplot Digitizer resulted in a comma-separated values (*.csv) file that contained the coordinate of every point in a rectangular (Cartesian) coordinate system (x,y); therefore, transformation into a polar coordinate system (r,θ) [17] was necessary to apply non-linear regression following the Equation (1) model. For simplicity, the center of the first ring translated as moving coincide with the pith position; therefore, the center of the first annual tree ring is the origin (0,0).…”

“…The goodness of fit of each regression analysis was measured using its coefficient of determination (R 2 ) and mean square error (MSE). The R 2 and MSE are familiar in simple linear [34][35][36][37][38][39], multiple linear [40][41][42][43], and non-linear [17,[44][45][46][47][48][49] regression to determine its goodness of fit. Higher R 2 and smaller MSE indicate the better-fit model.…”

“…If influenced by genetics only, the cambium differentiation rate of a healthy tree is similar along the stem circumference and generates the same width of the xylem layer; therefore, the stem's cross-section forms a circle [17]. Figure 1 shows that the cross-section shape is more like an ellipse than a circle, and the distance between two consecutive rings along the circumferential position is not constant.…”

“…Figure 1 shows the data point plots of each tree ring with its actual photograph background. The (x,y) rectangular (Cartesian) transformation into the (r,θ) polar coordinate system enabled the curve-fitting procedure using non-linear regression based on Equation ( 5) model [17]. The non-linear regression procedure resulted in wellfit ellipse curves with high coefficient determination (0.7842 < R 2 < 0.9971).…”

“…The gradation of earlywood and latewood generates the growth rings, and their numbers counted from the pith may represent a tree's age. Denih et al [17] proposed the model in Equation (1) to curve-fit raw data resembling an elliptical shape located in an arbitrary position in a polar coordinate system. Since it has been proven reliable for curve-fitting a tree stem's cross-sectional shape, it can also potentially estimate the annual tree ring shape and location.…”