A comparison of different methods for fitting the self-thinning equation

ObjectiveFor evaluating the self-thinning theory, selecting the optimum fitting method that can truly describe self-thinning dynamics is a hot topic and a difficult task.

MethodIn this study, we compared different approaches for fitting self-thinning equations, including three traditional methods, i.e., hand-fitting method, interval method and relative density (RD) method; three regression methods, i.e., ordinary least squares regression (OLS) method, reduced major axis regression (RMA) method and quantile regression (QR) method; and three frontier model methods, i.e., corrected OLS (COLS) method, deterministic frontier function (DFF) method and stochastic frontier function (SFF) method. The data from 553 sample plots of even-aged Chinese fir (Cunninghamia lanceolata) plantations in Fujian Province, eastern China was employed.

ResultThe results indicated that hand-fitting method was easy but subjective. The coefficients estimated by interval method can be influenced significantly by interval length, and the estimated slope tended to be flatter than the real slope. The RD method can avoid influence of independent-density mortality, but the result would be affected by the predetermined theoretical constant for the slope. The maximum size-density lines fitted by the OLS, RMA and COLS methods tended to inaccurately match the actual boundaries of data points, and differed from the stand self-thinning upper boundary line. The maximum size-density line fitted by QR method can be close to the stand self-thinning upper boundary line when the quantile value approached 100%. The maximum size-density line fitted by linear programming approach was more suitable than the line fitted by quadratic programming approach. However, statistical inference was very difficult with the DFF and QR methods. SFF method was relatively objective, however, the fitted maximum size-density line can truly describe self-thinning process only when variance of stochastic error terms was small enough and close to zero.

ConclusionFinally, the optimum self-thinning equation for Chinese fir plantations in Fujian Provence is determined as ln(QMD) = 7.795 − 0.620ln(N), which can provide a reference for developing efficient measures of stand density control.