Prediction model of base diameter of primary branch for Larix olgensis based on mixed effects

  Objective   This paper aims to establish an age group-individual tree two-level primary branch diameter model for Larix olgensis using the nonlinear mixed effect model approach, and provide theoretical basis for the research on the characteristics and differences of branch diameter growth in different age groups.

  Method   Out of four improved basic models, one was selected as the basic model after referring to the adjusting coefficient (R_\mathrma\mathrmd\mathrmj ^2) and root mean square error (RMSE). Nonlinear mixed-effect model of the branch diameter was constructed based on the selected model. Independent data were used to verify the model fitting results, while mean absolute error (MAE) and mean relative absolute error (MRAE) were used to evaluate the model prediction ability. The prediction values of the basic model and the mixed model were compared. Simulation of the branch diameter distribution in each age group was done using the random parameters from age group level.

  Result   Based on the improved Gompertz equation, the model performed the best when the age group random effects acted on parameter b, the individual tree random effects acted on parameters b, c and d at the same time, the variance-covariance structure of the random effects was generalized positive definite matrix, and the heteroscedasticity structure was power function. The adjustment coefficient (R_\mathrma\mathrmd\mathrmj ^2) of the mixed model was improved, and RMSE, MAE and MRAE were all significantly reduced. The values of the adjustment coefficient, RMSE, MAE and MRAE of the final model were 0.699 8, 4.768 4 mm, 3.705 8 mm and 0.391 6 mm, respectively. The predicted values of the mixed model reflected the differences between individual trees. The distribution range of branch diameter was found increasing with the increase of age groups, and the branch diameter growth showed difference between the age groups.

  Conclusion   The accuracy of the mixed effect model of branch diameter can be improved by incorporating the random effects of age groups and individual trees into the model. Simulation of the branch diameter growth using the random effect parameters in the age group level can reasonably reflect their growth patterns and differences, and also conform to the significance of the tree physiology. Therefore, the mixed effect model based on age group and individual tree level can reasonably predict the growth of primary branch diameter of Larix olgensis at different ages.