A growth model system based on nonlinear mixed effects

Abstract

The application discloses a kind of growth model systems based on nonlinear mixed effect, it is related to tree growth model technical field, including data source and processing module, basic growth model screening module, climate factor screening and model reparameterization module, nonlinear mixed effect model module, climate scenario prediction module, model evaluation and diagnosis module, the present application is with single tree as random effect, constructs climate sensitive type Slogistic1 nonlinear mixed effect model, by introducing the modeling method capable of simultaneously processing fixed effect and random effect, climate factor is regarded as fixed effect, single tree individual difference is regarded as random effect into unified framework, constructs the nonlinear mixed effect model capable of reflecting individual difference, the overall effect of quantifying climate factor and its individual differentiation response, for ginkgo artificial forest under the growth prediction and refinement management of climate change provides reliable model tool.

Description

Technical Field

[0001] This invention relates to the field of tree growth model technology, specifically a growth model system based on nonlinear mixed effects. Background Technology

[0002] Ginkgo trees possess ecological, economic, and ornamental value, and their plantations play a vital role in forestry production, ecological restoration, wetland protection, and urban greening. Therefore, in the context of climate change, in-depth research into the mechanisms by which climatic factors influence the growth of ginkgo plantations and the establishment of scientific growth prediction models can not only enrich the theoretical system of forestry growth model research but also provide important theoretical basis and technical support for forestry workers to formulate adaptive management strategies, address potential risks brought about by climate change, and achieve sustainable management of ginkgo plantations. This has significant theoretical and practical value. Nonlinear mixed-effects models are statistical models that simultaneously include fixed and random effects, effectively handling non-independent data with hierarchical structures or repeated observation characteristics. In forestry growth modeling, fixed effects are typically used to describe the research... The study investigates the average response of the population under specific independent variables (such as tree age and climate factors), while random effects are used to characterize the individual heterogeneity caused by unobserved factors among different levels of units (such as plots and individual trees). This decomposes data variation into two parts: structural variation explained by fixed effects and random variation captured by random effects. The purpose of constructing a climate-sensitive nonlinear mixed-effects growth model is twofold: on the one hand, by incorporating climate factors as fixed effects, it reveals the driving mechanism of environmental variables such as temperature, precipitation, and CO2 concentration on tree growth; on the other hand, by introducing random effects at the individual tree or plot level, it controls the interference of individual differences on model estimation, avoids misattributing unobserved heterogeneity to residuals, and prevents distortion of the fixed effect significance test, ultimately improving the model's explanatory power and prediction accuracy of the actual growth process. Traditional theoretical growth models primarily use tree age, diameter at breast height (DBH), and tree height as independent variables, employing fixed-parameter regression methods to characterize the average trend of tree growth. These models have played a fundamental role in describing the age-response patterns of single indicators such as DBH, tree height, and volume. However, traditional models have the following significant shortcomings: First, traditional models typically assume that all sample data are independent and identically distributed, failing to consider the individual heterogeneity between individual trees caused by factors such as microsite, genetic differences, and local competition, resulting in residuals often exhibiting heteroscedasticity. Second, traditional models often use tree age as the only or main explanatory variable, lacking a direct characterization of environmental driving mechanisms such as climate factors, and are difficult to reflect the dynamic response of tree growth under the background of climate change. Third, traditional models can only provide growth curves in the sense of population average, and cannot describe the unique growth trajectories of different individual trees at the same time, which limits their ability to make precise management predictions. The above shortcomings indicate that relying solely on the fixed-effects structure is insufficient to fully explain the multi-level variation characteristics of Ginkgo plantation growth data. Summary of the Invention

[0003] This invention provides a growth model system based on nonlinear hybrid effects, which can effectively solve the problems mentioned in the background art.

[0004] To achieve the above objectives, the present invention provides the following technical solution: a growth model system based on nonlinear mixed effects, comprising a data source and processing module, a basic growth model screening module, a climate factor screening and model reparameterization module, a nonlinear mixed effects model module, a climate scenario prediction module, and a model evaluation and diagnosis module; The data source and processing module includes standard tree trunk analysis data and annual climate factor data. The basic growth model screening module uses five basic tree growth models and some variations, namely Logistic, Richards, Gompertz, Korf and Schumacher, to perform nonlinear fitting on the volume V of Ginkgo. The climate factor screening and model reparameterization module expresses the parameters in the basic model as a nonlinear combination containing climate variables, i.e., reparameterization. At the same time, the introduction and removal of each variable must pass the F test. The nonlinear mixed-effects model module studies the growth of individual trees under random climate influences by constructing a nonlinear mixed-effects model, simulating the growth of trees in a natural environment. The climate scenario prediction module further analyzes the changing trend of ginkgo plantation volume growth under the background of climate change. The quantitative evaluation indicators for the model evaluation and diagnosis module follow the AIC, BIC, and R from the model screening stage. 2 The five indicators are RMSE and MAES.

[0005] According to the above technical solution, the standard tree trunk analysis data is obtained by combining UAV technology with manual field measurement to comprehensively acquire the stand growth data of Ginkgo plantation in the study area. Based on the acquired growth data, the tree height-diameter-at-breast-thickness curve of the standard tree is fitted to clarify the correlation between tree height and diameter at breast height, and the standard tree is selected as the core sample tree. Based on the acquired data, trunk analysis and volume table construction were carried out. Trunk analysis was performed on standard trees, and discs were cut from felled trees. In the laboratory, indoor measurements were conducted to obtain more detailed disc data. The tree age of the discs was obtained by the annual ring method, and the diameter of each annual ring was measured at the same time. After obtaining the analytical wood disc data of each standard tree, a standardized analytical wood volume table was constructed by combining the forest stand survey data. The analytical wood volume of each age group was calculated by interpolation, and the annual growth and average growth of each analytical tree were calculated to clarify the volume growth characteristics of Ginkgo at different growth stages and provide core parameters for the construction of the growth model. The annual climate factor data is collected by long-term monitoring of climate factors by meteorological stations in and around the study area, including key climate indicators such as annual average temperature, annual precipitation, extreme maximum temperature, extreme minimum temperature, and carbon dioxide index. After data collection, meteorological data needs to be screened, verified, and organized to remove abnormal data. At the same time, climate factor data and ginkgo growth data need to be matched on a time scale.

[0006] According to the above technical solution, the basic growth model screening module evaluates and compares the performance of each equation in volume curve fitting during the model fitting process. The Akaike information criterion, Bayesian information criterion, coefficient of determination, and root mean square error (RMSE) were used as evaluation indicators for the growth model to assess its fitting accuracy from multiple perspectives. Specifically, smaller AIC, BIC, and RMSE values ​​indicate better model fitting. 2 The closer the model is to 1, the better the fit. Finally, the optimal model that best reflects the volume growth of ginkgo trees is selected.

[0007] According to the above technical solution, the climate factors screened by the climate factor screening and model reparameterization module are used to establish a nonlinear expression of tree volume change and tree age and climate variables using a linear regression model. Through the reparameterization process, the key parameters in the model are replaced with linear expressions containing climate factors. This reparameterization process expands the original basic growth equation by selecting relevant climate factors, including: , CO2, replacing the parameters in the original basic growth equation with a linear combination of climatic factors.

[0008] According to the above technical solution, in the process of analyzing the model reparameterization results, the variable with the largest VIF is removed from the model by stepwise regression screening and variance inflation factor, and then refitted. Repeat this process until the VIF of all remaining variables is less than 5. A significance test should be performed on the parameters, and variables should be gradually eliminated. For the largest insignificant variable, refit until all variables are significant. Until then, the variables to be retained are carbon dioxide concentration (CO2) and annual average maximum temperature. .

[0009] According to the above technical solution, the standard form of the nonlinear mixed-effect model module is as follows: (15); In the formula: and The first The first object The 1-dimensional dependent variable values ​​of the observed times and Dimensional independent variable value, for 3D formal parameter vector, for 3D fixed effects parameter vector, To construct variables In the generated on an object dimensional random effects parameter vector, for The covariance matrix, for different , Independent of each other and They are respectively and The design matrix For different error terms, , They are independent of each other, and at the same time, it is assumed that they are independent of each other. With all They are independent of each other; Formal parameter components The model is affected by only one random effect factor, contains only one construct variable, and has no nested, interleaved, or interactive random effects. Therefore, the model can be defined as a single-level nonlinear mixed-effects model. The fitting method for the growth model is the standard nonlinear fitting algorithm, LM optimization algorithm. The reparameterization process is based on the optimal model selected from Origin and fitted using the least squares method.

[0010] Based on the above technical solution, random effects parameters are added to the Slogistic1 reparameterization model, resulting in a mixed-effects model with a total of 63 different combinations of random effects parameters. =63, fit all parameter combinations and select the optimal combination of evaluation indicators as the final Slogistic1 nonlinear mixed effect model, which is used to construct a nonlinear mixed effect model of ginkgo growth affected by climate factors. The model uses individual trees as the random effects level and only applies to... Four parameters are used to set the random effects, using a pdDiag diagonal structure. The random effects are independent of each other, with no covariance, and only the variance term is retained. The random effects follow a multivariate normal distribution.

[0011] According to the above technical solution, the climate scenario prediction module, after completing the construction of the basic growth model, the reparameterized model and the nonlinear mixed effect model, carries out volume growth prediction under different climate scenarios. The climate scenario settings include the current climate scenario and three typical concentration pathways: RCP2.6, RCP4.5, and RCP8.5, which are used to compare the differences in the predicted volume growth of Ginkgo under different climate change conditions. Climate data for all climate scenarios were predicted using the ClimateAP climate model. Predicted carbon dioxide concentrations for different climate scenarios were obtained from the ISIMIP Repository. Temperature climate factors were predicted using the ClimateAP climate model. Annual average maximum / minimum temperatures were obtained by weighted averaging of monthly average maximum temperatures over the years, using the following weighting formula: (17); In the formula: The annual weighted average temperature Let i be the predicted temperature for month i. Let i be the number of days in the i-th month; Instead of directly constructing a regional climate prediction model, we input climate factors under different scenarios into an established growth model to predict volume based on existing climate scenario data.

[0012] According to the above technical solution, in the prediction process, the basic growth model is first used as a control model. Then, the climate factors under different climate scenarios are input into the reparameterized model to calculate the prediction curve of the change of Ginkgo volume with tree age under each scenario. Finally, based on the reparameterized model, the random effect of individual trees is introduced, and the nonlinear mixed effect model is used to predict the volume growth process of different individual trees under each climate scenario.

[0013] According to the above technical solution, the five indicators of the model evaluation and the diagnostic module quantitative evaluation have different focuses. In the model iteration comparison, the AIC and BIC of the nonlinear mixed effect model are the lowest among the three models, and the RMSE and MAE are also far better than the basic model and the reparameterized model. Ri of the mixed-effects model 2 Although lower than the former two, R0.05 changes the error decomposition method due to the introduction of random effects. 2 It is no longer suitable as a single evaluation criterion, but should be judged in combination with AIC, BIC, RMSE, MAE and residual diagnosis results; Residual diagnosis is achieved through residual-prediction graphs, residual histograms, residual observation order graphs, and normal probability. Figure 4 The study systematically examines whether the assumptions of homoscedasticity, independence, and normality of the residuals are satisfied across several dimensions.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The basic growth model screening module systematically determines the optimal basic model form from the candidate equation family and simultaneously exposes the inherent limitations of the pure fixed effect structure. The climate factor screening and model reparameterization module realizes the transformation of the climate driving mechanism from a black box to an explicit expression, and achieves a structural upgrade from a pure time-dependent model to a climate-sensitive model through reparameterization. The nonlinear mixed effect model module, based on the introduction of random effects, realizes the leap from a single-level to a hierarchical model framework at the technical level, and achieves accurate preservation of the growth trajectory of individual trees. The climate scenario prediction module upgrades the model from a tool for fitting historical data to a tool for extrapolating future climate scenarios, giving the entire modeling system the ability to predict the future. This allows forestry operators to assess the long-term volume yield of Ginkgo plantations under different climate assumptions, providing quantitative basis for stand structure adjustment, harvesting cycle optimization, and the formulation of adaptive management strategies. The model evaluation and diagnosis module provides quantifiable, repeatable, and visualized quality verification for each model iteration, ensuring the accuracy of the final delivered model. The entire modeling system provides a complete quality assurance mechanism.

[0015] 2. This invention first compared five basic growth equations and determined Slogistic1 as the optimal model. Then, through stepwise regression, it screened out two key climate factors: carbon dioxide concentration and annual average maximum temperature. The basic model was then reparameterized. Finally, taking individual trees as random effects, a climate-sensitive Slogistic1 nonlinear mixed-effects model was constructed. By introducing a modeling method that can simultaneously handle fixed and random effects, climate factors are treated as fixed effects and individual tree differences are treated as random effects, all within a unified framework. This constructs a nonlinear mixed-effects model that can reflect individual differences, quantifying the overall effect of climate factors and their individual differentiated responses. This provides a reliable model tool for the growth prediction and refined management of Ginkgo plantations under climate change. Attached Figure Description

[0016] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0017] In the attached diagram: Figure 1 This is a schematic diagram of the composition of the growth model system of the present invention; Figure 2 This is a technical roadmap of the present invention; Figure 3 These are the model fitting curve and the line graphs of annual growth and average growth of this invention. Figure 4 This is the residual plot of the Slogistic1 basic model of this invention; Figure 5 This is the residual plot of the Slogistic1 reparameterized model of the present invention; Figure 6 This is the residual plot of the Slogistic1 nonlinear mixed-effects model of the present invention. Detailed Implementation

[0018] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0019] Example: Figure 1-2 As shown, the present invention provides a technical solution, a growth model system based on nonlinear mixed effects, including a data source and processing module, a basic growth model screening module, a climate factor screening and model reparameterization module, a nonlinear mixed effects model module, a climate scenario prediction module, and a model evaluation and diagnosis module. The data source and processing module includes standard tree trunk analysis data and annual climate factor data; The basic growth model screening module uses five basic tree growth models and some variations, namely Logistic, Richards, Gompertz, Korf and Schumacher, to perform nonlinear fitting on the volume V of Ginkgo. The climate factor screening and model reparameterization module represents the parameters in the basic model as a nonlinear combination of climate variables, i.e., reparameterization. At the same time, the introduction and removal of each variable must pass the F test. The nonlinear mixed-effects model module studies the growth of individual trees under random climatic influences by constructing a nonlinear mixed-effects model, thus better simulating the growth of trees in the natural environment. The climate scenario prediction module further analyzes the changing trend of ginkgo plantation volume growth under the background of climate change; The quantitative evaluation indicators for the model evaluation and diagnosis module are the same as those used in the model screening phase: AIC, BIC, and R. 2 The five indicators are RMSE and MAES.

[0020] Based on the above technical solution, the standard tree trunk analysis data adopts a combination of UAV technology and manual field measurement to comprehensively obtain the stand growth data of Ginkgo plantations in the study area. Specifically, UAV scanning technology is used to quickly obtain the spatial distribution, canopy structure and preliminary tree height data of Ginkgo in the entire study area. Combined with manual field measurement, the growth indicators of diameter at breast height, tree height and crown width of each individual Ginkgo tree at 1.3m above the ground are accurately measured to ensure the accuracy and completeness of the data. Based on the acquired growth data, the tree height-diameter-at-breast-thickness curve of the standard trees was fitted to clarify the correlation between tree height and diameter at breast height. Then, in accordance with the principles of stand uniformity and representativeness, standard trees with good growth status, no pests and diseases, and that can reflect the overall growth level of the stand were selected as core sample trees in each diameter class to provide basic data for subsequent trunk analysis and model construction. Based on the acquired data, trunk analysis and volume table construction were carried out. Trunk analysis was performed on standard timbers, and discs were obtained from felled timbers. For felled timbers with a height of less than 10 meters, discs were cut into 1-meter segments, and for felled timbers with a height of more than 10 meters, discs were cut into 2-meter segments. All the felled timber discs were further measured in the laboratory to obtain more detailed disc data. The tree age of the discs was obtained by the annual ring method, and the diameter of each annual ring was measured. The diameter of the disc was obtained by taking the diameter of the annual rings in the east-west and north-south directions and taking the average value as the diameter of each annual ring in the disc. After obtaining the analytical wood disc data for each standard tree, a standardized analytical wood volume table was constructed by combining it with forest stand survey data. The analytical wood volume for each age group was calculated using interpolation, and the annual growth and average growth of each analytical tree were calculated to clarify the volume growth characteristics of Ginkgo at different growth stages, providing core parameters for the construction of the growth model, as shown in the following formula: (1); (2); In the formula: Let be the volume of timber in year t, where t is the tree age. For annual growth, This represents the average growth. The annual climate factor data is collected by collecting long-term climate factor data from meteorological stations in and around the study area, including key climate indicators such as annual average temperature, annual precipitation, extreme maximum temperature, extreme minimum temperature, and carbon dioxide index. The climate data are all from the monthly data of the historical ground meteorological observation dataset of the meteorological data center, and the average value is taken as the annual observation data. After data collection, meteorological data needs to be screened, verified, and organized to remove abnormal data and ensure the reliability and continuity of climate factor data. At the same time, the climate factor data is matched with ginkgo growth data on time scale to provide data support for model coupling.

[0021] Based on the above technical solution, the basic growth model screening module evaluates and compares the performance of each equation in volume curve fitting during the model fitting process. Some variations are shown in the table below:

[0022] In the formula: For the volume of wood, Tree age It is the natural logarithm. These are parameters to be determined. To ensure the rationality and scientific validity of the model selection, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and the Coefficient of Determination (R²) are adopted. 2 Coefficient of Determination (AIC) and Root Mean Square Error (RMSE) are used as indicators to evaluate the fitting accuracy of a growth model from multiple perspectives. The smaller the values ​​of AIC, BIC, and RMSE, the better the model's fit. 2 The closer the model is to 1, the better the fit. Finally, the optimal model that best reflects the volume growth of Ginkgo biloba is selected, as shown in the following formula: (3); (4) (5); (6); (7); (8); (9); In the formula: SST is the total sum of squares, and SSE is the residual sum of squares. These are measured values. Model predictions Average of measured values Sample size Number of model parameters The maximum likelihood function value of the model; The evaluation results of the basic growth model for Ginkgo biloba are shown in the table below:

[0023] The results in the table show that the differences in BIC and AIC between the basic models are relatively small, with the models Slogistic1 and Schumacher having the smallest BIC and AIC. The evaluation index R2 can be seen from the data in the table: ; Based on the evaluation results of the above models, Slogistic1 was ultimately selected as the basic Ginkgo growth model: ; In the formula: V is the volume of a single piece of wood. Let t be the natural logarithm, and t be the age of the tree. The Slogistic 1 model fitting curve for ginkgo volume growth and the line graphs of annual growth and average growth are shown below. Figure 3 According to the S-shaped growth trend of ginkgo volume, the volume growth is relatively slow in the 0-10 year range, and enters a period of rapid growth in the 15-25 year range, with the annual growth continuously increasing. After 25 years, the annual growth of volume gradually decreases, the average growth is relatively flat, and the maximum annual growth is 24.6. Although none of the standard ginkgo trees obtained were actually over 30 years old, their maturity age can be predicted to be around 36 years from the line graph of annual growth and average growth.

[0024] Based on the above technical solution, in the basic model, the climate factor screening and model reparameterization module shows that the growth of forest volume depends entirely on tree age. In order to quantify the impact of climate change on forest volume growth, the introduction and removal of each variable are carried out through F test to ensure that the variance contribution of all variables in the final regression equation is statistically significant. The selected climate factors were used to establish a nonlinear expression for the relationship between tree volume change and tree age and climate variables using a linear regression model: (10); In the formula: These are the parameters in the original formula. For constant terms, These are the coefficients of the regression equation. These are the filtered climate factor variables; To further explore the impact of climate change on the volume growth of Ginkgo biloba, a reparameterization process was used to replace the key parameters in the model with linear expressions containing climate factors. This reparameterization process expanded the original basic growth equation and enhanced its explanatory power for climate factors. Taking Slogistic1 as an example, the model was reparameterized, and the equation form is as follows: (11); In the formula: For the volume of wood, Tree age The age of the growth inflection point The parameter to be estimated; By screening relevant climate factors, including: , CO2, the parameters in the original basic growth equation are replaced with a linear combination of climatic factors, as follows: (12); (13); In the formula: For constant terms, , These are the coefficients of the regression equation. This is the annual average maximum temperature. The lowest annual average temperature is represented by , and CO2 represents the average annual carbon dioxide concentration. After reparameterization, the reparameterized Slogistic1 model containing climate variables and with statistically significant parameters is as follows: (14); In the formula: These are the parameters to be estimated.

[0025] Based on the above technical solution, in the process of analyzing the model reparameterization results, the variable with the largest VIF is removed from the model by stepwise regression screening and variance inflation factor, and then refitted. Repeat this process until the VIF of all remaining variables is less than 5. Since not all climatic factors have a significant impact on ginkgo volume growth, it is necessary to perform significance tests on the parameters and gradually eliminate them. For the largest insignificant variable, refit until all variables are significant. Until then, the variables to be retained are carbon dioxide concentration (CO2) and annual average maximum temperature. The test results are shown in the table below:

[0026] After processing, the Slogistic1 reparameterized model containing climate variables and with statistically significant parameters is as follows: (1101); (1201); (1301); In the formula: For the first Tree No. Annual volume, For the first Tree No. Annual carbon dioxide concentration For the first Tree No. The average annual maximum temperature of the year.

[0027] Based on the above technical solution, the standard form of the nonlinear mixed-effects model module is as follows: (15); In the formula: and The first The first object The 1-dimensional dependent variable values ​​of the observed times and Dimensional independent variable value, for 3D formal parameter vector, for 3D fixed effects parameter vector, To construct variables In the generated on an object dimensional random effects parameter vector, for The covariance matrix, for different , Independent of each other and They are respectively and The design matrix For different error terms, , They are independent of each other, and at the same time, it is assumed that they are independent of each other. With all They are independent of each other; When various random effect parameters and When all data follow a normal distribution, the model is called a normal NLMEM. Formal parameter components Influenced by only one random effect factor, the model contains only one constructive variable, with no nested, interleaved, or interactive random effects. Therefore, the model can be defined as a single-level nonlinear mixed-effects model, which can be simplified to: (16); The initial growth model fitting was completed in Origin 2024. The fitting method for the growth model was the Levenberg-Marquardt optimization algorithm, the standard algorithm for nonlinear fitting in Origin 2024. The reparameterization process was based on the least squares fitting of the optimal model selected in Origin, which was completed by the R package "minpack.lm". The nonlinear mixed-effect model of Ginkgo volume growth took the single tree sample as the random effect. The model construction and prediction were completed by the R package "nlme".

[0028] Based on the above technical solutions, random effects parameters are added to the Slogistic1 reparameterization model, resulting in a mixed-effects model with a total of 63 different combinations of random effects parameters. =63, fit all parameter combinations and select the optimal combination of evaluation indicators as the final Slogistic1 nonlinear mixed effect model, which is used to construct a nonlinear mixed effect model of ginkgo growth affected by climate factors. The Slogistic1 nonlinear mixed-effects model with the best overall performance was selected: (19); (20); In the formula: For fixed effects parameters, For the first The random effect of the tree satisfies , This refers to the concentration of carbon dioxide. This is the annual average maximum temperature. For the first The first tree Year's volume of timber, For the first The age of the trees For the first Tree No. Annual climate factor parameters The variance-covariance matrix of random effects. This is the random error term; The model uses individual trees as the random effects level and only applies to... The four parameters define random effects, using a pdDiag diagonal structure. The random effects are independent, have no covariance, and only retain the variance term. The random effects follow a multivariate normal distribution. The covariance matrix and the overall random component structure of the model can be represented as: (twenty one); (twenty two); In the formula: Intercept variance covariates variance covariates The variance; The parameter composition and evaluation indicators of the mixed-effects model are shown in the table below:

[0029] Based on the above technical solutions, the climate scenario prediction module, after completing the construction of the basic growth model, the reparameterized model and the nonlinear mixed effect model, can carry out volume growth prediction under different climate scenarios. The climate scenario settings include the current climate scenario and three typical concentration pathways: RCP2.6, RCP4.5, and RCP8.5. RCP2.6, RCP4.5, and RCP8.5 represent future low-emission, medium-emission, and high-emission climate scenarios, respectively, and are used to compare the differences in the predicted volume growth of Ginkgo under different climate change conditions. Climate data for all climate scenarios were predicted using the ClimateAP climate model. Predicted carbon dioxide concentrations for different climate scenarios were obtained from the ISIMIP Repository. Temperature climate factors were predicted using the ClimateAP climate model. Annual average maximum / minimum temperatures were obtained by weighted averaging of monthly average maximum temperatures over the years, using the following weighting formula: (17); In the formula: The annual weighted average temperature Let i be the predicted temperature for month i. Let i be the number of days in the i-th month; Instead of directly constructing a regional climate prediction model, we input climate factors under different scenarios into an established growth model to predict volume based on existing climate scenario data. Since the climate variables ultimately retained during the model reparameterization process are carbon dioxide concentration and annual average maximum temperature, the carbon dioxide concentration and annual average maximum temperature corresponding to different climate scenarios are mainly used as input variables in climate scenario prediction to calculate the predicted values ​​of the reparameterized model and the nonlinear mixed-effects model.

[0030] Based on the above technical solution, in the prediction process, the basic growth model is first used as a control model. This model only uses tree age as an independent variable and does not introduce climate factors. Therefore, its prediction results do not change with the climate scenario. It is mainly used to reflect the overall average growth trend of ginkgo volume with tree age. Secondly, the climate factors under different climate scenarios are input into the reparameterized model to calculate the prediction curve of ginkgo volume with tree age under each scenario. This is used to analyze the impact of climate factor changes on the overall volume growth trend. Finally, based on the reparameterized model, the stochastic effect of individual trees is introduced. The nonlinear mixed effect model is used to predict the volume growth process of different individual trees under each climate scenario to reflect the differences in growth response between individual trees under climate change conditions.

[0031] Based on the above technical solutions, the five indicators for quantitative evaluation of the model evaluation and diagnostic modules have different focuses. In the comparison of model iterations, the AIC and BIC of the nonlinear mixed effects model are the lowest among the three models, and the RMSE and MAE are also far better than the basic model and the reparameterized model. Ri of the mixed-effects model 2 Although lower than the former two, R0.05 changes the error decomposition method due to the introduction of random effects. 2 It is no longer suitable as a single evaluation criterion, but should be judged in combination with AIC, BIC, RMSE, MAE and residual diagnosis results; Residual diagnosis is achieved through residual-prediction graphs, residual histograms, residual observation order graphs, and normal probability. Figure 4 The study systematically examines whether the assumptions of homoscedasticity, independence, and normality of the residuals are satisfied across multiple dimensions. Through diagnosis, it reveals why the basic model and the reparameterized model have defects: the residuals of the former exhibit a funnel-shaped diffusion, while the residuals of the latter still show a trend of fluctuating expansion. In contrast, the residual diagnostic plot of the nonlinear mixed effects model basically satisfies all the assumptions, thus verifying the effectiveness of the modeling strategy of introducing random effects. Specifically, the nonlinear mixed-effects model of Ginkgo biloba was constructed step by step from the basic Slogistic1 model to the Slogistic1 reparameterized model and the Slogistic1 nonlinear mixed-effects model. The model evaluation indicators of each model are shown in the table below:

[0032] The table contains three models R. 2 All three models are above 0.75, indicating that they all have good explanatory power for volumetric growth data. Among them, the Slogistic1 reparameterized model has the best explanatory power for data variation, followed by the Slogistic1 basic model. In the mixed effects model, part of the variance is explained by random effects, therefore the R-squared value of the fixed effects part is relatively high. 2 With boundary R 2The interpretation of this value differs from that of traditional regression. A lower value is a normal phenomenon. The comprehensive evaluation should combine AIC, BIC and residual diagnosis results to evaluate the quality of the model. The Slogistic1 nonlinear mixed-effects model has the lowest AIC and BIC values, which are significantly lower than the Slogistic1 reparameterized model and the basic Slogistic1 model. The AIC and BIC of the Slogistic1 reparameterized model are also significantly lower than those of the Slogistic1 base model. This result indicates that, compared to the base model, the Slogistic1 reparameterized model reparameterizes the parameters of the original Slogistic1 model, adds climate factors, and optimizes the model structure through parameter reconstruction, thereby improving the goodness of fit without significantly increasing complexity. Although the Slogistic1 nonlinear mixed-effects model introduces random effects and increases model complexity, the improvement in goodness of fit far outweighs the penalty brought by the complexity, and the overall performance of the model is significantly better than the first two types of models. The RMSE and MAE of the Slogistic1 nonlinear mixed effects model are significantly lower than those of the Slogistic1 reparameterized model and the basic Slogistic1 model. The RMSE and MAE of the Slogistic1 reparameterized model are also lower than those of the basic Slogistic1 model. This indicates that the Slogistic1 nonlinear mixed effects model effectively captures the individual heterogeneity and overall heterogeneity in the data by introducing random effects, which greatly reduces the prediction error and the prediction accuracy is significantly better than the other two types of models. The Slogistic1 reparameterized model also effectively improves the prediction performance of the basic Slogistic1 model and reduces the prediction bias through parameter optimization. Classical regression analysis typically requires that the model error satisfy homoscedasticity; otherwise, it will affect the reliability of parameter estimation and model testing results. Figure 4-6 The residual diagnostic results of the Slogistic1 basic model, the Slogistic1 reparameterized model, and the Slogistic1 nonlinear mixed effects model are presented respectively, including the residual-fit value relationship graph, residual histogram, residual change with observation order graph, and normal probability graph. from Figure 4 It can be seen that the residual distribution of the Slogistic1 basic model exhibits obvious heteroscedasticity. In the graph of the relationship between residuals and fitted values, as the fitted values ​​increase, the range of residual dispersion gradually expands, showing a funnel-shaped distribution characteristic. The residuals in the low-fit value region are mainly concentrated near the zero value, while the high-fit value region shows large positive and negative deviations. In the residual change graph with observation order, some observation points fluctuate significantly and there are large abnormal residuals. In the normal probability graph, the sample points deviate significantly from the reference line at both ends, indicating that the residual distribution deviates from the normal distribution. Overall, it shows that the basic model fails to effectively describe the heterogeneity of the data and the model fitting effect is limited. Figure 5 The Slogistic1 reparameterized model shown has certain improvements compared to the basic Slogistic1 model. In the graph of the relationship between residuals and fitted values, the dispersion of residuals is reduced. However, as the fitted values ​​increase, the trend of residual fluctuation gradually increases can still be observed. Heteroscedasticity still exists. Although the residual histogram is more concentrated near the zero value, the distribution is still skewed. The sample points at both ends of the normal probability plot still deviate from the reference line, indicating that the improvement in the normality of the residuals is limited. The residuals change with the observation order plot still has obvious fluctuation peaks, indicating that the reparameterization process only partially reduced the model error and failed to fundamentally solve the problem of variance inequality. Figure 6 The residual distribution of the Slogistic1 nonlinear mixed-effects model is significantly better than the previous two models. In the graph of the relationship between residuals and predicted values, most residuals are evenly distributed on both sides of the zero line, without obvious diffusion trend. Overall, it shows good randomness and homoscedasticity. The residual histogram shows that the residuals are mainly concentrated near zero and the distribution is symmetrical. The sample points in the normal probability plot are basically distributed along the reference line, with deviations only at the tail, indicating that the residuals approximately follow a normal distribution. In the residuals change with the observation order plot, the residuals fluctuate little and have no obvious trend. Meanwhile, the residual range of this model is significantly smaller than that of the previous two types of models, indicating that the Slogistic1 nonlinear mixed-effects model effectively explains the heterogeneity among individuals by introducing random effects, greatly improving the model fitting accuracy and prediction stability, and the model assumptions are basically satisfied.

[0033] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A growth model system based on nonlinear hybrid effects, characterized in that: It includes modules for data sourcing and processing, basic growth model screening, climate factor screening and model reparameterization, nonlinear mixed-effects model, climate scenario prediction, and model evaluation and diagnosis. The data source and processing module includes standard tree trunk analysis data and annual climate factor data. The basic growth model screening module uses five basic tree growth models and some variations, namely Logistic, Richards, Gompertz, Korf and Schumacher, to perform nonlinear fitting on the volume V of Ginkgo. The climate factor screening and model reparameterization module represents the parameters in the basic model as a nonlinear combination containing climate variables, i.e., reparameterization. The introduction and removal of each variable must pass the F test. The nonlinear mixed-effects model module studies the growth of individual trees under random climate influences by constructing a nonlinear mixed-effects model, and the climate scenario prediction module further analyzes the changing trend of ginkgo plantation volume growth under the background of climate change. The quantitative evaluation indicators for the model evaluation and diagnosis module follow the AIC, BIC, and R from the model screening stage. 2 The five indicators are RMSE and MAES.

2. The growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: The standard tree trunk analysis data was obtained by combining drone technology with manual on-site measurement. Based on the acquired data, trunk analysis and volume table construction were carried out. Trunk analysis was performed on standard trees, and discs were cut from felled trees. In the laboratory, indoor measurements were conducted to obtain more detailed disc data. The tree age of the discs was obtained by the annual ring method, and the diameter of each annual ring was measured at the same time. After obtaining the analytical wood disc data of each standard tree, a standardized analytical wood volume table was constructed by combining the forest stand survey data. The analytical wood volume of each age group was calculated by interpolation, and the annual growth and average growth of each analytical tree were calculated to clarify the volume growth characteristics of Ginkgo at different growth stages and provide core parameters for the construction of the growth model. The annual climate factor data is collected by long-term monitoring of climate factors by meteorological stations in and around the study area, including key climate indicators such as annual average temperature, annual precipitation, extreme maximum temperature, extreme minimum temperature, and carbon dioxide index. After data collection, meteorological data needs to be screened, verified, and organized to remove abnormal data. At the same time, climate factor data and ginkgo growth data need to be matched on a time scale.

3. The growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: During the model fitting process, the basic growth model screening module evaluates and compares the performance of each equation in volume curve fitting. The Akaike information criterion, Bayesian information criterion, coefficient of determination, and root mean square error were used as evaluation indicators for the growth model to assess the model's fitting accuracy from multiple perspectives.

4. The growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: The climate factors selected by the climate factor screening and model reparameterization module are used to establish a nonlinear expression of tree volume change in relation to tree age and climate variables using a linear regression model. The reparameterization process replaces the key parameters in the model with linear expressions containing climate factors, thus expanding the original basic growth equations by selecting relevant climate factors.

5. The growth model system based on nonlinear hybrid effects according to claim 4, characterized in that: In the process of analyzing the model reparameterization results, the variable with the largest VIF was removed from the model by stepwise regression screening and variance inflation factor, and then the model was refitted. Repeat this process until the VIF of all remaining variables is less than 5. A significance test should be performed on the parameters, and variables should be gradually eliminated. For the largest insignificant variable, refit until all variables are significant. Until then, the variables to be retained are carbon dioxide concentration (CO2) and annual average maximum temperature. .

6. The growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: The standard form of the nonlinear mixed-effects model module is as follows: （15）； In the formula: and The first The first object The 1-dimensional dependent variable values ​​of the observed times and Dimensional independent variable value, for 3D formal parameter vector, for 3D fixed effects parameter vector, To construct variables In the generated on an object dimensional random effects parameter vector, for The covariance matrix, for different , Independent of each other and They are respectively and The design matrix For different error terms, , They are independent of each other, and at the same time, it is assumed that they are independent of each other. With all They are independent of each other; Formal parameter components The model is affected by only one random effect factor, contains only one construct variable, and has no nested, interleaved, or interactive random effects. Therefore, the model can be defined as a single-level nonlinear mixed-effects model. The fitting method for the growth model is the standard nonlinear fitting algorithm, LM optimization algorithm. The reparameterization process is based on the optimal model selected from Origin and fitted using the least squares method.

7. The growth model system based on nonlinear hybrid effects according to claim 6, characterized in that: By adding random effects parameters to the Slogistic1 reparameterized model, there are a total of 63 different combinations of random effects parameters in the mixed-effects model. =63, fit all parameter combinations and select the optimal combination of evaluation indicators as the final Slogistic1 nonlinear mixed effect model, which is used to construct a nonlinear mixed effect model of ginkgo growth affected by climate factors. The model uses individual trees as the random effects level and only applies to... Four parameters are used to set the random effects, using a pdDiag diagonal structure. The random effects are independent of each other, with no covariance, and only the variance term is retained. The random effects follow a multivariate normal distribution.

8. The growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: Based on the construction of the basic growth model, the reparameterized model and the nonlinear mixed effect model, the climate scenario prediction module performs volume growth prediction under different climate scenarios. The climate scenario settings include the current climate scenario and three typical concentration pathways: RCP2.6, RCP4.5, and RCP8.5, which are used to compare the differences in the predicted volume growth of Ginkgo under different climate change conditions. Climate data for all climate scenarios were predicted using the ClimateAP climate model. Predicted carbon dioxide concentrations for different climate scenarios were obtained from the ISIMIP Repository. Temperature climate factors were predicted using the ClimateAP climate model. Annual average maximum / minimum temperatures were obtained by weighted averaging of monthly average maximum temperatures over the years, using the following weighting formula: （17）； In the formula: The annual weighted average temperature. Let i be the predicted temperature for month i. Let i be the number of days in the i-th month; Instead of directly constructing a regional climate prediction model, we input climate factors under different scenarios into an established growth model to predict volume based on existing climate scenario data.

9. A growth model system based on nonlinear hybrid effects according to claim 8, characterized in that: In the prediction process, the basic growth model is first used as a control model. Then, the climate factors under different climate scenarios are input into the reparameterized model to calculate the prediction curve of the change of Ginkgo volume with tree age under each scenario. Finally, based on the reparameterized model, the random effect of individual trees is introduced, and the nonlinear mixed effect model is used to predict the volume growth process of different individual trees under each climate scenario.

10. A growth model system based on nonlinear hybrid effects according to claim 1, characterized in that: The five indicators of the model evaluation and the diagnostic module quantitative evaluation have different focuses. In the model iteration comparison, the AIC and BIC of the nonlinear mixed effects model are the lowest among the three models, and the RMSE and MAE are also far better than the basic model and the reparameterized model. Ri of the mixed-effects model 2 Although lower than the former two, R0.05 changes the error decomposition method due to the introduction of random effects. 2 It is no longer suitable as a single evaluation criterion, but should be judged in combination with AIC, BIC, RMSE, MAE and residual diagnosis results; Residual diagnosis systematically examines whether the assumptions of homoscedasticity, independence, and normality of residuals are satisfied through four dimensions: residual-predicted value relationship graph, residual histogram, residual observation order graph, and normal probability graph.

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