# Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted Pinus sylvestris var. Mongolica Trees

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

#### 2.2. Data Collection and Analysis

#### 2.3. Model Selection for the Crown Profile

_{1}, a

_{2}, and a

_{3}are the parameters to be estimated. To make the curve reasonable in describing the outer crown profile, the parameters should be restricted as a

_{1}> 0, a

_{2}> 0, and a

_{3}< 0. By analyzing the relationship between the parameters and tree variables, DBH, CR, and HD were at last introduced into the equation and the model is finally defined as Equation (2):

_{1}–a

_{5}are the model parameters to be estimated. DBH is the diameter at breast height, CR is the crown ratio, and HD is the ratio of tree height to DBH.

_{1}–a

_{6}are the model parameters.

_{1}–a

_{4}are the model parameters to be estimated.

^{2}, root mean square error (RMSE), and Akaike information criterion (AIC) were used to compare the goodness-of-fit of the three basic models. The mean error and mean absolute error for the upper crown, shade crown, and the total tree were calculated. The model with the largest Ra

^{2}, the smallest RMSE and AIC, the smallest mean error and mean absolute error was selected as the best model to develop the outer crown profile model.

#### 2.4. Quantile Regression for the Mixed-Effects Outer Crown Profile Model

_{ij}is the jth observation from the ith subject; ${x}_{ij}^{T}$ is design matrix with the dimension of N × k; β is the fixed effects parameters with the dimension of k × 1; z

_{ij}is the design matrix with the dimension of q × 1; and b

_{i}are the random effects with the dimension of q × 1. Thus, the qth quantile regression for linear mixed-effects model is written as Equation (6):

^{2}, Var(ε

_{p}), [E(ε

_{p})]

^{2}, predicted mean error (MEP), and predicted mean absolute error (MAEP); and the function can be found in Gao et al. [11]. Based on the results of the dummy variable approach, stand age and stand density were introduced into the quantile regression for the mixed-effects outer crown profile model to analyze the effect of stand variables on the crown profile.

## 3. Results

#### 3.1. Best Model Selection for the Crown Profile Model

^{2}of 0.79, RMSE of 0.3544 m, and AIC of 762. Additionally, Equations (2)–(4) were further used to predict the crown radius for the total tree, upper crown, and shade crown. For the total tree, the power-exponential equation and modified Kozak equation produced an equivalent mean error, and the simple polynomial equation produced a relatively larger mean error. As for the light crown, the crown radius was overestimated by all three equations. The mean error for the power-exponential equation was 0.0224 m, which was the largest compared to the two other equations, but still lower than 0.03 m. For the shade crown, all three equations underestimated the crown radius. The modified Kozak equation had the smallest mean error, and the power-exponential equation was slightly larger, while the simple polynomial equation was the largest. In comparison, the power-exponential equation produced the smallest mean absolute error for the total tree, upper crown, and lower crown, and the simple polynomial equation was the largest. Due to the fact that the error produced by the individual trees was mainly derived from the shade crown, the comparison between the shade crowns was used as the criteria to select the best model. Thus, the power-exponential equation was selected as the best equation to model the outer crown profile for the planted Pinus sylvestris var. mongolica trees. The parameters in Equation (2) were restricted as a

_{1}> 0, a

_{2}> 0, and a

_{3}< 0 so that the curve could approach the true shape of the crown. By reparametrization, $DB{H}^{{a}_{1}}$ > 0, (a

_{2}+ a

_{3})·CR > 0, and (a

_{4}+ a

_{5})·HD < 0, which meets the requirement of the symbol of the parameters.

_{3}as the dummy variable for both stand age and stand density showed the best performance. The models are shown as Equations (10) and (11).

_{6}and a

_{7}(Table 4). On the whole, this indicated that the crown radius for the tree with the same size increased with the increasing of age. As for stand density (Figure 2B), the predicted crown radius of the tree from a stand density of larger than 3000 trees ha

^{−1}(Density group 4) had the smallest crown radius, followed by the density of between 2000 and 3000 trees ha

^{−1}(Density group 3). The tree from the stand density between 1000 and 2000 trees ha

^{−1}(Density group 2) and the stand density of lower than 1000 trees ha

^{−1}(Density group 1) had the largest crown radius. However, the crown radius of the two groups was almost equivalent.

#### 3.2. Quantile Regression for the Linear Mixed-Effects Crown Profile Model

_{1}–b

_{2}, b

_{31}–b

_{33}, b

_{4}–b

_{5}are model parameters.

_{1}, b

_{2}were assumed to be the random effects on the individual tree level. The following is the process of calculating the fixed effects parameter and variance-covariance structure for the random effects of qrLMM with the same equation form Equation (13):

_{1}, b

_{31}, and b

_{5}as the random effects achieved the smallest AIC, BIC, and HQ, and the largest logLike and was selected as the best model. Compared to the 0.95th linear quantile regression model, the R

^{2}of the qrLMM, considering b

_{1}, b

_{31}, and b

_{5}as the random effects parameters, increased and [E(ε

_{p})]

^{2}decreased (Table 5). The mean error and mean absolute error of qrLMM were also decreased when compared to the linear quantile regression (Table 5). Therefore, the precision and predicted ability of quantile regression for the linear mixed-effects model was improved when compared to the quantile regression model. The estimates for the parameters and variance-covariance structure are listed in Table 7 and all parameters were stable.

_{1}, b

_{31}, and b

_{5}at the tree level, respectively. The predicted largest crown radius by linear quantile regression and qrLMM were plotted against the observed values as shown in Figure 3. It was well documented that the largest crown radius for most trees was overestimated by the linear quantile regression model, while this situation was improved for the qrLMM. The predicted largest crown radius by qrLMM against the observed value was almost distributed around the two sides of y = x. The predicted largest crown radius by quantile regression and qrLMM was also regressed with the observed largest crown radius, respectively. The R

^{2}for the linear quantile regression model was 0.70, and the quantile regression for the linear mixed-effects model was 0.95 (Figure 3). Thus, the prediction for the largest crown radius by qrLMM was improved by adding the random effects into the quantile regression.

#### 3.3. Effect of Stand Age and Stand Density on the Crown Profile

## 4. Discussion

^{2}of 0.70. Furthermore, the R

^{2}of the largest crown radius increased to 0.95 by including the random effects into the quantile regression model. Thus, the quantile regression for the linear mixed-effects model for the outer crown profile showed good performance in both describing the crown shape and largest crown radius prediction. Thus, the largest crown radius is very useful in canopy closure estimation with higher precision. Besides, it is also helpful to study the competition between the subject trees and neighbor trees. The biological diversity and the habitat of the wildlife in the stand are also closely related to the crown structure and the crown layer. It indicated that the crown profile model is not only useful in the individual tree growth and yield model, but also very useful in the biological research area. This is why we used the quantile regression for linear mixed-effects models to increase the prediction precision for the largest crown radius.

_{3}, used as the dummy variable for both stand age and stand density, showed the best performance. Therefore, we constructed a crown profile model including stand age and stand density as Equation (12). The crown radius of the trees with the same size increased with increasing stand age and our results conformed to the study of Baldwin et al. [6]. The reason is that the trees occupy a relatively larger space to grow in a larger age group than in a lower age group. The stand age and stand density also provide a biological explanation about the effects on the crown profile. Due to the stand environment variations and the differences of competition level between the sample trees, the trees with the same stand age might show different crown profile. It could explain the growth history of the individual trees and it could provide effective measurement for forest management. The trees with the same age that achieved different crown profiles also showed the biological variations between the individual trees. The trees will grow into the upper layer or occupy the released space due to the change of the stand density. Thus, the trees with the same stand age but different growth environment might show different crown profiles. We also analyzed the effect of both stand density and HD on the crown profile. Stand density had a weaker effect on the crown profile when compared to HD. The possible reason is that the range of stand density is narrow within each group so variations could not be reflected. Another reason might be that the change of crown profile lags when responding to the change of stand density. Thus, we propose that the method of using HD to substitute the effect of stand density is encouraging if the data of stand density is missing.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Individual tree, crown, and branch attributes. HT is total tree height; CL is crown length; HBLC is height to the first live branch of the whorl; DINC is the depth into the crown radius of interest; BR is branch radius; BC is branch chord length; VA is branch angle; and L is absolute depth into the crown from the tree tip down to the branch basis.

**Figure 2.**Effects of stand age and stand density on the crown profile based on the dummy variable approach. (

**A**) reflects the effect of stand age on the crown profile and (

**B**) reflects the effect of stand density on the crown profile. Age group 1 represents the stand age lower than 20 years, Age group 2 between 21 and 30 years, Age group 3 of 31–40 years, and Age group 4 between 41 and 60 years. Density group 1 represents the stand density lower than 1000 trees ha

^{−1}; Density group 2 between 1000 and 2000 trees ha

^{−1}; Density group 3 between 2000 and 3000 trees ha

^{−1}, and Density group 4 of larger than 3000 trees ha

^{−1}.

**Figure 3.**Observed largest crown radius (LCR) plotted against the predicted crown radius by the quantile regression (

**A**) and quantile regression for linear mixed-effects model (

**B**).

**Figure 4.**The fitted outer crown profiles against the observed crown radius for three sample trees from three same plots (stand age is 20 years and stand density is 3950 trees ha

^{−1}for tree (

**A1**–

**A3**); stand age is 33 years and stand density is 1883 trees ha

^{−1}for trees (

**B1**–

**B3**); stand age is 44 years and stand density is 1640 trees ha

^{−1}for trees (

**C1**–

**C3**)) for the planted Pinus sylvestris var. mongolica trees.

**Figure 5.**Effect of stand age, stand density and ratio of tree height to diameter at the breast height (HD) on the crown profile keeping other variables as constant.

Plot Number | Stand Age (Year) | Stand Density (Trees ha^{−1}) | Mean DBH (cm) | Mean Tree Height (m) | Stand Volume (m ^{3} ha^{−1}) | Stand Basal Area (m ^{2} ha^{−1}) | Numbers of Sample Trees |
---|---|---|---|---|---|---|---|

1 | 42 | 385 | 29.1 | 18.2 | 221.9 | 25.81 | 5 |

2 | 42 | 650 | 23.6 | 16.9 | 227.2 | 28.86 | 5 |

3 | 26 | 783 | 15.2 | 8.6 | 100.1 | 14.99 | 5 |

4 | 33 | 1883 | 13.3 | 13.9 | 168.8 | 27.11 | 5 |

5 | 18 | 3633 | 8.6 | 5.7 | 117.0 | 22.37 | 5 |

6 | 38 | 1025 | 19.1 | 15.4 | 220.4 | 30.33 | 5 |

7 | 44 | 1640 | 19.0 | 21.1 | 347.7 | 47.98 | 5 |

8 | 33 | 3840 | 10.6 | 9.9 | 163.9 | 28.62 | 5 |

9 | 43 | 1100 | 20.1 | 16.5 | 264.3 | 35.73 | 5 |

10 | 31 | 2025 | 13.4 | 11.7 | 192.5 | 30.33 | 5 |

11 | 46 | 450 | 29.6 | 20.7 | 287.0 | 32.86 | 5 |

12 | 38 | 1220 | 19.2 | 16.0 | 267.9 | 36.65 | 5 |

13 | 45 | 1217 | 21.0 | 19.5 | 344.0 | 44.36 | 5 |

14 | 20 | 3950 | 9.7 | 7.7 | 77.7 | 31.01 | 5 |

15 | 12 | 2350 | 7.1 | 4.3 | 36.2 | 9.62 | 6 |

**Table 2.**Descriptive statistics of sample trees and all branches for 76 planted Pinus sylvestris var. mongolica trees.

Statistics | Tree Variables (N = 76) | Branch Variables (N = 3658) | |||||
---|---|---|---|---|---|---|---|

DBH (cm) | HT (m) | CL | BL (cm) | BC (cm) | VA (°) | BD (mm) | |

Mean | 17.9 | 14.3 | 5.4 | 130 | 121 | 47 | 2.02 |

Std | 7.2 | 5.2 | 1.6 | 88 | 82 | 16 | 1.10 |

Min | 6.0 | 3.5 | 2.6 | 3 | 3 | 10 | 0.09 |

Max | 34.5 | 22.5 | 10.9 | 536 | 521 | 150 | 7.16 |

**Table 3.**Mean error and mean absolute error produced by power-exponential equation, modified Kozak equation and simple polynomial equation.

Crown | Models | Power-Exponential Equation | Modified Kozak Equation | Simple Polynomial Equation |
---|---|---|---|---|

Total tree | R_{a}^{2} | 0.79 | 0.77 | 0.76 |

RMSE | 0.3544 | 0.3675 | 0.3722 | |

AIC | 762 | 835 | 858 | |

Mean error | 0.0065 | 0.0065 | 0.0079 | |

Mean absolute error | 0.2579 | 0.2664 | 0.2717 | |

Light crown | Mean error | 0.0224 | 0.0209 | 0.0206 |

Mean absolute error | 0.2467 | 0.2511 | 0.2588 | |

Shade crown | Mean error | −0.1155 | −0.1039 | −0.1357 |

Mean absolute error | 0.3436 | 0.3835 | 0.3713 bottom boder |

**Table 4.**Estimates for the parameters of the dummy variable regression model for the stand age and stand density (Equations (10) and (11)).

Dummy Variables | Parameters | a_{1} | a_{2} | a_{3}_{0} | a_{31} | a_{32} | a_{33} | a_{4} | a_{5} |
---|---|---|---|---|---|---|---|---|---|

Stand age | Estimates | 0.3283 | 0.8064 | −0.1450 | −0.4671 | −0.4477 | −0.6057 | 0.4686 | −1.3186 |

Std | 0.0296 | 0.0768 | 0.1175 | 0.1605 | 0.2065 | 0.1803 | 0.2818 | 0.2936 | |

Stand density | Estimates | 0.3385 | 0.7571 | −0.4034 | −0.3886 | −0.1364 | 0.1067 | 0.4734 | −1.3342 |

Std | 0.0296 | 0.0752 | 0.1539 | 0.2087 | 0.1148 | 0.1747 | 0.2752 | 0.2841 |

**Table 5.**The statistics for the third marginal model, quantile regression model (with the 0.90th, 0.95th, and 0.99th quantile), and the quantile regression for linear mixed-effects model for the 0.95th quantile.

Models | R^{2} | Var(ε_{p}) | [E(ε_{p})]^{2} | MEP | MAEP |
---|---|---|---|---|---|

M3 | 0.9380 | 0.2291 | 0.0017 | 0.0388 | 0.1626 |

q = 0.90 | 0.8789 | 0.2351 | 0.0510 | 0.2280 | 0.2312 |

q = 0.95 | 0.9342 | 0.2312 | 0.0046 | 0.0668 | 0.1635 |

q = 0.99 | 0.7987 | 0.3446 | 0.0577 | −0.2400 | 0.3187 |

q = 0.95 (qrLMM) | 0.9431 | 0.2190 | 0.0026 | −0.0545 | 0.1458 |

**Table 6.**Fitting statistics for the quantile regression for linear mixed-effects models of the 0.95th quantile with different random effects parameters.

Random Effect | AIC | BIC | HQ | logLike |
---|---|---|---|---|

b_{1} | 1091 | 1130 | 1106 | −537 |

b_{2} | 1399 | 1438 | 1414 | −691 |

b_{1}, b_{31} | 1064 | 1109 | 1081 | −523 |

b_{1}, b_{31}, b_{4} | 1036 | 1104 | 1062 | −504 |

**Table 7.**Estimates for the parameters and variance-covariance structure for the quantile regression for linear mixed-effects outer crown profile model.

Parameter | b_{1} | b_{2} | b_{31} | b_{32} | b_{33} | b_{4} | b_{5} |
---|---|---|---|---|---|---|---|

Estimates | 0.4084 | 0.6081 | 0.1240 | −0.0079 | 0.0001 | 1.2039 | −1.6362 |

Std | 0.0126 | 0.0358 | 0.0783 | 0.0029 | 0.0000 | 0.1353 | 0.1461 |

_{1}) = 0.0648, var(b

_{31}) = 0.8988, var(b

_{4}) = 0.0321, var(b

_{1}, b

_{31}) = 0.1875, var(b

_{1}, b

_{4}) = −0.2731, var(b

_{31}, b

_{4}) = −0.8317.

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Sun, Y.; Gao, H.; Li, F. Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted *Pinus sylvestris* var. *Mongolica* Trees. *Forests* **2017**, *8*, 446.
https://doi.org/10.3390/f8110446

**AMA Style**

Sun Y, Gao H, Li F. Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted *Pinus sylvestris* var. *Mongolica* Trees. *Forests*. 2017; 8(11):446.
https://doi.org/10.3390/f8110446

**Chicago/Turabian Style**

Sun, Yunxia, Huilin Gao, and Fengri Li. 2017. "Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted *Pinus sylvestris* var. *Mongolica* Trees" *Forests* 8, no. 11: 446.
https://doi.org/10.3390/f8110446