# Live Crown Ratio Models for Loblolly Pine (Pinus taeda) with Beta Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Methods and Models

#### 2.3. Beta Regression

#### 2.4. Model Evaluation and Additional Predictors

^{2}/ha) of trees larger than the subject tree in the given plot, were added to the model. The contribution of stand-level variables was assessed by comparing errors produced by enhanced models with the errors obtained from the base model.

## 3. Results and Discussion

^{2}) for the fitted models ranged from 0.65 for the exponential PTAEDA 4.0 model to 0.69 for the beta regression model (Table 3). Note that the number of parameters estimated in these models is different. The PTAEDA 4.0 model requires the estimation of only two parameters, the Weibull model has three parameters, and Richards and beta models have four parameters. However, since the number of samples used in this study is large (n = 20,330), the difference between R

^{2}and adjusted R

^{2}is negligible. Furthermore, the number of independent (predictor) variables required by all models is same, i.e., all models require age, dbh, and total height to predict LCR. All models had similar residual plots, with residuals forming a band around zero without any trend. The predicted values of LCR obtained from all models are plotted against measured (observed) values of loblolly pine LCR in Figure 4.

## 4. Conclusions

^{2}) compared to the other models and generally produced a smaller root mean square error. The beta regression model used in this study produced slightly smaller error statistics (performed better) than the nonlinear models fit using the ordinary least-squares method, as well as the crown ratio model used in a widely used individual tree growth and yield model (PTAEDA 4.0) for loblolly pine in the southern United States. Furthermore, the assumption of error distribution in beta regression makes it more appropriate for fitting LCR models. Family effects were significantly different but family NC3, which was characterized by its slow growth and small crown, was not significantly different from the local commercial check. The basal area of larger trees did not reduce error, but the addition of trees per hectare did slightly improve model performance. Crown ratio is measured in percentage unit and should be modeled using generalized linear models that assume a beta distribution for error terms.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationship between live crown ratio (LCR) and dbh for different families of loblolly pine. There is no obvious trend in LCR-dbh relationship for any family used in this study. This trend was consistent when plotted by family and spacing.

**Figure 2.**Relationship between live crown ratio (LCR) and total tree height for different families of loblolly pine. LCR slightly increased with increasing height and then declined and stabilized. This trend was consistent when plotted by family and spacing.

**Figure 4.**Fitted values for loblolly pine live crown ratio models evaluated within this study plotted against the observed values.

**Table 1.**Mean and standard deviation (in parenthesis) for tree-level variables by spacing and family. NC8 is fast growing with small crown, NC4 and NC7 are fast growing with large crown, NC3 and NC6 are slow growing with small crown, NC2 and NC5 are characterized by slow growth and large crown. LCK is a “local commercial check”.

Family | dbh (cm) | Height (m) | Live Crown Ratio | ||||||
---|---|---|---|---|---|---|---|---|---|

1.5 × 1.5 | 2.4 × 2.4 | 3.0 × 3.0 | 1.5 × 1.5 | 2.4 × 2.4 | 3.0 × 3.0 | 1.5 × 1.5 | 2.4 × 2.4 | 3.0 × 3.0 | |

NC1 | 13.26 (3.29) | 18.45 (4.06) | 21.24 (4.78) | 15.09 (3.83) | 16.79 (4.58) | 16.81 (4.70) | 0.37 (0.10) | 0.44 (0.13) | 0.48 (0.14) |

NC2 | 13.35 (3.58) | 18.19 (4.56) | 21.36 (5.42) | 14.82 (4.06) | 16.32 (4.76) | 16.28 (4.94) | 0.36 (0.11) | 0.44 (0.13) | 0.51 (0.15) |

NC3 | 13.33 (3.56) | 18.44 (4.47) | 20.97 (4.83) | 14.48 (3.94) | 15.83 (4.65) | 16.08 (4.88) | 0.37 (0.10) | 0.46 (0.13) | 0.51 (0.14) |

NC4 | 13.51 (3.59) | 18.27 (4.05) | 21.50 (4.79) | 14.86 (4.00) | 16.41 (4.51) | 16.74 (4.74) | 0.36 (0.10) | 0.41 (0.12) | 0.47 (0.14) |

NC5 | 13.16 (3.52) | 17.94 (4.35) | 20.76 (5.52) | 14.58 (4.11) | 15.91 (4.79) | 15.96 (5.14) | 0.35 (0.10) | 0.43 (0.13) | 0.49 (0.15) |

NC6 | 13.11 (3.53) | 18.17 (4.45) | 20.20 (4.89) | 14.16 (3.96) | 15.49 (4.56) | 15.27 (4.70) | 0.35 (0.10) | 0.43 (0.11) | 0.49 (0.13) |

NC7 | 13.69 (3.55) | 18.89 (4.43) | 20.89 (5.39) | 15.23 (4.15) | 16.72 (4.97) | 16.23 (5.02) | 0.36 (0.10) | 0.44 (0.12) | 0.50 (0.14) |

NC8 | 13.81 (3.72) | 18.67 (4.82) | 21.94 (6.10) | 14.98 (4.16) | 16.56 (4.91) | 16.29 (5.29) | 0.37 (0.11) | 0.44 (0.13) | 0.50 (0.14) |

LCK | 13.37 (3.86) | 18.52 (4.66) | 21.50 (5.18) | 13.87 (3.95) | 15.14 (4.60) | 15.33 (4.66) | 0.38 (0.11) | 0.46 (0.14) | 0.52 (0.14) |

**Table 2.**Models and equations for predicting live crown ratio of loblolly pine trees used in this study.

Model | Prediction Equation | |
---|---|---|

PTAEDA 4.0 | $\widehat{\mathrm{LCR}}=1-\mathrm{exp}\left(\left({\mathrm{b}}_{0}-\frac{{\mathrm{b}}_{1}}{\mathrm{A}}\right)\left(\frac{\mathrm{D}}{\mathrm{H}}\right)\right)$ | (1) |

Richards | $\widehat{\mathrm{LCR}}=\frac{1}{{\left[1+\mathrm{exp}\left(-\left({\mathrm{b}}_{0}+\frac{{\mathrm{b}}_{1}}{\mathrm{A}}+{\mathrm{b}}_{2}\mathrm{D}+{\mathrm{b}}_{3}\mathrm{H}+{\mathrm{b}}_{4}\left(\frac{\mathrm{D}}{\mathrm{H}}\right)\right)\right)\right]}^{\frac{1}{6}}}$ | (2) |

Weibull | $\widehat{\mathrm{LCR}}=1-\mathrm{exp}\left(-{\left({\mathrm{b}}_{0}+\frac{{\mathrm{b}}_{1}}{\mathrm{A}}+{\mathrm{b}}_{2}\mathrm{D}+{\mathrm{b}}_{3}\mathrm{H}\right)}^{10}\right)$ | (3) |

Beta | $\widehat{\mathrm{LCR}}=\frac{\mathrm{exp}\left({\mathrm{b}}_{0}+{\mathrm{b}}_{1}\mathrm{A}+{\mathrm{b}}_{2}\mathrm{D}+{\mathrm{b}}_{3}\mathrm{H}+{\mathrm{b}}_{4}\left(\frac{\mathrm{D}}{\mathrm{H}}\right)\right)}{1+\mathrm{exp}\left({\mathrm{b}}_{0}+{\mathrm{b}}_{1}\mathrm{A}+{\mathrm{b}}_{2}\mathrm{D}+{\mathrm{b}}_{3}\mathrm{H}+{\mathrm{b}}_{4}\left(\frac{\mathrm{D}}{\mathrm{H}}\right)\right)}$ | (4) |

**Table 3.**Parameter estimates, standard errors, and the coefficient of determination (R

^{2}) for live crown ratio models fitted within the study. Prediction equations are provided in Table 2.

Model | Parameter Estimate * (Standard Error) | R^{2} | ||||
---|---|---|---|---|---|---|

b_{0} | b_{1} | b_{2} | b_{3} | b_{4} | ||

PTAEDA 4.0 | −0.1823 (0.0029) | 3.9966 (0.0364) | - | - | - | 0.65 |

Richards | −15.9744 (0.1858) | 66.4144 (0.9222) | 0.0728 (0.0075) | 0.1010 (0.0100) | 2.3063 (0.099) | 0.66 |

Weibull | −0.7719 (0.0030) | −1.403 (0.0191) | −0.0048 (0.0001) | 0.0016 (0.0001) | - | 0.65 |

Beta | −1.3176 (0.0357) | −0.1175 (0.0014) | −0.0389 (0.0022) | 0.1006 (0.0028) | 1.5514 (0.0312) | 0.69 |

**Table 4.**Evaluation statistics produced by different models for predicting crown ratio of loblolly pine. Smallest values are preferred, and the best results are given in

**bold**.

Model | MAE | MAPE | RMSE | RMSE % |
---|---|---|---|---|

PTAEDA 4.0 | 0.06 | 16.84 | 0.08 | 18.79 |

Weibull | 0.06 | 16.72 | 0.08 | 18.81 |

Richards | 0.06 | 16.58 | 0.08 | 18.44 |

Beta | 0.05 | 15.45 | 0.07 | 17.58 |

**Table 5.**Root mean square error percent and mean absolute percent error produced by four LCR models at different ages.

Method | RMSE Percent | MAPE | ||||||
---|---|---|---|---|---|---|---|---|

Age 9 | Age 13 | Age 17 | Age 21 | Age 9 | Age 13 | Age 17 | Age 21 | |

Beta | 15.78 | 17.70 | 18.41 | 20.91 | 13.51 | 16.45 | 15.77 | 16.97 |

PTAEDA 4.0 | 16.11 | 20.00 | 19.75 | 22.26 | 14.48 | 17.90 | 16.54 | 19.71 |

Richards | 15.93 | 19.44 | 18.26 | 23.51 | 14.30 | 16.90 | 15.19 | 21.74 |

Weibull | 16.85 | 18.94 | 18.32 | 24.16 | 15.43 | 16.72 | 15.01 | 20.98 |

**Table 6.**RMSE percent and MAPE by age by incorporating additional stand-level variables (BAL and TPH) in the beta regression model.

Method | RMSE Percent | MAPE | ||||||
---|---|---|---|---|---|---|---|---|

Age 9 | Age 13 | Age 17 | Age 21 | Age 9 | Age 13 | Age 17 | Age 21 | |

Base Model | 15.78 | 17.7 | 18.41 | 20.91 | 13.51 | 16.45 | 15.77 | 16.97 |

Base + BAL | 15.78 | 17.68 | 18.35 | 20.91 | 13.52 | 16.45 | 15.72 | 16.97 |

Base + TPH | 14.23 | 16.91 | 18.08 | 21.33 | 12.30 | 15.94 | 15.68 | 16.97 |

Base + BAL + TPH | 13.99 | 16.76 | 18.26 | 21.43 | 12.14 | 15.72 | 15.98 | 16.99 |

**Table 7.**Parameter estimates and their standard errors (in parenthesis) for family-specific beta regression models for predicting live crown ratio of loblolly pine.

Family | b_{0} | b_{1} | b_{2} | b_{3} | b_{4} |
---|---|---|---|---|---|

NC1 | −1.5167 (0.1031) | −0.1252 (0.0038) | −0.0619 (0.0063) | 0.1274 (0.0082) | 1.8206 (0.0927) |

NC2 | −1.7774 (0.1138) | −0.1186 (0.0045) | −0.0671 (0.0069) | 0.1245 (0.0091) | 2.1049 (0.1011) |

NC3 | −1.5013 (0.1101) | −0.1059 (0.0044) | −0.0487 (0.0070) | 0.1029 (0.0092) | 1.7306 (0.0959) |

NC4 | −0.9988 (0.1094) | −0.1433 (0.0042) | −0.0209 (0.0068) | 0.1057 (0.0088) | 1.1797 (0.0965) |

NC5 | −1.0764 (0.1047) | −0.1174 (0.0042) | −0.0234 (0.0065) | 0.082 (0.00840) | 1.3402 (0.0911) |

NC6 | −1.4602 (0.1102) | −0.0979 (0.0043) | −0.0368 (0.0071) | 0.0926 (0.0093) | 1.4788 (0.0955) |

NC7 | −1.3677 (0.1079) | −0.1072 (0.0039) | −0.0348 (0.0065) | 0.0882 (0.0082) | 1.5712 (0.0951) |

NC8 | −1.1268 (0.0975) | −0.1173 (0.0040) | −0.0357 (0.0059) | 0.0923 (0.0075) | 1.4463 (0.0856) |

LCK | −1.4793 (0.1087) | −0.1377 (0.0043) | −0.0477 (0.0067) | 0.1359 (0.0090) | 1.6572 (0.0900) |

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**MDPI and ACS Style**

Poudel, K.P.; Avery, S.C.; Granger, J.J.
Live Crown Ratio Models for Loblolly Pine (*Pinus taeda*) with Beta Regression. *Forests* **2021**, *12*, 1409.
https://doi.org/10.3390/f12101409

**AMA Style**

Poudel KP, Avery SC, Granger JJ.
Live Crown Ratio Models for Loblolly Pine (*Pinus taeda*) with Beta Regression. *Forests*. 2021; 12(10):1409.
https://doi.org/10.3390/f12101409

**Chicago/Turabian Style**

Poudel, Krishna P., Samantha C. Avery, and Joshua J. Granger.
2021. "Live Crown Ratio Models for Loblolly Pine (*Pinus taeda*) with Beta Regression" *Forests* 12, no. 10: 1409.
https://doi.org/10.3390/f12101409