# Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

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## Abstract

**:**

_{n}), number of acceptances by the KS test, the Cramér von Misses (W

^{2}) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·d

_{min}). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R

^{2}values of 0.750, 0.724 and 0.873 for d

_{g}, d

_{med}and d

_{max}. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset

^{2}, to achieve a minimum number of 30 trees per plot. Diameter at breast height (DBH at 1.3 m above the ground) of a total of 1685 trees was measured with callipers, to an accuracy of 0.1 cm. The following field and LiDAR variables were calculated from the inventory data: quadratic mean diameter, mean diameter, maximum diameter, number of trees per hectare, dominant height, basal area and total LiDAR return density within the plots (pulses·m

^{−2}). Summary statistics are shown in Table 1.

#### 2.2. Lidar Metrics

^{−2}, with a theoretical laser pulse density required for the PNOA project of 0.5 first returns per square meter. Summary statistics of the LiDAR return density per square meter within the plots are shown in Table 1. The LiDAR data were also processed with FUSION software [23]. As reported by the provider (https://pnoa.ign.es/) accessed on 1 April 2020, the vertical accuracy of the LiDAR metrics, given by the RMSE, is ≤0.20 m. The set of metrics from the points laid above 2 m was extracted for each plot (Table 2).

#### 2.3. Diameter Distribution Models and Fitting

#### 2.3.1. The Weibull Function

#### 2.3.2. The Beta Function

#### 2.3.3. The Generalized Beta Distribution (GBD)

#### 2.3.4. The Johnson’s SB Function

#### 2.3.5. The Gamma Function

_{min}was used for the location parameters, as [32] achieved good results with this value compared to other constraints. For the Weibull-2P and gamma-2P functions, the value of the location parameters was zero. Scale parameters and the upper limit of the four-parameter functions (Johnson’s SB, beta and generalized beta) were established as the maximum diameter of the distributions (d

_{max}) and the upper limit of the largest diameter class for the beta function [27,28] to improve parameters’ convergence. The size of the diameter classes was assumed to equal 1 cm.

#### 2.4. Goodness of Fit Evaluation

_{n}) and Cramér von Mises (W

^{2}) statistics, bias, mean square error (MSE) and number of acceptances by the KS test. For a given cumulative distribution function, F(x), ${D}_{n}=su{p}_{x}|{F}_{n}\left(x\right)-{F}_{0}\left(x\right)|$, where sup

_{x}is the supremum of the set of distances, calculated as follows [33]:

_{n}(x

_{i}) and F

_{0}(x

_{j}), are compared.

^{2}) is a measure of the square of the distance between the empirical and the cumulative theoretical distribution [35]:

_{i}is the observed relative frequency of trees in each diameter class, $\widehat{{Y}_{i}}$ is the theoretical value predicted by the model and N is the number of diameter classes. The bias and MSE were calculated for each fit as the mean relative frequency of trees.

#### 2.5. Recovering the Parameters of the Distributions from LiDAR Metrics

_{g}) and the mean diameter (d

_{m}), as follows:

_{g}and d

_{m}[10]. However, in the four-parameter functions (Johnson’s SB, beta and generalized beta), the range and the upper limit of the distributions can also be estimated. In this case, both values were used as the maximum diameter (d

_{max}), which was then related to LiDAR metrics. Location parameters of the recovered functions were predetermined as the minimum diameter inventory (7.5 cm) for the Weibull-3P, Johnson’s SB, beta and generalized beta functions, and as zero for the Weibull-2P and gamma-2P functions.

_{max}, d

_{g}and LiDAR metrics:

_{max}and d

_{g}are the dependent variables, X

_{1}, X

_{2}, …, X

_{m}represents the independent, potentially explanatory variables related to the LiDAR-derived height distribution and canopy closure, ${\alpha}_{0}$, ${\alpha}_{1}$, …${\alpha}_{n}$ and ${\beta}_{0}$, ${\beta}_{1}$, …${\beta}_{n}$ are the parameters to be estimated in the fitting process and $\epsilon $ is the additive error term, which is assumed to be independent and normally and identically distributed, with zero mean.

_{m}is always smaller than or equal to d

_{g}, and we therefore used the following model expression to take this restriction into account [9,10,11,36]:

_{1}, X

_{2}, …, X

_{m}are the potential explanatory variables related to the LiDAR-derived height distribution and canopy closure, ${\delta}_{0}$,${\delta}_{1}$,…${\delta}_{m}$ are the parameters to be estimated in the fitting process and $\epsilon $ is the additive error term.

_{g}was the dependent variable) and Equation (30) were fitted simultaneously with “seemingly unrelated regression” (SUR) to prevent cross-correlation between error components. Goodness of fits were evaluated with the coefficient of determination (R

^{2}) and the root mean square error (RMSE).

## 3. Results

_{n}), Cramér von Mises (W

^{2}), bias and mean squared error (MSE).

_{n}was yielded by the beta function (0.138560), followed by the Weibull-2P function (0.146922) and the Weibull-3P function (0.150761). The highest value was yielded by the gamma-2P function (0.164750), followed by the Johnson’s SB function (0.162578) and the generalized beta function (0.153656). Regarding the Cramér von Mises statistic (W

^{2}), the lowest value corresponded to the Johnson’s SB function (0.043088) and the highest to the Weibull-2P function (0.081863). The order of the functions in terms of W

^{2}was Johnson’s SB < Weibull-3P < generalized beta < gamma-2P < beta < Weibull-2P. The smallest MSE value corresponded to the beta function (0.001851), followed by the generalized beta function (0.001879). The highest MSE value was yielded by the Weibull-2P function (0.002009). The order for the six distributions studied was beta < generalized beta < Johnson’s SB < Weibull-3P < gamma-2P < Weibull-2P. The bias may be less important for comparison of the results because errors with different signs can compensate each other in the total mean value. Thus, considering the values of D

_{n}, W

^{2}and MSE for the six distributions studied, the beta function provided the most accurate fits to the observed data and the Weibull-2P and the Gamma-2P functions yielded the poorest fits.

^{2}, RMSE and % RMSE) of the simultaneous fitting of Equations (29) and (30) used to estimate the mean diameter (d

_{m}) and the quadratic mean diameter (d

_{g}) from LiDAR data and for Equation (28) used to estimate d

_{max}are shown in Table 5.

_{max}) was considered the scale parameter for the Weibull-3P, Johnson’s SB and generalized beta functions and for establishing the upper limit of the beta function. The maximum diameter was predicted by the following independent variables: mean absolute deviation for height (LH_AAD) and 95% height percentile (LH_P95). The linear model yielded an accuracy of R

^{2}= 0.873 and RMSE = 2.758. Models [29,30] used for simultaneous fitting of quadratic mean diameter (d

_{g}) and the mean diameter (d

_{med}) yielded R

^{2}values of 0.750 and 0.724 and RMSE values of 2.542 and 2.549, respectively. Good results were obtained for these key variables in terms of recovering the diameter distributions with the six functions.

_{n}was obtained by the beta function (0.254078), as in the fitting step, and then by the generalized beta function (0.264840) and the gamma-2P function (0.272107). The highest value was yielded by the Weibull-3P function (0.286170), followed by the Weibull-2P function (0.282500) and the Johnson’s SB function (0.273945).

^{2}) was also yielded by the beta function (0.381916) and the highest by the Weibull-2P function (0.498277). The order of the distributions for the W

^{2}value was beta < generalized beta < gamma-2P < Johnson’s SB < Weibull-3P < Weibull-2P. As for D

_{n}and W

^{2}, the smallest value for the MSE corresponded to the beta distribution (0.002649), followed by the generalized beta function (0.002711). The highest value of the MSE corresponded to the gamma-2P function (0.002792). The order in terms of MSE for the six distributions recovered was beta < generalized beta < Weibull-2P < Weibull-3P < gamma-2P < Johnson’s SB. As for the other statistics, bias was also higher for recovery of the distributions from LiDAR metrics than for fitting to the observed data. Results for the KS test are also consistent with the other statistics, with the following order for plots accepted: beta function (35 plots: 70%), gamma-2P function (34 plots: 68%), generalized beta function (33 plots: 66%), Johnson’s SB function (32 plots: 64%), Weibull-2P function (28 plots: 56%) and Weibull-3P function (26 plots: 52%). Thus, considering the values of D

_{n}, W

^{2}, MSE and KS test for the six distributions studied, the beta function was the most accurate for recovering the parameters from LiDAR data, followed by the generalized beta function. The Weibull-3P, Weibull-2P and the Johnson’s SB functions yielded poorer results.

## 4. Discussion

_{n}, W

^{2}and MSE) are similar to those obtained in [28] for Eucalyptus globulus stands in NW Spain. However, more accurate results were obtained in the same study for Pinus radiata stands by using the same functions. In a previous study, similar results were also obtained for Pinus sylvestris and better than those reported for Pinus pinaster stands in NW Spain [32].

_{g}, and LH_MAD_MEDIAN (median of the absolute deviations from the overall height median) for d

_{med}. In other studies, height percentiles were also used as independent variables for estimating d

_{g}(75% percentile and number of LiDAR last returns above a height of 1 m) and for d

_{med}(1% percentile) in the models obtained in [10] for recovering the Weibull-2P distribution in 25 plantation plots of Pinus radiata in Northwest Spain, with R

^{2}for the d

_{g}and d

_{med}models of 0.80 and 0.77, respectively. For Pinus halepensis, we obtained similar R

^{2}values (0.75 and 0.72 for d

_{g}and d

_{med}, respectively). The RMSE values (2.54 cm for d

_{g}and 2.55 cm for d

_{med}) are consistent with the values reported in international forestry literature. For example, for German forests dominated by Picea abies (L.) Karst., the authors of [39] used data from a 0.44 pulse m

^{−2}LiDAR flight and reported an RMSE of 2.44 cm for d

_{med}, while the authors of [40] studied a broad range of forest types (coniferous and hardwoods) and conditions across Ontario by using an artificially reduced LiDAR database of 0.5 pulses m

^{−2}, reporting RMSE values ranging from 0.76 to 4.3 cm for d

_{g}. The authors of [10] reported RMSE values of 3.42 for d

_{g}and 3.63 for d

_{med}, while the authors of [11] used exponential models instead of linear models to estimate the moments of the Johnson’s SB and the Weibull-2P functions, obtaining an R

^{2}of 0.82 and 0.86 for the d

_{med}of Pinus radiata and Eucalyptus globulus respectively, and 0.84 and 0.89 for d

_{g}of the same species. We compared the use of linear and exponential models for obtaining both variables and found that the results were similar, with the exponential model even including more independent variables. Thus, the linear models were considered more suitable.

_{min}, producing a similar mean value considering all plots of 7.18 cm (Table 3). The authors of [11] considered the location parameter for the Johnson’s SB to equal zero, thus avoiding modeling it with LiDAR metrics, while the scale parameter considered was also the maximum diameter (d

_{max}).

_{max}included, as LiDAR explanatory variables, LH_AAD (mean absolute deviation for height) and LH_P95 (95% height percentile), with R

^{2}= 0.87 and % RMSE = 10.56. The authors of [11] obtained similar values for Pinus radiata (R

^{2}= 0.93 and % RMSE = 8%) and Eucalyptus globulus (R

^{2}= 0.83 and % RMSE = 12%). The results for this variable modeled from LiDAR data are more accurate than those obtained in [27] and [9] with stand variables. The statistics obtained for recovery of the six functions were also reasonable. For example, the authors of [32] reported D

_{n}values of 0.1924 for the Weibull-2P fitted by Maximum Likelihood, 0.2285 for the Johnson’s SB fitted by conditional maximum likelihood (CML) and 0.1812 for the beta fitted by moments to observed distributions of Pinus pinaster in Northwest Spain. The authors of [33] obtained a D

_{n}value of 0.193 in the fits by maximum likelihood of the Weibull-3P to observed data of Pinus taeda plantations in the USA. The value obtained for the recovered beta from LiDAR metrics (0.2541) is close to these values. The number of KS acceptances are consistent with those found in previous studies. The percentage of acceptance (between 52% and 70%) was similar for Eucalyptus globulus in northwest Portugal [11] and higher than in Pinus radiata stands in northwest Spain [10,11].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**General location of the study area. Filled dots represent plots with available LiDAR information.

**Figure 2.**(

**a**–

**c**) Behavior of the MSE in each diameter class for the fitting and recovery steps: beta vs. Johnson’ SB; generalized beta vs. Weibull-3P and gamma-2P vs. Weibull-2P.

**Figure 3.**(

**a**–

**f**). Paired comparisons: beta vs. Johnson’s SB (

**a**,

**b**), generalized beta vs. Weibull-3P (

**c**,

**d**) and gamma-2P vs. Weibull-2P (

**e**,

**f**) for the observed and described (fitted) distributions recovered from LiDAR data.

Species | Variable | Mean | Max | Min | SD |
---|---|---|---|---|---|

Pinus halepensis | d_{g} | 17.8 | 30.6 | 11.7 | 5.1 |

d_{med} | 17.3 | 29.6 | 11.4 | 4.8 | |

d_{max} | 25.7 | 49.6 | 18.8 | 7.4 | |

N | 1054 | 3200 | 176 | 588.4 | |

Ho | 10.1 | 19.1 | 6.2 | 3.2 | |

G | 24.1 | 58.9 | 6.3 | 11.9 | |

LRD | 1.070 | 2.222 | 0.453 | 0.421 |

_{g}: quadratic mean diameter; d

_{med}: mean diameter; d

_{max}: maximum diameter; N: number of trees per hectare; Ho: dominant height; G: basal area; LRD: total LiDAR return density within the plots (pulses·m

^{−2}).

Variables Related to Height Distribution (m) | Description |

LH_MIN, LH_MAX, LH_MEAN | Minimum, maximum and mean height |

LH_MODE, LH_MEDIAN, LH_SD, LH_CV | Mode, median, standard deviation and height’s coefficient of variation |

LH_SK, LH_KUR | Skewness and kurtosis |

LH_IQ | Interquartile amplitude |

LH_AAD | Mean absolute deviation |

LH_MAD_MEDIAN, LH_MAD_MODE | Median of the absolute deviations from the overall height median (LH_MAD_MEDIAN) and mode (LH_MAD_MODE) |

LH_L1, LH_L2…, LH_L4 | L moments |

INT_L_SK, INT_L_KUR | Linear combinations of L moments (skewness and kurtosis) |

LH_P05,…, LH_P95 | Percentiles |

LH_P25; LH_P75 | First and third quartiles |

Variables Related to Canopy Closure (%) | Description |

LFCC | Percentage of first returns above 2 m |

LFCC_MEAN | Percentage of first returns above LH_MEAN |

LFCC_MODE | Percentage of first returns above LH_MODE |

LFCC_ALL | Percentage of all returns above 2 m |

LFCC_ALL_MEAN | Percentage of all returns above LH_MEAN |

LFCC_ALL_MODE | Percentage of all returns above LH_MODE |

ALL_MEAN_FIRST | 100* all returns above LH_MEAN / total first returns |

ALL_FIRST | 100* all returns above 2 m / total first returns |

R2_COUNT | Number of first returns above 2 m |

CANOPY | Canopy relief ratio: (h _{mean} − h_{min})/(h_{max} − h_{min}) |

**Table 3.**Descriptive statistics for the parameters of the functions in the fitting and recovery steps.

Function | Step | Param | Mean | SD | Min | Max |
---|---|---|---|---|---|---|

Weibull-2P | Fitting | b | 18.757 | 5.072 | 12.415 | 32.084 |

c | 4.905 | 1.125 | 2.163 | 8.020 | ||

Recovery | b | 18.918 | 4.614 | 13.699 | 33.740 | |

c | 4.873 | 0.825 | 2.215 | 6.577 | ||

Weibull-3P | Fitting | a | 7.184 | 1.750 | 5.625 | 14.100 |

b | 11.222 | 4.005 | 6.312 | 24.255 | ||

c | 2.598 | 0.460 | 1.535 | 3.560 | ||

Recovery | a | 7.500 | - | 7.500 | 7.500 | |

b | 11.008 | 4.637 | 5.513 | 25.319 | ||

c | 2.489 | 0.619 | 1.245 | 3.646 | ||

beta | Fitting | c | 0.006 | 0.015 | 1.47 × 10^{–6} | 0.107 |

L | 7.184 | 1.750 | 5.625 | 14.100 | ||

U | 25.739 | 7.403 | 16.700 | 49.600 | ||

α | 1.191 | 0.633 | 0.250 | 2.574 | ||

γ | 1.015 | 0.720 | 0.166 | 3.178 | ||

Recovery | c | 0.010 | 0.015 | 4.34 × 10^{–7} | 0.069 | |

L | 7.500 | - | 7.500 | 7.500 | ||

U | 26.100 | 7.229 | 18.000 | 55.000 | ||

α | 0.972 | 0.690 | 0.017 | 3.023 | ||

γ | 0.734 | 0.488 | 0.046 | 2.003 | ||

Generalized beta | Fitting | $\complement $ | 471.180 | 925.744 | 1.251 | 5827.044 |

B1 | 7.184 | 1.750 | 5.625 | 14.100 | ||

B2 | 25.739 | 7.403 | 16.700 | 49.600 | ||

B3 | 2.253 | 1.034 | 0.543 | 4.885 | ||

B4 | 4.285 | 1.806 | 0.358 | 9.363 | ||

Recovery | $\complement $ | 365.657 | 868.723 | 1.428 | 5125.128 | |

B1 | 7.500 | - | 7.500 | 7.500 | ||

B2 | 26.104 | 7.222 | 18.286 | 55.318 | ||

B3 | 2.057 | 1.100 | 0.435 | 4.647 | ||

B4 | 4.116 | 1.015 | 0.974 | 6.354 | ||

Johnson’s SB | Fitting | ε | 7.184 | 1.750 | 5.625 | 14.100 |

λ | 25.739 | 7.403 | 16.700 | 49.600 | ||

γ | 0.727 | 0.344 | 0.027 | 1.379 | ||

δ | 1.327 | 0.235 | 0.630 | 1.879 | ||

Recovery | ε | 7.500 | - | 7.500 | 7.500 | |

λ | 26.104 | 7.222 | 18.286 | 55.318 | ||

γ | 0.801 | 0.409 | −0.152 | 1.457 | ||

δ | 1.295 | 0.190 | 0.698 | 1.724 | ||

gamma-2P | Fitting | α | 19.906 | 8.426 | 4.636 | 48.331 |

β | 1.062 | 0.815 | 0.382 | 5.329 | ||

Recovery | α | 18.594 | 5.410 | 4.398 | 31.558 | |

β | 1.089 | 0.802 | 0.562 | 5.093 |

Function | D_{n} | W^{2} | Bias | MSE |
---|---|---|---|---|

Weibull-2P | 0.146922 | 0.081863 | 0.003424 | 0.002009 |

Weibull-3P | 0.150761 | 0.046157 | 0.002637 | 0.001904 |

beta | 0.138560 | 0.062044 | 0.001643 | 0.001851 |

Generalized beta | 0.153656 | 0.048215 | 0.002563 | 0.001879 |

Johnson’s SB | 0.162578 | 0.043088 | 0.002276 | 0.001892 |

gamma-2P | 0.164750 | 0.054360 | 0.002736 | 0.001996 |

_{n}: Kolmogorov–Smirnov statistic; W

^{2}: Cramér von Mises statistic; MSE: mean squared error.

**Table 5.**Parameter estimates and goodness-of-fit statistics for the simultaneous fitting of Equations (29) and (30) used to estimate d

_{med}and d

_{g}from LiDAR data and for Equation (28) used to estimate d

_{max}.

Equation | Dep var | Independent Variable | Param | Param Estim | $\mathbf{P}>|\mathit{t}|$ | R2 | RMSE | RMSE% |
---|---|---|---|---|---|---|---|---|

(28) | d_{max} | Intercept | ${\alpha}_{0}$ | 6.733 | <0.0001 | 0.873 | 2.758 | 10.56 |

LH_AAD | ${\alpha}_{1}$ | 8.143 | <0.0001 | |||||

LH_P95 | ${\alpha}_{2}$ | 7.728 | <0.0001 | |||||

(29) | d_{g} | Intercept | ${\beta}_{0}$ | 5.329 | <0.0001 | 0.750 | 2.542 | 14.24 |

LH_P90 | ${\beta}_{1}$ | 1.335 | <0.0001 | |||||

(30) | d_{med} | Intercept | ${\delta}_{0}$ | −1.789 | <0.0001 | 0.724 | 2.549 | 14.72 |

LH_MAD_MEDIAN | ${\delta}_{1}$ | 1.049 | <0.0001 |

Function | D_{n} | W^{2} | Bias | MSE | KS Acceptance (%) |
---|---|---|---|---|---|

Weibull-2P | 0.282500 | 0.498277 | 0.004303 | 0.002732 | 28 (56%) |

Weibull-3P | 0.286170 | 0.497091 | 0.003619 | 0.002746 | 26 (52%) |

beta | 0.254078 | 0.381916 | 0.003744 | 0.002649 | 35 (70%) |

Generalized beta | 0.264840 | 0.389917 | 0.003673 | 0.002711 | 33 (66%) |

Johnson’s SB | 0.273945 | 0.411657 | 0.003578 | 0.002851 | 32 (64%) |

Gamma-2P | 0.272107 | 0.398073 | 0.004047 | 0.002792 | 34 (68%) |

_{n}: Kolmogorov–Smirnov statistic; W

^{2}: Cramer von Mises statistic; MSE: mean squared error.

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## Share and Cite

**MDPI and ACS Style**

Gorgoso-Varela, J.J.; Ponce, R.A.; Rodríguez-Puerta, F.
Modeling Diameter Distributions with Six Probability Density Functions in *Pinus halepensis* Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain). *Remote Sens.* **2021**, *13*, 2307.
https://doi.org/10.3390/rs13122307

**AMA Style**

Gorgoso-Varela JJ, Ponce RA, Rodríguez-Puerta F.
Modeling Diameter Distributions with Six Probability Density Functions in *Pinus halepensis* Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain). *Remote Sensing*. 2021; 13(12):2307.
https://doi.org/10.3390/rs13122307

**Chicago/Turabian Style**

Gorgoso-Varela, J. Javier, Rafael Alonso Ponce, and Francisco Rodríguez-Puerta.
2021. "Modeling Diameter Distributions with Six Probability Density Functions in *Pinus halepensis* Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)" *Remote Sensing* 13, no. 12: 2307.
https://doi.org/10.3390/rs13122307