# Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data

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## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Material

#### 4.2. Methods

#### 4.2.1. Mixture Models and Inference

#### 4.2.2. Application of Mixture Models to Sample Plots

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A. A Brief Introduction to the EM Algorithm for Mixture Models Fitted to Ungrouped Data

#### Appendix A.1. M-Step of the EM Algorithm for Mixture of Gamma Distributions

#### Appendix A.2. M-Step of the EM Algorithm for Mixture of Log-Normal Distributions

#### Appendix A.3. M-Step of the EM Algorithm for Mixture of Weibull Distributions

## Appendix B. A Brief Introduction to the EM Algorithm for Mixture Models Fitted to Grouped Data

## Appendix C. Initial Values for Implementing the EM Algorithm Applied to Samples 1, 2, and 3

**Table A1.**Initial values of the EM algorithm for sample 1 obtained using the K-means clustering approach when DBH data are ungrouped (UG) and grouped (G) in classes of width 5 cm. It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | ||||
---|---|---|---|---|

K | Type | Family | ${\mathbf{\alpha}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\alpha}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\alpha}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ | ${\mathbf{\beta}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\beta}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\beta}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ |

2 | G | log-normal | ${(2.787,4.189)}^{{}^{\prime}}$ | ${(0.258,0.483)}^{{}^{\prime}}$ |

gamma | ${(38.508,13.458)}^{{}^{\prime}}$ | ${(1.771,1.356)}^{{}^{\prime}}$ | ||

Weibull | ${(7.1000,3.873)}^{{}^{\prime}}$ | ${(72.890,20.173)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(2.850,4.190)}^{{}^{\prime}}$ | ${(0.367,0.264)}^{{}^{\prime}}$ | |

gamma | ${(37.567,11.605)}^{{}^{\prime}}$ | ${(1.820,1.594)}^{{}^{\prime}}$ | ||

Weibull | ${(4.807,2.636)}^{{}^{\prime}}$ | ${(73.300,20.588)}^{{}^{\prime}}$ | ||

3 | G | log-normal | ${(4.411,4.120,2.694)}^{{}^{\prime}}$ | ${(0.277,0.155,0.683)}^{{}^{\prime}}$ |

gamma | ${(127.705,172.082,13.422)}^{{}^{\prime}}$ | ${(0.621,0.353,1.393)}^{{}^{\prime}}$ | ||

Weibull | ${(35.232,12.082,3.499)}^{{}^{\prime}}$ | ${(85.467,66.073,20.783)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(0.068,0.075,0.367)}^{{}^{\prime}}$ | ${(4.440,4.138,2.851)}^{{}^{\prime}}$ | |

gamma | ${(0.561,0.299,1.820)}^{{}^{\prime}}$ | ${(43.036,48.327,37.577)}^{{}^{\prime}}$ | ||

Weibull | ${(87.668,65.810,20.588)}^{{}^{\prime}}$ | ${(7.317,7.078,2.635)}^{{}^{\prime}}$ |

**Table A2.**Initial values of the EM algorithm for sample 2 obtained using the K-means clustering approach when DBH data are ungrouped (UG) and grouped (G) in classes of width 5 cm. It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | ||||
---|---|---|---|---|

K | Type | Family | ${\mathbf{\alpha}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\alpha}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\alpha}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ | ${\mathbf{\beta}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\beta}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\beta}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ |

2 | G | log-normal | ${(4.228,3.361)}^{{}^{\prime}}$ | ${(0.221,0.096)}^{{}^{\prime}}$ |

gamma | ${(28.139,15.983)}^{{}^{\prime}}$ | ${(0.699,2.291)}^{{}^{\prime}}$ | ||

Weibull | ${(29.191,4.347)}^{{}^{\prime}}$ | ${(68.226,31.488)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(4.257,3.417)}^{{}^{\prime}}$ | ${(0.209,0.349)}^{{}^{\prime}}$ | |

gamma | ${(484.296,12.085)}^{{}^{\prime}}$ | ${(0.142,2.374)}^{{}^{\prime}}$ | ||

Weibull | ${(10.724,3.512)}^{{}^{\prime}}$ | ${(70.542,31.526)}^{{}^{\prime}}$ | ||

3 | G | log-normal | ${(4.241,2.957,3.460)}^{{}^{\prime}}$ | ${(0.174,0.313,0.322)}^{{}^{\prime}}$ |

gamma | ${(1403.329,22.372,86.368)}^{{}^{\prime}}$ | ${(0.048,0.903,0.388)}^{{}^{\prime}}$ | ||

Weibull | ${(47.700,5.661,10.943)}^{{}^{\prime}}$ | ${(69.290,21.868,35.109)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(4.275,3.030,3.449)}^{{}^{\prime}}$ | ${(0.290,0.257,0.290)}^{{}^{\prime}}$ | |

gamma | ${(484.296,17.989,69.850)}^{{}^{\prime}}$ | ${(0.1442,1.113,0.470)}^{{}^{\prime}}$ | ||

Weibull | ${(10.724,3.356,7.815)}^{{}^{\prime}}$ | ${(70.542,21.847,34.643)}^{{}^{\prime}}$ |

**Table A3.**Initial values of the EM algorithm for sample 3 obtained using the K-means clustering approach when DBH data are ungrouped (UG) and grouped (G) in classes of width 2.5 cm. It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | ||||
---|---|---|---|---|

K | Type | Family | ${\mathbf{\alpha}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\alpha}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\alpha}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ | ${\mathbf{\beta}}^{\left(\mathbf{0}\right)}={\left({\mathbf{\beta}}_{\mathbf{1}}^{\left(\mathbf{0}\right)},\dots ,{\mathbf{\beta}}_{\mathrm{K}}^{\left(\mathbf{0}\right)}\right)}^{{}^{\prime}}$ |

1 | G | log-normal | 2.868 | 0.838 |

gamma | 2.544 | 9.841 | ||

Weibull | 1.217 | 26.718 | ||

UG | log-normal | 2.803 | 0.911 | |

gamma | 2.536 | 9.853 | ||

Weibull | 1.686 | 27.749 | ||

2 | G | log-normal | ${(4.076,2.710)}^{{}^{\prime}}$ | ${(0.349,0.408)}^{{}^{\prime}}$ |

gamma | ${(14.734,5.464)}^{{}^{\prime}}$ | ${(4.250,2.990)}^{{}^{\prime}}$ | ||

Weibull | ${(4.182,2.290)}^{{}^{\prime}}$ | ${(68.919,18.445)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(4.085,2.660)}^{{}^{\prime}}$ | ${(0.318,0.511)}^{{}^{\prime}}$ | |

gamma | ${(14.740,5.447)}^{{}^{\prime}}$ | ${(4.246,2.991)}^{{}^{\prime}}$ | ||

Weibull | ${(4.096,2.606)}^{{}^{\prime}}$ | ${(69.054,18.529)}^{{}^{\prime}}$ | ||

3 | G | log-normal | ${(4.237,3.330,2.251)}^{{}^{\prime}}$ | ${(0.171,0.477,0.215)}^{{}^{\prime}}$ |

gamma | ${(12.866,22.071,17.321)}^{{}^{\prime}}$ | ${(4.797,1.857,0.955)}^{{}^{\prime}}$ | ||

Weibull | ${(5.202,4.403,4.124)}^{{}^{\prime}}$ | ${(73.599,30.952,13.272)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(4.257,3.488,2.561)}^{{}^{\prime}}$ | ${(0.181,0.342,0.300)}^{{}^{\prime}}$ | |

gamma | ${(30.998,19.398,8.810)}^{{}^{\prime}}$ | ${(1.790,2.315,1.537)}^{{}^{\prime}}$ | ||

Weibull | ${(5.726,4.821,3.383)}^{{}^{\prime}}$ | ${(77.389,37.974,15.182)}^{{}^{\prime}}$ | ||

4 | G | log-normal | ${(4.309,3.883,3.125,2.521)}^{{}^{\prime}}$ | ${(0.295,0.257,0.232,0.435)}^{{}^{\prime}}$ |

gamma | ${(51.062,35.001,25.532,15.781)}^{{}^{\prime}}$ | ${(1.521,1.343,0.916,0.717)}^{{}^{\prime}}$ | ||

Weibull | ${(8.125,6.965,5.706,4.664)}^{{}^{\prime}}$ | ${(82.462,50.264,25.293,12.383)}^{{}^{\prime}}$ | ||

UG | log-normal | ${(4.310,3.899,3.234,2.463)}^{{}^{\prime}}$ | ${(0.303,0.077,0.199,0.210)}^{{}^{\prime}}$ | |

gamma | ${(52.172,47.226,28.442,12.874)}^{{}^{\prime}}$ | ${(1.495,1.048,0.911,0.933)}^{{}^{\prime}}$ | ||

Weibull | ${(7.217,7.001,5.565,4.124)}^{{}^{\prime}}$ | ${(82.857,52.662,28.022,13.272)}^{{}^{\prime}}$ |

## Appendix D. Figures A1–A5

**Figure A1.**Histograms of DBH data in sample 1. (

**a**): 2-component mixture models fitted to grouped data corresponding to classes of width 2.5 cm, (

**b**): 2-component mixture models fitted to ungrouped data, (

**c**): 2-component mixture models fitted to grouped data corresponding to classes of width 5 cm, (

**d**): 2-component mixture models fitted to ungrouped data, (

**e**): 3-component mixture models fitted to grouped data corresponding to classes of width 2.5 cm, (

**f**): 3-component mixture models fitted to ungrouped data, (

**g**): 3-component mixture models fitted to grouped data corresponding to classes of width 5 cm, (

**h**): 3-component mixture models fitted to ungrouped data. Superimposed are estimated pdf of the mixture of Weibull (solid line), gamma (dashed line), and log-normal (dotted line) distributions.

**Figure A2.**Histograms of DBH data in sample 2. (

**a**): 2-component mixture models fitted to grouped data corresponding to classes of width 2.5 cm, (

**b**): 2-component mixture models fitted to ungrouped data, (

**c**): 2-component mixture models fitted to grouped data corresponding to classes of width 5 cm, (

**d**): 2-component mixture models fitted to ungrouped data, (

**e**): 3-component mixture models fitted to grouped data corresponding to classes of width 2.5 cm, (

**f**): 3-component mixture models fitted to ungrouped data, (

**g**): 3-component mixture models fitted to grouped data corresponding to classes of width 5 cm, (

**h**): 3-component mixture models fitted to ungrouped data. Superimposed in each subfigure are estimated pdf of the mixture of Weibull (solid line), gamma (dashed line), and log-normal (dotted line) distributions.

**Figure A3.**Histograms of DBH data in sample 3 corresponding to classes of width 2.5 cm. Superimposed in each subfigure are estimated pdf of the mixture of Weibull (solid line), gamma (dashed line), and log-normal (dotted line) distributions. The fitted pdf in subfigures are related to: (

**a**) 2-component mixture models fitted to grouped data, (

**b**) 2-component mixture models fitted to ungrouped data, (

**c**) 3-component mixture models fitted to grouped data, (

**d**) 3-component mixture models fitted to ungrouped data, (

**e**) 4-component mixture models fitted to grouped data, and (

**f**) 4-component mixture models fitted to ungrouped data.

**Figure A4.**Histograms of DBH data in sample 3 corresponding to classes of width 5 cm. Superimposed in each subfigure are estimated pdf of the mixture of Weibull (solid line), gamma (dashed line), and log-normal (dotted line) distributions. The fitted pdf in subfigures are related to: (

**a**) 2-component mixture models fitted to grouped data, (

**b**) 2-component mixture models fitted to ungrouped data, (

**c**) 3-component mixture models fitted to grouped data, (

**d**) 3-component mixture models fitted to ungrouped data, (

**e**) 4-component mixture models fitted to grouped data, and (

**f**) 4-component mixture models fitted to ungrouped data.

**Figure A5.**Histogram of DBH observations for sample 1 (

**a**), sample 2 (

**b**), and sample 3 (

**c**). Superimposed are pdfs of NK, GSM, and mixture of log-normal models. The pdf of the gamma mixture model has been not shown for the ease of comparison.

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**Table 1.**Parameter estimates and goodness-of-fit statistics for the mixture of log-normal, gamma, and Weibull distributions fitted to sample 1 when DBH data are ungrouped (UG), grouped in classes of width 2.5 cm (G2.5), and grouped in classes of width 5 cm (G5). It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | Statistic | |||||
---|---|---|---|---|---|---|

K | Type | Family | $\mathbf{\alpha}={({\mathbf{\alpha}}_{\mathbf{1}},\dots ,{\mathbf{\alpha}}_{\mathrm{K}})}^{{}^{\prime}}$ | $\mathbf{\beta}={({\mathbf{\beta}}_{\mathbf{1}},\dots ,{\mathbf{\beta}}_{\mathrm{K}})}^{{}^{\prime}}$ | AIC | LL |

2 | G2.5 | log-normal | ${(4.210,2.894)}^{{}^{\prime}}$ | ${(0.153,0.288)}^{{}^{\prime}}$ | 73.476 | −31.738 |

gamma | ${(11.848,40.816)}^{{}^{\prime}}$ | ${(1.592,1.671)}^{{}^{\prime}}$ | 74.028 | −32.014 | ||

Weibull | ${(3.553,6.460)}^{{}^{\prime}}$ | ${(20.971,73.501)}^{{}^{\prime}}$ | 76.675 | −33.337 | ||

G5 | log-normal | ${(4.216,2.890)}^{{}^{\prime}}$ | ${(0.266,0.148)}^{{}^{\prime}}$ | 45.312 | −17.656 | |

gamma | ${(13.399,44.042)}^{{}^{\prime}}$ | ${(1.395,1.556)}^{{}^{\prime}}$ | 46.154 | −18.077 | ||

Weibull | ${(3.487,6.855)}^{{}^{\prime}}$ | ${(20.796,73.218)}^{{}^{\prime}}$ | 49.115 | −19.557 | ||

UG | log-normal | ${(2.876,4.213)}^{{}^{\prime}}$ | ${(0.287,0.157)}^{{}^{\prime}}$ | 207.174 | −98.587 | |

gamma | ${(39.902,12.080)}^{{}^{\prime}}$ | ${(1.714,1.532)}^{{}^{\prime}}$ | 207.815 | −98.907 | ||

Weibull | ${(6.397,3.629)}^{{}^{\prime}}$ | ${(73.265,20.551)}^{{}^{\prime}}$ | 211.038 | −100.519 | ||

3 | G2.5 | log-normal | ${(2.600,3.108,4.203)}^{{}^{\prime}}$ | ${(0.0371,0.1722,0.153)}^{{}^{\prime}}$ | 65.557 | −24.778 |

gamma | ${(334.213,108.835,12.499)}^{{}^{\prime}}$ | ${(0.251,0.575,1.512)}^{{}^{\prime}}$ | 70.185 | −27.092 | ||

Weibull | ${(22.398,12.600,3.724)}^{{}^{\prime}}$ | ${(86.140,64.990,20.979)}^{{}^{\prime}}$ | 70.969 | −27.484 | ||

G5 | log-normal | ${(4.110,4.135,2.947)}^{{}^{\prime}}$ | ${(0.023,0.104,0.253)}^{{}^{\prime}}$ | 46.263 | −15.131 | |

gamma | ${(15.288,90.470,0.122)}^{{}^{\prime}}$ | ${(1.288,0.694,0.122)}^{{}^{\prime}}$ | 46.670 | −15.335 | ||

Weibull | ${(45.341,10.354,3.836)}^{{}^{\prime}}$ | ${(83.511,63.075,21.076)}^{{}^{\prime}}$ | 48.099 | −16.049 | ||

UG | log-normal | ${(4.441,4.136,2.876)}^{{}^{\prime}}$ | ${(0.049,0.094,0.287)}^{{}^{\prime}}$ | 210.274 | −97.137 | |

gamma | ${(416.582,112.260,12.080)}^{{}^{\prime}}$ | ${(0.204,0.560,1.532)}^{{}^{\prime}}$ | 210.673 | −97.336 | ||

Weibull | ${(8.048,7.846,6.397)}^{{}^{\prime}}$ | ${(25.679,15.343,73.265)}^{{}^{\prime}}$ | 214.205 | −99.102 |

**Table 2.**Parameter estimates and goodness-of-fit statistics for the mixture of log-normal, gamma, and Weibull distributions fitted to sample 2 when DBH data are ungrouped (UG), grouped in classes of width 2.5 cm (G2.5), and grouped in classes of width 5 cm (G5). It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | Statistic | |||||
---|---|---|---|---|---|---|

K | Type | Family | $\mathbf{\alpha}={({\mathbf{\alpha}}_{\mathbf{1}},\dots ,{\mathbf{\alpha}}_{\mathrm{K}})}^{{}^{\prime}}$ | $\mathbf{\beta}={({\mathbf{\beta}}_{\mathbf{1}},\dots ,{\mathbf{\beta}}_{\mathrm{K}})}^{{}^{\prime}}$ | AIC | LL |

2 | G2.5 | log-normal | ${(4.230,3.318)}^{{}^{\prime}}$ | ${(0.025,0.289)}^{{}^{\prime}}$ | 72.305 | −31.152 |

gamma | ${(972.111,13.375)}^{{}^{\prime}}$ | ${(0.071,2.142)}^{{}^{\prime}}$ | 69.215 | −29.607 | ||

Weibull | ${(52.200,4.698)}^{{}^{\prime}}$ | ${(69.976,31.530)}^{{}^{\prime}}$ | 63.122 | −26.561 | ||

G5 | log-normal | ${(4.203,3.322)}^{{}^{\prime}}$ | ${(0.028,0.276)}^{{}^{\prime}}$ | 41.739 | −15.869 | |

gamma | ${(278.370,14.189)}^{{}^{\prime}}$ | ${(0.240,2.022)}^{{}^{\prime}}$ | 40.463 | −15.232 | ||

Weibull | ${(38.627,4.643)}^{{}^{\prime}}$ | ${(68.320,31.567)}^{{}^{\prime}}$ | 35.990 | −12.995 | ||

UG | log-normal | ${(4.234,3.318)}^{{}^{\prime}}$ | ${(0.037,0.289)}^{{}^{\prime}}$ | 304.256 | −147.128 | |

gamma | ${(732.216,13.388)}^{{}^{\prime}}$ | ${(0.094,2.143)}^{{}^{\prime}}$ | 300.953 | −145.476 | ||

Weibull | ${(39.990,4.706)}^{{}^{\prime}}$ | ${(70.189,31.421)}^{{}^{\prime}}$ | 294.357 | −142.178 | ||

3 | G2.5 | log-normal | ${(4.223,3.018,3.481)}^{{}^{\prime}}$ | ${(0.032,0.240,0.133)}^{{}^{\prime}}$ | 58.660 | −21.330 |

gamma | ${(638.393,18.451,59.230)}^{{}^{\prime}}$ | ${(0.106,1.128,0.553)}^{{}^{\prime}}$ | 58.522 | −21.261 | ||

Weibull | ${(41.376,4.926,7.611)}^{{}^{\prime}}$ | ${(69.697,21.784,35.645)}^{{}^{\prime}}$ | 60.877 | −22.438 | ||

G5 | log-normal | ${(4.200,2.943,3.469)}^{{}^{\prime}}$ | ${(0.020,0.168,0.135)}^{{}^{\prime}}$ | 38.490 | −11.245 | |

gamma | ${(632.997,37.412,53.350)}^{{}^{\prime}}$ | ${(0.105,0.507,0.605)}^{{}^{\prime}}$ | 38.658 | −11.329 | ||

Weibull | ${(38.628,6.351,7.438)}^{{}^{\prime}}$ | ${(68.320,21.144,35.285)}^{{}^{\prime}}$ | 42.186 | −13.0932 | ||

UG | log-normal | ${(4.234,2.972,3.477)}^{{}^{\prime}}$ | ${(0.037,0.228,0.123)}^{{}^{\prime}}$ | 300.263 | −142.131 | |

gamma | ${(732.216,20.710,65.696)}^{{}^{\prime}}$ | ${(0.094,0.967,0.496)}^{{}^{\prime}}$ | 299.940 | −141.970 | ||

Weibull | ${(39.990,5.932,8.117)}^{{}^{\prime}}$ | ${(70.189,21.657,35.461)}^{{}^{\prime}}$ | 300.493 | −142.246 |

**Table 3.**Parameter estimates and goodness-of-fit statistics for the mixture of log-normal, gamma, and Weibull distributions fitted to sample 3 when DBH data are ungrouped (UG) and grouped (G) in classes of width 2.5 cm. It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | Statistics | |||||
---|---|---|---|---|---|---|

K | Type | Family | $\mathbf{\alpha}={({\mathbf{\alpha}}_{\mathbf{1}},\dots ,{\mathbf{\alpha}}_{\mathrm{K}})}^{{}^{\prime}}$ | $\mathbf{\beta}={({\mathbf{\beta}}_{\mathbf{1}},\dots ,{\mathbf{\beta}}_{\mathrm{K}})}^{{}^{\prime}}$ | AIC | LL |

1 | G | log-normal | 2.956 | 0.693 | 214.360 | −105.180 |

gamma | 2.046 | 12.233 | 259.537 | −127.768 | ||

Weibull | 1.407 | 35.634 | 329.422 | −162.7114 | ||

UG | log-normal | 2.954 | 0.693 | 3201.910 | −1598.955 | |

gamma | 2.045 | 12.223 | 3272.889 | −1634.444 | ||

Weibull | 1.368 | 27.634 | 3307.548 | −1651.774 | ||

2 | G | log-normal | ${(4.000,2.668)}^{{}^{\prime}}$ | ${(0.350,0.435)}^{{}^{\prime}}$ | 175.664 | −82.832 |

gamma | ${(7.531,5.853)}^{{}^{\prime}}$ | ${(7.269,2.606)}^{{}^{\prime}}$ | 181.498 | −85.749 | ||

Weibull | ${(3.421,2.377)}^{{}^{\prime}}$ | ${(70.085,18.450)}^{{}^{\prime}}$ | 202.398 | −96.199 | ||

UG | log-normal | ${(4.101,2.689)}^{{}^{\prime}}$ | ${(0.263,0.448)}^{{}^{\prime}}$ | 3140.958 | −1565.479 | |

gamma | ${(14.141,5.081)}^{{}^{\prime}}$ | ${(4.403,3.207)}^{{}^{\prime}}$ | 3159.619 | −1574.517 | ||

Weibull | ${(3.796,2.308)}^{{}^{\prime}}$ | ${(67.822,18.481)}^{{}^{\prime}}$ | 3194.808 | −1592.176 | ||

3 | G | log-normal | ${(4.165,2.376,3.062)}^{{}^{\prime}}$ | ${(0.248,0.268,0.423)}^{{}^{\prime}}$ | 174.497 | −79.248 |

gamma | ${(12.858,13.141,12.418)}^{{}^{\prime}}$ | ${(4.797,1.857,0.955)}^{{}^{\prime}}$ | 176.244 | −80.122 | ||

Weibull | ${(4.704,2.496,3.875)}^{{}^{\prime}}$ | ${(75.592,31.643,13.306)}^{{}^{\prime}}$ | 193.224 | −88.612 | ||

UG | log-normal | ${(4.238,3.520,2.556)}^{{}^{\prime}}$ | ${(0.189,0.228,0.352)}^{{}^{\prime}}$ | 3139.964 | −1561.982 | |

gamma | ${(27.672,18.870,8.462)}^{{}^{\prime}}$ | ${(2.551,1.840,1.610)}^{{}^{\prime}}$ | 3152.290 | −1566.975 | ||

Weibull | ${(5.448,4.466,3.152)}^{{}^{\prime}}$ | ${(77.088,38.036,15.169)}^{{}^{\prime}}$ | 3188.805 | −1586.149 | ||

4 | G | log-normal | ${(4.331,3.922,3.146,2.428)}^{{}^{\prime}}$ | ${(0.167,0.218,0.306,0.288)}^{{}^{\prime}}$ | 180.238 | −79.119 |

gamma | ${(40.550,29.524,21.736,12.860)}^{{}^{\prime}}$ | ${(1.860,1.583,1.124,0.935)}^{{}^{\prime}}$ | 184.617 | −81.308 | ||

Weibull | ${(7.234,4.908,3.170,3.418)}^{{}^{\prime}}$ | ${(87.458,59.458,32.177,14.167)}^{{}^{\prime}}$ | 199.350 | −88.675 | ||

UG | log-normal | ${(4.329,3.892,3.236,2.462)}^{{}^{\prime}}$ | ${(0.143,0.144,0.188,0.292)}^{{}^{\prime}}$ | 3138.762 | −1558.381 | |

gamma | ${(47.339,48.153,28.019,12.138)}^{{}^{\prime}}$ | ${(1.621,1.028,0.924,1.007)}^{{}^{\prime}}$ | 3142.216 | −1560.108 | ||

Weibull | ${(7.656,6.784,5.427,4.000)}^{{}^{\prime}}$ | ${(52.661,82.269,28.017,13.269)}^{{}^{\prime}}$ | 3172.867 | −1575.433 |

**Table 4.**Parameter estimates and goodness-of-fit statistics for the mixture of log-normal, gamma, and Weibull distributions fitted to sample 3 in grouped case when class width is 5 cm. It should be noted that the estimated vector of mixing parameters is not given for the sake of saving space.

Estimated Parameters | Statistics | ||||
---|---|---|---|---|---|

K | Model | AIC | LL | ||

1 | log-normal | 2.950 | 0.700 | 136.568 | −66.284 |

gamma | 2.017 | 12.376 | 179.552 | −87.776 | |

Weibull | 1.403 | 34.300 | 237.321 | −116.660 | |

2 | log-normal | ${(3.531,2.481)}^{{}^{\prime}}$ | ${(0.568,0.303)}^{{}^{\prime}}$ | 103.765 | −46.882 |

gamma | ${(3.259,9.486)}^{{}^{\prime}}$ | ${(12.567,1.366)}^{{}^{\prime}}$ | 107.470 | −48.735 | |

Weibull | ${(3.638,2.276)}^{{}^{\prime}}$ | ${(70.807,18.911)}^{{}^{\prime}}$ | 132.321 | −61.160 | |

3 | log-normal | ${(4.145,3.104,2.379)}^{{}^{\prime}}$ | ${(0.257,0.368,0.198)}^{{}^{\prime}}$ | 95.901 | −39.950 |

gamma | ${(9.607,12.329,19.674)}^{{}^{\prime}}$ | ${(6.160,1.843,0.569)}^{{}^{\prime}}$ | 96.210 | −40.105 | |

Weibull | ${(6.104,2.586,2.626)}^{{}^{\prime}}$ | ${(83.387,48.790,17.167)}^{{}^{\prime}}$ | 129.290 | −56.645 | |

4 | log-normal | ${(4.320,3.881,3.098,2.388)}^{{}^{\prime}}$ | ${(0.170,0.227,0.296,0.200)}^{{}^{\prime}}$ | 101.564 | −39.782 |

gamma | ${(46.628,23.178,18.438,16.192)}^{{}^{\prime}}$ | ${(0.935,3.0233,1.298,0.716)}^{{}^{\prime}}$ | 103.357 | −40.678 | |

Weibull | ${(18.360,9.149,2.697,2.630)}^{{}^{\prime}}$ | ${(95.022,71.635,46.513,17.165)}^{{}^{\prime}}$ | 132.513 | −55.256 |

**Table 5.**Goodness-of-fit statistics for fitting mixture of gamma, mixture of log-normal, GSM, and NK models to samples 1 and 2 in ungrouped case. The number of components used for fitting the log-normal and gamma mixture models to samples 1 and 2 are 2 and 3, respectively. The number of components used for fitting GSM model to both samples is 250.

Sample | Measure | Family | |||
---|---|---|---|---|---|

Gamma | Log-Normal | NK | GSM | ||

1 | KS | 0.090 | $\mathbf{0}.\mathbf{088}$ | 0.143 | 0.106 |

AD | 0.214 | $\mathbf{0}.\mathbf{198}$ | 0.457 | 0.399 | |

CVM | 0.024 | $\mathbf{0}.\mathbf{022}$ | 0.056 | 0.045 | |

2 | KS | 0.110 | $\mathbf{0}.\mathbf{104}$ | 0.100 | 0.128 |

AD | $\mathbf{0}.\mathbf{218}$ | 0.221 | 0.255 | 0.671 | |

CVM | 0.035 | $\mathbf{0}.\mathbf{034}$ | 0.046 | 0.124 | |

3 | KS | 0.039 | $\mathbf{0}.\mathbf{039}$ | 0.026 | 0.094 |

AD | 0.519 | $\mathbf{0}.\mathbf{475}$ | 0.328 | 4.397 | |

CVM | 0.069 | 0.074 | 0.021 | 0.436 |

Sample | Sampling | Number of | Min. | 1st | Median | Mean | 3rd | Max. |
---|---|---|---|---|---|---|---|---|

Area | Trees (n) | Quartile | Quartile | |||||

1 | USA | 24 | 12.20 | 17.30 | 41.30 | 43.46 | 65.42 | 90.20 |

2 | USA | 40 | 11.40 | 24.43 | 31.10 | 31.73 | 34.65 | 71.10 |

3 | Iran | 399 | 6.00 | 11.25 | 16.50 | 25.00 | 29.12 | 101.50 |

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## Share and Cite

**MDPI and ACS Style**

Zenner, E.K.; Teimouri, M. Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data. *Forests* **2021**, *12*, 1196.
https://doi.org/10.3390/f12091196

**AMA Style**

Zenner EK, Teimouri M. Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data. *Forests*. 2021; 12(9):1196.
https://doi.org/10.3390/f12091196

**Chicago/Turabian Style**

Zenner, Eric K., and Mahdi Teimouri. 2021. "Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data" *Forests* 12, no. 9: 1196.
https://doi.org/10.3390/f12091196