# Approximating Forest Resource Dynamics in Peninsular Malaysia Using Parametric and Nonparametric Models, and Its Implications for Establishing Forest Reference (Emission) Levels under REDD+

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Stages of Forest Transition Curves

#### 2.2. Study Site and Data

#### 2.3. Methods

#### 2.3.1. Parametric Models

#### 2.3.2. Nonparametric Models

_{i}are a random sample from a continuous population that has median 0 [53,54,55]. For a linear model or a nonlinear model, the function m(

**x**

_{i}) was specified in advance. However, for a nonparametric regression model, the function is not specified in advance. Fox and Weisberg [55] emphasized that “the object of nonparametric regression is to estimate the regression function m(

**x**

_{i}) directly, rather than to estimate parameters.” In this research, three smoothing approaches introduced by Hollander et al. [53], namely, Friedman local averaging, Nadaraya–Watson kernel smoothing, and Cleveland local regression models, were applied.

#### 2.3.3. Model Comparations

_{i}− fitted value

_{i})

^{2}is the residual sum of squares, and K is the total number of free parameters, including intercept and σ

^{2}. When the sample size, n, is small, AIC may perform poorly. Therefore, a bias-corrected version of AIC, AICc was suggested [58,59,60,61,62]). AICc was defined as

_{j}) were calculated as

## 3. Results

#### 3.1. Parametric Models

_{j}, RMSE and MAE. First, eight models are ranked in the same order in their dimensions of RSS, log (L), AICc, RMSE, and MAE. The biexponential model is at the top with the lowest AICc. However, Δ

_{2}(AICc) is 0.95, which is less than 2, thereby showing that the second model, which is the Gompertz growth model, can be considered one of the best models. The quartic model ranked third and is less plausible compared with the best model by Δ

_{3}(AICc) of 9.182. However, the quartic model outperformed five other parametric models. The linear model is the poorest among the eight models. Second, the results by Akaike weights, w

_{j}, show that the weight of evidence or probability for the biexponential model, being the best model, was 0.612; that of the Gompertz growth model was 0.382; that of the quartic model was 0.006; and that of all the others was 0. The evidence ratio was calculated by w

_{i}/w

_{j}, and we obtained w

_{1}/w

_{2}= 1.60, w

_{1}/w

_{3}= 102, w

_{2}/w

_{3}= 63.67, thereby showing that the first model is not the best model. However, both the first and second models are much better than the third model. Given that 0.612 + 0.382 = 0.994, the evidence shows that the best is among the first two models. For polynomial functions, the order is as follows: quartic > cubic > quadratic > linear.

#### 3.2. Nonparametric Models

#### 3.3. Cross-Validation

## 4. Discussion

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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No. | Models | Equations |
---|---|---|

1 | Linear | $y=a+bx$ |

2 | Quadratic | $y=a+bx+c{x}^{2}$ |

3 | Cubic | $y=a+bx+c{x}^{2}+d{x}^{3}$ |

4 | Quartic | $y=a+bx+c{x}^{2}+d{x}^{3}+e{x}^{4}$ |

5 | Power | $y=a{x}^{b}$ |

6 | Michaelis–Menten | $y=({V}_{max}x)/({K}_{m}+x)$ |

7 | Biexponential | $y={A}_{1}\mathrm{exp}(-{e}^{lrc1}\ast x)+{A}_{2}\mathrm{exp}(-{e}^{lrc2}\ast x)$ |

8 | Gompertz growth | $y=a{b}^{x}+c$ |

No. | Models | RSS | Log (L) | No. Par | AICc | Δ_{j} | w_{j} | RMSE | MAE |
---|---|---|---|---|---|---|---|---|---|

1 | Biex. | 419,622 | −275.00 | 5 | 561.49 | 0.00 | 0.612 | 96 | 67 |

2 | Gom. | 452,497 | −276.73 | 4 | 562.44 | 0.95 | 0.382 | 99 | 74 |

3 | Quartic | 483,607 | −278.26 | 6 | 570.67 | 9.18 | 0.006 | 103 | 74 |

4 | Cubic | 833,612 | −290.78 | 5 | 593.07 | 31.58 | 0.000 | 135 | 107 |

5 | M-M | 902,040 | −292.60 | 3 | 591.77 | 30.28 | 0.000 | 140 | 114 |

6 | Qua. | 2,246,443 | −313.58 | 4 | 636.14 | 74.65 | 0.000 | 221 | 177 |

7 | Power | 2,698,227 | −317.80 | 3 | 642.17 | 80.68 | 0.000 | 242 | 201 |

8 | Linear | 5,017,771 | −332.07 | 3 | 670.70 | 109.21 | 0.000 | 330 | 271 |

No. | Models | RSS | RMSE | MAE |
---|---|---|---|---|

9 | Friedman | 200,926 | 66 | 46 |

10 | Nadaraya–Watson | 285,149 | 79 | 52 |

11 | Cleveland | 569,620 | 111 | 75 |

No. | Models | RMSE | MAE |
---|---|---|---|

1 | Biexponential | 25 | 22 |

2 | Nadaraya–Watson kernel smoothing | 25 | 23 |

3 | Gompertz growth | 76 | 73 |

4 | Michaelis–Menten | 132 | 131 |

5 | Power | 322 | 320 |

6 | linear | 533 | 525 |

7 | Quadratic | 789 | 749 |

8 | Cubic | 849 | 754 |

9 | Quartic | 895 | 758 |

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**MDPI and ACS Style**

Michinaka, T.
Approximating Forest Resource Dynamics in Peninsular Malaysia Using Parametric and Nonparametric Models, and Its Implications for Establishing Forest Reference (Emission) Levels under REDD+. *Land* **2018**, *7*, 70.
https://doi.org/10.3390/land7020070

**AMA Style**

Michinaka T.
Approximating Forest Resource Dynamics in Peninsular Malaysia Using Parametric and Nonparametric Models, and Its Implications for Establishing Forest Reference (Emission) Levels under REDD+. *Land*. 2018; 7(2):70.
https://doi.org/10.3390/land7020070

**Chicago/Turabian Style**

Michinaka, Tetsuya.
2018. "Approximating Forest Resource Dynamics in Peninsular Malaysia Using Parametric and Nonparametric Models, and Its Implications for Establishing Forest Reference (Emission) Levels under REDD+" *Land* 7, no. 2: 70.
https://doi.org/10.3390/land7020070