# Applicability of Common Algorithms in Species–Area Relationship Model Fitting

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. The Gauss–Newton Method

- (1)
- Set the initial parameter ${x}_{0}$ manually.
- (2)
- Find the descent direction $h$ by solving $J{\left(x\right)}^{t}J\left(x\right)h=-J{\left(x\right)}^{t}f\left(x\right)$, then update ${x}_{0}$ by ${x}_{1}={x}_{0}+h$.
- (3)
- Update $x$ continuously until $x$ is close to ${x}_{T}$.

#### 2.2. The Levenberg–Marquardt Method

- (1)
- Set the initial parameter ${x}_{0}$.
- (2)
- Find the descent direction ${h}_{lm}$ by solving $(J{\left(x\right)}^{T}J\left(x\right)+\mu I){h}_{lm}=-J{\left(x\right)}^{T}f\left(x\right)$.
- (3)
- Decide whether to accept ${h}_{lm}$ according to φ.
- (4)
- Iterate until $x$ is close to ${x}_{T}$.

#### 2.3. The Nelder–Mead Method

#### 2.4. Data Simulation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The number of steps required for the algorithms to iterate from the initial parameters ${x}_{0}$ to ${x}_{T}$. GN represents the Gauss–Newton method, LM represents the Levenberg–Marquardt method, and NM represents the Nelder–Mead method.

**Figure 2.**The time consumption in seconds for the algorithms to iterate from the initial parameters ${x}_{0}$ to ${x}_{T}$. GN represents the Gauss–Newton method, LM represents the Levenberg–Marquardt method, and NM represents the Nelder–Mead method.

**Figure 3.**The differences in the sensitivity of different algorithms to the initial parameters when fitting the various models. GN represents the Gauss–Newton method, LM represents the Levenberg–Marquardt method, and NM represents the Nelder–Mead method. C indicates that the algorithm converges; NC indicates that the algorithm does not converge.

**Table 1.**SAR models used to evaluate the applicability of the different algorithms, and the function with white noise. In this case, the independent variable $\left(A\right)$ is area and the dependent variable $f\left(x\right)$ is the number of species, ${b}_{0}$, ${b}_{1}$ and ${b}_{2}$ are the parameters.

No. | Curve Name | Model | Parameters | Date Simulation | Shape Type |
---|---|---|---|---|---|

1 | Power | ${b}_{0}{A}^{{b}_{1}}$ | 2 | ${b}_{0}{A}^{{b}_{1}}+noise$ | Convex |

2 | Gompertz | ${b}_{0}\mathrm{exp}\left(-\mathrm{exp}\left(-{b}_{1}A+{b}_{2}\right)\right)$ | 3 | ${b}_{0}\mathrm{exp}\left(-\mathrm{exp}\left(-{b}_{1}A+{b}_{2}\right)\right)+noise$ | Sigmod |

3 | Logarithmic | ${b}_{0}+{b}_{1}\mathrm{log}A$ | 2 | ${b}_{0}+{b}_{1}\mathrm{log}A+noise$ | Convex |

4 | Negative exponential | ${b}_{0}-{b}_{0}\mathrm{exp}\left(-{b}_{1}A\right)$ | 2 | ${b}_{0}-{b}_{0}\mathrm{exp}\left(-{b}_{1}A\right)+noise$ | Convex |

5 | Weibull | ${b}_{0}\left(1-\mathrm{exp}\left(-{b}_{1}{A}^{{b}_{2}}\right)\right)$ | 3 | ${b}_{0}\left(1-\mathrm{exp}\left(-{b}_{1}{A}^{{b}_{2}}\right)\right)+noise$ | Sigmod |

6 | Persistence function 2 (P2) | ${b}_{0}{A}^{{b}_{1}}\mathrm{exp}\left(-{b}_{2}/A\right)$ | 3 | ${b}_{0}{A}^{{b}_{1}}\mathrm{exp}\left(-{b}_{2}/A\right)+noise$ | Sigmod |

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Liu, Z.; Liu, X.; Shen, T.-J.
Applicability of Common Algorithms in Species–Area Relationship Model Fitting. *Diversity* **2022**, *14*, 212.
https://doi.org/10.3390/d14030212

**AMA Style**

Liu Z, Liu X, Shen T-J.
Applicability of Common Algorithms in Species–Area Relationship Model Fitting. *Diversity*. 2022; 14(3):212.
https://doi.org/10.3390/d14030212

**Chicago/Turabian Style**

Liu, Zhidong, Xiaoke Liu, and Tsung-Jen Shen.
2022. "Applicability of Common Algorithms in Species–Area Relationship Model Fitting" *Diversity* 14, no. 3: 212.
https://doi.org/10.3390/d14030212