# An Adaptive Multipath Linear Interpolation Method for Sample Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Hypothesis and Methodology Statement

Algorithm 1:AMLI |

## 3. Simulation Experiments

#### 3.1. Monte Carlo Simuations

**Remark**

**1.**

#### 3.2. Analysis of Hyperparameter Taking Values

#### 3.3. Comparison with Other Interpolation Methods

## 4. Application of AMLI Method in Machine Learning

#### 4.1. Simulated Data Prediction

#### 4.2. Actual Data Prediction

#### 4.2.1. Demand Forecast for Shared Bike Rental

#### 4.2.2. Concentration Forecast for $P{M}_{2.5}$

## 5. Proof

## 6. Extension

#### 6.1. AMLI Plus

Algorithm 2: AMLI plus |

#### 6.2. The Proof of AMLI Plus

**Remark**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Simulation | MSE | ${\mathit{p}}_{(0.5)}$ | ${\mathit{p}}_{(1)}$ | ${\mathit{p}}_{(1.5)}$ | ${\mathit{p}}_{(2)}$ | ${\mathit{p}}_{(2.5)}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Before | After | Before | After | Before | After | Before | After | Before | After | Before | After | |

1 | 0.957 | 0.762 | 0.604 | 0.570 | 0.316 | 0.251 | 0.130 | 0.083 | 0.036 | 0.022 | 0.010 | 0.006 |

2 | 0.980 | 0.774 | 0.614 | 0.567 | 0.313 | 0.258 | 0.132 | 0.089 | 0.043 | 0.023 | 0.011 | 0.005 |

3 | 0.977 | 0.789 | 0.619 | 0.571 | 0.308 | 0.254 | 0.131 | 0.090 | 0.043 | 0.025 | 0.011 | 0.005 |

4 | 0.990 | 0.701 | 0.609 | 0.549 | 0.317 | 0.232 | 0.133 | 0.073 | 0.044 | 0.017 | 0.011 | 0.003 |

5 | 0.982 | 0.756 | 0.611 | 0.552 | 0.306 | 0.245 | 0.127 | 0.085 | 0.038 | 0.024 | 0.010 | 0.005 |

6 | 0.987 | 0.742 | 0.706 | 0.632 | 0.411 | 0.297 | 0.126 | 0.057 | 0.000 | 0.000 | 0.000 | 0.000 |

Simulation | MSE | |||
---|---|---|---|---|

AMLI | Linear Interpolation | Quadratic Spline | Cubic Spline | |

1 | 0.762 | 1.385 | 5.542 | 9.452 |

2 | 0.774 | 1.618 | 6.434 | 8.976 |

3 | 0.789 | 1.812 | 6.441 | 8.338 |

4 | 0.701 | 1.578 | 18.156 | 42.997 |

5 | 0.756 | 1.203 | 10.322 | 29.542 |

6 | 0.742 | 0.959 | 4.802 | 6.619 |

MSE | KNN | FNN | GBDT | RF |
---|---|---|---|---|

Before AMLI processing | 1.70 | 0.942 | 1.210 | 1.507 |

After AMLI processing | 1.07 | 0.713 | 1.008 | 1.320 |

Variable Name | Variable Definition |
---|---|

Season | 1 = Spring |

2 = Summer | |

3 = Autumn | |

4 = Winter | |

Holiday | 1 = Holiday |

0 = Non-holiday | |

Workdays | 1 = Working day |

0 = Weekend | |

Weather | 1 = Sunny, cloudy |

2 = Foggy, overcast | |

3 = Light snow, drizzle | |

4 = Heavy rain, heavy snow, heavy fog | |

Temp | Temperature in Celsius |

Atemp | Apparent temperature |

Humidity | Humidity |

Windspeed | Wind speed |

Casual | Number of nonregistered users |

Registered | Number of registered users |

Count | Total number of car rentals |

N | Hyper-Parameter | N (After Processing) | AMLI Processing | KNN | FNN | GBDT | RF |
---|---|---|---|---|---|---|---|

1000 | $K=25$ | 191,057 | Before | 208.230 | 0.9677 | 88.152 | 231.210 |

$\eta =5$ | After | 46.340 | 0.0791 | 68.394 | 158.940 | ||

3000 | $K=40$ | 290,856 | Before | 42.757 | 0.3342 | 22.794 | 43.679 |

$\eta =3$ | After | 26.020 | 0.0084 | 14.301 | 25.451 | ||

7620 | $K=65$ | 428,277 | Before | 17.464 | 0.1168 | 9.879 | 23.521 |

$\eta =1$ | After | 6.798 | 0.0051 | 6.981 | 2.472 |

N | Hyperparameter | N (After Processing) | AMLI Processing | $\mathit{KNN}$ | $\mathit{FNN}$ | $\mathit{GBDT}$ | $\mathit{RF}$ |
---|---|---|---|---|---|---|---|

500 | $K=17$ | 1089 | Before | 3.16 | 4.45 | 3.14 | 3.39 |

$\eta =10$ | After | 2.89 | 2.55 | 2.36 | 3.01 | ||

1500 | $K=23$ | 7003 | Before | 2.69 | 2.54 | 2.33 | 2.62 |

$\eta =30$ | After | 2.18 | 2.07 | 1.86 | 2.33 | ||

3193 | $K=50$ | 24,559 | Before | 1.77 | 1.63 | 1.48 | 1.54 |

$\eta =60$ | After | 1.41 | 1.13 | 1.07 | 1.16 |

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**MDPI and ACS Style**

Du, Y.; Jin, X.; Wang, H.; Lu, M.
An Adaptive Multipath Linear Interpolation Method for Sample Optimization. *Mathematics* **2023**, *11*, 768.
https://doi.org/10.3390/math11030768

**AMA Style**

Du Y, Jin X, Wang H, Lu M.
An Adaptive Multipath Linear Interpolation Method for Sample Optimization. *Mathematics*. 2023; 11(3):768.
https://doi.org/10.3390/math11030768

**Chicago/Turabian Style**

Du, Yukun, Xiao Jin, Hongxia Wang, and Min Lu.
2023. "An Adaptive Multipath Linear Interpolation Method for Sample Optimization" *Mathematics* 11, no. 3: 768.
https://doi.org/10.3390/math11030768