# Polynomial Chaos Expansion-Based Enhanced Gaussian Process Regression for Wind Velocity Field Estimation from Aircraft-Derived Data

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art

## 2. Methods

#### 2.1. Data Derivation and Exploratory Analysis

#### 2.1.1. Data Source

#### 2.1.2. ADS-B and Mode S Systems

#### 2.1.3. Wind Velocity Derivation from ADS-B and Mode S Data

#### 2.1.4. Exploratory Data Analysis

#### 2.2. Gaussian Process Regression

#### 2.3. Polynomial Chaos Expansion

**x**is assumed to be affected by uncertainties, it can be represented by a random vector $\mathbf{X}=({X}_{1},{X}_{2},\dots ,{X}_{d})$ with a joint Probability Density Function (PDF) ${f}_{\mathbf{X}}=({f}_{{X}_{1}},{f}_{{X}_{2}},\dots ,{f}_{{X}_{d}})$, and then $Y=\mathcal{M}\left(\mathbf{X}\right)$ is an output random variable, which is obtained by propagating the input vector uncertainties through the mapping $\mathcal{M}$.

#### 2.4. Polynomial Chaos Expansion-Based Enhanced Gaussian Process Regression

#### 2.5. Adaptation of PCE-GPR to the Wind Velocity Output

## 3. Results

#### 3.1. Model Set Up

**x**are the coordinates of the spatiotemporal position of the aircraft. The kernel function (11) produces continuous and smooth GP samples, thus providing a smooth regression capable of uniformly approximating any continuous function on a compact subset contained in the input space [41].

#### 3.2. Wind Velocity Field Reconstruction

- By randomly choosing sets of individual observations, which will be referred to as data set randomly split by observation.
- By randomly selecting sets of flights, employing the individual observations gathered from them, which will be referred to as a data set randomly split by flight.

#### 3.3. Wind Velocity Field Short-Term Prediction

#### 3.4. Validation of the PCE-GPR Model

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Wind velocity field reconstruction: Reconstructed wind velocity field obtained using the Day 1 data set and the PCE-GPR method for different altitudes (A). A selection of members of the training and test data sets, along with the mean wind speed (${\overline{s}}_{w}$), are also included.

**Figure 6.**Rose diagrams of the wind velocity estimation errors, segmented by altitude, for the Day 1 data set split by flight.

**Figure 7.**Wind speed reconstruction from 14:10 to 15:00 UTC at cruise altitude for the Day 2 data set.

**Figure 8.**Wind velocity field prediction: Predicted wind velocity field at cruise altitude obtained using the Day 2 data set and the PCE-GPR method for different instants in time. A selection of members of the test data sets, along with the mean wind speed (${\overline{s}}_{w}$), are also included.

Wind Speed (m/s) | Wind Direction (Deg) | |||
---|---|---|---|---|

Day 1 | Day 2 | Day 1 | Day 2 | |

Min. | 0 | 0.013 | 0.01 | 163.79 |

Max. | 56.04 | 100.75 | 359.99 | 351.55 |

Mean | 17.80 | 60.56 | 307.16 | 166.66 |

Dispersion | 11.30 | 16.67 | 19.40 (%) | 2.11 (%) |

**Table 2.**Wind velocity field reconstruction: Estimation errors for the u and v components of the wind velocity.

Data Set Split by Observation | Data Set Split by Flight | ||||
---|---|---|---|---|---|

Measure of Error | Component | Day 1 | Day 2 | Day 1 | Day 2 |

RMSE (m/s) | u | 2.26 (18%) | 1.50 (21%) | 5.84 (1%) | 6.06 (−4%) |

v | 1.46 (44%) | 1.45 (22%) | 4.79 (14%) | 4.84 (1%) | |

MAE (m/s) | u | 1.17 (22%) | 0.99 (22%) | 4.46 (−1%) | 4.37 (−2%) |

v | 0.83 (43%) | 1.05 (19%) | 3.45 (13%) | 3.59 (3%) | |

MAD (m/s) | u | 0.53 (23%) | 0.64 (22%) | 3.60 (−6%) | 3.03 (−2%) |

v | 0.49 (31%) | 0.80 (15%) | 2.44 (9%) | 2.70 (5%) |

**Table 3.**Wind velocity field prediction: Estimation errors for the u and v components of the wind velocity field.

Measure of Error | Component | Day 1 | Day 2 |
---|---|---|---|

RMSE (m/s) | u | 5.28 (6%) | 6.37 (13%) |

v | 5.16 (6%) | 5.80 (8%) | |

MAE (m/s) | u | 4.00 (12%) | 4.19 (29%) |

v | 3.93 (12%) | 4.40 (15%) | |

MAD (m/s) | u | 3.00 (4%) | 3.25 (12%) |

v | 3.07 (3%) | 3.52 (4%) |

**Table 4.**Validation of the PCE-GPR model: Comparison between the estimates obtained using the PCE-GPR method and the ECMWF ERA5 reanalysis data.

Measure | Variable | Day 1 | Day 2 |
---|---|---|---|

Bias (m/s) | Wind speed | −2.75 | −0.24 |

MAE (m/s) | Wind speed | 4.5 | 5.79 |

Bias (deg) | Wind direction | 2.06 | −1.36 |

Dispersion (%) | Wind direction | 8.5 | 0.33 |

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**MDPI and ACS Style**

Marinescu, M.; Olivares, A.; Staffetti, E.; Sun, J.
Polynomial Chaos Expansion-Based Enhanced Gaussian Process Regression for Wind Velocity Field Estimation from Aircraft-Derived Data. *Mathematics* **2023**, *11*, 1018.
https://doi.org/10.3390/math11041018

**AMA Style**

Marinescu M, Olivares A, Staffetti E, Sun J.
Polynomial Chaos Expansion-Based Enhanced Gaussian Process Regression for Wind Velocity Field Estimation from Aircraft-Derived Data. *Mathematics*. 2023; 11(4):1018.
https://doi.org/10.3390/math11041018

**Chicago/Turabian Style**

Marinescu, Marius, Alberto Olivares, Ernesto Staffetti, and Junzi Sun.
2023. "Polynomial Chaos Expansion-Based Enhanced Gaussian Process Regression for Wind Velocity Field Estimation from Aircraft-Derived Data" *Mathematics* 11, no. 4: 1018.
https://doi.org/10.3390/math11041018