# Modeling Multivariate Spray Characteristics with Gaussian Mixture Models

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Training Data

#### 2.2. Gaussian Mixture Model

- $\mathit{\theta}=\{{\mathit{\mu}}_{1},{\mathit{\mu}}_{2},\dots ,{\mathit{\mu}}_{N},{\mathbf{\Sigma}}_{1},{\mathbf{\Sigma}}_{2},\dots ,{\mathbf{\Sigma}}_{N},{w}_{1},{w}_{2},\dots ,{w}_{N}\}$ are the parameters of the GMM,
- ${\mathit{\mu}}_{n}$ is the mean vector of the n-th Gaussian component,
- ${\mathbf{\Sigma}}_{n}$ is the covariance matrix of the n-th Gaussian component, and
- ${w}_{n}$ is the weight of the n-th Gaussian component, representing the probability of selecting that component.

#### Model Selection, Initialization, and Evaluation

## 3. Results and Discussions

#### 3.1. Assessment of Model Accuracy and Complexity

#### 3.2. Conservation of Feature Correlations

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDF | Cumulative distribution function |

GMM | Gaussian mixture model |

GT | Ground truth |

Probability density function | |

SPH | Smoothed particle hydrodynamics |

## References

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**Figure 1.**Impact of operating conditions (OP) on the primary atomization process and the formed droplet size distributions. Inlet and outlet boundary conditions are denoted as red and green, respectively. (

**a**) $OP1$, (

**b**) $OP4$.

**Figure 2.**Univariate PDFs for 4 different $OP$’s with varying load for the equivalent diameter D, the radial distance to the rotation axis r, and axial and radial velocities ${u}_{\mathrm{ax}}$ and ${u}_{\mathrm{rad}}$.

**Figure 3.**Univariate CDFs for 4 different $OP$s for the equivalent diameter D, the radial distance to the rotation axis r, and axial and radial velocities ${u}_{\mathrm{ax}}$ and ${u}_{\mathrm{rad}}$.

**Figure 4.**Pearson correlation coefficients between the different features for every operating point.

**Figure 6.**Visualization of a GMM with n components for a bimodal univariate distribution. (

**a**) $n=2$; (

**b**) $n=5$.

**Figure 7.**Evolution of Hellinger distances H as a measure of dissimilarity between the model predictions and the ground truth (GT) with increasing number of Gaussians (n) in GMM for $OP4$. (

**a**) Diameter D; (

**b**) radial coordinate r; (

**c**) axial velocity ${u}_{ax}$; (

**d**) radial velocity ${u}_{rad}$; (

**e**) 4D droplet data.

**Figure 8.**Normalized loss function $L/L\left(1\right)$ using multivariate Hellinger distance. (

**a**) Exact values; (

**b**) rolling mean.

**Figure 9.**Deviation of the cumulative distribution functions of the training data and the sampled data with increased model complexity. (

**a**) $n=12$; (

**b**) $n=25$.

**Figure 10.**Comparison of the Pearson correlation coefficients of the training data and sampled data. Solid bars denote the ground truth, while dashed bars give the GMM predictions. Colors denote the operating points. (

**a**) $n=12$; (

**b**) $n=25$.

**Figure 11.**Bivariate joint PDFs of radial coordinate and axial velocity (

**left**) and radial coordinate and radial velocity (

**right**) of the GMM and the training data for $OP4$. (

**a**) $n=12$; (

**b**) $n=25$; (

**c**) training data.

$OP$ | 1 | 2 | 3 | 4 |

N | 6094 | 15,501 | 44,484 | 44,572 |

**Table 2.**Analytical univariate distribution functions for each droplet feature. These distributions are used as a benchmark model to assess the predictive capabilities of the GMM.

Feature | D | r | ${u}_{ax}$ | ${u}_{rad}$ |

Distribution | Exponentiated Weibull | Johnson SB | Log-normal | Log-normal |

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**MDPI and ACS Style**

Wicker, M.; Ates, C.; Okraschevski, M.; Holz, S.; Koch, R.; Bauer, H.-J.
Modeling Multivariate Spray Characteristics with Gaussian Mixture Models. *Energies* **2023**, *16*, 6818.
https://doi.org/10.3390/en16196818

**AMA Style**

Wicker M, Ates C, Okraschevski M, Holz S, Koch R, Bauer H-J.
Modeling Multivariate Spray Characteristics with Gaussian Mixture Models. *Energies*. 2023; 16(19):6818.
https://doi.org/10.3390/en16196818

**Chicago/Turabian Style**

Wicker, Markus, Cihan Ates, Max Okraschevski, Simon Holz, Rainer Koch, and Hans-Jörg Bauer.
2023. "Modeling Multivariate Spray Characteristics with Gaussian Mixture Models" *Energies* 16, no. 19: 6818.
https://doi.org/10.3390/en16196818