Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation
Abstract
:1. Introduction
2. Methods
2.1. The Employed Statistical Models
2.1.1. The Monte Carlo Method
2.1.2. The Copula-Based Method
2.2. The Employed Generative Deep Learning Models
2.2.1. GAN
2.2.2. DDPM
- Forward diffusion process
- Reverse diffusion process
- Loss function
2.3. Metrics for Comparative Evaluation
3. Results and Analysis
3.1. Experimental Data
3.1.1. Dataset 1: Structural Plane Set of Valle Study Area
3.1.2. Dataset 2: Structural Plane Set of Tunhovd Study Area
3.1.3. Dataset 3: Structural Plane Set of Straumklumpen Study Area
3.1.4. Dataset 4: Structural Plane Set of Tunhovd Study Area
3.1.5. Dataset 5: Structural Plane Set of Tunhovd Study Area
3.2. Experimental Environments
3.3. Experimental Results
3.3.1. Generated Results of Dataset 1
3.3.2. Generated Results of Dataset 2
3.3.3. Generated Results of Dataset 3
3.3.4. Generated Results of Dataset 4
3.3.5. Generated Results of Dataset 5
3.4. Comparative Analysis
4. Discussion
4.1. Performance Analysis of the Statistical Methods
4.2. Performance Analysis of the Deep Generative Models
4.3. Comparative Analysis of Statistical Methods and Deep Generative Models
4.4. Future Work
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Metrics | Principles of the Metrics | |
---|---|---|
Qualitative analysis | Comparison of histograms of probability distributions | The histogram illustrates the distribution of values of a variable across a specified range. Each bar in the graph corresponds to the frequency of values within that range, presenting a visual depiction of the data distribution through a series of vertical bars or lines of different heights. A histogram serves as a precise graphical portrayal of the numerical data distribution. |
Scatter plot Comparison | A scatter plot represents the distribution of data points on the Cartesian coordinate system in regression analysis. It provides a visual assessment of the potential correlation strength between two variables, aiding in the selection of a suitable function to model the data points. | |
Quantitative calculation | Comparison of mean value and variance | Mean and variance are important statistics that describe the characteristics of the data and can provide a visual representation of the overall characteristics and distribution of the dataset. |
Pearson correlation coefficient | The Pearson correlation coefficient is the most commonly used method for conducting correlation analysis. It serves the purpose of the strength of correlation between two variables. A larger absolute value of the correlation coefficient indicates a stronger correlation, while a value closer to zero suggests a weaker correlation. |
Groups | Location | Number | Distribution Type | Correlation | Pearson Correlation Coefficient | ||
---|---|---|---|---|---|---|---|
Dip Direction | Dip Angle | Trace Length | |||||
Dataset 1 | Valle study area | 40 | Normal | Normal | Log-normal | Negative | −0.374 |
Dataset 2 | Tunhovd study area | 157 | Normal | Normal | Log-normal | Positive | 0.361 |
Dataset 3 | Straumklumpen study area | 257 | Normal | Normal | Log-normal | Negative | −0.342 |
Dataset 4 | Tunhovd study area | 325 | Normal | Normal | Log-normal | Negative | −0.252 |
Dataset 5 | Oernlia study area | 766 | Normal | Normal | Log-normal | Negative | −0.685 |
Dip Direction (°) | Dip Angle (°) | Trace Length (m) | |||||||
---|---|---|---|---|---|---|---|---|---|
Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | |
Measured factual | Normal | 74.80 | 6.04 | Normal | 46.72 | 4.19 | Log-normal | 1.88 | 2.24 |
Monte Carlo method | Normal | 73.30 | 6.56 | Normal | 46.73 | 3.69 | Log-normal | 1.75 | 1.61 |
Copula-based method | Normal | 75.23 | 6.32 | Normal | 46.40 | 4.59 | Log-normal | 1.79 | 1.63 |
GAN | Normal | 77.22 | 5.34 | Normal | 46.88 | 4.65 | Log-normal | 1.98 | 1.90 |
DDPM | Normal | 75.73 | 8.94 | Normal | 45.97 | 4.67 | Log-normal | 4.01 | 3.00 |
Dip Direction (°) | Dip Angle (°) | Trace Length (m) | |||||||
---|---|---|---|---|---|---|---|---|---|
Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | |
Measured factual | Normal | 262.29 | 10.56 | Normal | 50.29 | 5.15 | Log-normal | 1.77 | 1.18 |
Monte Carlo method | Normal | 261.83 | 9.93 | Normal | 50.08 | 5.45 | Log-normal | 1.75 | 1.61 |
Copula-based method | Normal | 263.72 | 10.69 | Normal | 50.49 | 4.99 | Log-normal | 1.68 | 1.10 |
GAN | Normal | 258.88 | 12.01 | Normal | 46.99 | 5.14 | Log-normal | 1.13 | 0.57 |
DDPM | Normal | 261.41 | 12.07 | Normal | 49.49 | 5.59 | Log-normal | 1.78 | 1.17 |
Dip Direction (°) | Dip Angle (°) | Trace Length (m) | |||||||
---|---|---|---|---|---|---|---|---|---|
Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | |
Measured factual | Normal | 217.12 | 7.84 | Normal | 38.13 | 4.58 | Log-normal | 6.17 | 3.24 |
Monte Carlo method | Normal | 217.23 | 8.27 | Normal | 38.05 | 4.78 | Log-normal | 5.88 | 2.64 |
Copula-based method | Normal | 218.22 | 7.85 | Normal | 37.45 | 4.52 | Log-normal | 5.94 | 2.72 |
GAN | Normal | 216.87 | 5.62 | Normal | 38.06 | 4.27 | Log-normal | 6.21 | 3.14 |
DDPM | Normal | 217.54 | 8.49 | Normal | 37.70 | 5.52 | Log-normal | 6.39 | 3.80 |
Dip Direction (°) | Dip Angle (°) | Trace Length (m) | |||||||
---|---|---|---|---|---|---|---|---|---|
Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | |
Measured factual | Normal | 80.50 | 6.35 | Normal | 54.63 | 4.04 | Log-normal | 2.25 | 1.41 |
Monte Carlo method | Normal | 80.1 | 6.34 | Normal | 54.39 | 4.20 | Log-normal | 2.31 | 1.43 |
Copula-based method | Normal | 76.69 | 6.49 | Normal | 54.72 | 4.13 | Log-normal | 2.24 | 1.14 |
GAN | Normal | 80.73 | 5.70 | Normal | 54.70 | 3.06 | Log-normal | 2.22 | 1.03 |
DDPM | Normal | 81.63 | 6.12 | Normal | 54.71 | 3.83 | Log-normal | 2.35 | 1.53 |
Dip Direction (°) | Dip Angle (°) | Trace Length (m) | |||||||
---|---|---|---|---|---|---|---|---|---|
Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | Distribution Type | Mean Value | Standard Deviation | |
Measured factual | Normal | 258.78 | 12.46 | Normal | 36.03 | 9.02 | Log-normal | 3.95 | 2.80 |
Monte Carlo method | Normal | 258.78 | 12.44 | Normal | 36.02 | 9.01 | Log-normal | 3.85 | 1.67 |
Copula-based method | Normal | 259.30 | 12.14 | Normal | 35.74 | 8.96 | Log-normal | 4.06 | 1.80 |
GAN | Normal | 259.77 | 11.23 | Normal | 35.38 | 8.14 | Log-normal | 3.88 | 2.77 |
DDPM | Normal | 258.59 | 11.94 | Normal | 35.53 | 8.41 | Log-normal | 4.57 | 2.92 |
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Meng, H.; Xu, N.; Zhu, Y.; Mei, G. Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation. Mathematics 2024, 12, 2545. https://doi.org/10.3390/math12162545
Meng H, Xu N, Zhu Y, Mei G. Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation. Mathematics. 2024; 12(16):2545. https://doi.org/10.3390/math12162545
Chicago/Turabian StyleMeng, Han, Nengxiong Xu, Yunfu Zhu, and Gang Mei. 2024. "Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation" Mathematics 12, no. 16: 2545. https://doi.org/10.3390/math12162545