Title Page

Contents

Abstract 11

Chapter 1. Introduction 12

Chapter 2. Background 15

2.1. Autoencoder 15

2.2. Mixture model 18

2.2.1. Gaussian mixture model (GMM) 18

2.3. Deep autoencoding Gaussian mixture model 23

Chapter 3. Skew-normal distribution 27

3.1. Univariate skew-normal distribution 27

3.2. Multivariate skew-normal distribution (MSN) 28

3.2.1. Multivariate truncated normal distribution 29

3.2.2. Parameter estimation 31

Chapter 4. Deep auotoencoding skew-normal mixture model (DASKNM-EM) 34

4.1. Introduction 34

4.2. Compression network 35

4.3. Estimation network 36

Chapter 5. Simulation 42

5.1. Simulation environment 42

5.1.1. Accuracy 44

5.1.2. Baseline model 47

5.2. Simulation results 48

5.2.1. Case 1 48

5.2.2. Case 2 52

5.2.3. Case 3 54

5.2.4. Case 4 58

Chapter 6. Real data examples 61

6.1. Credit card 61

6.2. Satellite 65

Chapter 7. Conclusion 69

References 71

논문요약 74

Table 5.1. The average absolute skewness and kurtosis for 50 times for case 1 and case 2 49

Table 5.2. The information criterion for component selection in case 1 and case 2 50

Table 5.3. The average accuracy over 50 simulations for case 1 50

Table 5.4. The average accuracy over 50 simulations for case 2 53

Table 5.5. The average absolute skewness and kurtosis for 50 times for case 3 and case 4 55

Table 5.6. The information criteria for component selection in case 3 and case 4 56

Table 5.7. The average accuracy over 50 simulations for case 3 56

Table 5.8. The average accuracy over 50 simulations for case 4 59

Table 6.1. The information criterion for component selection in credit card 62

Table 6.2. The average accuracy of simulation 20 times for the credit card data 64

Table 6.3. The information criterion for component selection in satellite data 66

Table 6.4. The average accuracy of simulation 20 times for the satellite data 67

Figure 2.1. Architecture of Autoencoder 16

Figure 5.1. The z-space for each simulation case. 43

Figure 5.2. Confusion matrix 45

Figure 5.3. Box plots for absolute skewness and kurtosis of z-vector for case 1 and case 2 49

Figure 5.4. The box-plot of F₁ when n is 1500 (left) and when n is 3000 (right) for case 1 51

Figure 5.5. The box-plot of F₁ when n is 1500 (left) and when n is 3000 (right) for case 2 54

Figure 5.6. Box plots for skewness and kurtosis of z-vector for case 3 and case 4 55

Figure 5.7. The box-plot of F₁ when n is 1500 (left) and when n is 3000 (right) for case 3 57

Figure 5.8. The box-plot of F₁ when n is 1500 (left) and when n is 3000 (right) for case 4 59

Figure 6.1. z-space (left) and CDF of energy value(right) of credit card data 62

Figure 6.2. The box plots of F₁ score with 50 times for credit card 64

Figure 6.3. z-space (left) and CDF of energy value (right) of satellite data 67

Figure 6.4. The box plots of F₁ score with 50 times for satellite 67