FragmGAN: generative adversarial nets for fragmentary data imputation and prediction

Figures & data

Table 1. The response patterns of the Internet Loan data.

Response	Data source
Pattern	Card	Shopping	Mobile	Bureau	Fraud
1	√	√	√	√	√
2	√	√	√	√
3	√	√	√
4	√	√		√	√
5	√	√		√
6	√	√
7	√		√
8	√			√	√
9	√			√
10	√

Table 2. The response patterns of the ADNI data.

Response	Data source
Pattern	CSF	PET	MRI	Gene
1	√	√	√	√
2	√	√	√
3	√	√		√
4		√	√	√
5		√		√
6		√	√
7				√
8			√

Table 3. Imputation performance for UCI datasets in terms of RMSE (Average $\pm$ Std) of imputation error.

Algorithm	Breast	Spam	Letter	Credit	News
MCAR, miss rate=20%
FragmGAN	0.0599 $\pm$ .0021	0.0537 $\pm$ .0014	0.1251 $\pm$ .0026	0.1781 $\pm$ .0057	0.1484 $\pm$ .0008
FragmGAN no hint	0.0715 $\pm$ .0022	0.0545 $\pm$ .0007	0.1313 $\pm$ .0078	0.1833 $\pm$ .0020	0.1595 $\pm$ .0049
GAIN	0.0658 $\pm$ .0030	0.0544 $\pm$ .0005	0.1295 $\pm$ .0032	0.1814 $\pm$ .0033	0.1580 $\pm$ .0063
GAIN no hint	0.0736 $\pm$ .0036	0.0574 $\pm$ .0004	0.1338 $\pm$ .0058	0.1899 $\pm$ .0056	0.1606 $\pm$ .0051
MICE	0.0872 $\pm$ .0019	0.0715 $\pm$ .0011	0.1611 $\pm$ .0045	0.1875 $\pm$ .0051	0.2152 $\pm$ .0095
MissForest	0.0608 $\pm$ .0012	0.0594 $\pm$ .0003	0.1371 $\pm$ .0012	0.2033 $\pm$ .0080	0.1932 $\pm$ .0049
Matrix	0.1148 $\pm$ .0021	0.0562 $\pm$ .0012	0.1530 $\pm$ .0035	0.2449 $\pm$ .0033	0.2291 $\pm$ .0061
AE	0.0727 $\pm$ .0011	0.0620 $\pm$ .0007	0.1361 $\pm$ .0020	0.2137 $\pm$ .0026	0.1963 $\pm$ .0004
EM	0.0754 $\pm$ .0027	0.0680 $\pm$ .0002	0.1679 $\pm$ .0005	0.2312 $\pm$ .0008	0.2687 $\pm$ .0003
MisGAN	0.0707 $\pm$ .0016	0.0582 $\pm$ .0004	0.1347 $\pm$ .0020	0.1913 $\pm$ .0008	0.1746 $\pm$ .0034
MAR, miss rate=20%
FragmGAN	0.0667 $\pm$ .0084	0.0512 $\pm$ .0007	0.1364 $\pm$ .0072	0.1844 $\pm$ .0029	0.1630 $\pm$ .0043
FragmGAN no hint	0.0730 $\pm$ .0063	0.0519 $\pm$ .0007	0.1495 $\pm$ .0052	0.1966 $\pm$ .0046	0.1737 $\pm$ .0060
GAIN	0.0671 $\pm$ .0092	0.0526 $\pm$ .0008	0.1457 $\pm$ .0084	0.1909 $\pm$ .0040	0.1690 $\pm$ .0047
GAIN no hint	0.0756 $\pm$ .0099	0.0526$\pm$ 0010	0.1505 $\pm$ .0055	0.1901 $\pm$ .0040	0.1747 $\pm$ .0097
MICE	0.0931 $\pm$ .0060	0.0705 $\pm$ .0008	0.1531 $\pm$ .0059	0.2487 $\pm$ .0078	0.2159 $\pm$ .0096
MissForest	0.0625 $\pm$ .0003	0.0576 $\pm$ .0004	0.1331 $\pm$ .0059	0.2626 $\pm$ .0025	0.1977 $\pm$ .0086
Matrix	0.1146 $\pm$ .0031	0.0556 $\pm$ .0008	0.1488 $\pm$ .0048	0.2268 $\pm$ .0056	0.2337 $\pm$ .0086
AE	0.0843 $\pm$ .0038	0.0618 $\pm$ .0017	0.1427 $\pm$ .0018	0.2110 $\pm$ .0024	0.2048 $\pm$ .0008
EM	0.0869 $\pm$ .0016	0.0678 $\pm$ .0030	0.1784 $\pm$ .0050	0.2412 $\pm$ .0006	0.2659 $\pm$ .0010
MisGAN	0.0713 $\pm$ .0014	0.0577 $\pm$ .0007	0.1428 $\pm$ .0037	0.2014 $\pm$ .0046	0.1883 $\pm$ .0035

Figure 1. RMSE of imputation error of the Credit data under different miss rates. Left: MCAR. Right: MAR.

Table 4. Prediction performance for UCI datasets in terms of AUC (Average ± Std).

Algorithm	Breast	Spam	Credit	News
MCAR, miss rate=20%
FragmGAN $\gamma =0.5$	0.9932 ± .0035	0.9534 ± .0029	0.7643 ± .0034	0.9709 ± .0020
FragmGAN $\gamma =1$	0.9920 ± .0056	0.9528 ± .0030	0.7557 ± .0021	0.9620 ± .0017
GAIN	0.9912 ± .0055	0.9513 ± .0037	0.7521 ± .0022	0.9607 ± .0026
MICE	0.9809 ± .0045	0.9444 ± .0032	0.7492 ± .0038	0.9294 ± .0021
MissForest	0.9892 ± .0060	0.9466 ± .0069	0.7495 ± .0026	0.9409 ± .0029
Matrix	0.9827 ± .0062	0.9021 ± .0057	0.7273 ± .0074	0.8438 ± .0054
AE	0.9850 ± .0078	0.9392 ± .0053	0.7463 ± .0044	0.9211 ± .0025
EM	0.9853 ± .0029	0.9172 ± .0059	0.7418 ± .0068	0.8754 ± .0028
MisGAN	0.9858 ± .0025	0.9485 ± .0042	0.7488 ± .0017	0.9505 ± .0012
MAR, miss rate=20%
FragmGAN $\gamma =0.5$	0.9936 ± .0056	0.9530 ± .0034	0.7622 ± .0027	0.9696 ± .0024
FragmGAN $\gamma =1$	0.9928 ± .0045	0.9521 ± .0029	0.7518 ± .0015	0.9598 ± .0017
GAIN	0.9914 ± .0040	0.9511 ± .0032	0.7505 ± .0021	0.9592 ± .0023
MICE	0.9878 ± .0063	0.9375 ± .0036	0.7366 ± .0033	0.9325 ± .0040
MissForest	0.9839 ± .0035	0.9519 ± .0042	0.7355 ± .0026	0.9405 ± .0026
Matrix	0.9815 ± .0083	0.9033 ± .0045	0.7342 ± .0028	0.8596 ± .0036
AE	0.9895 ± .0056	0.9347 ± .0041	0.7485 ± .0056	0.9291 ± .0041
EM	0.9892 ± .0064	0.9134 ± .0036	0.7427 ± .0063	0.8828 ± .0061
MisGAN	0.9863 ± .0023	0.9499 ± .0050	0.7483 ± .0021	0.9492 ± .0020

Table 5. Prediction performance for the two motivation examples (Average ± Std).

Algorithm	Internet Loan (RMSE)	ADNI (RMSE)	ADNI (AUC)
FragmGAN ${\gamma }_{\mathrm{cv}}$	0.8865 ± .0015	0.0851 ± .0019	0.7823 ± .0026
FragmGAN $\gamma =1$	0.9267 ± .0036	0.0897 ± .0022	0.7701 ± .0036
FragmGAN $\gamma =0.75$	0.9151 ± .0029	0.0883 ± .0023	0.7721 ± .0028
FragmGAN $\gamma =0.5$	0.8928 ± .0026	0.0871 ± .0020	0.7773 ± .0023
FragmGAN $\gamma =0.25$	0.9286 ± .0044	0.0895 ± .0028	0.7719 ± .0018
GAIN	0.9246 ± .0034	0.0921 ± .0014	0.7622 ± .0028
MICE	0.9934 ± .0036	0.1034 ± .0029	0.6587 ± .0029
MissForest	0.9982 ± .0041	0.1124 ± .0019	0.7583 ± .0022
Matrix	0.9913 ± .0039	0.1134 ± .0016	0.7343 ± .0018
AE	0.9884 ± .0035	0.0994 ± .0017	0.7400 ± .0036
EM	0.9896 ± .0034	0.1042 ± .0023	0.7020 ± .0048
MisGAN	0.9889 ± .0021	0.0997 ± .0024	0.7384 ± .0026
Model Averaging	0.9831 ± .0071	0.1000 ± .0095	not applicable
FR-FI	1.0560 ± .0112	0.1057 ± .0087	not applicable

Table B1. The number of variables at each data source and sample size of each response pattern for the Internet Loan data.

Response	Data source					Sample
Pattern	Card (5)	Shopping (4)	Mobile (5)	Bureau (7)	Fraud (4)	Size
1	√	√	√	√	√	115
2	√	√	√	√		29
3	√	√	√			220
4	√	√		√	√	232
5	√	√		√		113
6	√	√				222
7	√		√			11
8	√			√	√	38
9	√			√		102
10	√					302
Total						1384

Table B2. The number of variables at each data source and sample size of each response pattern for the ADNI data.

Response	Data source				Sample
Pattern	CSF (3)	PET (10)	MRI (10)	Gene (10)	Size
1	√	√	√	√	413
2	√	√	√		367
3	√	√		√	34
4		√	√	√	109
5		√		√	81
6		√	√		54
7				√	51
8			√		58
Total					1167