Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery
Abstract
:1. Introduction
2. Sufficient Screening Utility
2.1. A Quantile-Adaptive Correlation Test Statistic
- (I)
- (Quantile-Heterogeneity) the index set of sufficient active predictors satisfies that may be different for different ;
- (II)
- (Sparsity) the dimensionality for some constant , but , where is the cardinality of , and n is the sample size.
2.2. Asymptotic Properties of the Test Statistic
- (C1)
- There exists constants and , s.t. ;
- (C2)
- There exists , s.t. ;
- (C3)
- The grids number of response satisfies , where and .
3. False Discovery Control Model
3.1. Adaptive False Discovery Control Model
Algorithm 1 QA-SVS-AFD algorithm. |
Input: Observation sample and the number of grids K Output: The screened sufficient variable set () Step 1 Calculate of Equation (5) for different ; Step 2 Compute by ; Step 3 Search for the screened sufficient active variable set in Equation (10). |
3.2. False Discovery Rate Control Model
Algorithm 2 QA-SVS-FDR(K) algorithm. |
Input: Observation sample , the number of grids K, and the prespecified level Output: The screened sufficient variable sets () Step 1 Calculate of Equation (5) for different ; Step 2 Compute each of Equation (11) for by taking each value of ; Step 3 For given , search for the set in Equation (12); Step 4 Find and let ; Step 5 Separate the screened sufficient active set of Equation (11) by . |
Algorithm 3 QA-SVS-FDR-S algorithm. |
Input: Observation sample , the number of grids K, and the prespecified level Output: The screened sufficient variable sets Step 1 Calculate in Remark 5; Step 2 Compute each in Remark 5 for taking each value of ; Step 3 For given , separate for the set in Remark 5; Step 4 Find and let ; Step 5 Separate the screened sufficient active set in Remark 5 by . |
4. Simulation Studies
4.1. Performance of QA-SVS-A
4.2. Performance of QA-SVS-FD
- : the average number of screened variables;
- FDR: the average of empirical FDP;
- : the average of .
5. Real Dataset Research
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Main Proof
Appendix A.1. Proof of Remark 1
Appendix A.2. Proof of Lemma 1
Appendix A.3. Proof of Lemma 2
- For all , we obtain that
- For all , we obtain that
- For all , we obtain that
- For all , we have
Appendix A.4. Proof of Corollary 1
Appendix A.5. Proof of Theorem 1
Appendix A.6. Proof of Theorem 2
Appendix A.7. Proof of Theorem 3
Appendix A.8. Proof of Corollary 2
Appendix A.9. Proof of Theorem 4
Appendix B
Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 5.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 5.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 5.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 5.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 | 5.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 7.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.5 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 | 3.0 | 3.0 | 3.0 | 4.0 | 11.5 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 4.0 | 12.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 | 3.0 | 3.0 | 3.0 | 4.0 | 7.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 | 3.0 | 3.0 | 3.0 | 4.0 | 10.0 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 | 3.0 | 3.0 | 4.0 | 5.0 | 6.0 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.5 |
QA-SIS(0.1) | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 | 3.0 | 3.0 | 3.0 | 3.5 | 16.5 |
QA-SIS(0.3) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QA-SIS(0.7) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 4.0 | 4.0 | 10.5 |
QA-SIS(0.9) | 4.0 | 15.0 | 32.5 | 74.5 | 217.0 | 4.5 | 14.5 | 36.5 | 81.5 | 299.0 |
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 5.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 6.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 8.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 6.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 | 3.0 | 3.0 | 3.0 | 4.0 | 7.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 4.0 | 6.5 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 4.0 | 6.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 | 3.0 | 3.0 | 3.0 | 4.0 | 6.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 4.0 | 54.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 4.0 | 32.0 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 8.0 | 3.0 | 3.0 | 3.0 | 4.0 | 60.5 |
QCS(9) | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 | 3.0 | 3.0 | 3.0 | 4.0 | 19.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 | 3.0 | 3.0 | 4.0 | 6.0 | 92.0 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 18.5 | 3.0 | 3.0 | 4.0 | 5.0 | 11.5 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QA-SIS(0.1) | 3.0 | 3.0 | 3.0 | 3.0 | 31.0 | 3.0 | 3.0 | 3.0 | 4.0 | 28.0 |
QA-SIS(0.3) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 7.5 |
QA-SIS(0.7) | 3.0 | 3.0 | 3.0 | 4.0 | 18.5 | 3.0 | 3.0 | 3.0 | 7.0 | 27.5 |
QA-SIS(0.9) | 16.0 | 58.5 | 130.5 | 323.5 | 1382.0 | 6.0 | 31.0 | 124.5 | 421.5 | 1844.0 |
Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 8.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 |
SIS | 414.0 | 686.0 | 781.5 | 894.0 | 979.5 | 558.5 | 706.5 | 824.5 | 932.0 | 988.0 |
DC-SIS | 441.5 | 619.0 | 742.5 | 840.0 | 962.0 | 341.5 | 601.0 | 747.5 | 895.0 | 971.0 |
QA-SIS(0.1) | 135.5 | 223.0 | 321.5 | 428.5 | 626.5 | 82.5 | 134.5 | 201.5 | 303.0 | 543.0 |
QA-SIS(0.3) | 14.0 | 28.0 | 49.5 | 93.5 | 237.5 | 10.0 | 24.5 | 65.0 | 116.0 | 593.5 |
QA-SIS(0.5) | 8.0 | 20.0 | 32.0 | 54.5 | 162.5 | 3.0 | 5.0 | 6.0 | 8.0 | 15.0 |
QA-SIS(0.7) | 37.5 | 73.0 | 146.5 | 223.0 | 394.5 | 9.0 | 15.5 | 20.0 | 32.5 | 56.5 |
QA-SIS(0.9) | 152.5 | 291.0 | 418.0 | 562.0 | 816.0 | 60.5 | 145.0 | 215.5 | 307.5 | 548.5 |
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
SIS | 2126.0 | 3353.0 | 3956.5 | 4469.5 | 4874.5 | 3038.5 | 3633.5 | 4071.0 | 4528.5 | 4954.5 |
DC-SIS | 1949.0 | 3324.0 | 4045.5 | 4387.5 | 4832.0 | 1485.5 | 3097.5 | 3959.0 | 4368.0 | 4702.0 |
QA-SIS(0.1) | 507.5 | 1137.5 | 1470.0 | 1981.5 | 3183.0 | 433.0 | 747.5 | 1177.5 | 1573.5 | 2852.5 |
QA-SIS(0.3) | 54.5 | 97.0 | 177.0 | 348.5 | 1122.5 | 39.0 | 126.0 | 288.0 | 595.5 | 2742.0 |
QA-SIS(0.5) | 33.5 | 95.0 | 184.5 | 351.5 | 711.0 | 6.0 | 11.0 | 15.0 | 29.0 | 62.5 |
QA-SIS(0.7) | 143.0 | 419.5 | 690.0 | 1263.5 | 2217.5 | 34.5 | 60.5 | 100.0 | 175.0 | 402.0 |
QA-SIS(0.9) | 670.5 | 1140.0 | 1775.5 | 2586.5 | 3707.5 | 439.5 | 717.0 | 1022.0 | 1547.0 | 2662.5 |
Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 4.0 | 9.0 | 28.5 | 317.0 | 3.0 | 3.0 | 6.0 | 24.0 | 156.5 |
QA-SVS-A(5) | 3.0 | 3.0 | 8.0 | 40.0 | 179.5 | 3.0 | 3.0 | 4.0 | 8.0 | 42.5 |
QA-SVS-A(6) | 3.0 | 3.0 | 5.0 | 12.0 | 93.5 | 3.0 | 3.0 | 4.0 | 5.0 | 23.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 5.0 | 13.0 | 82.0 | 3.0 | 3.0 | 4.0 | 9.5 | 59.5 |
QA-SVS-A(8) | 3.0 | 3.0 | 4.0 | 11.5 | 52.0 | 3.0 | 3.0 | 3.0 | 7.5 | 71.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 4.0 | 11.0 | 47.5 | 3.0 | 3.0 | 3.0 | 6.0 | 53.5 |
QA-SVS-A(10) | 3.0 | 3.0 | 4.0 | 10.5 | 89.5 | 3.0 | 3.0 | 4.0 | 6.5 | 59.0 |
QCS(4) | 3.0 | 3.0 | 4.0 | 10.0 | 46.0 | 3.0 | 3.0 | 3.0 | 6.0 | 60.5 |
QCS(5) | 3.0 | 3.0 | 4.0 | 8.0 | 58.0 | 3.0 | 3.0 | 4.0 | 6.5 | 66.5 |
QCS(6) | 3.0 | 3.0 | 4.0 | 11.0 | 251.0 | 3.0 | 3.0 | 3.0 | 6.0 | 44.0 |
QCS(7) | 3.0 | 3.0 | 4.0 | 11.5 | 76.5 | 3.0 | 3.0 | 4.0 | 6.0 | 58.5 |
QCS(8) | 3.0 | 3.0 | 6.0 | 17.5 | 118.0 | 3.0 | 3.0 | 4.0 | 9.5 | 76.5 |
QCS(9) | 3.0 | 4.0 | 6.0 | 22.5 | 118.0 | 3.0 | 3.0 | 4.0 | 15.0 | 66.5 |
QCS(10) | 3.0 | 4.0 | 7.0 | 20.0 | 156.5 | 3.0 | 3.0 | 5.0 | 9.5 | 72.5 |
SIS | 260.0 | 559.0 | 802.5 | 922.5 | 981.5 | 227.5 | 574.5 | 769.5 | 907.0 | 988.0 |
DC-SIS | 3.0 | 3.0 | 6.0 | 28.0 | 450.0 | 3.0 | 3.0 | 4.0 | 19.5 | 350.0 |
QA-SIS(0.1) | 82.0 | 168.5 | 300.0 | 521.5 | 849.5 | 70.0 | 132.0 | 241.5 | 395.0 | 737.5 |
QA-SIS(0.3) | 10.5 | 21.0 | 42.5 | 98.0 | 195.0 | 4.5 | 12.0 | 21.0 | 53.0 | 306.0 |
QA-SIS(0.5) | 5.5 | 20.5 | 56.0 | 166.5 | 517.5 | 3.0 | 9.5 | 24.5 | 90.5 | 439.5 |
QA-SIS(0.7) | 4.0 | 11.0 | 30.5 | 85.5 | 309.0 | 7.5 | 15.5 | 35.5 | 114.5 | 355.0 |
QA-SIS(0.9) | 93.5 | 191.0 | 382.0 | 587.0 | 798.5 | 111.5 | 259.5 | 467.5 | 660.5 | 852.5 |
QA-SVS-A(4) | 3.0 | 12.0 | 58.5.0 | 200.0 | 831.0 | 3.0 | 5.0 | 13.0 | 81.0 | 615.5 |
QA-SVS-A(5) | 3.0 | 6.0 | 18.0 | 98.0 | 791.0 | 3.0 | 4.0 | 8.0 | 33.0 | 242.5 |
QA-SVS-A(6) | 3.0 | 4.0 | 9.0 | 45.0 | 322.0 | 3.0 | 4.0 | 8.0 | 31.5 | 241.0 |
QA-SVS-A(7) | 3.0 | 4.0 | 6.5 | 42.0 | 549.5 | 3.0 | 4.0 | 7.0 | 29.0 | 165.0 |
QA-SVS-A(8) | 3.0 | 4.0 | 7.0 | 26.0 | 152.5 | 3.0 | 3.0 | 6.0 | 19.5 | 183.5 |
QA-SVS-A(9) | 3.0 | 4.0 | 12.5 | 70.0 | 552.0 | 3.0 | 3.0 | 5.5 | 19.0 | 423.5 |
QA-SVS-A(10) | 3.0 | 5.0 | 12.5 | 44.0 | 539.0 | 3.0 | 3.0 | 6.0 | 28.5 | 334.5 |
QCS(4) | 3.0 | 3.0 | 6.0 | 14.5 | 166.0 | 3.0 | 3.0 | 4.0 | 14.5 | 163.5 |
QCS(5) | 3.0 | 4.0 | 6.5 | 30.0 | 209.5 | 3.0 | 3.0 | 4.0 | 8.5 | 73.5 |
QCS(6) | 3.0 | 4.0 | 9.5 | 40.0 | 594.5 | 3.0 | 3.0 | 5.0 | 17.0 | 160.5 |
QCS(7) | 3.0 | 4.0 | 12.0 | 50.0 | 417.0 | 3.0 | 3.0 | 7.0 | 25.0 | 125.5 |
QCS(8) | 3.0 | 5.0 | 20.0 | 64.5 | 619.0 | 3.0 | 3.0 | 7.0 | 25.0 | 94.0 |
QCS(9) | 3.0 | 6.0 | 22.0 | 97.0 | 642.5 | 3.0 | 4.0 | 9.0 | 39.5 | 324.0 |
QCS(10) | 3.0 | 6.0 | 21.5 | 163.0 | 1131.5 | 3.0 | 4.0 | 8.0 | 37.5 | 431.5 |
SIS | 1341.0 | 3294.5 | 4114.0 | 4586.0 | 4968.0 | 1252.5 | 3027.5 | 3972.0 | 4524.0 | 4937.5 |
DC-SIS | 3.0 | 5.0 | 12.0 | 110.0 | 1952.0 | 3.0 | 4.0 | 10.0 | 57.5 | 1297.5 |
QA-SIS(0.1) | 423.0 | 1111.5 | 1827.0 | 3080.0 | 4736.5 | 350.0 | 709.5 | 1175.5 | 1655.0 | 3101.0 |
QA-SIS(0.3) | 19.0 | 77.0 | 198.0 | 556.0 | 1755.5 | 14.0 | 45.5 | 128.0 | 402.0 | 1319.0 |
QA-SIS(0.5) | 13.5 | 82.5 | 377.5 | 787.0 | 2140.0 | 4.0 | 32.0 | 177.5 | 469.5 | 2936.0 |
QA-SIS(0.7) | 16.5 | 48.0 | 177.5 | 472.0 | 1475.0 | 22.5 | 96.5 | 336.0 | 777.5 | 2261.5 |
QA-SIS(0.9) | 549.0 | 1025.5 | 1727.0 | 2413.0 | 3997.5 | 434.5 | 1132.5 | 2093.5 | 3309.0 | 4599.5 |
Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 11.0 | 41.5 | 211.0 | 717.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.0 |
QA-SVS-A(5) | 3.0 | 4.5 | 26.5 | 126.5 | 380.5 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 13.0 | 69.5 | 319.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 6.0 | 18.0 | 309.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 |
QA-SVS-A(8) | 3.0 | 3.0 | 6.0 | 16.5 | 107.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 4.5 | 10.0 | 94.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 4.0 | 13.0 | 112.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 4.0 | 31.5 | 95.3 | 329.5 | 730.5 | 3.0 | 3.0 | 3.0 | 5.5 | 71.0 |
QCS(5) | 8.0 | 39.0 | 148.5 | 437.0 | 742.0 | 3.0 | 3.0 | 3.0 | 4.0 | 10.5 |
QCS(6) | 4.0 | 28.5 | 88.0 | 287.0 | 664.5 | 3.0 | 3.0 | 3.0 | 4.0 | 46.0 |
QCS(7) | 3.5 | 22.5 | 95.5 | 256.0 | 563.5 | 3.0 | 3.0 | 3.0 | 3.0 | 6.0 |
QCS(8) | 3.0 | 19.0 | 67.0 | 201.0 | 535.0 | 3.0 | 3.0 | 3.0 | 3.0 | 11.0 |
QCS(9) | 3.0 | 8.0 | 50.0 | 165.5 | 494.5 | 3.0 | 3.0 | 3.0 | 3.0 | 11.0 |
QCS(10) | 3.0 | 9.5 | 38.0 | 124.0 | 539.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SIS(0.1) | 4.0 | 7.5 | 14.5 | 28.0 | 152.0 | 3.0 | 4.0 | 5.0 | 7.5 | 15.0 |
QA-SIS(0.3) | 4.0 | 35.0 | 110.0 | 247.0 | 840.5 | 3.0 | 3.0 | 3.0 | 5.0 | 17.0 |
QA-SIS(0.5) | 27.5 | 141.0 | 262.0 | 516.5 | 880.5 | 3.0 | 4.0 | 7.0 | 35.5 | 258.0 |
QA-SIS(0.7) | 49.5 | 185.0 | 425.0 | 736.5 | 929.0 | 8.5 | 32.0 | 160.5 | 385.5 | 865.0 |
QA-SIS(0.9) | 241.5 | 456.5 | 648.5 | 878.0 | 985.0 | 67.0 | 275.0 | 541.5 | 754.5 | 972.0 |
QA-SVS-A(4) | 6.0 | 52.0 | 347.0 | 1262.5 | 3451.5 | 3.0 | 3.0 | 3.0 | 3.0 | 5.5 |
QA-SVS-A(5) | 3.0 | 17.5 | 149.0 | 472.0 | 2285.5 | 3.0 | 3.0 | 3.0 | 3.5 | 6.0 |
QA-SVS-A(6) | 3.0 | 11.0 | 47.5 | 293.0 | 1288.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(7) | 3.0 | 5.0 | 19.5 | 89.5 | 1288.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 10.0 | 95.0 | 923.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(9) | 3.0 | 4.0 | 8.0 | 33.5 | 783.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(10) | 3.0 | 4.0 | 11.0 | 58.0 | 725.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 40.5 | 329.0 | 853.5 | 1965.0 | 3801.0 | 3.0 | 3.0 | 4.0 | 32.0 | 277.0 |
QCS(5) | 12.5 | 164.5 | 515.0 | 1677.5 | 3445.5 | 3.0 | 3.0 | 3.0 | 3.0 | 112.0 |
QCS(6) | 6.5 | 62.5 | 262.0 | 917.0 | 2845.5 | 3.0 | 3.0 | 3.0 | 5.5 | 35.0 |
QCS(7) | 18.5 | 105.0 | 404.5 | 937.0 | 2855.0 | 3.0 | 3.0 | 3.0 | 4.0 | 31.5 |
QCS(8) | 5.5 | 84.0 | 333.0 | 788.5 | 3004.0 | 3.0 | 3.0 | 3.0 | 4.0 | 16.0 |
QCS(9) | 3.0 | 46.0 | 173.5 | 596.5 | 1803.5 | 3.0 | 3.0 | 3.0 | 4.0 | 28.5 |
QCS(10) | 6.5 | 40.5 | 213.5 | 939.0 | 2590.5 | 3.0 | 3.0 | 3.0 | 5.0 | 13.0 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 43.5 | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SIS(0.1) | 8.0 | 23.0 | 52.5 | 149.0 | 919.0 | 5.0 | 11.0 | 17.5 | 27.0 | 55.5 |
QA-SIS(0.3) | 3.0 | 117.5 | 562.0 | 1466.0 | 3196.5 | 3.0 | 3.0 | 4.0 | 6.5 | 109.0 |
QA-SIS(0.5) | 69.5 | 611.5 | 1754.5 | 3333.5 | 4554.0 | 3.5 | 8.0 | 59.0 | 552.5 | 1786.5 |
QA-SIS(0.7) | 184.5 | 1131.0 | 2307.0 | 3419.0 | 4573.0 | 17.0 | 287.5 | 886.5 | 1798.0 | 3692.5 |
QA-SIS(0.9) | 867.0 | 1881.5 | 3007.0 | 3863.5 | 4763.5 | 841.5 | 1862.5 | 2873.0 | 3834.0 | 4603.0 |
Method | QA-SVS-AFD(K) | QA-SVS-FDR(K) | QCS-FDR(K) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
K | 2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 |
Scenario 2.1 | |||||||||||||||
12.08 | 10.69 | 10.52 | 10.23 | 10.04 | 11.68 | 11.69 | 11.92 | 11.88 | 11.53 | 11.17 | 11.22 | 11.25 | 11.29 | 11.10 | |
FDR | 0.16 | 0.06 | 0.05 | 0.02 | 0.00 | 0.14 | 0.14 | 0.16 | 0.15 | 0.13 | 0.10 | 0.10 | 0.10 | 0.11 | 0.09 |
F-score | 0.91 | 0.97 | 0.98 | 0.99 | 1.00 | 0.92 | 0.92 | 0.91 | 0.92 | 0.93 | 0.95 | 0.94 | 0.94 | 0.94 | 0.95 |
Scenario 2.2 | |||||||||||||||
50.33 | 48.23 | 45.02 | 38.21 | 27.37 | 52.35 | 51.98 | 52.22 | 52.19 | 52.05 | 50.22 | 50.67 | 49.77 | 49.41 | 48.43 | |
FDR | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.04 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 | 0.04 |
F-score | 0.98 | 0.98 | 0.95 | 0.86 | 0.70 | 0.97 | 0.98 | 0.97 | 0.97 | 0.97 | 0.96 | 0.96 | 0.95 | 0.95 | 0.94 |
Scenario 2.3 | |||||||||||||||
12.09 | 10.67 | 10.40 | 10.10 | 10.02 | 11.74 | 11.58 | 11.52 | 11.60 | 11.57 | 11.08 | 10.90 | 11.16 | 10.90 | 10.90 | |
FDR | 0.17 | 0.06 | 0.04 | 0.01 | 0.00 | 0.14 | 0.13 | 0.13 | 0.13 | 0.13 | 0.09 | 0.08 | 0.10 | 0.08 | 0.08 |
F-score | 0.91 | 0.97 | 0.98 | 1.00 | 1.00 | 0.92 | 0.93 | 0.93 | 0.93 | 0.93 | 0.95 | 0.96 | 0.95 | 0.96 | 0.96 |
Scenario 2.4 | |||||||||||||||
50.05 | 45.91 | 39.33 | 25.62 | 15.27 | 52.23 | 51.96 | 51.91 | 52.07 | 51.56 | 49.53 | 48.03 | 47.22 | 44.47 | 43.53 | |
FDR | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 |
F-score | 0.98 | 0.96 | 0.88 | 0.67 | 0.46 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.95 | 0.93 | 0.92 | 0.90 | 0.89 |
2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 | |
Scenario 2.5 | |||||||||||||||
10.89 | 9.88 | 9.30 | 7.95 | 5.90 | 10.47 | 10.43 | 10.54 | 10.47 | 10.40 | 10.00 | 10.09 | 9.88 | 9.62 | 9.48 | |
FDR | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.04 | 0.05 | 0.04 | 0.04 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 |
F-score | 0.96 | 0.99 | 0.96 | 0.88 | 0.73 | 0.98 | 0.98 | 0.97 | 0.98 | 0.98 | 0.94 | 0.95 | 0.94 | 0.93 | 0.92 |
Scenario 2.6 | |||||||||||||||
14.81 | 3.65 | 0.49 | 0.08 | 0.00 | 12.65 | 13.93 | 12.88 | 10.88 | 9.95 | 1.83 | 1.30 | 0.94 | 0.72 | 0.45 | |
FDR | 0.05 | NaN | NaN | NaN | NaN | 0.04 | 0.04 | 0.05 | 0.04 | 0.04 | NaN | NaN | NaN | NaN | NaN |
F-score | 0.43 | 0.13 | 0.02 | 0.00 | 0.00 | 0.38 | 0.41 | 0.38 | 0.33 | 0.31 | 0.06 | 0.05 | 0.03 | 0.02 | 0.02 |
Scenario 2.7 | |||||||||||||||
10.92 | 10.02 | 9.99 | 9.99 | 9.90 | 10.39 | 10.51 | 10.46 | 10.52 | 10.60 | 10.66 | 10.72 | 10.63 | 10.53 | 10.50 | |
FDR | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.03 | 0.04 | 0.04 | 0.05 | 0.05 | 0.06 | 0.06 | 0.05 | 0.05 | 0.04 |
F-score | 0.96 | 1.00 | 1.00 | 1.00 | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | 0.97 | 0.97 | 0.97 | 0.97 | 0.98 | 0.98 |
Scenario 2.8 | |||||||||||||||
38.60 | 18.94 | 6.36 | 1.24 | 0.19 | 43.52 | 41.46 | 40.05 | 38.34 | 36.21 | 22.75 | 17.21 | 11.87 | 8.08 | 6.15 | |
FDR | 0.02 | 0.00 | 0.00 | NaN | NaN | 0.05 | 0.04 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05 | NaN | NaN | NaN |
F-score | 0.85 | 0.55 | 0.22 | 0.05 | 0.01 | 0.89 | 0.86 | 0.84 | 0.83 | 0.80 | 0.59 | 0.48 | 0.35 | 0.26 | 0.20 |
Appendix C
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Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 3.0 | 8.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 3.0 | 10.5 | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 4.0 | 7.5 | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SIS(0.1) | 3.0 | 3.0 | 5.0 | 10.0 | 60.0 | 3.0 | 3.0 | 3.0 | 5.0 | 23.0 |
QA-SIS(0.3) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 |
QA-SIS(0.7) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QA-SIS(0.9) | 3.0 | 3.0 | 4.0 | 12.0 | 56.0 | 3.0 | 3.0 | 3.0 | 7.5 | 38.5 |
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.5 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 6.0 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 3.0 | 8.5 |
QCS(7) | 3.0 | 3.0 | 3.0 | 3.0 | 10.0 | 3.0 | 3.0 | 3.0 | 3.0 | 14.5 |
QCS(8) | 3.0 | 3.0 | 3.0 | 3.0 | 30.0 | 3.0 | 3.0 | 3.0 | 3.0 | 24.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 4.0 | 28.5 | 3.0 | 3.0 | 3.0 | 4.0 | 46.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 5.0 | 36.5 | 3.0 | 3.0 | 3.0 | 5.5 | 71.5 |
SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SIS(0.1) | 3.0 | 4.0 | 11.5 | 56.5 | 241.0 | 3.0 | 3.0 | 5.0 | 18.0 | 254.0 |
QA-SIS(0.3) | 3.0 | 3.0 | 3.0 | 3.0 | 8.5 | 3.0 | 3.0 | 3.0 | 4.0 | 6.5 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.5 |
QA-SIS(0.7) | 3.0 | 3.0 | 3.0 | 3.0 | 14.0 | 3.0 | 3.0 | 3.0 | 3.0 | 9.0 |
QA-SIS(0.9) | 3.0 | 4.0 | 13.0 | 55.0 | 160.0 | 3.0 | 3.0 | 6.0 | 14.0 | 222.5 |
Method | 5% | 25% | 50% | 75% | 95% | 5% | 25% | 50% | 75% | 95% |
---|---|---|---|---|---|---|---|---|---|---|
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.0 | 11.0 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 3.0 | 17.0 | 3.0 | 3.0 | 3.0 | 3.0 | 7.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 4.0 | 12.5 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 4.0 | 15.5 | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 |
QCS(8) | 3.0 | 3.0 | 3.0 | 5.0 | 35.0 | 3.0 | 3.0 | 3.0 | 3.0 | 8.0 |
QCS(9) | 3.0 | 3.0 | 3.0 | 9.0 | 127.0 | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 |
QCS(10) | 3.0 | 3.0 | 3.0 | 6.0 | 32.5 | 3.0 | 3.0 | 3.0 | 3.0 | 7.5 |
SIS | 287.0 | 486.5 | 697.5 | 870.0 | 986.5 | 127.5 | 330.5 | 573.5 | 824.0 | 971.0 |
DC-SIS | 3.0 | 3.0 | 3.0 | 3.0 | 260.0 | 3.0 | 3.0 | 3.0 | 3.0 | 17.0 |
QA-SIS(0.1) | 176.5 | 262.0 | 394.5 | 576.5 | 814.5 | 61.5 | 147.5 | 257.0 | 394.0 | 630.5 |
QA-SIS(0.3) | 3.0 | 3.0 | 4.0 | 6.0 | 21.5 | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 3.0 | 6.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SIS(0.7) | 3.0 | 5.0 | 8.5 | 23.5 | 67.0 | 3.0 | 3.0 | 3.0 | 4.0 | 5.5 |
QA-SIS(0.9) | 100.5 | 238.0 | 368.0 | 517.5 | 866.5 | 35.5 | 85.5 | 143.5 | 301.0 | 601.0 |
QA-SVS-A(4) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(5) | 3.0 | 3.0 | 3.0 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(6) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(7) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(8) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(9) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SVS-A(10) | 3.0 | 3.0 | 3.0 | 3.0 | 4.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(4) | 3.0 | 3.0 | 3.0 | 3.5 | 46.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(5) | 3.0 | 3.0 | 3.0 | 8.0 | 72.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(6) | 3.0 | 3.0 | 3.0 | 6.5 | 38.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(7) | 3.0 | 3.0 | 3.0 | 10.5 | 171.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(8) | 3.0 | 3.0 | 4.0 | 12.0 | 124.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(9) | 3.0 | 3.0 | 4.0 | 15.5 | 353.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QCS(10) | 3.0 | 3.0 | 4.0 | 20.0 | 221.5 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
SIS | 1391.0 | 2628.5 | 3487.0 | 4254.5 | 4810.5 | 1165.5 | 2144.0 | 3288.5 | 4218.5 | 4937.5 |
DC-SIS | 3.0 | 3.0 | 3.0 | 4.0 | 1197.5 | 3.0 | 3.0 | 3.0 | 3.0 | 57.5 |
QA-SIS(0.1) | 439.0 | 1062.5 | 1653.5 | 2543.5 | 3717.0 | 260.5 | 629.0 | 1142.0 | 1813.5 | 3317.0 |
QA-SIS(0.3) | 4.0 | 6.0 | 9.5 | 20.5 | 110.5 | 3.0 | 3.0 | 4.0 | 5.0 | 9.5 |
QA-SIS(0.5) | 3.0 | 3.0 | 3.0 | 5.0 | 13.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
QA-SIS(0.7) | 5.0 | 11.0 | 25.0 | 61.0 | 377.0 | 3.0 | 3.0 | 4.0 | 5.0 | 9.5 |
QA-SIS(0.9) | 625.0 | 1510.5 | 2279.0 | 3218.0 | 4569.0 | 175.0 | 500.0 | 868.0 | 1592.5 | 2403.0 |
Method | QA-SVS-AFD(K) | QA-SVS-FDR(K) | QCS-FDR(K) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 6 | |
Scenario 2.1 | |||||||||||||||
12.17 | 10.81 | 10.57 | 10.23 | 10.05 | 11.70 | 11.74 | 11.76 | 11.75 | 11.52 | 11.33 | 11.24 | 11.29 | 11.09 | 11.10 | |
FDR | 0.17 | 0.07 | 0.05 | 0.02 | 0.00 | 0.14 | 0.14 | 0.14 | 0.14 | 0.13 | 0.11 | 0.10 | 0.11 | 0.09 | 0.09 |
F-score | 0.90 | 0.96 | 0.97 | 0.99 | 1.00 | 0.92 | 0.92 | 0.92 | 0.92 | 0.93 | 0.94 | 0.94 | 0.94 | 0.95 | 0.95 |
Scenario 2.2 | |||||||||||||||
50.40 | 48.14 | 44.87 | 37.95 | 26.82 | 52.16 | 52.29 | 52.28 | 52.21 | 52.35 | 50.53 | 50.52 | 50.16 | 49.59 | 48.59 | |
FDR | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
F-score | 0.98 | 0.98 | 0.95 | 0.86 | 0.69 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.96 | 0.95 | 0.95 | 0.94 | 0.94 |
Scenario 2.3 | |||||||||||||||
11.98 | 10.66 | 10.38 | 10.13 | 10.02 | 11.48 | 11.51 | 11.70 | 11.40 | 11.59 | 11.11 | 11.02 | 11.22 | 10.96 | 11.00 | |
FDR | 0.16 | 0.06 | 0.03 | 0.01 | 0.00 | 0.12 | 0.12 | 0.14 | 0.12 | 0.13 | 0.09 | 0.09 | 0.10 | 0.08 | 0.08 |
F-score | 0.91 | 0.97 | 0.98 | 0.99 | 1.00 | 0.93 | 0.93 | 0.92 | 0.94 | 0.93 | 0.95 | 0.95 | 0.94 | 0.96 | 0.95 |
Scenario 2.4 | |||||||||||||||
50.31 | 46.07 | 39.30 | 26.54 | 14.55 | 52.22 | 52.26 | 51.75 | 51.88 | 51.77 | 49.77 | 47.84 | 46.93 | 44.97 | 43.69 | |
FDR | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 | 0.05 | 0.05 |
F-score | 0.98 | 0.96 | 0.88 | 0.69 | 0.45 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.95 | 0.93 | 0.92 | 0.90 | 0.89 |
Scenario 2.5 | |||||||||||||||
10.82 | 9.93 | 9.16 | 7.84 | 5.71 | 10.36 | 10.73 | 10.50 | 10.44 | 10.55 | 9.93 | 9.99 | 9.76 | 9.57 | 9.09 | |
FDR | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.06 | 0.05 | 0.04 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 | 0.05 |
F-score | 0.96 | 0.99 | 0.95 | 0.87 | 0.72 | 0.98 | 0.97 | 0.97 | 0.98 | 0.97 | 0.94 | 0.95 | 0.94 | 0.93 | 0.90 |
Scenario 2.6 | |||||||||||||||
14.87 | 3.63 | 0.53 | 0.07 | 0.01 | 12.86 | 15.04 | 12.49 | 10.79 | 9.23 | 1.54 | 1.35 | 1.07 | 0.61 | 0.48 | |
FDR | 0.06 | NaN | NaN | NaN | NaN | 0.05 | 0.05 | 0.03 | NaN | NaN | NaN | NaN | NaN | NaN | NaN |
F-score | 0.43 | 0.13 | 0.02 | 0.00 | 0.00 | 0.38 | 0.43 | 0.38 | 0.33 | 0.29 | 0.05 | 0.04 | 0.03 | 0.02 | 0.02 |
Scenario 2.7 | |||||||||||||||
11.11 | 10.00 | 10.00 | 9.99 | 9.93 | 10.66 | 10.44 | 10.65 | 10.45 | 10.54 | 10.62 | 10.60 | 10.64 | 10.45 | 10.61 | |
FDR | 0.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.06 | 0.04 | 0.06 | 0.04 | 0.05 | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 |
F-score | 0.95 | 1.00 | 1.00 | 1.00 | 1.00 | 0.97 | 0.98 | 0.97 | 0.98 | 0.98 | 0.97 | 0.97 | 0.97 | 0.98 | 0.97 |
Scenario 2.8 | |||||||||||||||
38.86 | 19.20 | 5.93 | 1.26 | 0.29 | 43.22 | 42.34 | 40.35 | 37.54 | 36.33 | 23.00 | 16.93 | 11.91 | 8.62 | 6.20 | |
FDR | 0.03 | 0.00 | 0.00 | NaN | NaN | 0.05 | 0.05 | 0.05 | 0.04 | 0.05 | 0.05 | 0.04 | 0.04 | 0.05 | NaN |
F-score | 0.85 | 0.55 | 0.21 | 0.05 | 0.01 | 0.88 | 0.87 | 0.85 | 0.82 | 0.80 | 0.59 | 0.48 | 0.36 | 0.27 | 0.20 |
K | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|
5% PD | |||||
QA-SVS-AFD(K) | 20,152 | 2435 | 28 | 0 | 0 |
QA-SVS-FDR(K) | 89,353 | 85,426 | 83,991 | 76,492 | 74,106 |
QCS-FDR(K) | 262,144 | 262,144 | 262,144 | 262,144 | 262,144 |
95% PD | |||||
QA-SVS-AFD(K) | 5151 | 502 | 15 | 0 | 0 |
QA-SVS-FDR(K) | 76,800 | 76,442 | 75,863 | 78,157 | 66,473 |
QCS-FDR(K) | 262,144 | 262,144 | 262,144 | 262,144 | 262,144 |
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Yuan, Z.; Chen, J.; Qiu, H.; Huang, Y. Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery. Entropy 2023, 25, 524. https://doi.org/10.3390/e25030524
Yuan Z, Chen J, Qiu H, Huang Y. Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery. Entropy. 2023; 25(3):524. https://doi.org/10.3390/e25030524
Chicago/Turabian StyleYuan, Zihao, Jiaqing Chen, Han Qiu, and Yangxin Huang. 2023. "Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery" Entropy 25, no. 3: 524. https://doi.org/10.3390/e25030524
APA StyleYuan, Z., Chen, J., Qiu, H., & Huang, Y. (2023). Quantile-Adaptive Sufficient Variable Screening by Controlling False Discovery. Entropy, 25(3), 524. https://doi.org/10.3390/e25030524