RiskLogitboost Regression for Rare Events in Binary Response: An Econometric Approach
Abstract
:1. Introduction
- (i)
- Some models exhibit bias towards the majority class or underestimate the minority class. Some classifiers are suitable for balanced data [28,29] or treat the minority class as noise [30]. Moreover, some popular tree-based and boosting-based algorithms have been shown to have a high predictive performance measured only with evaluation metrics that consider all observations equally important [31].
- (ii)
- Unlike econometric methods, several machine learning methods are considered as black boxes in terms of interpretation. They are frequently interpreted using single metrics such as classification accuracy as unique descriptions of complex tasks [32], and they are not able to provide robust explanations for high-risk environments.
2. Background
2.1. Boosting Methods
Algorithm 1.Gradient Boosting Machine |
1. |
2. For d = 1 to D do: |
2.1 |
2.2 |
2.3. . |
2.4 . |
3. End for |
Algorithm 2. Tree Gradient Boost |
1. , where is the mean of . |
2. For d = 1 to D do: |
2.1 |
2.2 ) |
2.3 . |
2.4 |
3. End for |
2.2. Penalized Regression Methods
Algorithm 3. Ridge Logistic Regression. |
1. Minimizing the negative likelihood function: |
2. Penalizing: + |
Algorithm 4. Lasso Logistic Regression. |
1. Minimizing the negative likelihood function: |
2. Penalizing: + . |
2.3. Interpretable Machine Learning
3. The Rare Event Problem with RiskLogitboost Regression
Algorithm 5. Logitboost |
1. , |
are the probability estimates. |
2. For d= 1 to D do: |
2.1 |
2.2 |
2.3 |
2.4 and |
= |
3. End for |
3.1. RiskLogitboost Regression Weighting Mechanism to Improve Rare-Class Learning
3.2. Bias Correction with Weights
3.3. RiskLogitboost Regression
Algorithm 6. RiskLogitboost regression |
1. = 0, |
are the probability estimates. |
2. For d = 1 to D do: |
2.1 |
2.2 |
2.3 |
2.4 |
2.5 |
2.6 . |
2.7 |
3. End For |
4. |
5. p = 1, …, P. |
6. Correcting Bias: . |
4. Illustrative Data
5. Discussion of Results
5.1. Predictive Performance of Extremes
5.2. Interpretable RiskLogitboost Regression
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Computation of as Transformed Response
Appendix B. Computation of Weights
References
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Training Data Set (RMSE Y = 1) | ||||||||||||
Lower Extreme | Upper Extreme | |||||||||||
0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | 0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | |
RiskLogitboost regression | 0.2454 | 0.1825 | 0.1496 | 0.1132 | 0.0927 | 0.0803 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Ridge Logistic | 0.9629 | 0.9629 | 0.9629 | 0.9629 | 0.9628 | 0.9628 | 0.9627 | 0.9627 | 0.9627 | 0.9627 | 0.9627 | 0.9627 |
Lasso Logistic | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 |
Boosting Tree | 0.9787 | 0.9747 | 0.9727 | 0.9700 | 0.9679 | 0.9665 | 0.9162 | 0.9293 | 0.9417 | 0.9495 | 0.9522 | 0.9539 |
Logitboost | 0.9829 | 0.9799 | 0.9781 | 0.9736 | 0.9707 | 0.9688 | 0.9416 | 0.9479 | 0.9505 | 0.9530 | 0.9545 | 0.9557 |
SMOTEBoost | 0.6963 | 0.6901 | 0.6852 | 0.6800 | 0.6761 | 0.6725 | 0.6046 | 0.6090 | 0.6117 | 0.6178 | 0.6222 | 0.6264 |
RUSBoost | 0.5811 | 0.5742 | 0.562 | 0.5517 | 0.5447 | 0.5391 | 0.4466 | 0.4727 | 0.4853 | 0.4931 | 0.4970 | 0.5001 |
WLR | 0.9992 | 0.9982 | 0.9973 | 0.9961 | 0.9950 | 0.9939 | 0.4788 | 0.7092 | 0.7961 | 0.8676 | 0.8996 | 0.9183 |
PLR (PSWa) | 0.9992 | 0.9982 | 0.9973 | 0.9961 | 0.9950 | 0.9939 | 0.4788 | 0.7092 | 0.7961 | 0.8676 | 0.8996 | 0.9183 |
PLR (PSWb) | 0.9820 | 0.9790 | 0.9771 | 0.9725 | 0.9697 | 0.9678 | 0.9407 | 0.9470 | 0.9496 | 0.9520 | 0.9536 | 0.9547 |
SyntheticPL | 0.9830 | 0.9803 | 0.9783 | 0.9736 | 0.9708 | 0.9689 | 0.9380 | 0.9467 | 0.9497 | 0.9523 | 0.9540 | 0.9552 |
WeiLogRFL | 0.3696 | 0.2860 | 0.2386 | 0.1826 | 0.1498 | 0.1297 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Testing Data Set (RMSE Y = 1) | ||||||||||||
Lower Extreme | Upper Extreme | |||||||||||
0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | 0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | |
RiskLogitboost regression | 0.4690 | 0.3725 | 0.3133 | 0.2421 | 0.1991 | 0.1724 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Ridge Logistic | 0.9629 | 0.9629 | 0.9629 | 0.9629 | 0.9628 | 0.9628 | 0.9627 | 0.9627 | 0.9627 | 0.9627 | 0.9627 | 0.9627 |
Lasso Logistic | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 | 0.9628 |
Boosting Tree | 0.9788 | 0.9750 | 0.9731 | 0.9705 | 0.9683 | 0.9669 | 0.9156 | 0.9297 | 0.9424 | 0.9498 | 0.9525 | 0.9542 |
Logitboost | 0.8745 | 0.8723 | 0.8710 | 0.8688 | 0.8674 | 0.8665 | 0.8558 | 0.8577 | 0.8586 | 0.8595 | 0.8601 | 0.8606 |
SMOTEBoost | 0.6959 | 0.6901 | 0.6854 | 0.6801 | 0.6762 | 0.6727 | 0.6042 | 0.6088 | 0.6116 | 0.6180 | 0.6226 | 0.6270 |
RUSBoost | 0.5781 | 0.5600 | 0.5515 | 0.5425 | 0.5358 | 0.5312 | 0.4434 | 0.4539 | 0.4727 | 0.4858 | 0.4913 | 0.4948 |
WLR | 0.9993 | 0.9982 | 0.9973 | 0.9961 | 0.9950 | 0.9938 | 0.4523 | 0.7057 | 0.7959 | 0.8664 | 0.8989 | 0.9178 |
PLR (PSWa) | 0.9993 | 0.9982 | 0.9973 | 0.9961 | 0.9950 | 0.9938 | 0.4523 | 0.7057 | 0.7959 | 0.8664 | 0.8989 | 0.9178 |
PLR (PSWb) | 0.9822 | 0.9792 | 0.9773 | 0.9729 | 0.9700 | 0.9681 | 0.9409 | 0.9471 | 0.9497 | 0.9522 | 0.9537 | 0.9549 |
SyntheticPL | 0.8745 | 0.8721 | 0.8708 | 0.8686 | 0.8673 | 0.8664 | 0.8559 | 0.8577 | 0.8587 | 0.8596 | 0.8602 | 0.8607 |
WeiLogRFL | 0.4690 | 0.3725 | 0.3133 | 0.2421 | 0.1991 | 0.1724 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Training Data Set (RMSE Y = 0) | ||||||||||||
Lower Extreme | Upper Extreme | |||||||||||
0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | 0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | |
RiskLogitboost regression | 0.7508 | 0.8219 | 0.8605 | 0.9062 | 0.9352 | 0.9514 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
Ridge Logistic | 0.0371 | 0.0371 | 0.0371 | 0.0371 | 0.0371 | 0.0372 | 0.0373 | 0.0373 | 0.0373 | 0.0373 | 0.0373 | 0.0373 |
Lasso Logistic | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 |
Boosting Tree | 0.0197 | 0.0221 | 0.0247 | 0.0273 | 0.0294 | 0.0313 | 0.0773 | 0.0583 | 0.0513 | 0.0470 | 0.0451 | 0.0436 |
Logitboost | 0.0157 | 0.0188 | 0.0202 | 0.0227 | 0.0264 | 0.0289 | 0.0574 | 0.0510 | 0.0485 | 0.0460 | 0.0445 | 0.0434 |
SMOTEBoost | 0.2978 | 0.3070 | 0.3116 | 0.3171 | 0.3206 | 0.3240 | 0.3958 | 0.3909 | 0.3865 | 0.3797 | 0.3752 | 0.3704 |
RUSBoost | 0.4219 | 0.4403 | 0.4488 | 0.4579 | 0.4646 | 0.4692 | 0.5566 | 0.5463 | 0.5281 | 0.5149 | 0.5094 | 0.5058 |
WLR | 0.0008 | 0.0020 | 0.0029 | 0.0043 | 0.0055 | 0.0067 | 0.5230 | 0.3106 | 0.2356 | 0.1733 | 0.1433 | 0.1250 |
PLR (PSWa) | 0.0008 | 0.0020 | 0.0029 | 0.0043 | 0.0055 | 0.0067 | 0.5230 | 0.3106 | 0.2356 | 0.1733 | 0.1433 | 0.1250 |
PLR (PSWb) | 0.0166 | 0.0198 | 0.0212 | 0.0237 | 0.0274 | 0.0299 | 0.0582 | 0.0519 | 0.0495 | 0.0470 | 0.0455 | 0.0444 |
SyntheticPL | 0.0157 | 0.0183 | 0.0198 | 0.0225 | 0.0262 | 0.0287 | 0.0601 | 0.0521 | 0.0493 | 0.0466 | 0.0450 | 0.0439 |
WeiLogRFL | 0.6258 | 0.7202 | 0.7758 | 0.8467 | 0.8945 | 0.9212 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
Testing Data Set (RMSE Y = 0) | ||||||||||||
Lower Extreme | Upper Extreme | |||||||||||
0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | 0.01 | 0.05 | 0.10 | 0.20 | 0.3 | 0.4 | |
RiskLogitboost regression | 0.5446 | 0.6488 | 0.7134 | 0.7988 | 0.8598 | 0.8957 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
Ridge Logistic | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 | 0.0374 |
Lasso Logistic | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 | 0.0372 |
Boosting Tree | 0.0197 | 0.0220 | 0.0246 | 0.0272 | 0.0293 | 0.0312 | 0.0774 | 0.0583 | 0.0512 | 0.0470 | 0.0451 | 0.0436 |
Logitboost | 0.1247 | 0.1269 | 0.1279 | 0.1295 | 0.1311 | 0.1322 | 0.1440 | 0.1420 | 0.1411 | 0.1401 | 0.1395 | 0.1390 |
SMOTEBoost | 0.2976 | 0.3069 | 0.3116 | 0.3171 | 0.3206 | 0.3240 | 0.3959 | 0.3909 | 0.3865 | 0.379 | 0.375 | 0.3705 |
RUSBoost | 0.4189 | 0.4259 | 0.4383 | 0.4487 | 0.4558 | 0.4614 | 0.5534 | 0.5280 | 0.5154 | 0.5074 | 0.5034 | 0.5003 |
WLR | 0.0009 | 0.0020 | 0.0029 | 0.0043 | 0.0055 | 0.0067 | 0.5294 | 0.3119 | 0.2364 | 0.1738 | 0.1438 | 0.1254 |
PLR (PSWa) | 0.0008 | 0.0020 | 0.0029 | 0.0043 | 0.0055 | 0.0067 | 0.5230 | 0.3106 | 0.2356 | 0.1733 | 0.1433 | 0.1250 |
PLR (PSWb) | 0.0167 | 0.0198 | 0.0212 | 0.0236 | 0.0273 | 0.0298 | 0.0582 | 0.0519 | 0.0495 | 0.0470 | 0.0455 | 0.0444 |
SyntheticPL | 0.1247 | 0.1271 | 0.1283 | 0.1298 | 0.1313 | 0.1323 | 0.1438 | 0.1418 | 0.1409 | 0.1400 | 0.1394 | 0.1389 |
WeiLogRFL | 0.5446 | 0.6488 | 0.7134 | 0.7988 | 0.8598 | 0.8957 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
Variables | Categories | RiskLogitboost Regression | RiskLogitboost Regression (Standard Error) | RiskLogitboost Regression (Confidence Intervals) |
---|---|---|---|---|
* Intercept | 20.874 | 7.4130 | (6.3445; 35.4035) | |
Power | e | −0.6527 | 3.5641 | (−7.6383; 6.3329) |
f | −1.3379 | 3.4769 | (−8.1526; 5.4768) | |
g | −0.8003 | 3.4506 | (−7.5635; 5.9629) | |
h | 4.9061 | 4.9344 | (−4.7653; 14.578) | |
i | 7.8770 | 5.4611 | (−2.8268; 18.5808) | |
j | 8.0675 | 5.5682 | (−2.8462; 18.9812) | |
* k | 18.1880 | 7.1178 | (4.2371; 32.1389) | |
* l | 45.3320 | 1.0540 | (43.2662; 47.3978) | |
* m | 99.6840 | 1.5136 | (96.7173; 102.6507) | |
* n | 144.1900 | 1.7590 | (140.7424; 147.6376) | |
* o | 145.8000 | 17.6033 | (111.2975; 180.3025) | |
Brand | Japanese (except Nissan) or Korean | −7.6774 | 5.7732 | (−18.9929; 3.6381) |
Mercedes, Chrysler or BMW | −2.0130 | 6.7667 | (−15.2757; 11.2497) | |
Opel, General Motors or Ford | −6.5298 | 5.7170 | (−17.7351; 4.6755) | |
other | 8.2048 | 7.9329 | (−7.3437; 23.7533) | |
* Renault, Nissan or Citroen | −10.3760 | 4.9954 | (−20.1669; −0.5850) | |
Volkswagen, Audi, Skoda or Seat | −5.5055 | 5.8621 | (−16.9952; 5.9842) | |
Region | Basse-Normandie | 10.279 | 7.1850 | (−3.8036; 24.3616) |
Bretagne | −3.4953 | 4.9434 | (−13.1844; 6.1938) | |
Centre | −6.5749 | 4.2924 | (−14.9880; 1.8382) | |
* Haute-Normandie | 27.6060 | 9.3055 | (9.3672; 45.8448) | |
Ile-de-France | −4.1033 | 5.12264 | (−14.1437; 5.9371) | |
* Limousin | 34.5520 | 10.0028 | (14.9465; 54.1575) | |
Nord-Pas-de-Calais | 0.0897 | 5.7443 | (−11.1691; 11.3485) | |
Pays-de-la-Loire | −2.7310 | 5.0910 | (−12.7094; 7.2474) | |
Poitou-Charentes | 2.4523 | 5.9926 | (−9.2932; 14.1978) | |
Density | 0.0003 | 0.00025 | (−0.0003; 0.0009) | |
Gas Regular | 0.0187 | 2.1895 | (−4.2727; 4.3101) | |
Car Age | 0.1053 | 0.1969 | (−0.2806; 0.4912) | |
Driver Age | 0.0217 | 0.0712 | (−0.1179; 0.1613) |
Order | RiskLogitboost | Boosting Tree | Ridge Logistic | Logitboost |
---|---|---|---|---|
First | Power o | Driver Age | Brand Japanese (except Nissan) or Korean | Region Limousin |
Second | Power n | Brand Japanese (except Nissan) or Korean | Region Haute-Normandie | Power m |
Third | Power m | Car Age | Brand Opel, General Motors or Ford | Power l |
Fourth | Power l | Density | Brand Volkswagen, Audi, Skoda or Seat | Power n |
Fifth | Region Limousin | Brand Opel, General Motors or Ford | Region Nord-Pas-de-Calais | Region Haute-Normandie |
Sixth | Region Haute-Normandie | Region Haute-Normandie | Brand Mercedes, Chrysler or BMW | Power k |
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Pesantez-Narvaez, J.; Guillen, M.; Alcañiz, M. RiskLogitboost Regression for Rare Events in Binary Response: An Econometric Approach. Mathematics 2021, 9, 579. https://doi.org/10.3390/math9050579
Pesantez-Narvaez J, Guillen M, Alcañiz M. RiskLogitboost Regression for Rare Events in Binary Response: An Econometric Approach. Mathematics. 2021; 9(5):579. https://doi.org/10.3390/math9050579
Chicago/Turabian StylePesantez-Narvaez, Jessica, Montserrat Guillen, and Manuela Alcañiz. 2021. "RiskLogitboost Regression for Rare Events in Binary Response: An Econometric Approach" Mathematics 9, no. 5: 579. https://doi.org/10.3390/math9050579