Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. ·...

20
Tree-based Cost Sensitive Methods for Fraud Detection in Imbalanced Data Guillaume Metzler, X. Badiche, B. Belkasmi, E. Fromont, A. Habrard and M. Sebban The Seventeenth International Symposium on Intelligent Data Analysis, ’s-Hertogenbocsh, 24-26 October 1

Transcript of Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. ·...

Page 1: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

Tree-based Cost Sensitive Methods for Fraud Detection

in Imbalanced DataGuillaume Metzler, X. Badiche, B. Belkasmi, E. Fromont, A. Habrard and M. Sebban

The Seventeenth International Symposium on Intelligent Data Analysis,’s-Hertogenbocsh, 24-26 October

�1

Page 2: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�2

Imbalanced learning

Balanced dataset (left)VS.

Imbalanced one (right)

Negative (N: common, majority, genuine) Positive (P: rare, minority, fraud)

In fraud detection (or imbalanced learning): few number of frauds or anomalies,P << N and IR = P/N < 0.5% in real cases.Examples: spam detection, medical diagnosis, intrusion detection, bank fraud detection, …

Most of the classical Machine Learning techniques do not work well in such context

they focus on the majority class and predict all instances as negative

Page 3: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�3

How to deal with imbalanced data?

Data level: use sampling methods like under/over-sampling or SMOTE

Algorithms: • use metric learning based algorithm (e.g. LMNN)• cost-sensitive learning• combine several models together (e.g. boosting and stacking)

Post-Process: tune the decision threshold (class probability estimator)

All strategies present advantages and drawbacks

Page 4: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

!4

Context and Outline

Context: Bank Fraud detection on check transactions, <0.5% of fraudRetailers maximize their profit and avoid frauds

Target: Build a model which focuses on retailers’ desires cost-sensitive model

Outline 1. Cost-sensitive decision trees2. Ensemble of cost-sensitive decision trees

1. Random forest 2. Gradient boosting-based model

3. Experiments on a real dataset

Page 5: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�5

Usual decision tree splitting criterion

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Gini impurity of the node (binary case):

Split is made by maximizing:

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9+c450HruOI6FffypFw9N6MuYh8HOKJ5nqKKGupokvcIj3jCs1WzIiu37j5TrYLR7OHbsh4+AEIBkDc=</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpHZoXk4lSiuAPuNVPE/9A/8I7YwpqEZ2Q5My595yZe6+fBCJVjvNasBYWl5ZXiqtr6xubW9ulnd1WGmeS8SaLg1h2fC/lgYh4UwkV8E4iuRf6AW/74wsdb99ymYo4ulKThPdCbxSJoWCeIqrR6Mt+qexUHLPseeDmoIx81ePSC64xQAyGDCE4IijCATyk9HThwkFCXA9T4iQhYeIc91gjbUZZnDI8Ysf0HdGum7MR7bVnatSMTgnolaS0cUiamPIkYX2abeKZcdbsb95T46nvNqG/n3uFxCrcEPuXbpb5X52uRWGIM1ODoJoSw+jqWO6Sma7om9tfqlLkkBCn8YDikjAzylmfbaNJTe26t56Jv5lMzeo9y3MzvOtb0oDdn+OcB63jiutU3MuTcvU8H3UR+zjAEc3zFFXUUEeTvEd4xBOerZoVWZl195lqFXLNHr4t6+EDUEGQPQ==</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Children nodes

Parent node

X

v2Children

�S � ↵v�Sv

<latexit sha1_base64="HvXRDI8HVmwCYre11sT3I6kWEe8=">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</latexit><latexit sha1_base64="HvXRDI8HVmwCYre11sT3I6kWEe8=">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</latexit><latexit sha1_base64="HvXRDI8HVmwCYre11sT3I6kWEe8=">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</latexit><latexit sha1_base64="HvXRDI8HVmwCYre11sT3I6kWEe8=">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</latexit>

[1] Breiman, L., Friedman, J. Olshen, R., Stone, C.: Classification and Regression Trees. Wadsworth and Brooks, CA (1984).

� = 1�X

c

p2c = 1� p2+ � p2� = 2p+p�<latexit sha1_base64="m/U7d1zP8ngUmeOTyRdmeH12z+I=">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</latexit><latexit sha1_base64="m/U7d1zP8ngUmeOTyRdmeH12z+I=">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</latexit><latexit sha1_base64="m/U7d1zP8ngUmeOTyRdmeH12z+I=">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</latexit><latexit sha1_base64="m/U7d1zP8ngUmeOTyRdmeH12z+I=">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</latexit>

Page 6: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

Pred. fraud Pred. genuineActual fraud cTPi cFNi

Actual genuine cFPi cTNi<latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">AAADs3icjVFLb9QwEJ40QNvwWuDIxWIL4hQlFYhKXApIFSe0lXbbSptq5TjerdW85NiV2jQn+I+IfwAXfgNjx1u2VAgcJZn55vvm4UnrXDQqir55a/6t23fWNzaDu/fuP3g4ePT4oKm0ZHzCqrySRylteC5KPlFC5fyolpwWac4P09MPJn54xmUjqnKszmt+XNBFKeaCUYXQbPAzSflClK2iqc6p7Fp2iU9HAvKCbCVvt8hI8iwkc0l11gMOd4EFLzWW7rEkCUhyYlohQU96x5Sm+ap8E/Vs1o5HM9HZZOjsfbIOynt10FOd+FqJpX5vVT/+redldjULSc6amjLeRuE2KzoMB7PBMAoje8hNI3bGENwZVYOvkEAGFTDQUACHEhTaOVBo8JlCDBHUiB1Di5hES9g4hw4C1GpkcWRQRE/xu0Bv6tASfZOzsWqGVXJ8JSoJPEdNhTyJtqlGbFzbzAb9W+7W5jS9neM/dbkKRBWcIPov3ZL5vzozi4I57NgZBM5UW8RMx1wWbW/FdE5WplKYoUbM2BnGJdrMKpf3TKymsbObu6U2/t0yDWp85rgafpguccHxn+u8aRxsh3EUxvuvhrvv3ao34Ck8g5e4zzewCx9hBBNg3ti78D57X/zX/tRP/aynrnlO8wSuHb/4BQJ615g=</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit>

A cost sensitive model

where:

m<latexit sha1_base64="dOK6BAI6cS+sWCskd9XD2I1tIh8=">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</latexit><latexit sha1_base64="dOK6BAI6cS+sWCskd9XD2I1tIh8=">AAACxHicjVHLSsNAFD2Nr/quunQTLIKrkoigy6IgLluwD9AiSTqtQycPZiZCKfoDbvXbxD/Qv/DOmIJaRCckOXPuPWfm3htmgivtea8lZ25+YXGpvLyyura+sVnZ2m6rNJcRa0WpSGU3DBQTPGEtzbVg3UyyIA4F64SjMxPv3DGpeJpc6nHGenEwTPiAR4EmqhnfVKpezbPLnQV+AaooViOtvOAafaSIkCMGQwJNWCCAoucKPjxkxPUwIU4S4jbOcI8V0uaUxSgjIHZE3yHtrgo2ob3xVFYd0SmCXklKF/ukSSlPEjanuTaeW2fD/uY9sZ7mbmP6h4VXTKzGLbF/6aaZ/9WZWjQGOLE1cKops4ypLipcctsVc3P3S1WaHDLiDO5TXBKOrHLaZ9dqlK3d9Daw8TebaVizj4rcHO/mljRg/+c4Z0H7sOZ7Nb95VK2fFqMuYxd7OKB5HqOOCzTQst6PeMKzc+4IRzn5Z6pTKjQ7+Lachw9Q/I9y</latexit><latexit sha1_base64="dOK6BAI6cS+sWCskd9XD2I1tIh8=">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</latexit><latexit sha1_base64="dOK6BAI6cS+sWCskd9XD2I1tIh8=">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</latexit>

cTPi = 0 cFNi = (r � c) ·mcFPi = ⇢ · r ·m� ⇣ cTNi = r ·m

<latexit sha1_base64="pP2o5d7UpKVtOTjCR9mqP8fAu0s=">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</latexit><latexit sha1_base64="pP2o5d7UpKVtOTjCR9mqP8fAu0s=">AAADWnicjVFdS9xAFL3ZtFVjP9bqW1+m3W2xFpdECvWlIBWkT2ULuyoYWSaz4zo4yYTJRLBhf4a/TQr+AP0NffHO7Ci2UtoJSe4995wzc+dmpRSVieOfQSt89PjJ3PxCtPj02fMX7aWXu5WqNeNDpqTS+xmtuBQFHxphJN8vNad5JvledrJt63unXFdCFQNzVvLDnE4KcSQYNQiN2udRmvGJKBpDs1pSPW0kkVMSkS4bNYP+SEw/x13yjrh855vNV/U6e5+ysTIk75I0nXF3HDfVx2pW0sRTyDpJf3BDyZ3NwNkQfd8j5cX47gzRqN2Je7Fb5GGQ+KADfvVV+wJSGIMCBjXkwKEAg7EEChU+B5BADCVih9AgpjESrs5hChFqa2RxZFBET/A7wezAowXm1rNyaoa7SHw1Kgm8RY1CnsbY7kZcvXbOFv2bd+M87dnO8J95rxxRA8eI/kt3y/xfne3FwBFsuh4E9lQ6xHbHvEvtbsWenNzryqBDiZiNx1jXGDOnvL1n4jSV693eLXX1K8e0qM2Z59ZwbU+JA07+HOfDYHejl8S95PvHztYXP+p5eAVvYBXn+Qm24Cv0YYjev4LXwVrwoXUZtsKFcHFGbQVeswy/rXDlBuEPvlE=</latexit><latexit sha1_base64="pP2o5d7UpKVtOTjCR9mqP8fAu0s=">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</latexit><latexit sha1_base64="pP2o5d7UpKVtOTjCR9mqP8fAu0s=">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</latexit>

amount of the transaction

profit rate

loss rate (after insurance) of an unpaid transaction

probability of finding another source of paymentcustomer dissatisfaction costr

<latexit sha1_base64="NQLme2DjYhpxjzVmYR57QeHGz7Y=">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</latexit><latexit sha1_base64="NQLme2DjYhpxjzVmYR57QeHGz7Y=">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</latexit><latexit sha1_base64="NQLme2DjYhpxjzVmYR57QeHGz7Y=">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</latexit><latexit sha1_base64="NQLme2DjYhpxjzVmYR57QeHGz7Y=">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</latexit> ⇣<latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I64qmLbQFpmk0zo0TUIyUWpx4w+41S8T/0D/wjvjFNQiOiHJmXPvOTP3Xj8JRSYd57Vgzc0vLC4Vl0srq2vrG+XNrUYW52nAvSAO47Tls4yHIuKeFDLkrSTlbOSHvOkPT1W8ecPTTMTRpRwnvDtig0j0RcAkUV7njkt2Va44VUcvexa4BlRgVj0uv6CDHmIEyDECRwRJOARDRk8bLhwkxHUxIS4lJHSc4x4l0uaUxSmDETuk74B2bcNGtFeemVYHdEpIb0pKG3ukiSkvJaxOs3U8186K/c17oj3V3cb0943XiFiJa2L/0k0z/6tTtUj0caxrEFRTohlVXWBcct0VdXP7S1WSHBLiFO5RPCUcaOW0z7bWZLp21Vum4286U7FqH5jcHO/qljRg9+c4Z0HjoOo6VffisFI7MaMuYge72Kd5HqGGM9ThkbfAI57wbJ1biXVrjT9TrYLRbOPbsh4+AM+AkT0=</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I64qmLbQFpmk0zo0TUIyUWpx4w+41S8T/0D/wjvjFNQiOiHJmXPvOTP3Xj8JRSYd57Vgzc0vLC4Vl0srq2vrG+XNrUYW52nAvSAO47Tls4yHIuKeFDLkrSTlbOSHvOkPT1W8ecPTTMTRpRwnvDtig0j0RcAkUV7njkt2Va44VUcvexa4BlRgVj0uv6CDHmIEyDECRwRJOARDRk8bLhwkxHUxIS4lJHSc4x4l0uaUxSmDETuk74B2bcNGtFeemVYHdEpIb0pKG3ukiSkvJaxOs3U8186K/c17oj3V3cb0943XiFiJa2L/0k0z/6tTtUj0caxrEFRTohlVXWBcct0VdXP7S1WSHBLiFO5RPCUcaOW0z7bWZLp21Vum4286U7FqH5jcHO/qljRg9+c4Z0HjoOo6VffisFI7MaMuYge72Kd5HqGGM9ThkbfAI57wbJ1biXVrjT9TrYLRbOPbsh4+AM+AkT0=</latexit>

⇢<latexit sha1_base64="JU7YHLTvpw87Z3x1jn/MttlKjrw=">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</latexit><latexit sha1_base64="JU7YHLTvpw87Z3x1jn/MttlKjrw=">AAACx3icjVHLSsNAFD2Nr/quunQTLIKrkoigy6Ib3VWwD2iLJOm0HZpkwmRSLMWFP+BW/0z8A/0L74wpqEV0QpIz595zZu69fhLyVDnOa8FaWFxaXimurq1vbG5tl3Z2G6nIZMDqgQiFbPleykIes7riKmStRDIv8kPW9EcXOt4cM5lyEd+oScK6kTeIeZ8HntJURw7FbansVByz7Hng5qCMfNVE6QUd9CAQIEMEhhiKcAgPKT1tuHCQENfFlDhJiJs4wz3WSJtRFqMMj9gRfQe0a+dsTHvtmRp1QKeE9EpS2jgkjaA8SVifZpt4Zpw1+5v31Hjqu03o7+deEbEKQ2L/0s0y/6vTtSj0cWZq4FRTYhhdXZC7ZKYr+ub2l6oUOSTEadyjuCQcGOWsz7bRpKZ23VvPxN9Mpmb1PshzM7zrW9KA3Z/jnAeN44rrVNzrk3L1PB91Efs4wBHN8xRVXKKGOnkP8YgnPFtXlrDG1t1nqlXINXv4tqyHD6nPkMg=</latexit><latexit sha1_base64="JU7YHLTvpw87Z3x1jn/MttlKjrw=">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</latexit><latexit sha1_base64="JU7YHLTvpw87Z3x1jn/MttlKjrw=">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</latexit>

Goal: maximize the overall profit of the retailer

�6

Cost-sensitive matrix [2] with expert criteria

c<latexit sha1_base64="fn13+bgcNWsCxwLFqHgm2cbshSg=">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</latexit><latexit sha1_base64="fn13+bgcNWsCxwLFqHgm2cbshSg=">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</latexit><latexit sha1_base64="fn13+bgcNWsCxwLFqHgm2cbshSg=">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</latexit><latexit sha1_base64="fn13+bgcNWsCxwLFqHgm2cbshSg=">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</latexit>

[2] Bahnsen, A.C., Villegas, S., Aouada, D., Ottersten, B., Correa, A.M.: Fraud Detection by Stacking Cost-sensitive Decision Trees. DSCS (2017).

Page 7: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�7

�S<latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit>

Cost sensitive decision trees

Parent node

Pred. fraud Pred. genuineActual fraud cTPi cFNi

Actual genuine cFPi cTNi<latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit><latexit sha1_base64="1bS6IZqWmN+c7AAKCjaP67uC4Ew=">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</latexit>

Use the « cost » matrix

Splitting criterion:

�S =1

|S|X

i2S�

⇣m+

mcFPi(xi) +

m�m

cTNi(xi)⌘+

1

|S|X

i2S+

⇣m+

mcTPi(xi) +

m�m

cFNi(xi)⌘,

<latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit>

If and , standard Gini Impurity.cTPi = cTNi = 0<latexit sha1_base64="6Z3C5SlEEHYYBWkDduVh0+uaEgU=">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</latexit><latexit sha1_base64="6Z3C5SlEEHYYBWkDduVh0+uaEgU=">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</latexit><latexit sha1_base64="6Z3C5SlEEHYYBWkDduVh0+uaEgU=">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</latexit><latexit sha1_base64="6Z3C5SlEEHYYBWkDduVh0+uaEgU=">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</latexit>

cFPi = cFNi = 1<latexit sha1_base64="q5tX3uVGiQQViX/Ao+MUW3ur3Nk=">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</latexit><latexit sha1_base64="q5tX3uVGiQQViX/Ao+MUW3ur3Nk=">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</latexit><latexit sha1_base64="q5tX3uVGiQQViX/Ao+MUW3ur3Nk=">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</latexit><latexit sha1_base64="q5tX3uVGiQQViX/Ao+MUW3ur3Nk=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Page 8: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�8

Cost sensitive decision trees

�S<latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit>

�Sl<latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">AAACz3icjVHLSsNAFD3GV62vqks3wSK4KokIuiy60GWL9gFtKZN0WkPzYjJRSqm49Qfc6l+Jf6B/4Z0xBbWITkhy5tx7zsy914l9L5GW9TpnzC8sLi3nVvKra+sbm4Wt7XoSpcLlNTfyI9F0WMJ9L+Q16UmfN2PBWeD4vOEMz1S8ccNF4kXhlRzFvBOwQej1PZdJotrtcxYErDu+7PqTbqFolSy9zFlgZ6CIbFWiwgva6CGCixQBOEJIwj4YEnpasGEhJq6DMXGCkKfjHBPkSZtSFqcMRuyQvgPatTI2pL3yTLTapVN8egUpTeyTJqI8QVidZup4qp0V+5v3WHuqu43o72ReAbES18T+pZtm/lenapHo40TX4FFNsWZUdW7mkuquqJubX6qS5BATp3CP4oKwq5XTPptak+jaVW+Zjr/pTMWqvZvlpnhXt6QB2z/HOQvqhyXbKtnVo2L5NBt1DrvYwwHN8xhlXKCCGnnHeMQTno2qcWvcGfefqcZcptnBt2U8fAAytJQn</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">AAACz3icjVHLSsNAFD3GV62vqks3wSK4KokIuiy60GWL9gFtKZN0WkPzYjJRSqm49Qfc6l+Jf6B/4Z0xBbWITkhy5tx7zsy914l9L5GW9TpnzC8sLi3nVvKra+sbm4Wt7XoSpcLlNTfyI9F0WMJ9L+Q16UmfN2PBWeD4vOEMz1S8ccNF4kXhlRzFvBOwQej1PZdJotrtcxYErDu+7PqTbqFolSy9zFlgZ6CIbFWiwgva6CGCixQBOEJIwj4YEnpasGEhJq6DMXGCkKfjHBPkSZtSFqcMRuyQvgPatTI2pL3yTLTapVN8egUpTeyTJqI8QVidZup4qp0V+5v3WHuqu43o72ReAbES18T+pZtm/lenapHo40TX4FFNsWZUdW7mkuquqJubX6qS5BATp3CP4oKwq5XTPptak+jaVW+Zjr/pTMWqvZvlpnhXt6QB2z/HOQvqhyXbKtnVo2L5NBt1DrvYwwHN8xhlXKCCGnnHeMQTno2qcWvcGfefqcZcptnBt2U8fAAytJQn</latexit>

�Sr<latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">AAACz3icjVHLSsNAFD3GV62vqks3wSK4KokIuiy60GWL9gFtKZN0WkPzYjJRSqm49Qfc6l+Jf6B/4Z0xBbWITkhy5tx7zsy914l9L5GW9TpnzC8sLi3nVvKra+sbm4Wt7XoSpcLlNTfyI9F0WMJ9L+Q16UmfN2PBWeD4vOEMz1S8ccNF4kXhlRzFvBOwQej1PZdJotrtcxYErDu+7IpJt1C0SpZe5iywM1BEtipR4QVt9BDBRYoAHCEkYR8MCT0t2LAQE9fBmDhByNNxjgnypE0pi1MGI3ZI3wHtWhkb0l55Jlrt0ik+vYKUJvZJE1GeIKxOM3U81c6K/c17rD3V3Ub0dzKvgFiJa2L/0k0z/6tTtUj0caJr8KimWDOqOjdzSXVX1M3NL1VJcoiJU7hHcUHY1cppn02tSXTtqrdMx990pmLV3s1yU7yrW9KA7Z/jnAX1w5JtlezqUbF8mo06h13s4YDmeYwyLlBBjbxjPOIJz0bVuDXujPvPVGMu0+zg2zIePgBA+pQt</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit>

maxattributes

�Sl + �Sr � �S<latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit>

Parent node

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Children nodes

Compute the splitting criterion

�S =1

|S|X

i2S�

⇣m+

mcFPi(xi) +

m�m

cTNi(xi)⌘+

1

|S|X

i2S+

⇣m+

mcTPi(xi) +

m�m

cFNi(xi)⌘,

<latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">AAADtnicjVFdb9MwFL1ZgI3y1cEjLxYVUqewKkGTthekaUiDJ1TUdp20jMhx3c5anESOg5giv46fiPgH8Mg/4Nqk2qCamKMkx+fcc6+vb1pmotJh+N1b8+/cvbe+cb/z4OGjx0+6m0+PqqJWjE9YkRXqOKUVz0TOJ1rojB+XilOZZnyanr+1+vQzV5Uo8rG+KPmppItczAWjGqmk+yt+R6WkyYi8IfFsrihrItPEaNFkRNzfxFUtk0bEIiejZNuQOONz3V+GyyQwjTQsaQ6HiTD95otJxBYJrvTtVh9/uNJjJRZneiu4Vc3A3FRyfK1ksFLwcLXgq07S7YWD0C2yCqIW9KBdw6L7DWKYQQEMapDAIQeNOAMKFT4nEEEIJXKn0CCnEAmnczDQQW+NURwjKLLn+F3g7qRlc9zbnJVzM6yS4avQSeAlegqMU4htNeL02mW27E25G5fTnu0C/2mbSyKr4QzZ//mWkbf12V40zGHP9SCwp9IxtjvWZqndrdiTk2tdacxQImfxDHWFmDnn8p6J81Sud3u31Ok/XKRl7Z61sTX8tKfEAUf/jnMVHL0eROEg+rjT2z9oR70Bz+EF9HGeu7AP72EIE2De1DPepffV3/M/+dxf/Ald81rPM/hr+eVvxZ/pQA==</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit><latexit sha1_base64="2RM2rzTRdRGdX5WBtZBoUP2W4dQ=">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</latexit>

Look for the best (attribute, value) which is solution of:

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit>

maxattributes

�Sl + �Sr � �S<latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">AAADCnicjVFdS9xAFD0bbbvdfrjqY1+CS6FQuiQi6KMoYh+32FVhsyyT7KwdzBeTSVFC/oH/xDffxNf+AZ/E+gf0X3hnOqLtUtoJSc49954zc+eGeSwK5XnXDWdm9tnzF82XrVev37yda88v7BZZKSPej7I4k/shK3gsUt5XQsV8P5ecJWHM98LDTZ3f+85lIbL0qzrO+TBhB6mYiIgpokbtraBMx5TnqgoUP1IVU0qKsFS8qOsqSNhR7QbbLEnYqNoZxfXHx0DWn2ywM2p3vK5nljsNfAs6sKuXta8QYIwMEUok4EihCMdgKOgZwIeHnLghKuIkIWHyHDVapC2pilMFI/aQvgcUDSybUqw9C6OOaJeYXklKF+9Jk1GdJKx3c02+NM6a/Zt3ZTz12Y7pH1qvhFiFb8T+S/dQ+b863YvCBGumB0E95YbR3UXWpTS3ok/uPulKkUNOnMZjykvCkVE+3LNrNIXpXd8tM/lbU6lZHUe2tsSdPiUN2P9znNNgd7nre13/y0pnfcOOuol3WMIHmucq1vEZPfTJ+xSX+Ikb58Q5c86di1+lTsNqFvHbcn7cA0HmrW0=</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit>

Page 9: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�9

Cost sensitive decision trees

�S<latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit><latexit sha1_base64="BIXbODMpeWG7dB8Gx+GczYUpOEY=">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</latexit>

Parent node

Children nodesSl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9+c450HruOI6FffypFw9N6MuYh8HOKJ5nqKKGupokvcIj3jCs1WzIiu37j5TrYLR7OHbsh4+AEIBkDc=</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

�Sl<latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit><latexit sha1_base64="rcL6SciQX/3h8oHCBPPhqa5X21k=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

�Sr<latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">AAACz3icjVHLSsNAFD3GV62vqks3wSK4KokIuiy60GWL9gFtKZN0WkPzYjJRSqm49Qfc6l+Jf6B/4Z0xBbWITkhy5tx7zsy914l9L5GW9TpnzC8sLi3nVvKra+sbm4Wt7XoSpcLlNTfyI9F0WMJ9L+Q16UmfN2PBWeD4vOEMz1S8ccNF4kXhlRzFvBOwQej1PZdJotrtcxYErDu+7IpJt1C0SpZe5iywM1BEtipR4QVt9BDBRYoAHCEkYR8MCT0t2LAQE9fBmDhByNNxjgnypE0pi1MGI3ZI3wHtWhkb0l55Jlrt0ik+vYKUJvZJE1GeIKxOM3U81c6K/c17rD3V3Ub0dzKvgFiJa2L/0k0z/6tTtUj0caJr8KimWDOqOjdzSXVX1M3NL1VJcoiJU7hHcUHY1cppn02tSXTtqrdMx990pmLV3s1yU7yrW9KA7Z/jnAX1w5JtlezqUbF8mo06h13s4YDmeYwyLlBBjbxjPOIJz0bVuDXujPvPVGMu0+zg2zIePgBA+pQt</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit><latexit sha1_base64="4zlCNNzmZu4JmnY8VAkSdwVZBUA=">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</latexit>

Until maximum depth or other stopping criterion

Sll<latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit>

Slr<latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit>

Srl<latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit>

Srr<latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit>

maxattributes

�Sl + �Sr � �S<latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">AAADCnicjVFdS9xAFD0bbbvdfrjqY1+CS6FQuiQi6KMoYh+32FVhsyyT7KwdzBeTSVFC/oH/xDffxNf+AZ/E+gf0X3hnOqLtUtoJSc49954zc+eGeSwK5XnXDWdm9tnzF82XrVev37yda88v7BZZKSPej7I4k/shK3gsUt5XQsV8P5ecJWHM98LDTZ3f+85lIbL0qzrO+TBhB6mYiIgpokbtraBMx5TnqgoUP1IVU0qKsFS8qOsqSNhR7QbbLEnYqNoZxfXHx0DWn2ywM2p3vK5nljsNfAs6sKuXta8QYIwMEUok4EihCMdgKOgZwIeHnLghKuIkIWHyHDVapC2pilMFI/aQvgcUDSybUqw9C6OOaJeYXklKF+9Jk1GdJKx3c02+NM6a/Zt3ZTz12Y7pH1qvhFiFb8T+S/dQ+b863YvCBGumB0E95YbR3UXWpTS3ok/uPulKkUNOnMZjykvCkVE+3LNrNIXpXd8tM/lbU6lZHUe2tsSdPiUN2P9znNNgd7nre13/y0pnfcOOuol3WMIHmucq1vEZPfTJ+xSX+Ikb58Q5c86di1+lTsNqFvHbcn7cA0HmrW0=</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">AAADCnicjVFdS9xAFD0bbbvdfrjqY1+CS6FQuiQi6KMoYh+32FVhsyyT7KwdzBeTSVFC/oH/xDffxNf+AZ/E+gf0X3hnOqLtUtoJSc49954zc+eGeSwK5XnXDWdm9tnzF82XrVev37yda88v7BZZKSPej7I4k/shK3gsUt5XQsV8P5ecJWHM98LDTZ3f+85lIbL0qzrO+TBhB6mYiIgpokbtraBMx5TnqgoUP1IVU0qKsFS8qOsqSNhR7QbbLEnYqNoZxfXHx0DWn2ywM2p3vK5nljsNfAs6sKuXta8QYIwMEUok4EihCMdgKOgZwIeHnLghKuIkIWHyHDVapC2pilMFI/aQvgcUDSybUqw9C6OOaJeYXklKF+9Jk1GdJKx3c02+NM6a/Zt3ZTz12Y7pH1qvhFiFb8T+S/dQ+b863YvCBGumB0E95YbR3UXWpTS3ok/uPulKkUNOnMZjykvCkVE+3LNrNIXpXd8tM/lbU6lZHUe2tsSdPiUN2P9znNNgd7nre13/y0pnfcOOuol3WMIHmucq1vEZPfTJ+xSX+Ikb58Q5c86di1+lTsNqFvHbcn7cA0HmrW0=</latexit><latexit sha1_base64="Ftp3N/l3ExGJ/gWsa7RArXLvpHQ=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">AAACxnicjVHLSsNAFD3GV42vqks3wSK4KokbXRbddFmpfUAtkkynNTQvJhOlFMEfcKufJv6B/oV3ximoRXRCkjPn3nNm7r1BFoW5dN3XBWtxaXlltbRmr29sbm2Xd3bbeVoIxlssjVLRDfycR2HCWzKUEe9mgvtxEPFOMD5X8c4tF3mYJpdykvF+7I+ScBgyXxLVbNr2dbniVl29nHngGVCBWY20/IIrDJCCoUAMjgSScAQfOT09eHCREdfHlDhBKNRxjnvYpC0oi1OGT+yYviPa9Qyb0F555lrN6JSIXkFKB4ekSSlPEFanOTpeaGfF/uY91Z7qbhP6B8YrJlbihti/dLPM/+pULRJDnOoaQqop04yqjhmXQndF3dz5UpUkh4w4hQcUF4SZVs767GhNrmtXvfV1/E1nKlbtmckt8K5uSQP2fo5zHrSPq55b9S7cSu3MjLqEfRzgiOZ5ghrqaKBF3iM84gnPVt1KrMK6+0y1FoxmD9+W9fABja6PfA==</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Page 10: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�10

Cost sensitive decision trees

How to label the leaves ?

Compute the profits if all the examples are predicted positive and negative

Choose the label which is solution of:

�+<latexit sha1_base64="sTKBxCkrrVc/s2K0VqHbiMBeK6o=">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</latexit><latexit sha1_base64="sTKBxCkrrVc/s2K0VqHbiMBeK6o=">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</latexit><latexit sha1_base64="sTKBxCkrrVc/s2K0VqHbiMBeK6o=">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</latexit><latexit sha1_base64="sTKBxCkrrVc/s2K0VqHbiMBeK6o=">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</latexit>

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

��<latexit sha1_base64="+6Lqn6iiiDIV68Ge1pdkwnYJG+s=">AAACy3icjVHLSsNAFD2Nr1pfVZdugkVwY0lE0GXRjRuhgn1AW2QyndbQvJhMhFpd+gNu9b/EP9C/8M6Ygg9EJyQ5c+45d+be6yWBnyrHeSlYM7Nz8wvFxdLS8srqWnl9o5nGmeSiweMglm2PpSLwI9FQvgpEO5GChV4gWt7oRMdb10KmfhxdqHEieiEbRv7A50wR1e4OWRiyy73LcsWpOmbZP4GbgwryVY/Lz+iijxgcGUIIRFCEAzCk9HTgwkFCXA8T4iQh38QF7lAib0YqQQpG7Ii+Q9p1cjaivc6ZGjenUwJ6JTlt7JAnJp0krE+zTTwzmTX7W+6JyanvNqa/l+cKiVW4IvYv31T5X5+uRWGAI1ODTzUlhtHV8TxLZrqib25/qkpRhoQ4jfsUl4S5cU77bBtPamrXvWUm/mqUmtV7nmszvOlb0oDd7+P8CZr7VdepuucHldpxPuoitrCNXZrnIWo4RR0NM8cHPOLJOrNS68a6/ZBahdyziS/Lun8HT/6SNg==</latexit><latexit sha1_base64="+6Lqn6iiiDIV68Ge1pdkwnYJG+s=">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</latexit><latexit sha1_base64="+6Lqn6iiiDIV68Ge1pdkwnYJG+s=">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</latexit><latexit sha1_base64="+6Lqn6iiiDIV68Ge1pdkwnYJG+s=">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</latexit>

j<latexit sha1_base64="g8asREj2W0iG7iAbBPStAl7fpYA=">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</latexit><latexit sha1_base64="g8asREj2W0iG7iAbBPStAl7fpYA=">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</latexit><latexit sha1_base64="g8asREj2W0iG7iAbBPStAl7fpYA=">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</latexit><latexit sha1_base64="g8asREj2W0iG7iAbBPStAl7fpYA=">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</latexit>

��(l) =1

|l|

0

@X

i:xi2l\S�

cTNi +X

i:xi2l\S+

cFNi

1

A

<latexit sha1_base64="VuUFhL40S7BVB9Jz9WzVp9hCNrs=">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</latexit><latexit sha1_base64="VuUFhL40S7BVB9Jz9WzVp9hCNrs=">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</latexit><latexit sha1_base64="VuUFhL40S7BVB9Jz9WzVp9hCNrs=">AAADPHicjVFdSxwxFL0zfm23atf2sS/BRVgRlxkpKEJBFEqfikVXBUeGTDa7BjMfZDK2yzB/zb8hfehbH1pKX/vcmxjBD6RmmMnJueecyU2SQopSB8E3z5+anpmda71ov5xfWHzVWXp9VOaVYnzAcpmrk4SWXIqMD7TQkp8UitM0kfw4udgz9eNLrkqRZ4d6UvCzlI4zMRKMaqTizpdoTNOUxus9ufo+Go4UZXXY1BF6NJHEzk0k+Uj3orJK41psf41FJDJTZLQgB/F6w+L68FMsGrL2UOMka0bywUgiJcbnerUdd7pBP7CDPAahA11wYz/vXEMEQ8iBQQUpcMhAI5ZAocTnFEIIoEDuDGrkFCJh6xwaaKO3QhVHBUX2Ar9jXJ06NsO1ySytm+FfJL4KnQRW0JOjTiE2fyO2Xtlkwz6VXdtMs7cJzonLSpHVcI7s/3y3yuf6TC8aRrBlexDYU2EZ0x1zKZU9FbNzcqcrjQkFcgYPsa4QM+u8PWdiPaXt3ZwttfWfVmlYs2ZOW8Evs0u84PDhdT4GRxv9MOiHn991d3bdVbfgLSxDD+9zE3bgI+zDALO/e1PevLfgX/k//N/+nxup7znPG7g3/L//AMHEvA4=</latexit><latexit sha1_base64="VuUFhL40S7BVB9Jz9WzVp9hCNrs=">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</latexit>

�+(l) =1

|l|

0

@X

i:xi2l\S�

cFPi +X

i:xi2l\S�

cTPi

1

A

<latexit sha1_base64="+O/3afGns0K59kwgsWOLrpNVvWQ=">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</latexit><latexit sha1_base64="+O/3afGns0K59kwgsWOLrpNVvWQ=">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</latexit><latexit sha1_base64="+O/3afGns0K59kwgsWOLrpNVvWQ=">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</latexit><latexit sha1_base64="+O/3afGns0K59kwgsWOLrpNVvWQ=">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</latexit>

maxj2{+,�}

�j<latexit sha1_base64="GGq1sQJ9eRUJvbJxhTZcY2AagSI=">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</latexit><latexit sha1_base64="GGq1sQJ9eRUJvbJxhTZcY2AagSI=">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</latexit><latexit sha1_base64="GGq1sQJ9eRUJvbJxhTZcY2AagSI=">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</latexit><latexit sha1_base64="GGq1sQJ9eRUJvbJxhTZcY2AagSI=">AAAC7XicjVHLSsNAFD2N7/qqunQTLIKglkQEBTeiG5cK1haMlEk61tG8SCZiCf0Fd+7ErT/gVn9D/AP9C++MKfhAdEKSM+fec2buvW7si1Ra1kvJGBgcGh4ZHSuPT0xOTVdmZo/SKEs8XvciP0qaLku5L0Jel0L6vBknnAWuzxvuxa6KNy55koooPJTdmJ8ErBOKU+ExSVSrsuRkYZviXObnjghNJ19eWXV6vTxgVz1ny3Q6LAhY67zcqlStmqWX+RPYBaiiWPtR5RkO2ojgIUMAjhCSsA+GlJ5j2LAQE3eCnLiEkNBxjh7KpM0oi1MGI/aCvh3aHRdsSHvlmWq1R6f49CakNLFImojyEsLqNFPHM+2s2N+8c+2p7talv1t4BcRKnBH7l66f+V+dqkXiFJu6BkE1xZpR1XmFS6a7om5ufqpKkkNMnMJtiieEPa3s99nUmlTXrnrLdPxVZypW7b0iN8ObuiUN2P4+zp/gaK1mWzX7YL26vVOMehTzWMASzXMD29jDPurkfY0HPOLJiIwb49a4+0g1SoVmDl+Wcf8OFRmfkg==</latexit>

……………………………………….

Page 11: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�11

Cost sensitive random forest

Build a collection of trees

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9+c450HruOI6FffypFw9N6MuYh8HOKJ5nqKKGupokvcIj3jCs1WzIiu37j5TrYLR7OHbsh4+AEIBkDc=</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Sll<latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit>

Slr<latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit>

Srl<latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xqbl5OJWEtX/oBb/THxD/QvvDOmoBbRCUnOnHvPmbn3urHvJdKyXnPGzOzc/EJ+sbC0vLK6VlzfqCdRKhivsciPRNN1Eu57Ia9JT/q8GQvuBK7PG+7gWMUbt1wkXhReyGHM24HTD72exxxJVP28MxL+uFMsWWVLL3Ma2BkoIVvVqPiCS3QRgSFFAI4QkrAPBwk9LdiwEBPXxog4QcjTcY4xCqRNKYtThkPsgL592rUyNqS98ky0mtEpPr2ClCZ2SBNRniCsTjN1PNXOiv3Ne6Q91d2G9Hczr4BYiSti/9JNMv+rU7VI9HCoa/CoplgzqjqWuaS6K+rm5peqJDnExCncpbggzLRy0mdTaxJdu+qto+NvOlOxas+y3BTv6pY0YPvnOKdBfa9sW2X7bL9UOcpGnccWtrFL8zxABSeookbe13jEE56NU+PGuDPuP1ONXKbZxLdlPHwAFCaRvw==</latexit>

Srr<latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9+c450HruOI6FffypFw9N6MuYh8HOKJ5nqKKGupokvcIj3jCs1WzIiu37j5TrYLR7OHbsh4+AEIBkDc=</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpHZoXk4lSiuAPuNVPE/9A/8I7YwpqEZ2Q5My595yZe6+fBCJVjvNasBYWl5ZXiqtr6xubW9ulnd1WGmeS8SaLg1h2fC/lgYh4UwkV8E4iuRf6AW/74wsdb99ymYo4ulKThPdCbxSJoWCeIqrR6Mt+qexUHLPseeDmoIx81ePSC64xQAyGDCE4IijCATyk9HThwkFCXA9T4iQhYeIc91gjbUZZnDI8Ysf0HdGum7MR7bVnatSMTgnolaS0cUiamPIkYX2abeKZcdbsb95T46nvNqG/n3uFxCrcEPuXbpb5X52uRWGIM1ODoJoSw+jqWO6Sma7om9tfqlLkkBCn8YDikjAzylmfbaNJTe26t56Jv5lMzeo9y3MzvOtb0oDdn+OcB63jiutU3MuTcvU8H3UR+zjAEc3zFFXUUEeTvEd4xBOerZoVWZl195lqFXLNHr4t6+EDUEGQPQ==</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpHZoXk4lSiuAPuNVPE/9A/8I7YwpqEZ2Q5My595yZe6+fBCJVjvNasBYWl5ZXiqtr6xubW9ulnd1WGmeS8SaLg1h2fC/lgYh4UwkV8E4iuRf6AW/74wsdb99ymYo4ulKThPdCbxSJoWCeIqrR6Mt+qexUHLPseeDmoIx81ePSC64xQAyGDCE4IijCATyk9HThwkFCXA9T4iQhYeIc91gjbUZZnDI8Ysf0HdGum7MR7bVnatSMTgnolaS0cUiamPIkYX2abeKZcdbsb95T46nvNqG/n3uFxCrcEPuXbpb5X52uRWGIM1ODoJoSw+jqWO6Sma7om9tfqlLkkBCn8YDikjAzylmfbaNJTe26t56Jv5lMzeo9y3MzvOtb0oDdn+OcB63jiutU3MuTcvU8H3UR+zjAEc3zFFXUUEeTvEd4xBOerZoVWZl195lqFXLNHr4t6+EDUEGQPQ==</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Sll<latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xo7eZhMxFq68gfc6o+Jf6B/4Z0xBbWITkhy5tx7zsy9142El0jLes0ZM7Nz8wv5xcLS8srqWnF9o56Eacx4jYUijJuuk3DhBbwmPSl4M4q547uCN9zBsYo3bnmceGFwIYcRb/tOP/B6HnMkUfXzzkiIcadYssqWXuY0sDNQQraqYfEFl+giBEMKHxwBJGEBBwk9LdiwEBHXxoi4mJCn4xxjFEibUhanDIfYAX37tGtlbEB75ZloNaNTBL0xKU3skCakvJiwOs3U8VQ7K/Y375H2VHcb0t/NvHxiJa6I/Us3yfyvTtUi0cOhrsGjmiLNqOpY5pLqrqibm1+qkuQQEadwl+IxYaaVkz6bWpPo2lVvHR1/05mKVXuW5aZ4V7ekAds/xzkN6ntl2yrbZ/ulylE26jy2sI1dmucBKjhBFTXyvsYjnvBsnBo3xp1x/5lq5DLNJr4t4+EDBdqRuQ==</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit>

Slr<latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xqbl5OJWEtX/oBb/THxD/QvvDOmoBbRCUnOnHvPmbn3urHvJdKyXnPGzOzc/EJ+sbC0vLK6VlzfqCdRKhivsciPRNN1Eu57Ia9JT/q8GQvuBK7PG+7gWMUbt1wkXhReyGHM24HTD72exxxJVP28M/LFuFMsWWVLL3Ma2BkoIVvVqPiCS3QRgSFFAI4QkrAPBwk9LdiwEBPXxog4QcjTcY4xCqRNKYtThkPsgL592rUyNqS98ky0mtEpPr2ClCZ2SBNRniCsTjN1PNXOiv3Ne6Q91d2G9Hczr4BYiSti/9JNMv+rU7VI9HCoa/CoplgzqjqWuaS6K+rm5peqJDnExCncpbggzLRy0mdTaxJdu+qto+NvOlOxas+y3BTv6pY0YPvnOKdBfa9sW2X7bL9UOcpGnccWtrFL8zxABSeookbe13jEE56NU+PGuDPuP1ONXKbZxLdlPHwAFCCRvw==</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit>

Srl<latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit>

Srr<latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xqbl5OJWEtX/oBb/THxD/QvvDOmoBbRCUnOnHvPmbn3urHvJdKyXnPGzOzc/EJ+sbC0vLK6VlzfqCdRKhivsciPRNN1Eu57Ia9JT/q8GQvuBK7PG+7gWMUbt1wkXhReyGHM24HTD72exxxJVP28MxJi3CmWrLKllzkN7AyUkK1qVHzBJbqIwJAiAEcISdiHg4SeFmxYiIlrY0ScIOTpOMcYBdKmlMUpwyF2QN8+7VoZG9JeeSZazegUn15BShM7pIkoTxBWp5k6nmpnxf7mPdKe6m5D+ruZV0CsxBWxf+kmmf/VqVokejjUNXhUU6wZVR3LXFLdFXVz80tVkhxi4hTuUlwQZlo56bOpNYmuXfXW0fE3nalYtWdZbop3dUsasP1znNOgvle2rbJ9tl+qHGWjzmML29ileR6gghNUUSPvazziCc/GqXFj3Bn3n6lGLtNs4tsyHj4AImyRxQ==</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">AAACu3icjVLLTsMwEJyGVwkFAlcuERUSpyrhAkckLj0WlT6kUqHEdYvVvOQ4oKriE7jCpyH+AP6CtUkloELgKMl4dmfs9TrMIpErz3utWCura+sb1U17q7a9s+vs1bp5WkjGOyyNUtkPg5xHIuEdJVTE+5nkQRxGvBdOL3S8d8dlLtLkSs0yPoyDSSLGggWKqHbbtm+cutfwzHCXgV+COsrRSp0XXGOEFAwFYnAkUIQjBMjpGcCHh4y4IebESULCxDkeYJO2oCxOGQGxU/pOaDYo2YTm2jM3akarRPRKUro4Ik1KeZKwXs018cI4a/Y377nx1Hub0T8svWJiFW6J/Uu3yPyvTteiMMaZqUFQTZlhdHWsdCnMqeidu1+qUuSQEafxiOKSMDPKxTm7RpOb2vXZBib+ZjI1q+eszC3wrndJDfZ/tnMZdE8avtfwLz1UcYBDHFMbT3GOJlrokOUEj3jCs9W0Eqv4vApWpbwT+/g2rPsPV5+OTg==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpHZoXk4lSiuAPuNVPE/9A/8I7YwpqEZ2Q5My595yZe6+fBCJVjvNasBYWl5ZXiqtr6xubW9ulnd1WGmeS8SaLg1h2fC/lgYh4UwkV8E4iuRf6AW/74wsdb99ymYo4ulKThPdCbxSJoWCeIqrR6Mt+qexUHLPseeDmoIx81ePSC64xQAyGDCE4IijCATyk9HThwkFCXA9T4iQhYeIc91gjbUZZnDI8Ysf0HdGum7MR7bVnatSMTgnolaS0cUiamPIkYX2abeKZcdbsb95T46nvNqG/n3uFxCrcEPuXbpb5X52uRWGIM1ODoJoSw+jqWO6Sma7om9tfqlLkkBCn8YDikjAzylmfbaNJTe26t56Jv5lMzeo9y3MzvOtb0oDdn+OcB63jiutU3MuTcvU8H3UR+zjAEc3zFFXUUEeTvEd4xBOerZoVWZl195lqFXLNHr4t6+EDUEGQPQ==</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Sll<latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xo7eZhMxFq68gfc6o+Jf6B/4Z0xBbWITkhy5tx7zsy9142El0jLes0ZM7Nz8wv5xcLS8srqWnF9o56Eacx4jYUijJuuk3DhBbwmPSl4M4q547uCN9zBsYo3bnmceGFwIYcRb/tOP/B6HnMkUfXzzkiIcadYssqWXuY0sDNQQraqYfEFl+giBEMKHxwBJGEBBwk9LdiwEBHXxoi4mJCn4xxjFEibUhanDIfYAX37tGtlbEB75ZloNaNTBL0xKU3skCakvJiwOs3U8VQ7K/Y375H2VHcb0t/NvHxiJa6I/Us3yfyvTtUi0cOhrsGjmiLNqOpY5pLqrqibm1+qkuQQEadwl+IxYaaVkz6bWpPo2lVvHR1/05mKVXuW5aZ4V7ekAds/xzkN6ntl2yrbZ/ulylE26jy2sI1dmucBKjhBFTXyvsYjnvBsnBo3xp1x/5lq5DLNJr4t4+EDBdqRuQ==</latexit>

Slr<latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit>

Srl<latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xqbl5OJWEtX/oBb/THxD/QvvDOmoBbRCUnOnHvPmbn3urHvJdKyXnPGzOzc/EJ+sbC0vLK6VlzfqCdRKhivsciPRNN1Eu57Ia9JT/q8GQvuBK7PG+7gWMUbt1wkXhReyGHM24HTD72exxxJVP28MxL+uFMsWWVLL3Ma2BkoIVvVqPiCS3QRgSFFAI4QkrAPBwk9LdiwEBPXxog4QcjTcY4xCqRNKYtThkPsgL592rUyNqS98ky0mtEpPr2ClCZ2SBNRniCsTjN1PNXOiv3Ne6Q91d2G9Hczr4BYiSti/9JNMv+rU7VI9HCoa/CoplgzqjqWuaS6K+rm5peqJDnExCncpbggzLRy0mdTaxJdu+qto+NvOlOxas+y3BTv6pY0YPvnOKdBfa9sW2X7bL9UOcpGnccWtrFL8zxABSeookbe13jEE56NU+PGuDPuP1ONXKbZxLdlPHwAFCaRvw==</latexit>

Srr<latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

Sl<latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit><latexit sha1_base64="hHqSiW65pfnOC4i7H3XXvwQUgDM=">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</latexit>

Sr<latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit><latexit sha1_base64="xH0W9KFFRKfSdEweTZgVbLN8qDw=">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</latexit>

Sll<latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit><latexit sha1_base64="EL8c8F3pXeBdtLNI+ySTRWgzljg=">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</latexit>

Slr<latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipaB9QS0mm0xqbl5OJWEtX/oBb/THxD/QvvDOmoBbRCUnOnHvPmbn3urHvJdKyXnPGzOzc/EJ+sbC0vLK6VlzfqCdRKhivsciPRNN1Eu57Ia9JT/q8GQvuBK7PG+7gWMUbt1wkXhReyGHM24HTD72exxxJVP28M/LFuFMsWWVLL3Ma2BkoIVvVqPiCS3QRgSFFAI4QkrAPBwk9LdiwEBPXxog4QcjTcY4xCqRNKYtThkPsgL592rUyNqS98ky0mtEpPr2ClCZ2SBNRniCsTjN1PNXOiv3Ne6Q91d2G9Hczr4BYiSti/9JNMv+rU7VI9HCoa/CoplgzqjqWuaS6K+rm5peqJDnExCncpbggzLRy0mdTaxJdu+qto+NvOlOxas+y3BTv6pY0YPvnOKdBfa9sW2X7bL9UOcpGnccWtrFL8zxABSeookbe13jEE56NU+PGuDPuP1ONXKbZxLdlPHwAFCCRvw==</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit><latexit sha1_base64="nabiqJdiahKYdvguCrP/fS1UrQ8=">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</latexit>

Srl<latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit><latexit sha1_base64="V1xwESfXAFDOaxR+YzuHay5XXt4=">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</latexit>

Srr<latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit><latexit sha1_base64="rsMihxogVcfoeWvQcvCJzPxZn/4=">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</latexit>

S<latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="lAbwfH9FUZ+g2YcxIBVKJqBpQTo=">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</latexit><latexit sha1_base64="I/fGdljkeLzSB+a3n+Eqe50X1vs=">AAACxnicjVHLSsNAFD3GV42vqks3wSK4KokbXRbddFmpfUAtkkynNTQvJhOlFMEfcKufJv6B/oV3ximoRXRCkjPn3nNm7r1BFoW5dN3XBWtxaXlltbRmr29sbm2Xd3bbeVoIxlssjVLRDfycR2HCWzKUEe9mgvtxEPFOMD5X8c4tF3mYJpdykvF+7I+ScBgyXxLVbNr2dbniVl29nHngGVCBWY20/IIrDJCCoUAMjgSScAQfOT09eHCREdfHlDhBKNRxjnvYpC0oi1OGT+yYviPa9Qyb0F555lrN6JSIXkFKB4ekSSlPEFanOTpeaGfF/uY91Z7qbhP6B8YrJlbihti/dLPM/+pULRJDnOoaQqop04yqjhmXQndF3dz5UpUkh4w4hQcUF4SZVs767GhNrmtXvfV1/E1nKlbtmckt8K5uSQP2fo5zHrSPq55b9S7cSu3MjLqEfRzgiOZ5ghrqaKBF3iM84gnPVt1KrMK6+0y1FoxmD9+W9fABja6PfA==</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002Wl9gG1SDKd1tC8mEyUUgR/wK1+mvgH+hfeGaegFtEJSc6ce8+Zuff6aRhk0nFeC9bC4tLySnG1tLa+sblV3t5pZ0kuGG+xJExE1/cyHgYxb8lAhrybCu5Ffsg7/vhcxTu3XGRBEl/KScr7kTeKg2HAPElUs1kqXZcrTtXRy54HrgEVmNVIyi+4wgAJGHJE4IghCYfwkNHTgwsHKXF9TIkThAId57hHibQ5ZXHK8Igd03dEu55hY9orz0yrGZ0S0itIaeOANAnlCcLqNFvHc+2s2N+8p9pT3W1Cf994RcRK3BD7l26W+V+dqkViiFNdQ0A1pZpR1THjkuuuqJvbX6qS5JASp/CA4oIw08pZn22tyXTtqreejr/pTMWqPTO5Od7VLWnA7s9xzoP2UdV1qu7FcaV2ZkZdxB72cUjzPEENdTTQIu8RHvGEZ6tuxVZu3X2mWgWj2cW3ZT18AI7uj4A=</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit><latexit sha1_base64="7e4I5KlLQSmUxfbnj2l99qbIWlI=">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</latexit>

•••••••••••••••••

Combine the output of each tree: average profit or predicted labelto build several model (see experimental setting)

Page 12: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

!12

Gradient Boosting based model

ri = gt = �@L(y, Ft�1(xi))

@Ft�1(xi)

<latexit sha1_base64="Ru5/DuvoUx+Y+f23nGEHF/RBRZc=">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</latexit><latexit sha1_base64="Ru5/DuvoUx+Y+f23nGEHF/RBRZc=">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</latexit><latexit sha1_base64="Ru5/DuvoUx+Y+f23nGEHF/RBRZc=">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</latexit><latexit sha1_base64="Ru5/DuvoUx+Y+f23nGEHF/RBRZc=">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</latexit>

(ft,↵t) = argmin↵,f

mX

i=1

(ri � ↵f(xi))2

<latexit sha1_base64="OLLUTAb/FqT+g2PtIoMBT0lLB3c=">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</latexit><latexit sha1_base64="OLLUTAb/FqT+g2PtIoMBT0lLB3c=">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</latexit><latexit sha1_base64="OLLUTAb/FqT+g2PtIoMBT0lLB3c=">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</latexit><latexit sha1_base64="OLLUTAb/FqT+g2PtIoMBT0lLB3c=">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</latexit>

Idea of boosting: combine several weak learners into a single strong model ft<latexit sha1_base64="1HpfWcOCeHV5KyyjaDXCExPXSaE=">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</latexit><latexit sha1_base64="1HpfWcOCeHV5KyyjaDXCExPXSaE=">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</latexit><latexit sha1_base64="1HpfWcOCeHV5KyyjaDXCExPXSaE=">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</latexit><latexit sha1_base64="1HpfWcOCeHV5KyyjaDXCExPXSaE=">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</latexit>

F<latexit sha1_base64="7M3HwT+X6Zmz27ofY1cdtVwzhV4=">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</latexit><latexit sha1_base64="7M3HwT+X6Zmz27ofY1cdtVwzhV4=">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</latexit><latexit sha1_base64="7M3HwT+X6Zmz27ofY1cdtVwzhV4=">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</latexit><latexit sha1_base64="7M3HwT+X6Zmz27ofY1cdtVwzhV4=">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</latexit>

FT =TX

t=0

↵tft<latexit sha1_base64="c49+/bqY7zspN5GtWFk1FvN1xwo=">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</latexit><latexit sha1_base64="c49+/bqY7zspN5GtWFk1FvN1xwo=">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</latexit><latexit sha1_base64="c49+/bqY7zspN5GtWFk1FvN1xwo=">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</latexit><latexit sha1_base64="c49+/bqY7zspN5GtWFk1FvN1xwo=">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</latexit>

Using:

Gradient Boosting model

Usual loss function for boosting: exponential loss

Gradient boosting: work in the function space rather than the parameter space [3]

Update rule: Ft = Ft�1 + ↵tft<latexit sha1_base64="r+ZvE+6uLzgnMnOHBSuA3CMRg2g=">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</latexit><latexit sha1_base64="r+ZvE+6uLzgnMnOHBSuA3CMRg2g=">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</latexit><latexit sha1_base64="r+ZvE+6uLzgnMnOHBSuA3CMRg2g=">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</latexit><latexit sha1_base64="r+ZvE+6uLzgnMnOHBSuA3CMRg2g=">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</latexit>

L(xi, yi) = yi exp(�F (xi)) + (1� yi) exp(F (xi))<latexit sha1_base64="TeUQCQpX3Xh6sO1XvUQ79xmlv9o=">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</latexit><latexit sha1_base64="TeUQCQpX3Xh6sO1XvUQ79xmlv9o=">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</latexit><latexit sha1_base64="TeUQCQpX3Xh6sO1XvUQ79xmlv9o=">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</latexit><latexit sha1_base64="TeUQCQpX3Xh6sO1XvUQ79xmlv9o=">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</latexit>

[3] Friedman, J.H.: Greedy Function Approximation: A Gradient Boosting Machine. Annals of Statistics 29 (2000).

Page 13: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�13

Cost sensitive gradient boosting

L<latexit sha1_base64="LcQyulsoniF+z1qJUXdPdjxJeqg=">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</latexit><latexit sha1_base64="LcQyulsoniF+z1qJUXdPdjxJeqg=">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</latexit><latexit sha1_base64="LcQyulsoniF+z1qJUXdPdjxJeqg=">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</latexit><latexit sha1_base64="LcQyulsoniF+z1qJUXdPdjxJeqg=">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</latexit>

Idea: include the cost matrix into the loss function

pi >cTNi � cFPi

cTPi � cFNi + cTNi � cFPi

= si<latexit sha1_base64="YMAeYTtzX0JLmlDKC1yzaEvQey0=">AAADFHicjVFBa9swGH1xtzbLti7tjruIhsFgJNhl0F5aQgtjpy2FJC00wdiK0ok4tpHlQjD+G/0nvfVWeu21jF032P7FPmkObCtjk7H99L73nvRJYRrJTLvup5qz8uDh6lr9UePxk6frz5obm8MsyRUXA55EiToJg0xEMhYDLXUkTlIlgnkYieNwdmjqx+dCZTKJ+3qRivE8OIvlVPJAE+U3PxRp6ct9NppMVcAL7hf9974sWZsRfNsjWFrSoLahbPX1UtdeqljJ9jJfNvxmy+24drD7wKtAC9XoJc07jDBBAo4ccwjE0IQjBMjoOYUHFylxYxTEKULS1gVKNMibk0qQIiB2Rt8zmp1WbExzk5lZN6dVInoVORlekichnSJsVmO2nttkw/4tu7CZZm8L+odV1pxYjY/E/su3VP6vz/SiMcWu7UFST6llTHe8SsntqZids1+60pSQEmfwhOqKMLfO5Tkz68ls7+ZsA1v/ZpWGNXNeaXN8N7ukC/b+vM77YLjd8dyOd/Sm1T2orrqOF9jCK7rPHXTxDj0MKPsSn/EFX50L58q5dm5+Sp1a5XmO34Zz+wNeW66k</latexit><latexit sha1_base64="YMAeYTtzX0JLmlDKC1yzaEvQey0=">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</latexit><latexit sha1_base64="YMAeYTtzX0JLmlDKC1yzaEvQey0=">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</latexit><latexit sha1_base64="YMAeYTtzX0JLmlDKC1yzaEvQey0=">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</latexit>

Our loss:

How to use the output of a gradient tree boosting model?

For a model which return a probability , predict fraud ( ) if [4]: pi<latexit sha1_base64="WNgaVcFvpCpEfe8SIOTDQV7Gw50=">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</latexit><latexit sha1_base64="WNgaVcFvpCpEfe8SIOTDQV7Gw50=">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</latexit><latexit sha1_base64="WNgaVcFvpCpEfe8SIOTDQV7Gw50=">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</latexit><latexit sha1_base64="WNgaVcFvpCpEfe8SIOTDQV7Gw50=">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</latexit>

Rewrite loss as a minimization problem:

[4] Elkan, C.: The Foundations of Cost-Sensitive Learning. Proceedings of the 17th International Joint Conference on Artificial Intelligence (2001).

yi = 1<latexit sha1_base64="1hkeUZyWN08EF+vw4/nHAMcPuA8=">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</latexit><latexit sha1_base64="1hkeUZyWN08EF+vw4/nHAMcPuA8=">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</latexit><latexit sha1_base64="1hkeUZyWN08EF+vw4/nHAMcPuA8=">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</latexit><latexit sha1_base64="1hkeUZyWN08EF+vw4/nHAMcPuA8=">AAACz3icjVHLSsNAFD3GV62vqks3wSK4KokIuhGKbly2YB/QlpKk03ZoXiQTpZSKW3/Arf6V+Af6F94Zp6AW0QlJzpx7z5m597qxz1NhWa8LxuLS8spqbi2/vrG5tV3Y2a2nUZZ4rOZFfpQ0XSdlPg9ZTXDhs2acMCdwfdZwR5cy3rhhScqj8FqMY9YJnEHI+9xzBFHt9tARk/G0y8/tfLdQtEqWWuY8sDUoQq9KVHhBGz1E8JAhAEMIQdiHg5SeFmxYiInrYEJcQoirOMMUedJmlMUowyF2RN8B7VqaDWkvPVOl9ugUn96ElCYOSRNRXkJYnmaqeKacJfub90R5yruN6e9qr4BYgSGxf+lmmf/VyVoE+jhTNXCqKVaMrM7TLpnqiry5+aUqQQ4xcRL3KJ4Q9pRy1mdTaVJVu+yto+JvKlOycu/p3Azv8pY0YPvnOOdB/bhkWyW7elIsX+hR57CPAxzRPE9RxhUqqJF3jEc84dmoGrfGnXH/mWosaM0evi3j4QM5QpO9</latexit>

yi = 1 yi = 0yi = 1 cTPi cFNi

yi = 0 cFPi cTNi<latexit sha1_base64="STC13MOQFUxcph5orxIGlwofRK0=">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</latexit><latexit sha1_base64="STC13MOQFUxcph5orxIGlwofRK0=">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</latexit><latexit sha1_base64="STC13MOQFUxcph5orxIGlwofRK0=">AAADj3icjVHRatswFL2uu611uzXdHvcimrTsKdhj0MLoCBuU7WWkkKSFuniyoiaijm1kuWA8/2B/oOwPtr/YlaIUd2VsMrbvPeeeo3ulOE9EoXz/h7Pmrj95+mxj09vafv5ip7P7clJkpWR8zLIkk+cxLXgiUj5WQiX8PJecLuKEn8XXnzR/dsNlIbJ0pKqcXy7oLBVXglGFUNS5DWM+E2mtaFwmVDY1+85YQzxyQHrh+x5+5lTVVROJ46C3hCzTpnxLhaFHwrnuhXi2qGopN1HKono0jERjfDA5+WoSrVxuWLX8VoqTtmJ0rwh5Or3v3Is6Xb/vm0UeB4ENumDXMOvcQQhTyIBBCQvgkILCOAEKBT4XEIAPOWKXUCMmMRKG59CAh9oSqzhWUESv8TvD7MKiKebaszBqhrsk+EpUEthHTYZ1EmO9GzF8aZw1+jfv2njq3ir8x9ZrgaiCOaL/0q0q/1enZ1FwBUdmBoEz5QbR0zHrUppT0Z2T1lQKHXLEdDxFXmLMjHJ1zsRoCjO7Pltq+J+mUqM6Z7a2hF+6S7zg4M/rfBxM3vYDvx+cvusOPtqr3oDXsAdv8D4PYQCfYQhjYI7vTJzI+ebuuofuB3ewLF1zrOYVPFjul99BqseC</latexit><latexit sha1_base64="STC13MOQFUxcph5orxIGlwofRK0=">AAADj3icjVHRatswFL2uu611uzXdHvcimrTsKdhj0MLoCBuU7WWkkKSFuniyoiaijm1kuWA8/2B/oOwPtr/YlaIUd2VsMrbvPeeeo3ulOE9EoXz/h7Pmrj95+mxj09vafv5ip7P7clJkpWR8zLIkk+cxLXgiUj5WQiX8PJecLuKEn8XXnzR/dsNlIbJ0pKqcXy7oLBVXglGFUNS5DWM+E2mtaFwmVDY1+85YQzxyQHrh+x5+5lTVVROJ46C3hCzTpnxLhaFHwrnuhXi2qGopN1HKono0jERjfDA5+WoSrVxuWLX8VoqTtmJ0rwh5Or3v3Is6Xb/vm0UeB4ENumDXMOvcQQhTyIBBCQvgkILCOAEKBT4XEIAPOWKXUCMmMRKG59CAh9oSqzhWUESv8TvD7MKiKebaszBqhrsk+EpUEthHTYZ1EmO9GzF8aZw1+jfv2njq3ir8x9ZrgaiCOaL/0q0q/1enZ1FwBUdmBoEz5QbR0zHrUppT0Z2T1lQKHXLEdDxFXmLMjHJ1zsRoCjO7Pltq+J+mUqM6Z7a2hF+6S7zg4M/rfBxM3vYDvx+cvusOPtqr3oDXsAdv8D4PYQCfYQhjYI7vTJzI+ebuuofuB3ewLF1zrOYVPFjul99BqseC</latexit>

L(y, y) =mX

i=1

[yi(yicTPi + (1� yi)cFNi) + (1� yi)(yicFPi + (1� yi)cTNi)]<latexit sha1_base64="XmDzFA4COcRS+DVICxu4qVgcGEM=">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</latexit><latexit sha1_base64="XmDzFA4COcRS+DVICxu4qVgcGEM=">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</latexit><latexit sha1_base64="XmDzFA4COcRS+DVICxu4qVgcGEM=">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</latexit><latexit sha1_base64="XmDzFA4COcRS+DVICxu4qVgcGEM=">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</latexit>

L(y, p) = �mX

i=1

(yicTPi + (1� yi)cFPi) Ipi>si + (yicFNi + (1� yi)cTNi)Ipisi

<latexit sha1_base64="fN6fpwE7YiFnpj7oYxWhEZUpqVs=">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</latexit><latexit sha1_base64="fN6fpwE7YiFnpj7oYxWhEZUpqVs=">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</latexit><latexit sha1_base64="fN6fpwE7YiFnpj7oYxWhEZUpqVs=">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</latexit><latexit sha1_base64="fN6fpwE7YiFnpj7oYxWhEZUpqVs=">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</latexit>

Page 14: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

!14

Cost sensitive gradient boosting

L(xi, yi) = (1� si)yie�F (xi) + si(1� yi)e

F (xi)<latexit sha1_base64="IBbHdlIofgkkUm++w/ICTJ9GJAA=">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</latexit><latexit sha1_base64="IBbHdlIofgkkUm++w/ICTJ9GJAA=">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</latexit><latexit sha1_base64="IBbHdlIofgkkUm++w/ICTJ9GJAA=">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</latexit><latexit sha1_base64="IBbHdlIofgkkUm++w/ICTJ9GJAA=">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</latexit>

The loss can be rewritten (using the analytical value of ):

L(xi, yi) = (1� si)yiIpi<si + si(1� yi)Ipi>si<latexit sha1_base64="KkxGhmiAoVwvw6lLPTkJ9BPaTAM=">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</latexit><latexit sha1_base64="KkxGhmiAoVwvw6lLPTkJ9BPaTAM=">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</latexit><latexit sha1_base64="KkxGhmiAoVwvw6lLPTkJ9BPaTAM=">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</latexit><latexit sha1_base64="KkxGhmiAoVwvw6lLPTkJ9BPaTAM=">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</latexit>

We show, because , it is enough to minimizeIpi>si exp(F (xi))<latexit sha1_base64="zWThB1tHjIh/sBYKh1oiEJefEto=">AAAC63icjVHLSsNAFD3GV62vqks3waK0m5KIoCspCqK7ClaFtoQknepgXiQTaSn9A3fuxK0/4Fb/Q/wD/QvvjCmoRXRCkjPn3nNm7r1O5PFEGMbrmDY+MTk1nZvJz87NLywWlpZPkzCNXVZ3Qy+Mzx07YR4PWF1w4bHzKGa273jszLnal/GzaxYnPAxORC9iLd++CHiHu7YgyipsNH1bXDpO/2hg9SOL7yYWHzQ9pjdZNyodlLoWL5fzeatQNCqGWvooMDNQRLZqYeEFTbQRwkUKHwwBBGEPNhJ6GjBhICKuhT5xMSGu4gwD5EmbUhajDJvYK/pe0K6RsQHtpWei1C6d4tEbk1LHOmlCyosJy9N0FU+Vs2R/8+4rT3m3Hv2dzMsnVuCS2L90w8z/6mQtAh3sqBo41RQpRlbnZi6p6oq8uf6lKkEOEXEStykeE3aVcthnXWkSVbvsra3ibypTsnLvZrkp3uUtacDmz3GOgtPNimlUzOOtYnUvG3UOq1hDiea5jSoOUUOdvG/wiCc8a752q91p95+p2limWcG3pT18APREne0=</latexit><latexit sha1_base64="zWThB1tHjIh/sBYKh1oiEJefEto=">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</latexit><latexit sha1_base64="zWThB1tHjIh/sBYKh1oiEJefEto=">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</latexit><latexit sha1_base64="zWThB1tHjIh/sBYKh1oiEJefEto=">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</latexit>

The gradient boosting model is computed using a specific solver, XGboost which only needs the first and second derivative of the loss function.

si<latexit sha1_base64="usNuiC8JmFrQV5DyAi1WGma1ZkE=">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</latexit><latexit sha1_base64="usNuiC8JmFrQV5DyAi1WGma1ZkE=">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</latexit><latexit sha1_base64="usNuiC8JmFrQV5DyAi1WGma1ZkE=">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</latexit><latexit sha1_base64="usNuiC8JmFrQV5DyAi1WGma1ZkE=">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</latexit>

Page 15: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�15

Experimental Setting

Baseline: standard random forest (RF) with 24 trees.

: each leaf is labeled according to the majority class of the examples that fall into the leaf + use of a majority vote.

: each leaf is labeled to maximize the profit over the set of all examplesin the leaf + use of a majority vote.

: same as before but vote is done with the concept of profits.

:gradient boosting model with a logistic loss, threshold is tuned for …criterium.

: gradient boosting model with the « profit loss »

RFlab�maj<latexit sha1_base64="9fF/6tyFNuaZSp+oeTVWi94ZaI0=">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</latexit><latexit sha1_base64="9fF/6tyFNuaZSp+oeTVWi94ZaI0=">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</latexit><latexit sha1_base64="9fF/6tyFNuaZSp+oeTVWi94ZaI0=">AAAC2HicjVHLSsNAFD2Nr1pf1S7dBIvgxpKIoMuiIC6r2Ae2pUzSaY3Ni2QilFJwJ279Abf6ReIf6F94Z0xBLaITkpw5954zc++1QteJhWG8ZrSZ2bn5hexibml5ZXUtv75Ri4MksnnVDtwgalgs5q7j86pwhMsbYcSZZ7m8bg2OZbx+w6PYCfwLMQx522N93+k5NhNEdfKFltUbnZ+MOyOXWbseux7ncp180SgZaunTwExBEemqBPkXtNBFABsJPHD4EIRdMMT0NGHCQEhcGyPiIkKOinOMkSNtQlmcMhixA/r2addMWZ/20jNWaptOcemNSKljmzQB5UWE5Wm6iifKWbK/eY+Up7zbkP5W6uURK3BF7F+6SeZ/dbIWgR4OVQ0O1RQqRlZnpy6J6oq8uf6lKkEOIXESdykeEbaVctJnXWliVbvsLVPxN5UpWbm309wE7/KWNGDz5zinQW2vZBol82y/WD5KR53FJrawQ/M8QBmnqKBK3kM84gnP2qV2q91p95+pWibVFPBtaQ8ftaCWnQ==</latexit><latexit sha1_base64="9fF/6tyFNuaZSp+oeTVWi94ZaI0=">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</latexit>

RFlab�pro<latexit sha1_base64="J9jIjvyZKocz0Lru9/tRtO4hd+c=">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</latexit><latexit sha1_base64="J9jIjvyZKocz0Lru9/tRtO4hd+c=">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</latexit><latexit sha1_base64="J9jIjvyZKocz0Lru9/tRtO4hd+c=">AAAC2HicjVHLSsNAFD2Nrxpf1S7dBIvgxpKIoMuiIC6r2Ae2pSTptIamSZhMhBIK7sStP+BWv0j8A/0L74wR1CI6IcmZc+85M/deJ/K9WJjmS06bmZ2bX8gv6kvLK6trhfWNehwm3GU1N/RD3nTsmPlewGrCEz5rRpzZI8dnDWd4LOONa8ZjLwwuxDhinZE9CLy+59qCqG6h2Hb66fnJpJv6trMb8XCi691CySybahnTwMpACdmqhoVntNFDCBcJRmAIIAj7sBHT04IFExFxHaTEcUKeijNMoJM2oSxGGTaxQ/oOaNfK2ID20jNWapdO8enlpDSwTZqQ8jhheZqh4olyluxv3qnylHcb09/JvEbEClwR+5fuM/O/OlmLQB+HqgaPaooUI6tzM5dEdUXe3PhSlSCHiDiJexTnhF2l/OyzoTSxql321lbxV5UpWbl3s9wEb/KWNGDr5zinQX2vbJll62y/VDnKRp3HJrawQ/M8QAWnqKJG3mM84BFP2qV2o91qdx+pWi7TFPFtaffv8WKWtg==</latexit><latexit sha1_base64="J9jIjvyZKocz0Lru9/tRtO4hd+c=">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</latexit>

RFmean�pro<latexit sha1_base64="cztvWkVGK+YFcbFV6O9TDbpcdWY=">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</latexit><latexit sha1_base64="cztvWkVGK+YFcbFV6O9TDbpcdWY=">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</latexit><latexit sha1_base64="cztvWkVGK+YFcbFV6O9TDbpcdWY=">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</latexit><latexit sha1_base64="cztvWkVGK+YFcbFV6O9TDbpcdWY=">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</latexit>

GBtune�...<latexit sha1_base64="rgh86IrOXcElPCaE0VY2qO8lEEw=">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</latexit><latexit sha1_base64="rgh86IrOXcElPCaE0VY2qO8lEEw=">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</latexit><latexit sha1_base64="rgh86IrOXcElPCaE0VY2qO8lEEw=">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</latexit><latexit sha1_base64="rgh86IrOXcElPCaE0VY2qO8lEEw=">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</latexit>

GBprofits<latexit sha1_base64="Wop/+WDTiyWTe2PWBhbdZmW4C0U=">AAAC2HicjVHLSsNAFD2Nr1pf1S7dBIvgqiQi6LLUhS4r2Ae2pSTptA5NkzCZCKUU3Ilbf8CtfpH4B/oX3hlTUIvohCRnzr3nzNx73cjnsbSs14yxsLi0vJJdza2tb2xu5bd36nGYCI/VvNAPRdN1YubzgNUklz5rRoI5I9dnDXd4quKNGyZiHgaXchyxzsgZBLzPPUcS1c0X2m5/clbpTiIR9rmMp9NcrpsvWiVLL3Me2CkoIl3VMP+CNnoI4SHBCAwBJGEfDmJ6WrBhISKugwlxghDXcYYpcqRNKItRhkPskL4D2rVSNqC98oy12qNTfHoFKU3skyakPEFYnWbqeKKdFfub90R7qruN6e+mXiNiJa6J/Us3y/yvTtUi0ceJroFTTZFmVHVe6pLorqibm1+qkuQQEadwj+KCsKeVsz6bWhPr2lVvHR1/05mKVXsvzU3wrm5JA7Z/jnMe1A9LtlWyL46K5Uo66ix2sYcDmucxyjhHFTXyHuMRT3g2roxb4864/0w1MqmmgG/LePgApI2XAQ==</latexit><latexit sha1_base64="Wop/+WDTiyWTe2PWBhbdZmW4C0U=">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</latexit><latexit sha1_base64="Wop/+WDTiyWTe2PWBhbdZmW4C0U=">AAAC2HicjVHLSsNAFD2Nr1pf1S7dBIvgqiQi6LLUhS4r2Ae2pSTptA5NkzCZCKUU3Ilbf8CtfpH4B/oX3hlTUIvohCRnzr3nzNx73cjnsbSs14yxsLi0vJJdza2tb2xu5bd36nGYCI/VvNAPRdN1YubzgNUklz5rRoI5I9dnDXd4quKNGyZiHgaXchyxzsgZBLzPPUcS1c0X2m5/clbpTiIR9rmMp9NcrpsvWiVLL3Me2CkoIl3VMP+CNnoI4SHBCAwBJGEfDmJ6WrBhISKugwlxghDXcYYpcqRNKItRhkPskL4D2rVSNqC98oy12qNTfHoFKU3skyakPEFYnWbqeKKdFfub90R7qruN6e+mXiNiJa6J/Us3y/yvTtUi0ceJroFTTZFmVHVe6pLorqibm1+qkuQQEadwj+KCsKeVsz6bWhPr2lVvHR1/05mKVXsvzU3wrm5JA7Z/jnMe1A9LtlWyL46K5Uo66ix2sYcDmucxyjhHFTXyHuMRT3g2roxb4864/0w1MqmmgG/LePgApI2XAQ==</latexit><latexit sha1_base64="Wop/+WDTiyWTe2PWBhbdZmW4C0U=">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</latexit>

Data: 10 months of transactions (6 training/validation set and 4 test set) ~ 2.7M of transactions and only 0.33% of frauds.

Page 16: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�16

Experimental Results Compare different procedures with the current model of the company

• Improve the benefit of the retailers with both models. • GB-based model gives better results.• Reduce the loss having a lower precision (< 30%) but higher recall.

Experiments Rate loss max profits Precision Recall F1

RF 2.99% 68.1% 5.66% 10.5%RFmaj 2.88% 73.8% 4.71% 8.86%RFmaj�mar 1.81% 30.2% 10.6% 15.7%RFmean�margin 1.87% 30.3% 9.52% 14.5%GBtune�Pre 3.01% 61.0% 6.49% 11.7%GBtune�mar 2.26% 19.1% 16.6% 17.8%GBtune�F1 2.70% 45.4% 9.24% 15.4%GBmargin 1.56% 18.8% 13.3% 15.6%

<latexit sha1_base64="2qnOrbRXGEGVyrI+sXgDj5ycTTk=">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</latexit><latexit sha1_base64="2qnOrbRXGEGVyrI+sXgDj5ycTTk=">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</latexit><latexit sha1_base64="2qnOrbRXGEGVyrI+sXgDj5ycTTk=">AAAFw3icjVPbbtNAEJ02BOpwa+GRlxVNIl5YeZ3Ecd6qohYeS0UvUl1FjrtpTX2TL6hV8A/wCv+G+AP4CCRm1nbatEWwVpLZM3PO3JxJ7Htppus/lpYb95r3H6xorYePHj95urr2bD+N8sSVe27kR8nhxEml74VyL/MyXx7GiXSCiS8PJudvyH/wSSapF4UfsstYHgfOaehNPdfJEBqvLf+2J/LUC2eZM8l9Jylm/me3fArWYoxpdOxMXmST6WzrIpaJF8gwSwtN69bwrpNJ5kdpygLngsVJNPUwoMvaNjp3Eul6lL9oaxW0K13H94s2Q4Du2+2xaBeabbeYfUadYOYycBtJdLqMGXw0sjtEwaJMiwu7g/iAmyYaXaYJnQ/QTyI1dzwLnI+UpuRblopE+rDHLUXv86Eo6Ra3zLvorwMcSiUhuCVqiZ7ODSWBeYlIFQz48LaEdELSwBmXDZMKhbFapqdkRnxgVDL9eSOL03i7iXpZHsrXOFNVUzmZHtfFXJDVSzEF1+0Orcnk/ZHya0IsVnilWHdJggY3zLmeGNGky/7n2sKkngtUvIYNaabFXeLb4lq1Bh/qV9X2B7yv+mdsxI2+mgCOSKE3pBaGOM/KB6qSqlir2isTPZqrglGs3CxGsJYtw5P5y94ar67rXFeH3TZEZaxDdXai1e9gwwlE4EIOAUgIIUPbBwdSfI5AgA4xYscwQyxBy1N+CQW0kJtjlMQIB9Fz/D7F21GFhngnzVSxXczi4ydBJoMOciKMS9CmbEz5c6VM6N+0Z0qTarvE30mlFSCawRmi/+LVkf/Lo14ymIKlevCwp1gh1J1bqeRqKlQ5u9ZVhgoxYmSfoD9B21XMes5McVLVO83WUf6fKpJQurtVbA6/qEpcsLi5ztvGvsHxbyze99c3NqtVr8ALeAmvcJ9D2IB3sAN74DZOGl8aXxvfmlvN82bSzMrQ5aWK8xwWTrP4A91EUqM=</latexit><latexit sha1_base64="2qnOrbRXGEGVyrI+sXgDj5ycTTk=">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</latexit>

Page 17: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

Evolution of Precision, Recall,F-Measure and the gap to the maximal profits with respect to the parameter ⇣

<latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">AAACyHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I64qmLbQFpmk0zo0TUIyUWpx4w+41S8T/0D/wjvjFNQiOiHJmXPvOTP3Xj8JRSYd57Vgzc0vLC4Vl0srq2vrG+XNrUYW52nAvSAO47Tls4yHIuKeFDLkrSTlbOSHvOkPT1W8ecPTTMTRpRwnvDtig0j0RcAkUV7njkt2Va44VUcvexa4BlRgVj0uv6CDHmIEyDECRwRJOARDRk8bLhwkxHUxIS4lJHSc4x4l0uaUxSmDETuk74B2bcNGtFeemVYHdEpIb0pKG3ukiSkvJaxOs3U8186K/c17oj3V3cb0943XiFiJa2L/0k0z/6tTtUj0caxrEFRTohlVXWBcct0VdXP7S1WSHBLiFO5RPCUcaOW0z7bWZLp21Vum4286U7FqH5jcHO/qljRg9+c4Z0HjoOo6VffisFI7MaMuYge72Kd5HqGGM9ThkbfAI57wbJ1biXVrjT9TrYLRbOPbsh4+AM+AkT0=</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit>

⇣<latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit>

Optimal value of for the retailers

NB: gap computed with ⇣ = 5<latexit sha1_base64="dwGeoal7m3frMGq5pepmoZF2Pvk=">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</latexit><latexit sha1_base64="dwGeoal7m3frMGq5pepmoZF2Pvk=">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</latexit><latexit sha1_base64="dwGeoal7m3frMGq5pepmoZF2Pvk=">AAACynicjVHLSsNAFD2Nr1pfVZdugkVwVRJRdCMU3bhwUcE+wBZJ0mkdmheTiVCLO3/ArX6Y+Af6F94Zp6AW0QlJzpx7zp259/ppyDPpOK8Fa2Z2bn6huFhaWl5ZXSuvbzSzJBcBawRJmIi272Us5DFrSC5D1k4F8yI/ZC1/eKrirVsmMp7El3KUsm7kDWLe54EniWp17pj0jg+uyxWn6uhlTwPXgArMqiflF3TQQ4IAOSIwxJCEQ3jI6LmCCwcpcV2MiROEuI4z3KNE3pxUjBQesUP6Dmh3ZdiY9ipnpt0BnRLSK8hpY4c8CekEYXWareO5zqzY33KPdU51txH9fZMrIlbihti/fBPlf32qFok+jnQNnGpKNaOqC0yWXHdF3dz+UpWkDClxCvcoLggH2jnps609ma5d9dbT8TetVKzaB0ab413dkgbs/hznNGjuVV2n6l7sV2onZtRFbGEbuzTPQ9RwhjoauspHPOHZOreENbLGn1KrYDyb+Lashw8uTZHD</latexit><latexit sha1_base64="dwGeoal7m3frMGq5pepmoZF2Pvk=">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</latexit>

• : customer dissatisfaction cost

• when increases the cost of a False Positive is decreasing

The Precision is increasing and the Recall is decreasing

⇣<latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit>

⇣<latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit><latexit sha1_base64="I8BPpJOVvKL6gQFJlaCoarhrogc=">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</latexit>

�17

Page 18: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

�18

Conclusion

• Provide an understandable model for our customer

• Reduce the gap of the maximal benefits from 2.99 % to 1.56 % (represents a gainof 60 k euros per 4 months

• Able to control the precision

• RF (with our decision rule) and GB-based model can give similar results [6], but GB-based models are trained one order of magnitude faster

Perspectives

• Improve the fraud detection models with currently unused informations:• loyalty cards• customers’ baskets• historical purchase information

[5] Nikolaou, N., Edakunni, N., Kull, M., Flach, P., Brown, G.: Cost-Sensitive Boosting Algorithms. Do we really need them ? Machine Learning 104 (2016).

Page 19: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

Thank you for your attention !

!19

Page 20: Tree-based Cost Sensitive Methods for Fraud Detection in … · 2020. 12. 11. · AAACxnicjVHLSsNAFD2Nr/quunQTLIKrkoigy6KbLiu1D6ilJNNpDZ08SCZKKYI/4FY/TfwD/QvvjFNQi+iEJGfOvefM3Hv9RASZdJzXgrWwuLS8UlxdW9/Y3Nou7ey2sjhPGW+yWMRpx/cyLoKIN2UgBe8kKfdCX/C2P75Q8fYtT7Mgjq7kJOG90BtFwTBgniSq0eiLfqnsVBy97HngGlCGWfW49IJrDBCDIUcIjgiSsICHjJ4uXDhIiOthSlxKKNBxjnuskTanLE4ZHrFj+o5o1zVsRHvlmWk1o1MEvSkpbRySJqa8lLA6zdbxXDsr9jfvqfZUd5vQ3zdeIbESN8T+pZtl/lenapEY4kzXEFBNiWZUdcy45Lor6ub2l6okOSTEKTygeEqYaeWsz7bWZLp21VtPx990pmLVnpncHO/qljRg9

Bibliography

[1] Breiman, L., Friedman, J. Olshen, R., Stone, C.: Classification and Regression Trees. Wadsworth and Brooks, CA (1984).

[2] Bahnsen, A.C., Villegas, S., Aouada, D., Ottersten, B., Correa, A.M.: Fraud Detection by Stacking Cost-sensitive Decision Trees. DSCS (2017).

[3] Friedman, J.H.: Greedy Function Approximation: A Gradient Boosting Machine. Annals of Statistics 29 (2000).

[4] Elkan, C.: The Foundations of Cost-Sensitive Learning. Proceedings of the 17th International Joint Conference on Artificial Intelligence (2001).

[5] Nikolaou, N., Edakunni, N., Kull, M., Flach, P., Brown, G.: Cost-Sensitive Boosting Algorithms. Do we really need them ? Machine Learning 104 (2016).

!20