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# Proportional association based roi model

• * Corresponding authors: Wenxue Huang and Lihong Zheng.
• Based on a local-to-global proportional association measure proposed by Huang, Shi and Wang , with cost and revenue information known, an association measure is proposed to maximize the expected RoI. A descriptive experiment with a synthetical data set is presented.

Mathematics Subject Classification: Primary: 62F07, 62H20; Secondary: 68T30.

 Citation: • • Table 1.  Contingency tables:$X_1$ vs $Y$ and $X_2$ vs $Y$

 $X_1|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $X_2|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $x_{1_1}$ 1000 100 500 400 $x_{2_1}$ 500 300 200 1500 $x_{1_2}$ 200 1500 500 300 $x_{2_2}$ 500 400 400 50 $x_{1_3}$ 400 50 500 500 $x_{2_3}$ 500 500 300 700 $x_{1_4}$ 300 700 500 400 $x_{2_4}$ 500 400 1000 100 $x_{1_5}$ 200 500 400 200 $x_{2_5}$ 200 400 500 200

Table 2.  Association matrices:$X_1$ vs $Y$ and $X_2$ vs $Y$

 $Y|\hat{Y}$ $\hat{y_1}|X_1$ $\hat{y_2}|X_1$ $\hat{y_3}|X_1$ $\hat{y_4}|X_1$ $Y|\hat{Y}$ $\hat{y_1}|X_2$ $\hat{y_2}|X_2$ $\hat{y_3}|X_2$ $\hat{y_4}X_2$ $y_1$ 0.34 0.18 0.27 0.22 $y_1$ 0.26 0.22 0.27 0.25 $y_2$ 0.13 0.48 0.24 0.15 $y_2$ 0.25 0.24 0.29 0.23 $y_{3}$ 0.24 0.28 0.27 0.21 $y_{3}$ 0.25 0.24 0.36 0.15 $y_{4}$ 0.25 0.25 0.28 0.22 $y_{4}$ 0.22 0.18 0.14 0.46

Table 3.  Contingency table for correct predictions: $W_1$ and $W_2$

 $X_1|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $X_2|Y$ $y_1$ $y_2$ $y_{3}$ $y_{4}$ $x_{1_1}$ 471 6 121 83 $x_{2_1}$ 98 34 19 926 $x_{1_2}$ 101 746 159 107 $x_{2_2}$ 177 114 113 1 $x_{1_3}$ 130 1 167 157 $x_{2_3}$ 114 124 42 256 $x_{1_4}$ 44 243 145 85 $x_{2_4}$ 109 81 489 6 $x_{1_5}$ 21 210 114 32 $x_{2_5}$ 36 119 206 28

Table 4.  Association measures: $\omega^{Y|X}$, and $\widehat{\omega}^{Y|X}$

 $X$ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ total revenue average revenue $X_1$ 0.3406 0.456 4313 0.4714 $X_2$ 0.3391 0.564 5178 0.5659

Table 5.  Association with/without cost vectors: $X_1$ and $X_2$

 $X$ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit $X_1$ 0.3406 0.3406 1.3057 12016.17 1.3132 $X_2$ 0.3391 0.3391 1.8546 17072.17 1.8658

Table 6.  Association with/without new cost vectors: $X_1$ and $X_2$

 $X$ $\omega^{Y|X}$ $\widehat{\omega}^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit $X_1$ 0.3406 0.3406 1.7420 15938.17 1.7419 $X_2$ 0.3391 0.3391 1.3424 12268.17 1.3408

Table 7.  Simulated feature selection: one variable

 $X$ $|Dmn(X)|$ $\omega^{Y|X}$ $\bar{\omega}^{Y|X}$ total profit average profit $V_1$ 7 0.3906 3.5381 35390 3.5390 $V_2$ 4 0.3882 3.8433 38771 3.8771 $V_{3}$ 4 0.3250 4.8986 48678 4.8678 $V_{4}$ 8 0.3274 3.7050 36889 3.6889

Table 8.  Simulated feature selection: two variables

 $X_1, X_2$ $|Dmn(X_1, X_2)|$ $\omega^{Y|(X_1, X_2)}$ $\bar{\omega}^{Y|(X_1, X_2)}$ total profit average profit $V_1,V_2$ 28 0.4367 1.8682 18971 1.8971 $V_1, V_{3}$ 28 0.4025 2.1106 20746 2.0746 $V_1, V_{4}$ 56 0.4055 1.8055 17915 1.7915 $V_{3}, V_2$ 16 0.4055 2.3585 24404 2.4404 $V_{3}, V_{4}$ 32 0.3385 2.0145 19903 1.9903
• Tables(8)

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