Proportional association based roi model

Wenxue Huang; Yuanyi Pan; Lihong Zheng

doi:10.3934/bdia.2017004

Article Contents

2017, Volume 2, Issue 2: 119-125. Doi: 10.3934/bdia.2017004

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Proportional association based roi model

1.
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China
2.
Clearpier Inc., 1300-121 Richmond St. W., Toronto, Ontario M5H 2K1 Canada
3.
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China

* Corresponding authors: Wenxue Huang and Lihong Zheng.
* Corresponding authors: Wenxue Huang and Lihong Zheng.

Published: March 2017

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Abstract

Based on a local-to-global proportional association measure proposed by Huang, Shi and Wang [9], 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.

Keywords:

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

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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

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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

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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

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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

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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

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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

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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

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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

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