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October  2016, 1(4): 341-347. doi: 10.3934/bdia.2016014

## Increase statistical reliability without losing predictive power by merging classes and adding variables

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

* Corresponding authors: Wenxue Huang and Xiaofeng Li

Revised  April 2017 Published  April 2017

It is usually true that adding explanatory variables into a probability model increases association degree yet risks losing statistical reliability. In this article, we propose an approach to merge classes within the categorical explanatory variables before the addition so as to keep the statistical reliability while increase the predictive power step by step.

Citation: Wenxue Huang, Xiaofeng Li, Yuanyi Pan. Increase statistical reliability without losing predictive power by merging classes and adding variables. Big Data & Information Analytics, 2016, 1 (4) : 341-347. doi: 10.3934/bdia.2016014
##### References:
 [1] H. L. Costner, Criteria for measure of association, American Sociology Review, 30 (1965), 341-353. [2] M. Dash and H. Liu, Feature selection for classification, Intell. Data. Anal., 1 (1997), 131-156.  doi: 10.1016/S1088-467X(97)00008-5. [3] R. L. Ebel, Estimation of the reliability of ratings, Psychomereika, 16 (1951), 407-424. [4] G. S. Fisher, Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, 1996. [5] P. Glasserman, Monte Carlo Method in Financial Engineering, (Stochastic Modelling and Applied Probability) (V. 53), Spinger, 2004. [6] L. A. Goodman and W. H. Kruskal, Measures of Associations for Cross Classification, With a foreword by Stephen E. Fienberg. Springer Series in Statistics, 1. Springer-Verlag, New York-Berlin, 1979. [7] L. Guttman, The test-retest reliability of qualitative data, Psychometrika, 11 (1946), 81-95.  doi: 10.1007/BF02288925. [8] I. Guyon and A. Elisseeff, An introduction to variable and feature selection, J. Mach. Learn. Res., 3 (2003), 1157-1182. [9] W. Huang and Y. Pan, On balancing between optimal and proportional categorical predictions, Big Data and Info. Anal., 1 (2016), 129-137.  doi: 10.3934/bdia.2016.1.129. [10] W. Huang, Y. Pan and J. Wu, Supervised Discretization with GK-τ, Proc. Comp. Sci., 17 (2013), 114-120. [11] W. Huang, Y. Pan and J. Wu, Supervised discretization for optimal prediction, Proc. Comp. Sci., 30 (2014), 75-80.  doi: 10.1016/j.procs.2014.05.383. [12] W. Huang, Y. Shi and X. Wang, A nominal association matrix with feature selection for categorical data, Communications in Statistics -Theory and Methods, 2017. [13] M. G. Kendall, The Advanced Theory of Statistics, London, Charles Griffin and Co. , Ltd, 1946. [14] C. J. Lloyd, Statistical Analysis of Categorical Data, John Wiley Sons, 1999. [15] K. Pearson and D. Heron, On Theories of association, Biometrika, 9 (1913), 159-315. [16] STATCAN, Survey of Family Expenditures -1996. (1998) [17] D. L. Streiner and G. R. Norman, On Theories of association, J. of Cli. Epid., 59 (2006), 327-330.  doi: 10.1016/j.jclinepi.2005.09.005.

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##### References:
 [1] H. L. Costner, Criteria for measure of association, American Sociology Review, 30 (1965), 341-353. [2] M. Dash and H. Liu, Feature selection for classification, Intell. Data. Anal., 1 (1997), 131-156.  doi: 10.1016/S1088-467X(97)00008-5. [3] R. L. Ebel, Estimation of the reliability of ratings, Psychomereika, 16 (1951), 407-424. [4] G. S. Fisher, Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, 1996. [5] P. Glasserman, Monte Carlo Method in Financial Engineering, (Stochastic Modelling and Applied Probability) (V. 53), Spinger, 2004. [6] L. A. Goodman and W. H. Kruskal, Measures of Associations for Cross Classification, With a foreword by Stephen E. Fienberg. Springer Series in Statistics, 1. Springer-Verlag, New York-Berlin, 1979. [7] L. Guttman, The test-retest reliability of qualitative data, Psychometrika, 11 (1946), 81-95.  doi: 10.1007/BF02288925. [8] I. Guyon and A. Elisseeff, An introduction to variable and feature selection, J. Mach. Learn. Res., 3 (2003), 1157-1182. [9] W. Huang and Y. Pan, On balancing between optimal and proportional categorical predictions, Big Data and Info. Anal., 1 (2016), 129-137.  doi: 10.3934/bdia.2016.1.129. [10] W. Huang, Y. Pan and J. Wu, Supervised Discretization with GK-τ, Proc. Comp. Sci., 17 (2013), 114-120. [11] W. Huang, Y. Pan and J. Wu, Supervised discretization for optimal prediction, Proc. Comp. Sci., 30 (2014), 75-80.  doi: 10.1016/j.procs.2014.05.383. [12] W. Huang, Y. Shi and X. Wang, A nominal association matrix with feature selection for categorical data, Communications in Statistics -Theory and Methods, 2017. [13] M. G. Kendall, The Advanced Theory of Statistics, London, Charles Griffin and Co. , Ltd, 1946. [14] C. J. Lloyd, Statistical Analysis of Categorical Data, John Wiley Sons, 1999. [15] K. Pearson and D. Heron, On Theories of association, Biometrika, 9 (1913), 159-315. [16] STATCAN, Survey of Family Expenditures -1996. (1998) [17] D. L. Streiner and G. R. Norman, On Theories of association, J. of Cli. Epid., 59 (2006), 327-330.  doi: 10.1016/j.jclinepi.2005.09.005.
Feature selection with merging: Occupation
 $X$ $\tau_b^{(Y|X)}$ $\lambda^{(Y|X)}$ $E(\mbox{Gini}(X|Y))$ $Age group'$+Sex 0.1484 0.0375 0.6688 (Age group'+Sex)'+Education' 0.1542 0.0447 0.6620
 $X$ $\tau_b^{(Y|X)}$ $\lambda^{(Y|X)}$ $E(\mbox{Gini}(X|Y))$ $Age group'$+Sex 0.1484 0.0375 0.6688 (Age group'+Sex)'+Education' 0.1542 0.0447 0.6620
Feature selection without merging: Occupation
 $X$ $\tau^{Y|X}$ $\lambda^{Y|X}$ $E(\mbox{Gini}(X|Y))$ Age group 0.1344 0.0311 0.8773 Age group + Sex 0.1511 0.0476 0.9228
 $X$ $\tau^{Y|X}$ $\lambda^{Y|X}$ $E(\mbox{Gini}(X|Y))$ Age group 0.1344 0.0311 0.8773 Age group + Sex 0.1511 0.0476 0.9228
Compare different merging threshold:Occupation
 $X$ $\phi^{st}(Y|X)$ $\lambda^{(Y|X)}$ $\tau^{(Y|X)}$ $E(Gini(X,Y))$ Age group - 0.0311 0.1344 0.8773 $Age group'$+Sex 0.0005 0.0414 0.1493 0.9222 $Age group'$+$Sex$ 0.0030 0.0375 0.1484 0.6688 $Age group'$+$Sex$ 0.0100 0.0000 0.0209 0.2710
 $X$ $\phi^{st}(Y|X)$ $\lambda^{(Y|X)}$ $\tau^{(Y|X)}$ $E(Gini(X,Y))$ Age group - 0.0311 0.1344 0.8773 $Age group'$+Sex 0.0005 0.0414 0.1493 0.9222 $Age group'$+$Sex$ 0.0030 0.0375 0.1484 0.6688 $Age group'$+$Sex$ 0.0100 0.0000 0.0209 0.2710
Compare different merging threshold
 $X$ $\lambda^{(Y|X)}$ $\tau^{(Y|X)}$ $E(\mbox{Gini}(X|Y))$ Rooms 0.3443598 0.3004656 0.8200656 $Rooms'$+$Tenure'$ 0.4255117 0.3583277 0.7911177 $(Rooms'$+$Tenure')'+bedroom'$ 0.4381247 0.3901767 0.7165204
 $X$ $\lambda^{(Y|X)}$ $\tau^{(Y|X)}$ $E(\mbox{Gini}(X|Y))$ Rooms 0.3443598 0.3004656 0.8200656 $Rooms'$+$Tenure'$ 0.4255117 0.3583277 0.7911177 $(Rooms'$+$Tenure')'+bedroom'$ 0.4381247 0.3901767 0.7165204
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