On balancing between optimal and proportional categorical predictions

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January 2016, 1(1): 129-137. doi: 10.3934/bdia.2016.1.129

On balancing between optimal and proportional categorical predictions

Wenxue Huang ^1, and Yuanyi Pan ^2,

1.	Department of Mathematics, Guangzhou University, Guangzhou, Guangdong 510006, China
2.	Kochava Inc, 414 Church Street, Suite 306, Sandpoint, Idaho 83864, United States

Received May 2015 Revised August 2015 Published September 2015

Abstract
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A bias-variance dilemma in categorical data mining and analysis is the fact that a prediction method can aim at either maximizing the overall point-hit accuracy without constraint or with the constraint of minimizing the distribution bias. However, one can hardly achieve both at the same time. A scheme to balance these two prediction objectives is proposed in this article. An experiment with a real data set is conducted to demonstrate some of the scheme's characteristics. Some basic properties of the scheme are also discussed.

Keywords: Bias-variance dilemma, point estimation, conditional distribution., proportional prediction, optimal prediction, categorical data.

Mathematics Subject Classification: Primary: 68T10, 62H20; Secondary: 62G8.

Citation: Wenxue Huang, Yuanyi Pan. On balancing between optimal and proportional categorical predictions. Big Data & Information Analytics, 2016, 1 (1) : 129-137. doi: 10.3934/bdia.2016.1.129

References:

[1]	A. C. Acock, Working with missing values, Journal of Marriage and Family, 67 (2005), 1012-1028. doi: 10.1111/j.1741-3737.2005.00191.x.
[2]	E. Acuna and C. Rodriguez, The treatment of missing values and its effect in the classifier accuracy, In Classification, Clustering and Data Mining Applications, (2004), 639-647.
[3]	G. E. Batista and M. C. Monard, An analysis of four missing data treatment methods for supervised learning, Applied Artificial Intelligence, 17 (2003), 519-533. doi: 10.1080/713827181.
[4]	J. Doak, An Evaluation of Feature Selection Methods and Their Application to Computer Security, UC Davis Department of Computer Science, 1992.
[5]	P. Domingos, A unified bias-variance decomposition, In Proceedings of 17th International Conference on Machine Learning. Stanford CA Morgan Kaufmann, 2000, 231-238.
[6]	, Survey of Family Expenditures - 1996,, STATCAN, (1998).
[7]	A. Farhangfar, L. Kurgan and J. Dy, Impact of imputation of missing values on classification error for discrete data, Pattern Recognition, 41 (2008), 3692-3705. doi: 10.1016/j.patcog.2008.05.019.
[8]	H. H. Friedman, On bias, variance, 0/1-loss, and the curse-of-dimensionality, Data mining and knowledge discovery, 1 (1997), 55-77.
[9]	S. Geman, E. Bienenstock and R. Doursaté, Neural networks and the bias/variance dilemma, Neural computation, 4 (1992), 1-58. doi: 10.1162/neco.1992.4.1.1.
[10]	L. A. Goodman and W. H. Kruskal, Measures of association for cross classification, J. American Statistical Association, 49 (1954), 732-764.
[11]	I. Guyon and A. Elisseeff, An introduction to variable and feature selection, J. Mach. Learn. Res., 3 (2003), 1157-1182.
[12]	L. Himmelspach and S. Conrad, Clustering approaches for data with missing values: Comparison and evaluation, In Digital Information Management (ICDIM), 2010 Fifth International Conference on,IEEE 2010, 19-28. doi: 10.1109/ICDIM.2010.5664691.
[13]	P. T. V. Hippel, Regression with missing Ys: An improved strategy for analyzing multiply imputed data, Sociological Methodology, 37 (2007), 83-117. doi: 10.1111/j.1467-9531.2007.00180.x.
[14]	W. Huang, Y. Shi and X. Wang, A nomminal association matrix with feature selection for categorical data, Communications in Statistics - Theory and Methods, to appear, 2015.
[15]	W. Huang, Y. Pan and J. Wu, Supervised Discretization for Optimal Prediction, Procedia Computer Science, 30 (2014), 75-80. doi: 10.1016/j.procs.2014.05.383.
[16]	G. James and T. Hastie, Generalizations of the Bias/Variance Decomposition for Prediction Error, Dept. Statistics, Stanford Univ., Stanford, CA, Tech. Rep, 1997.
[17]	S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86. doi: 10.1214/aoms/1177729694.
[18]	R. J. A. Little and D. B. Rubin, Statistical Analysis with Missing Data, John Wiley & Sons, Inc. 1987, New York, NY, USA.
[19]	H. Liu and H. Motoda, Feature Selection for Knowledge Discovery and Data Mining, Kluwer Academic Publishers 1998, Norwell, MA, USA. doi: 10.1007/978-1-4615-5689-3.
[20]	J. Luengo, S. García and F. Herrera, On the choice of the best imputation methods for missing values considering three groups of classification methods, Knowledge and information systems, 32 (2012), 77-108. doi: 10.1007/s10115-011-0424-2.
[21]	Z. Mark and Y. Baram, The bias-variance dilemma of the Monte Carlo method, Artificial Neural Networks,ICANN, 2130 (2001), 141-147. doi: 10.1007/3-540-44668-0_20.
[22]	R. Tibshirani, Bias, Variance and Prediction Error for Classification Rules, Citeseer 1996.
[23]	I. Yaniv and D. P. Foster, Graininess of judgment under uncertainty: An accuracy-informativeness trade-off, Journal of Experimental Psychology: General, 124 (1995), 424-432. doi: 10.1037/0096-3445.124.4.424.
[24]	L. Yu, K. K. Lai, S. Wang and W. Huang, A bias-variance-complexity trade-off framework for complex system modeling, In Computational Science and Its Applications-ICCSA 2006, Springer, 3980 (2006), 518-527. doi: 10.1007/11751540_55.
[25]	T. Zhou, Z. Kuscsik, J. Liu, M. Medo, J. R. Wakeling and Y. Zhang, Solving the apparent diversity-accuracy dilemma of recommender systems, Proceedings of the National Academy of Sciences, 107 (2010), 4511-4515. doi: 10.1073/pnas.1000488107.

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References:

[1]	A. C. Acock, Working with missing values, Journal of Marriage and Family, 67 (2005), 1012-1028. doi: 10.1111/j.1741-3737.2005.00191.x.
[2]	E. Acuna and C. Rodriguez, The treatment of missing values and its effect in the classifier accuracy, In Classification, Clustering and Data Mining Applications, (2004), 639-647.
[3]	G. E. Batista and M. C. Monard, An analysis of four missing data treatment methods for supervised learning, Applied Artificial Intelligence, 17 (2003), 519-533. doi: 10.1080/713827181.
[4]	J. Doak, An Evaluation of Feature Selection Methods and Their Application to Computer Security, UC Davis Department of Computer Science, 1992.
[5]	P. Domingos, A unified bias-variance decomposition, In Proceedings of 17th International Conference on Machine Learning. Stanford CA Morgan Kaufmann, 2000, 231-238.
[6]	, Survey of Family Expenditures - 1996,, STATCAN, (1998).
[7]	A. Farhangfar, L. Kurgan and J. Dy, Impact of imputation of missing values on classification error for discrete data, Pattern Recognition, 41 (2008), 3692-3705. doi: 10.1016/j.patcog.2008.05.019.
[8]	H. H. Friedman, On bias, variance, 0/1-loss, and the curse-of-dimensionality, Data mining and knowledge discovery, 1 (1997), 55-77.
[9]	S. Geman, E. Bienenstock and R. Doursaté, Neural networks and the bias/variance dilemma, Neural computation, 4 (1992), 1-58. doi: 10.1162/neco.1992.4.1.1.
[10]	L. A. Goodman and W. H. Kruskal, Measures of association for cross classification, J. American Statistical Association, 49 (1954), 732-764.
[11]	I. Guyon and A. Elisseeff, An introduction to variable and feature selection, J. Mach. Learn. Res., 3 (2003), 1157-1182.
[12]	L. Himmelspach and S. Conrad, Clustering approaches for data with missing values: Comparison and evaluation, In Digital Information Management (ICDIM), 2010 Fifth International Conference on,IEEE 2010, 19-28. doi: 10.1109/ICDIM.2010.5664691.
[13]	P. T. V. Hippel, Regression with missing Ys: An improved strategy for analyzing multiply imputed data, Sociological Methodology, 37 (2007), 83-117. doi: 10.1111/j.1467-9531.2007.00180.x.
[14]	W. Huang, Y. Shi and X. Wang, A nomminal association matrix with feature selection for categorical data, Communications in Statistics - Theory and Methods, to appear, 2015.
[15]	W. Huang, Y. Pan and J. Wu, Supervised Discretization for Optimal Prediction, Procedia Computer Science, 30 (2014), 75-80. doi: 10.1016/j.procs.2014.05.383.
[16]	G. James and T. Hastie, Generalizations of the Bias/Variance Decomposition for Prediction Error, Dept. Statistics, Stanford Univ., Stanford, CA, Tech. Rep, 1997.
[17]	S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86. doi: 10.1214/aoms/1177729694.
[18]	R. J. A. Little and D. B. Rubin, Statistical Analysis with Missing Data, John Wiley & Sons, Inc. 1987, New York, NY, USA.
[19]	H. Liu and H. Motoda, Feature Selection for Knowledge Discovery and Data Mining, Kluwer Academic Publishers 1998, Norwell, MA, USA. doi: 10.1007/978-1-4615-5689-3.
[20]	J. Luengo, S. García and F. Herrera, On the choice of the best imputation methods for missing values considering three groups of classification methods, Knowledge and information systems, 32 (2012), 77-108. doi: 10.1007/s10115-011-0424-2.
[21]	Z. Mark and Y. Baram, The bias-variance dilemma of the Monte Carlo method, Artificial Neural Networks,ICANN, 2130 (2001), 141-147. doi: 10.1007/3-540-44668-0_20.
[22]	R. Tibshirani, Bias, Variance and Prediction Error for Classification Rules, Citeseer 1996.
[23]	I. Yaniv and D. P. Foster, Graininess of judgment under uncertainty: An accuracy-informativeness trade-off, Journal of Experimental Psychology: General, 124 (1995), 424-432. doi: 10.1037/0096-3445.124.4.424.
[24]	L. Yu, K. K. Lai, S. Wang and W. Huang, A bias-variance-complexity trade-off framework for complex system modeling, In Computational Science and Its Applications-ICCSA 2006, Springer, 3980 (2006), 518-527. doi: 10.1007/11751540_55.
[25]	T. Zhou, Z. Kuscsik, J. Liu, M. Medo, J. R. Wakeling and Y. Zhang, Solving the apparent diversity-accuracy dilemma of recommender systems, Proceedings of the National Academy of Sciences, 107 (2010), 4511-4515. doi: 10.1073/pnas.1000488107.

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