Missing Data Estimation Using Rough Sets

Introduction

There are a number of general ways that have been used to approach the problem of missing data in databases (Little & Rubin, 1987: Rubin, 1976; Little & Rubin, 1989; Collins, Schafer Kam, 2001; Schafer & Graham, 2002). One of the simplest of these methods is the ‘list-wise deletion’, which simply deletes instances with missing values (Scheuren, 2005; King et al., 2001; Abdella, 2005). The major disadvantage of this method is the dramatic loss of information in data sets (King et al., 1988). Also, Enders and Peugh (2004) demonstrated that when there is a group of missing data, list-wise deletion gives biased parameters and standard errors. Another approach is ‘pair-wise deletion’ (Marsh, 1998).

Tsikritis (2005) observed that appropriately dealing with missing data has been underestimated by the operation management authors, unlike in other fields such as marketing, organizational behavior analysis, economics, statistics and psychometrics that have intensely attended to the issue. Tsikritis found from a review of 103 survey articles appearing in the Journal of Operations Management between 1993 and 2001 that list-wise deletion was the most widely used technique to deal with missing data. This is in spite of the fact that list-wise deletion is usually the least accurate method for handling missing data. This indicates, therefore, that missing data estimation methods that are simple to understand and implement still need to be developed and then promoted for wider usage.

Kim and Curry (1997) found that when 2% of the features are missing and the complete observation is deleted then up to 18% of the total data may be lost. The second common technique imputes the data by finding estimates of the values and missing entries are then replaced with these estimates. Various estimates have been used and these estimates include zeros, means and other statistical calculations. These estimations are then used as if they were the observed values. Xia et al. (1999) estimated missing data in climatological time and investigated six methods for imputing missing climatological data including daily maximum temperature, minimum temperature, air temperature, water vapor pressure, wind speed and precipitation. These researchers used the multiple regression analysis with the five closest weather stations and the results obtained from the six methods showed similar estimates for the averaged precipitation amount.

Another common technique assumes some models for the prediction of the missing values and uses the maximum likelihood approach to estimate the missing values (Nelwamondo, Mohamed, & Marwala, 2007; Dempster, Laird, & Rubin., 1977; Abdella & Marwala, 2006). In Chapter IV, the hybrid auto-associative neural networks and simple genetic algorithms are compared to the Gaussian mixture models trained with the expectation maximization (GMM-EM) algorithm and tested using data sets from an industrial power plant, an industrial winding process and HIV sero-prevalence survey data. The results obtained show that both methods perform well. The GMM-EM method was found to perform well in cases where there was little or no inter-dependency between variables, whereas the hybrid auto-associative neural network and genetic algorithm was found to be suited to problems where there were some inherent nonlinear relationships between some of the given variables.