A two step training sample optimization rule for the application of two and multiple group discriminant analysis.

The classification problem of assigning observations to one of several groups plays a key role in decision-making. When the criterion for classification involves one or more predictor variables along side with a categorical criterion, such prediction will call for the use of linear discriminant analysis (a.k.a discriminant analysis or DA). In DA, the task of maximizing the hit rate of the predictive discriminant function (PDF) usually begins with the researcher making choices about the variables that will be involved in the analysis. In order to achieve this, various variables selection strategies have been proposed. These methods search for subsets with greater classification capacity based on different criteria. However, uncertainty about the model which can be reduced if the subsets of variables meet a near optimal condition is completely neglected. In most cases, the derived PDF is not only obtained from a training sample that do not meet optimal condition, but that the training sample used is made up of subset of predictor variables that are indeed not unique to all other subsets from the same historical sample. Consequently, the obtained maximum hit rate of the derived PDF is often not statistically optimal even for the training sample that gave birth to it. Hence, we seek a rule for obtaining a near optimal training sample with the aim of building a PDF whose hit rate can be said to be statistically optimal for the training sample that gave birth to it as well as for future samples of any given data set from the same population.
In this thesis, a two-step training sample rule based on modified leave-one-out cross validation (LOOCV) and a symmetric trimming with modified winsorized mean is proposed. The rule fits a P+1 model, unlike the 2P-1 models in the literature (where P is the number of predictor variables). The first step of this rule is to select useful predictor variables that will yield greater classification accuracy with less computational time. The second step of this rule is to identify and remove legitimate contaminants or hidden influential observations in one or more interval independents, thereby resolving any significant difference in the variance for the group formed by the dependent in order to obtain a near optimal condition for the training sample.
Assessing the proposed rule using real life data sets, four predictive discriminant functions (PDFS) were built for each data set. The first three were built using the three notable classical variable selection strategies and the fourth was built using our proposed rule. The results obtained reveal that: (i) our proposed rule produces consistent high hit-rates with little variability; (ii) our approach is significantly better in terms of computational time; (iii) the risk of having two (or more) subsets of a given size that yields the same hit rate common with all-possible subset method was completely avoided; and (iv) the optimized PDF when tested on training and validation sets shows that overfitting problem was kept at minimum.