LINEAR DISCRIMINATE ANALYSIS. Linear discriminate analysis (LDA) provides a means by which the optimum separation, concerning the ratio of intra class to inter class distance can be realised. This allows the best possible decision boundaries to be chosen in graph space, thus giving the distance classifiers the best chance of a correct classification. There are two distinct types of LDA; the first, classdependant transformation, transforms the data sets independently using L optimising criteria for an Lclass problem. The second system, classindependent transformation, maximises the ratio of overall variance to within class variance. This approach uses only one optimising criteria, thus the entire set is transformed uniformly. To perform LDA, irrespective of the type as before mentioned, firstly the mean (centroid) must be calculated for each data set. The mean of the entire set must then be found; in the case of a twoclass example, this is found by: μ 3= p1×μ 1 p2×μ 2 (6.12) With μ 1 , μ 2 being the mean values of data set (class) 1 and 2 respectively, and p1 , p2 being the a priori probability of the classes. In this simple case, the probability is assumed to be 0.5 for each class. In LDA, the intra class and inter class scatter or variance is used to formulate the criteria upon which the separation transform is based. Intra class variation constitutes the covariance of each class, and the final measure is created such that: Sw=∑ x x ×cj
LINEAR DISCRIMINATE ANALYSIS. When used within this project, Linear Discriminate Analysis provided poor separation and did not aid the classification process (based upon the classification rates obtained in [85]). This is unfortunate, as this system should have increased accuracy by transforming to a best separation space. The fact that this was not done in a useful way suggests that the easily confusable objects (i.e. those with centroids in close proximity to each other) may not have an adequate best separation decision boundary due to the dominant direction of the data clustering. This report introduced two types of LDA transform, namely class dependant and class independent; however, only class dependent was used. It can be observed within the literature, that the two differing types of LDA have particular specialities. Class independent transform LDA allows the data set to generalise well, which in this instance is not desired. Class dependant LDA, on the other hand, aids in discrimination as it aims to find the best separation between two classes by linearly separating the classes individually. Figure 7.3 shows this idea: Figure 7.3 – Example of LDA Using Class Dependant Transform Superimposed Upon The Original Data. In can be seen, using the above figure for reference and the recreated right hand side of Figure 7.4 below, that the confusable objects (within the yellow square), may have no good separation based on LDA. The dominant directions coupled with their position make this kind of separation difficult. Figure 7.4 – Recreation of Figure 7.3, With Confusable Objects (Neighbouring Centroid Positions) and Dominant Directions Marked In Yellow. Due to the underlying model and its class distributions, the confusable objects are not resolved, and other centroids within the model have sufficient separation as to require no further separation. This technique does not improve the data model, and will not be included in the final classification system.
