Minimum likelihood threshold Quantisation errors

From figure 7-4 it can be seen that even on the ‘same file’ data sub-set, none of the classifiers achieve 100% correct classification. This shows that the different classes are not well separated. Figures 7-4 and 7-5 also demonstrate the ‘same ship/same day’ effect as discussed earlier. For each classifier, classification performance is highest on the ‘same file’ data, lower on the ‘same ship/same day’ data and lowest on the ‘different ship/different day’ data. Thus classification performance is inversely related to the independence of the training and test data sub-sets. This is in agreement with work by other authors [Tough & Ward].

Figure 7-4 shows that when the Gaussian distribution is used to model range cell fluctuations in a standard maximum likelihood classifier classification performance is higher than when the gamma distribution is used for both the ‘same file’ and ‘same day’ data sub-sets but for the ‘different’ data sub-set the converse is true. However, when even the lowest non-zero minimum likelihood threshold is introduced the classification performance of the gamma-based classifier rises above the performance of the Gaussian-based classifier. The data has been quantised to only a finite precision by the radar’s analogue-to-digital converter before being converted into a floating point number and rescaled by a CPU. Hence, even when contaminated by noise, very small signal values may be digitised to the value zero. It is thought that this difference in performance between the gamma and Gaussian forms of the maximum likelihood classifier is caused by the zero likelihood that is given to the occurrence of the event, X = 0 by any gamma distribution that has a shape parameter greater than one. Note that

because the Gaussian distribution has infinitely long tails it gives a non-zero value to the event, x = 0. As the multivariate distributions are modelled as the product of multiple independent univariate distributions, if any of the univariate distributions give an event zero likelihood of happening then the whole range profile will have a zero likelihood of being generated by that class and hence the class will not be selected by the classifier. However, if a minimum likelihood is used this situation does not occur and classes are not completely ruled out because of one single zero value. This explains the jum p in classification performance between a zero minimum likelihood threshold and a very

small threshold.

Although the use of a non-zero minimum likelihood prevents a class from having zero overall likelihood, its overall likelihood will be a lot less if it has had to make use of the minimum likelihood than if it has not. Hence, the correct class may still have a lower likelihood than an incorrect class because one range-cell is very different to expectation. As the minimum likelihood threshold is raised the probability of one range­ cell generating a likelihood sufficiently low so as to cause mis-classification is reduced. This results in a higher probability of correct classification: as the minimum likelihood threshold is increased the classification performance of the gamma-based classifier gradually rises for all three data sub-sets, however, this rise is very small and is difficult to see in figure 7-4.

As range-cell amplitudes cannot be negative the furthest a range-cell amplitude can be below the mean is to have a value zero. In addition, the standard deviation of the amplitude in a range-cell over a few degrees aspect tends to be of a size that is of the same order as the mean amplitude. Thus when a Gaussian distribution is being used, even if a range-cell has an amplitude that is as far below the mean as possible, it will

still be within a few standard deviations of the range-cell mean. Hence, it must have a likelihood that is at least within a few orders of magnitude of the likelihood of the mean amplitude. Thus low valued minimum likelihood thresholds will have no effect on the performance of the Gaussian-based maximum likelihood classifier. In contrast to the Gaussian distribution many forms of the gamma distribution would assign a zero likelihood to the possibility of generating a zero value even though zero is within a few standard deviations of the mean. Although this effect is difficult to see in figure 7-4, the classification performance of the Gaussian-based classifier is constant from minimum

likelihood thresholds of up to e'^^ in contrast to the steadily increasing

performance of the gamma-based classifier.

Since the ‘different’ data sub-set can be used to gauge how a classifier may perform in a more realistic scenario, and the performance of the garruna-based classifier is 10% higher than the Gaussian-based classifier on this data sub-set, the gamma distribution will be used to model range-cells fluctuations throughout this thesis.