\dbsp

Chapter \ref{conclusion} begins with Section \ref{overview} which is a broad overview of the previous content of this document.  
Section \ref{conclusions} describes conclusions reached based on the results of the experiments described previously in this document.
Section \ref{future-work} proposes additional open research questions which have been brought to light by the work and experimental results of this thesis.

\ssp

\section{Overview}
\label{overview}

\dbsp
In this thesis we have reviewed the broad field of data mining.  We identified classification as a useful tool to predict future occurrences based on current evidence and previous information.  We reviewed classification methods, including decision tree learners and \nbc in order to discuss the differences between available methods of performing this useful task.  We then identified \nbc as the ideal method for comparing discretization methods, as previous results from numerous sources have shown them to be prone to vast improvement when used with discretization.  

We discussed numerous methods of discretization, identifying their strengths and weaknesses and selecting a few of them for comparison against our newly proposed method, the DiscTree algorithm.  We introduced this algorithm in this paper and have implemented it.  We proposed comparing it to the previously described and selected methods in order to determine how it performed relative to them.  We discussed our experimental method, specifically using each of the selected discretization methods and the raw data from 25 data sets in a ten by ten-fold stratified cross-valdiation to obtain the classifier performance measures of \textit{acc}, \textit{bal}, \textit{pd}, \textit{!pf}, and \textit{prec}.  We then compared the results of each of these performance measures across each classifier to obtain the results provided in the form of win-loss tables.  

We will now discuss our conclusions based on the results collected and provide possible future avenues to continue this research.

\ssp

\section{Conclusions}
\label{conclusions}

\dbsp

The results of our discretization method comparison appear to align with the results from \cite{dou95} and \cite{det02}.  The entropy-minimization heuristic, encapsulated in the script \textit{fayyadIrani} (see \app{fayyadIranisource}) clearly wins when the $U$-test statistics are reviewed.  For each measure, it does at least twice as many times as any of the other methods to which it was compared.  

However, despite the results of the $U$-test statics, it is very interesting to note the data plots provided for each measure for each data set.  These plots show that, even in most of the cases where fayyadIrani wins, it is doing so almost marginally, with other methods plotting very close to it - so close the that fayyadIrani data points, plotted last, often plot over the other methods.  There are some clear instances where this isn't true - for example, the auto-mpg results which always show DiscTree as a clear winner and various results for each measure where fayyadIrani or another method clearly out pace the other methods - but in the majority of cases it appears that the differences between the scores for each method or just slight enough to qualify as different when compared using the $U$-test statistic.

Of interest to this particular thesis is the results for the DiscTree Algorithm.  For each measure, DiscTree performs second-best behind fayyadIrani and almost always better than all the other methods.  This is interesting because it often wins out over the PKID method, the most recently proposed of its competition.  That method had been shown to perform better in terms of minimized error than the fayyadIrani method in other cases \cite{yw01}.  We did not have similar results; PKID obtains fewer accuracy wins than fayyadIrani and does not even perform as well as DiscTree.  This leads us to believe that our method, while clearly not most superior in this comparison, is clearly not an idea that should just be discarded.

The data plots lead us to question whether there exists a continuing need to create better discretization methods.  While there may be ways yet unidentified to increase accuracy beyond the entropy-minimization heuristic, the plots of the data values for each performance measure are so close to one another in most cases that perhaps this goal of finding a "better" discretization method can end with a statement that, so long as a discretization method is used it should better the performance of the Na\"{i}ve Bayes classifier in most cases.  There are a few cases where one discretization method performs better than another, but overall, they perform very similarly.  We therefore conclude that discretization is a useful technique for aiding \nbc, but we can not put forth one particular method that we would always recommend, except to say that we believe that the methods compared herein perform very similarly in most cases and would recommend using simple methods when possible for easy of explanation.

\ssp

\section{Future Work}
\label{future-work}

\dbsp

Our research has led us to a few possible future paths for research in this area.  

An immediate possibility from our work would be the modification of DiscTree discretization method to create an incremental method of discretization.  Incremental discretization involves data being discretized as each new instance (or a very small group of instances) arrive, rather than requiring all the data as in the batch approach currently used.  Incremental discretization methods are very useful because they can be used on streams of data as well as on the files of instances used with the current batch method. For instance, incremental discretization methods could be used as part of a learning mechanism in a problem that involves collecting data online or from a real time source and analyzing it as it arrives, rather than performing the learning operations on static, previously created files stored on a disk. 

A second possible area of future research would be in expanding the DiscTree algorithm to examine other possible tree structures that may improve some aspect of the DiscTree algorithm's performance.  Specifically, we have identified splay trees and AVL trees as possible alternatives for the base data structure in the DiscTree algorithm.  The implementation of a Randomized Binary Search Tree was the easiest to implement at the inception of this idea of a base sorting data structure behind the discretization and performed fairly well when compared to some recommended methods.  It is likely that such other data structures would have similar or better performance.  Splay trees may be most interesting because their re-balancing of the tree in accordance with the most accessed values may help to prevent any useful nodes from being pruned during the garbage collection activities of the DiscTree algorithm.

A third possible area of future research would be combining the DiscTree algorithm with some additional forms of data preprocessing, such as attribute subset selection.  This technique involves using machine learning algorithm to determine the best combinations of attributes and pruning the data to only contain those attributes.  This might be useful because experimental results have shown that learners may become confused or distracted in their classification schemes by irrelevant attributes \cite{fw2}.  Pruning the data prior to running it through the DiscTree algorithm would not increase the method's performance in terms of classification results.  It would, however, save the algorithm time spent in building trees for the unused attributes.  Another option for preprocessing prior to use of the DiscTree algorithm would be a review of over-sampling and under-sampling techniques for the data provided to the discretization method.  In either case the data is examined for notable instances. In over-sampling, those instances are then repeated numerous times in the data used by the discretization method/classifier.  In under-sampling, the instances of interest are used to prune instances from the data set to avoid or minimize their affect on the discretization method/classifier.  Both could be potential boons for DiscTree if they could be harnessed to remove the worst or multiply the best examples that DiscTree should learn from.  Thus, improvements with additional preprocessing might be made to make DiscTree perform better.

A final possible area of future research would be examining the DiscTree Algorithm to determine if there is a way to best decide the "cludged" values of that algorithm, including:
\bi 
\item
The number of instances required between garbage collects
\item
The maximum allowable tree size
\item
The number of instances required between increments of maximum allowable tree size
\item
The number of instances per node/subtree required to make it worthy of being used in the discretization. 
\ei 
It is possible that finding optimum values for these items could improve the performance of DiscTree against its competition.

\ssp
