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diff --git a/aied2017/method.tex b/aied2017/method.tex index 3a14d53..c86af75 100644 --- a/aied2017/method.tex +++ b/aied2017/method.tex @@ -1 +1,82 @@ -\section{Method}
\ No newline at end of file +\section{Method} + +\subsection{Learning rules for correct and incorrect programs} +\begin{figure}[t] +\centering + \begin{enumerate} + \item Let $P$ be the data of all student programs, each described with a set of AST patterns + and classified as correct (it passes unit tests) or incorrect. + \item Let method $learn\_rules(target, P, P_1, sig, acc)$ be a method that learns + a set of rules for class $target$ from data $P$. The method needs to consider + two additional constraints: the significance of every attribute-value pair + in the condition part of the rule needs to be significant with respect to the likelihood-ratio + test ($p<sig$) and classification accuracy of each rule on data $P_1$ should be at least $acc$. + \item Let $I-rules = learn\_rules(incorrect, P, P, 0.05, 0.9)$ + \item Let $P_c$ be data $P$ without programs that are already covered by $I-rules$ + \item Let $C-rules = learn\_rules(correct, P, P_c, 0.05, 0.9)$ + \item Return $I-rules$ and $C-rules$ + \end{enumerate} + \caption{An outline of the algorithm for learning rules. The method $learn\_rules$, + which induces rules for a specific class, is a variant of the + CN2 algorithm~\cite{YYY} implemented within the Orange data-mining suite~\cite{XXX}. + In all our experiments, $sig$ was set to 0.05 and $acc$ was set to 0.9. } + \label{figure:algorithm} +\end{figure} + +As described in the previous section, the feature space contains all frequent AST patterns. +The submitted programs are then represented in this feature space and classified as either +correct, if they pass all prespecified tests, or incorrect otherwise. Such data +serves as the data set for machine learning. + +However, since we are always able to validate a program with unit tests, +the goal of machine learning is not to classify new programs, but to discover patterns +that are correlated with correctness of programs. This approach of machine learning is +referred to as descriptive induction, that is the automatic discovery of patterns describing +regularities in data. We use rule learning for this task since rule-based models are easy to comprehend. + +Before we can explain the algorithm, we need to discuss the reasons why a program +can be incorrect. From our previous pedagogical experience, a student program is +incorrect 1) if some incorrect pattern is present, which needs to be removed, or 2) if the +program lacks a certain relation (pattern) that should be included before the program can be correct. +We shall now explain how both types of errors can be identified with rules. + +To discover patterns related to the first point, the algorithm first learns rules that describe +incorrect programs. The conditions of these rules contain frequent patterns that symptom +incorrect programs. Since rules are used to generate hints and since hints should not be +presented to students unless they are probably correct, we require that each learned rule's +classification accuracy exceeds a certain threshould (in our experiments we used 90\%), +each conjunct in a condition is significant with respect to the likelihood-ratio test (with $p=0.05$ in our experiments), +and a conjunct can only specify the presence of a pattern. The former two constraints are needed +to induce good rules with significant patterns, while the latter constraint assures that rules mention +only presence (and not absence) of patterns as reasons for a program to be incorrect. + +With respect to the second type of error, we could try the same approach and learn rules for the class of +correct programs. Having accurate rules for correct programs, the conditional part of these rules would +define sufficient groups of patterns that render a program correct. However, it turns out that +it is difficult to learn accurate rules for correct programs, since these rules should contain all relevant patterns +and prevent incorrect patterns, yet a conjunct can only +specify the presence of a pattern. If specifying absence of patterns was allowed in rules' condition, +the learning problem would still be difficult, since usually there are many incorrect patterns. +A possible way to solve this problem is to learn from data set not containing programs that are covered by rules for incorrect class. +This way all known incorrect patterns are removed from the data and are not needed anymore in conditions of rules. +However, removing incorrect patterns also removes the need for relevant patterns. For example, if all incorrect programs were removed, +a rule ``IF True then class=correct'' would suffice. +Such rule does not contain any relevant patterns and could not be used to generate hints. +We achieved the best results by learning from both data sets. The original data set (with all programs) +is used to learn rules, while the filtered data are used to test whether a rule achieves the +required classification accuracy (90\%). + +Figure~\ref{figure:algorithm} contains an outline of the algorithm. The rules describing +incorrect programs are called $I-rules$ and the rules for correct programs are called $C-rules$. + +Even though our main interest is discovery of patterns, we could still use induced rules to classify +new programs, for example to evaluate the quality of rules. The classification procedure has three steps: +first check whether a $I-rule$ covers the program that needs to be classified, and if it +does, classify it as incorrect. Then, check whether a $C-rule$ covers the program and classify +it as correct if it does. Otherwise, if none of the induced rules cover this program, classify it as +incorrect. + +\subsection{Generating hints from rules} + + + |