From 1d56ef172fb2f85e055c780b625500ac06f425fb Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Fri, 22 Dec 2023 20:00:40 +0100 Subject: [PATCH] Add ML/DM association rules --- src/ainotes.cls | 1 + .../img/itemset_apriori.png | Bin 0 -> 283362 bytes .../img/rules_apriori.png | Bin 0 -> 230107 bytes src/machine-learning-and-data-mining/main.tex | 3 + .../sections/_association_rules.tex | 353 ++++++++++++++++++ 5 files changed, 357 insertions(+) create mode 100644 src/machine-learning-and-data-mining/img/itemset_apriori.png create mode 100644 src/machine-learning-and-data-mining/img/rules_apriori.png create mode 100644 src/machine-learning-and-data-mining/sections/_association_rules.tex diff --git a/src/ainotes.cls b/src/ainotes.cls index 3874f4d..794e6ed 100644 --- a/src/ainotes.cls +++ b/src/ainotes.cls @@ -63,6 +63,7 @@ \newtheorem*{example}{Example} \theoremstyle{definition} \newtheorem*{definition}{Def} +\newtheorem*{remark}{Remark} 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+\chapter{Association rules} + + +\section{Frequent itemset} + +\begin{description} + \item[Itemset] \marginnote{Itemset} + Collection of one or more items (e.g. $\{ \text{milk}, \text{bread}, \text{diapers} \}$). + + \item[K-itemset] \marginnote{K-itemset} + Itemset with $k$ items. + + \item[Support count] \marginnote{Support count} + Number of occurrences of an itemset in a dataset. + \begin{example} + \phantom{}\\ + \begin{minipage}{0.4\textwidth} + Given the following transactions: + \begin{center} + \begin{tabular}{|c|l|} + \hline + 1 & bread, milk \\ + 2 & beer, bread, diaper, eggs \\ + 3 & beer, coke, diaper, milk \\ + \textbf{4} & \textbf{beer, bread, diaper, milk} \\ + \textbf{5} & \textbf{bread, coke, diaper, milk} \\ + \hline + \end{tabular} + \end{center} + \end{minipage} + \begin{minipage}{0.5\textwidth} + The support count of the itemset containing bread, diapers and milk is: + \[ \sigma(\{ \text{bread}, \text{diapers}, \text{milk} \}) = 2 \] + \end{minipage} + \end{example} + + \item[Association rule] \marginnote{Association rule} + Given two itemsets $A$ and $C$, an association rule has form: + \[ A \rightarrow C \] + It means that there are transactions in the dataset where $A$ and $C$ co-occur. + Note that it is not strictly a logical implication. + + \item[Metrics] \phantom{} + \begin{description} + \item[Support] \marginnote{Support} + Given $N$ transactions, the support of an itemset $A$ is: + \[ \texttt{sup}(A) = \frac{\sigma(A)}{N} \] + The support of an association rule $A \rightarrow C$ is: + \[ \texttt{sup}(A \rightarrow C) = \texttt{sup}(A \cup C) = \frac{\sigma(A \cup C)}{N} \] + + Low support implies random associations. + + \begin{description} + \item[Frequent itemset] \marginnote{Frequent itemset} + Itemset whose support is at least a given threshold. + \end{description} + + \item[Confidence] \marginnote{Confidence} + Given an association rule $A \rightarrow C$, its confidence is given by: + \[ \texttt{conf}(A \rightarrow C) = \frac{\sigma(A \cup C)}{\sigma(A)} \in [0, 1] \] + + Low confidence implies low reliability. + + \begin{theorem} + The confidence of $A \rightarrow C$ can be computed given the supports of $A \rightarrow C$ and $A$: + \[ \texttt{conf}(A \rightarrow C) = \frac{\texttt{sup}(A \rightarrow C)}{\texttt{sup}(A)} \] + \end{theorem} + \end{description} + + \item[Association rule mining] \marginnote{Association rule mining} + Given $N$ transactions and two thresholds \texttt{min\_sup} and \texttt{min\_conf}, + association rule mining finds all the rules $A \rightarrow C$ such that: + \[ \begin{split} + \texttt{sup}(A \rightarrow C) &\geq \texttt{min\_sup} \\ + \texttt{conf}(A \rightarrow C) &\geq \texttt{min\_conf} + \end{split} \] + + This can be done in two steps: + \begin{enumerate} + \item \marginnote{Frequent itemset generation} + Determine the itemsets with $\text{support} \geq \texttt{min\_sup}$ (frequent itemsets). + \item \marginnote{Rule generation} + Determine the the association rules with $\text{confidence} \geq \texttt{min\_conf}$. + \end{enumerate} +\end{description} + + + +\section{Frequent itemset generation} + +\subsection{Brute force} +Given $D$ items, there are $2^D$ possible itemsets. +To compute the support of a single itemset, the complexity is $O(NW)$ where +$N$ is the number of transactions and $W$ is the width of the largest transaction. +Listing all the itemsets and computing their support have an exponential complexity of $O(NW2^D)$. + + +\subsection{Apriori principle} + +\begin{theorem} \marginnote{Apriori principle} + If an itemset is frequent, then all of its subsets are frequent. + + \begin{proof} + By the definition of support, it holds that: + \[ \forall X, Y: (X \subseteq Y) \Rightarrow (\texttt{sup}(X) \geq \texttt{sup}(Y)) \] + + In other words, the support metric is anti-monotone. + \end{proof} +\end{theorem} + +\begin{corollary} + If an itemset is infrequent, then all of its supersets are infrequent. +\end{corollary} + +\begin{example} \phantom{} + \begin{center} + \includegraphics[width=0.6\textwidth]{img/itemset_apriori.png} + \end{center} +\end{example} + +\begin{algorithm}[H] +\caption{Apriori principle} +\begin{lstlisting}[mathescape=true] +def candidatesGeneration(freq_itemsets$_k$): + candidate_itemsets$_{k+1}$ = selfJoin(freq_itemsets$_k$) + for itemset in candidate_itemsets$_{k+1}$: + for sub in subsetsOfSize($k$, itemset): + if sub not in freq_itemsets$_k$: + candidate_itemsets$_{k+1}$.remove(itemset) + return candidate_itemsets$_{k+1}$ + +def aprioriItemsetGeneration(transactions, min_sup): + freq_itemsets$_1$ = itemsetsOfSize(1, transactions) + k = 1 + while freq_itemsets$_1$ is not null: + candidate_itemsets$_{k+1}$ = candidatesGeneration(freq_itemsets$_k$) + freq_itemsets$_{k+1}$ = $\{ c \in \texttt{candidate\_itemsets}_{k+1} \mid \texttt{sup(}c\texttt{)} \geq \texttt{min\_sup} \}$ + k += 1 + return freq_itemsets$_k$ +\end{lstlisting} +\end{algorithm} + +\begin{description} + \item[Complexity] + The complexity of the apriori principle depends on: + \begin{itemize} + \item The choice of the support threshold. + \item The number of unique items. + \item The number and the width of the transactions. + \end{itemize} +\end{description} + + + +\section{Rule generation} + +\subsection{Brute force} +Given a frequent $k$-itemset $L$, there are $2^k-2$ possible association rules ($-2$ as $L \rightarrow \varnothing$ and $\varnothing \rightarrow L$ can be ignored). +For each possible rule, it is necessary to compute the confidence. The overall complexity is exponential. + +\subsection{Apriori principle} + +\begin{theorem} \marginnote{Apriori principle} + Without loss of generality, consider an itemset $\{ A, B, C, D \}$. + It holds that: + \[ \texttt{conf}(ABC \rightarrow D) \geq \texttt{conf}(AB \rightarrow CD) \geq \texttt{conf}(A \rightarrow BCD) \] +\end{theorem} + +\begin{example} \phantom{} + \begin{center} + \includegraphics[width=0.5\textwidth]{img/rules_apriori.png} + \end{center} +\end{example} + + + +\section{Interestingness measures} + +\begin{description} + \item[Contingency table] \marginnote{Contingency table} + Given an association rule $A \rightarrow C$, its contingency table is defined as: + \begin{center} + \def\arraystretch{1.1} + \begin{tabular}{c|c|c|c} + & $C$ & $\overline{C}$ & \\ + \hline + $A$ & $\prob{A \land C}$ & $\prob{A \land \overline{C}}$ & $\prob{A}$ \\ + \hline + $\overline{A}$ & $\prob{\overline{A} \land C}$ & $\prob{\overline{A} \land \overline{C}}$ & $\prob{\overline{A}}$ \\ + \hline + & $\prob{C}$ & $\prob{\overline{C}}$ & 100 \\ + \end{tabular} + \end{center} +\end{description} + +\begin{remark} + Confidence can be misleading. + \begin{example} \phantom{}\\ + \begin{minipage}[t]{0.36\textwidth} + Given the following contingency table: + \begin{center} + \begin{tabular}{c|c|c|c} + & coffee & $\overline{\text{coffee}}$ & \\ + \hline + tea & 15 & 5 & 20 \\ + \hline + $\overline{\text{tea}}$ & 75 & 5 & 80 \\ + \hline + & 90 & 10 & 100 \\ + \end{tabular} + \end{center} + \end{minipage} + \hspace{0.5cm} + \begin{minipage}[t]{0.6\textwidth} + We have that: + \[ \texttt{conf}(\text{tea} \rightarrow \text{coffee}) = \frac{\texttt{sup}(\text{tea}, \text{coffee})}{\texttt{sup}(\text{tea})} = \frac{15}{20} = 0.75 \] + But, we also have that: + \[ \prob{\text{coffee}} = 0.9 \hspace*{1cm} \prob{\text{coffee} \mid \overline{\text{tea}}} = \frac{75}{80} = 0.9375 \] + So, despite the high confidence of $(\text{tea} \rightarrow \text{coffee})$, + the probability of coffee increases in absence of tea. + \end{minipage} + \end{example} +\end{remark} + + +\subsection{Statistical-based measures} + +Measures that take into account the statistical independence of the items. + +\begin{description} + \item[Lift] \marginnote{Lift} + \[ \texttt{lift}(A \rightarrow C) = \frac{\texttt{conf}(A \rightarrow C)}{\texttt{sup}(C)} = \frac{\prob{A \land C}}{\prob{A}\prob{C}} \] + + If $\texttt{lift}(A \rightarrow C) = 1$, then $A$ and $C$ are independent. + + \item[Leverage] \marginnote{Leverage} + \[ \texttt{leve}(A \rightarrow C) = \texttt{sup}(A \cup C) - \texttt{sup}(A)\texttt{sup}(C) = \prob{A \land C} - \prob{A}\prob{C} \] + + If $\texttt{leve}(A \rightarrow C) = 0$, then $A$ and $C$ are independent. + + \item[Conviction] \marginnote{Conviction} + \[ \texttt{conv}(A \rightarrow C) = \frac{1 - \texttt{sup}(C)}{1 - \texttt{conf}(A \rightarrow C)} = \frac{\prob{A}(1-\prob{C})}{\prob{A}-\prob{A \land C}} \] +\end{description} + +\begin{table}[H] + \centering + \begin{tabular}{c|p{10cm}} + \hline + \textbf{Metric} & \textbf{Interpretation} \\ + \hline + High support & The rule applies to many transactions. \\ + \hline + High confidence & The chance that the rule is true for some transaction is high. \\ + \hline + High lift & Low chance that the rule is just a coincidence. \\ + \hline + High conviction & The rule is violated less often compared to the case when the antecedent and consequent are independent. \\ + \hline + \end{tabular} + \caption{Intuitive interpretation of the measures} +\end{table} + + + +\section{Multi-dimensional association rules} + +\begin{description} + \item[Mono-dimensional events] \marginnote{Mono-dimensional events} + Represented as transactions. Each event contains items that appear together. + + \item[Multi-dimensional events] \marginnote{Multi-dimensional events} + Represented as tuples. Each event contains the values of its attributes. + + \item[Mono/Multi-dimensional equivalence] \marginnote{Equivalence} + Mono-dimensional events can be converted into multi-dimensional events and vice versa. + + To transform quantitative attributes, it is usually useful to discretize them. + + \begin{example}[Multi to mono] \phantom{}\\ + \begin{minipage}{0.35\textwidth} + \begin{center} + \begin{tabular}{c|c|c} + \textbf{Id} & \textbf{co2} & \textbf{tin\_oxide} \\ + \hline + 1 & high & medium \\ + 2 & medium & low \\ + \end{tabular} + \end{center} + \end{minipage} + $\rightarrow$ + \begin{minipage}{0.48\textwidth} + \begin{center} + \begin{tabular}{c|c} + \textbf{Id} & \textbf{Transaction} \\ + \hline + 1 & $\{ \text{co2/high}, \text{tin\_oxide/medium} \}$ \\ + 2 & $\{ \text{co2/medium}, \text{tin\_oxide/low} \}$ \\ + \end{tabular} + \end{center} + \end{minipage} + \end{example} + + \begin{example}[Mono to multi] \phantom{}\\ + \begin{minipage}{0.35\textwidth} + \begin{center} + \begin{tabular}{c|c|c|c|c} + \textbf{Id} & \textbf{a} & \textbf{b} & \textbf{c} & \textbf{d} \\ + \hline + 1 & yes & yes & no & no \\ + 2 & yes & no & yes & no \\ + \end{tabular} + \end{center} + \end{minipage} + $\leftarrow$ + \begin{minipage}{0.30\textwidth} + \begin{center} + \begin{tabular}{c|c} + \textbf{Id} & \textbf{Transaction} \\ + \hline + 1 & $\{ \text{a}, \text{b} \}$ \\ + 2 & $\{ \text{a}, \text{c} \}$ \\ + \end{tabular} + \end{center} + \end{minipage} + \end{example} + +\end{description} + + + +\section{Multi-level association rules} +Organize items into an hierarchy. + +\begin{description} + \item[Specialized to general] \marginnote{Specialized to general} + Generally, the support of the rule increases. + \begin{example} + From $(\text{apple} \rightarrow \text{milk})$ to $(\text{fruit} \rightarrow \text{dairy})$ + \end{example} + + \item[General to specialized] \marginnote{General to specialized} + Generally, the support of the rule decreases. + \begin{example} + From $(\text{fruit} \rightarrow \text{dairy})$ to $(\text{apple} \rightarrow \text{milk})$ + \end{example} + + \item[Redundant level] \marginnote{Redundant level} + A more specialized rule in the hierarchy is redundant if its confidence is similar to the one of the more general rule. + + \item[Multi-level association rule mining] \marginnote{Multi-level association rule mining} + Run association rule mining on different levels of abstraction (general to specialized). + At each level, the support threshold is decreased. +\end{description} \ No newline at end of file