Association Rule

Goal: Find the most frequently appearing $X=(X_1,...,X_p)$. This problem can be viewed as to the problem finding the frequent subsets $(v_1,...v_L), \;v_j \subset X$, such that the probability density $P(v_l)$ evaluated at each of those values is relative large.

In most cases $X_j\in \{0,1\}$, where it is referred to as "market basket" analysis. For observation $i$, each variable $X_j$ is assigned one of two values; $x_{ij}=1$ if the $j_{th}$ item is purchased. In this setting of the goal, $X=v_l$ will nearly always be too small for reliable estimation. Thus we need to modify our goal as following way.

Modified Goal: Instead of seeking values $x$ where $P(x)$is large, We seeks regions of the $X$space with high probability content relative to their size or support. Then, the modified goal is to find subsets of variable $s_1,...,s_p$ such that the probability of each of the variables is relative large.

$$P\Big[\bigcap^p_{j=1} (X_j \in s_j)\Big]$$

The intersection part is called a conjunctive rule. The subsets $s_j$ are interval for quantitative $X_j$.

$$P\Big[\bigcap _{j \in J} (X_j=v_{0j})\Big]$$

$K\subset \{1,...,P\},\; P=\sum^p_{j=1}|S_j|$. $|S_j|$ is the number of distinct values attainable by $X_j$. $K$ is called an item set.

$$P\Big[\bigcap_{k\in K} (Z_k=1)\Big] =P\Big[\prod_{k \in K} Z_k=1\Big ] \\ \widehat{Pr}\Big[\prod _{k \in K} (Z_k=1) \Big]=\dfrac{1}{N} \sum^N_{i=1} \prod _{k \in K} z_{ik}$$

Market Basket Analysis