--- title: "Association Rules" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Association Rules} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ```{r, include=FALSE} options(tibble.width = Inf) ``` # Introduction **Association rules** are one of the most fundamental tools in data mining. An association rule has the form: > *antecedent* $\Rightarrow$ *consequent* where the *antecedent* (left-hand side) is a conjunction of predicates and the *consequent* (right-hand side) is a single predicate. The rule expresses that whenever the antecedent conditions are satisfied, the consequent tends to be satisfied as well. For example: > `middle_age & university_edu & IT_industry` $\Rightarrow$ `high_income` This rule states that middle-aged people with a university education working in the IT industry tend to have a high income. Association rules are evaluated using several quality measures: - **Support**: the relative frequency of rows satisfying both the antecedent and the consequent. Higher support means the rule applies to more data. - **Confidence**: the proportion of rows satisfying the antecedent that also satisfy the consequent. Higher confidence means the rule is more reliable. - **Coverage**: the relative frequency of rows satisfying the antecedent. - **Lift**: the ratio of the observed support to the expected support if the antecedent and consequent were independent. A lift greater than 1 indicates a positive association. The `nuggets` package supports searching for association rules in both **crisp** (Boolean) and **fuzzy** data through the `dig_associations()` function. Before using the package, the required libraries must be loaded: ```{r, message=FALSE} library(nuggets) library(dplyr) # for data manipulation ``` # Data Preparation For this tutorial, we use the built-in `CO2` dataset, which contains data from an experiment on the cold tolerance of the grass species *Echinochloa crus-galli*. The dataset has 84 observations and includes information about the plant's origin (`Type`), treatment (`Treatment`), ambient CO2 concentration (`conc`), and CO2 uptake rate (`uptake`). ```{r} head(CO2) ``` Before searching for association rules, data must be transformed into predicates (logical or fuzzy columns). We use the `partition()` function for this purpose. For a detailed explanation of data preparation techniques, see the `vignette("data-preparation")`. ## Crisp Data Preparation We prepare a crisp version of the dataset by transforming factors into dummy variables and numeric columns into interval-based logical predicates: ```{r} crisp_co2 <- CO2 |> select(-Plant) |> partition(Type, Treatment) |> partition(conc, .method = "crisp", .breaks = c(-Inf, 200, 500, Inf)) |> partition(uptake, .method = "crisp", .breaks = c(-Inf, 20, 35, Inf)) head(crisp_co2, n = 3) ``` ## Fuzzy Data Preparation For a fuzzy version, we transform numeric columns into fuzzy predicates using triangular membership functions: ```{r} fuzzy_co2 <- CO2 |> select(-Plant) |> partition(Type, Treatment) |> partition(conc, .method = "triangle", .breaks = c(-Inf, 95, 500, 1000, Inf)) |> partition(uptake, .method = "triangle", .breaks = c(-Inf, 10, 25, 40, Inf)) head(fuzzy_co2, n = 3) ``` # Basic Association Rule Search The simplest use of `dig_associations()` searches for all rules that meet given minimum support and confidence thresholds. ```{r} rules <- dig_associations(crisp_co2, min_support = 0.1, min_confidence = 0.8) head(rules) ``` The result is a tibble containing found rules with their quality measures: `antecedent`, `consequent`, `support`, `confidence`, `coverage`, `consequent_support`, `lift`, `length`, and the contingency table columns (`pp`, `pn`, `np`, `nn`). For instance, the first rule represents the following association rule: ```{r, include=FALSE} to_logical <- function(x) { x <- gsub("{", "", x, fixed = TRUE) x <- gsub("}", "", x, fixed = TRUE) x <- gsub(",", " & ", x, fixed = TRUE) } ``` > `r paste0("> ", to_logical(rules$antecedent[1]), " $\\Rightarrow$ ", to_logical(rules$consequent[1]))` # Specifying Antecedent and Consequent In many applications, you want to constrain which predicates can appear on each side of the rule. This is done with the `antecedent` and `consequent` arguments, which accept [tidyselect](https://tidyselect.r-lib.org/articles/syntax.html) expressions. For example, to find rules that predict the `uptake` rate from all other variables: ```{r} rules_uptake <- dig_associations(crisp_co2, antecedent = !starts_with("uptake"), consequent = starts_with("uptake"), min_support = 0.1, min_confidence = 0.8) head(rules_uptake, n = 3) ``` # Using the Disjoint Argument When data is prepared with `partition()`, a single original variable is often expanded into multiple predicates (e.g., `conc=(-Inf,200]` and `conc=(200,500]`). These predicates from the same variable should not appear together in the same antecedent, as their conjunction would be contradictory or redundant. The `disjoint` argument prevents this. It accepts a character vector of size equal to the number of columns in the input data such that each unique value in the vector corresponds to a group of mutually exclusive predicates. By default, `dig_associations()` uses `var_names()` on the column names to create the disjoint vector automatically. This works well for data prepared with `partition()`. You can also provide a custom disjoint vector: ```{r} disj <- var_names(colnames(crisp_co2)) disj rules <- dig_associations(crisp_co2, disjoint = disj, min_support = 0.1, min_confidence = 0.8) head(rules, n = 3) ``` # Controlling Rule Length The `min_length` and `max_length` arguments control the number of predicates in the antecedent: ```{r} # Find only rules with exactly 2 predicates in the antecedent rules <- dig_associations(crisp_co2, min_length = 2, max_length = 2, min_support = 0.1, min_confidence = 0.8) head(rules, n = 3) ``` Setting `min_length = 0` generates rules with an empty antecedent, which effectively computes the support of each consequent alone. # Limiting the Number of Results For large datasets, the number of possible rules can be enormous. The `max_results` argument limits the total number of rules generated: ```{r} rules <- dig_associations(crisp_co2, min_support = 0.05, min_confidence = 0.6, max_results = 5) nrow(rules) ``` # Fuzzy Association Rules When the data contains fuzzy predicates (numeric columns with values in [0, 1]), `dig_associations()` computes fuzzy support using a **t-norm** for conjunction. The `t_norm` argument specifies which t-norm to use: - `"goguen"` (default): the product t-norm - multiplies membership degrees - `"goedel"`: the minimum t-norm - takes the minimum of membership degrees - `"lukas"`: the Łukasiewicz t-norm = max(0, a + b - 1) ```{r} # Fuzzy rules using the product t-norm (default) fuzzy_rules <- dig_associations(fuzzy_co2, antecedent = !starts_with("uptake"), consequent = starts_with("uptake"), min_support = 0.05, min_confidence = 0.6, t_norm = "goguen") head(fuzzy_rules, n = 3) ``` The choice of t-norm affects how strictly the conjunction is evaluated. The Gödel t-norm is the least strict (produces higher support values), while the Łukasiewicz t-norm is the most strict. Note also that handling fuzzy data is generally much more slower and memory demanding than crisp data, especially for large datasets. # Computing Additional Interest Measures The `add_interest()` function computes additional interestingness measures for association rules beyond the basic support, confidence, and lift. It uses the contingency table columns (`pp`, `pn`, `np`, `nn`) that are automatically included in the output of `dig_associations()`. ```{r} # Add selected interest measures rules_enriched <- rules_uptake |> add_interest(measures = c("conviction", "leverage", "jaccard")) rules_enriched |> select(antecedent, consequent, confidence, conviction, leverage, jaccard) |> head(n = 3) ``` You can compute all available measures at once by omitting the `measures` argument: ```{r} rules_all_measures <- rules_uptake |> add_interest() colnames(rules_all_measures) ``` See the documentation of `add_interest()` for a complete list of supported quality measures: ```{r, eval=FALSE} ?add_interest ``` ## Smoothing Some interest measures may be undefined when contingency table counts are zero (e.g., division by zero). The `smooth_counts` argument applies Laplace smoothing to the counts before computing the measures: ```{r} rules_smoothed <- rules_uptake |> add_interest(measures = c("odds_ratio", "conviction"), smooth_counts = 0.5) rules_smoothed |> select(antecedent, consequent, confidence, odds_ratio, conviction) |> head(n = 3) ``` ## GUHA Quantifiers `add_interest()` also supports GUHA (General Unary Hypothesis Automaton) quantifiers, including statistical tests based on the binomial distribution: ```{r} rules_guha <- rules_uptake |> add_interest(measures = c("dfi", "fe", "lci"), p = 0.5) rules_guha |> select(antecedent, consequent, confidence, dfi, fe, lci) |> head(n = 3) ``` The `p` parameter represents the null-hypothesis probability used in the binomial-test-based quantifiers (`lci`, `uci`, `dlci`, `duci`, `lce`, `uce`). # Excluding Redundant and Entailed Rules In real-world datasets, some rules are trivially true or nearly so — for example, rules that follow directly from the structure of the data (e.g., `engine_type=electric => fuel_type=electricity`). Such near-certain rules are called **tautologies**, and are also referred to as **implications** (because they describe an `A => C` relationship that almost always holds) or **axioms** (because they can be assumed and used as starting assumptions for pruning). In classical logic, a *tautology* is strictly always true; here the term is used loosely for rules whose confidence is very high in the data. This distinction is a matter of logical philosophy and does not affect how the functions work. The `excluded` argument of `dig_associations()` accepts a list of known **implications (axioms)**. Each axiom is a character vector where all elements except the last form the antecedent and the last element is the consequent: `c(ant1, ant2, ..., antn, cons)`. The axioms are used to prune generated rules via the **modus ponens** inference rule: - A rule's consequent is excluded if it can be deduced from the rule's antecedent using the axioms (possibly via a chain of axiom applications). - A rule is pruned entirely if any predicate in the antecedent can be deduced from the remaining antecedent predicates via the axioms (it would be redundant). ## Finding Near-Tautologies with `dig_tautologies()` The `dig_tautologies()` function is specifically designed to find rules with very high confidence (near-tautologies / axioms). It searches iteratively, using rules found in earlier iterations to prune the search space in later iterations, so that only the most concise axioms are returned. ```{r} tautologies <- dig_tautologies(crisp_co2, antecedent = everything(), consequent = everything(), min_confidence = 0.95, min_support = 0.05, max_length = 2) tautologies ``` ## Using Axioms to Filter Association Rules Once axioms (near-tautologies) are identified, convert them to the format expected by the `excluded` argument using `parse_condition()`, passing both the antecedent and the consequent columns, and pass them to `dig_associations()`: ```{r} # Convert tautologies to the excluded (axioms) format excluded_conds <- parse_condition(tautologies$antecedent, tautologies$consequent) # Search for rules while excluding entailed patterns rules_filtered <- dig_associations(crisp_co2, antecedent = !starts_with("uptake"), consequent = starts_with("uptake"), excluded = excluded_conds, min_support = 0.1, min_confidence = 0.8) rules_filtered ``` By providing known axioms, the search skips rules whose consequent can be deduced from their antecedent, and also prunes rules that contain redundant antecedent predicates (deducible from the remaining antecedent predicates). This focuses the results on genuinely interesting patterns and can run significantly faster on large datasets. ## Manually Specifying Axioms You can also construct the `excluded` list manually. Each element is a character vector representing an implication (axiom): all elements except the last form the antecedent and the last element is the consequent: ```{r} # Axiom: "Treatment=chilled => Type=Mississippi" # Any rule whose consequent is "Type=Mississippi" and whose antecedent contains # "Treatment=chilled" will be excluded, because the consequent is deducible # from the antecedent via this axiom. manual_excluded <- list(c("Treatment=chilled", "Type=Mississippi")) rules_manual <- dig_associations(crisp_co2, antecedent = !starts_with("uptake"), consequent = starts_with("uptake"), excluded = manual_excluded, min_support = 0.1, min_confidence = 0.8) rules_manual ``` This is useful when you have domain knowledge about which implications are trivially true or undesirable, without needing to run `dig_tautologies()` first. # Summary This vignette demonstrated how to search for association rules using the `nuggets` package: 1. **Data preparation** with `partition()` transforms raw data into crisp or fuzzy predicates suitable for rule mining (see `vignette("data-preparation")` for details). 2. **Basic search** with `dig_associations()` finds rules meeting minimum support and confidence thresholds. 3. **Controlling the search space**: use `antecedent`/`consequent` to constrain which predicates appear on each side, `disjoint` to prevent contradictory combinations via `var_names()`, and `min_length`/`max_length` to control rule complexity. 4. **Fuzzy rules** extend the approach to graded membership using t-norms (`"goguen"`, `"goedel"`, `"lukas"`). 5. **Interest measures** can be added with `add_interest()` to evaluate rules from multiple perspectives (conviction, leverage, Jaccard, GUHA quantifiers, and many more). 6. **Excluding entailed rules** with the `excluded` argument and `dig_tautologies()` removes rules whose consequent is deducible from the antecedent via known axioms (near-tautologies / implications), and also prunes rules with redundant antecedent predicates (deducible from the remaining predicates). This speeds up the search and focuses results on genuinely interesting patterns. For further details, consult the function documentation: `dig_associations()`, `add_interest()`, `dig_tautologies()`, `parse_condition()`, `partition()`, `var_names()`.