Association Rules

In nuggets: Extensible Framework for Data Pattern Exploration

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

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:

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).

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:

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:

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.

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:

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 expressions.

For example, to find rules that predict the uptake rate from all other variables:

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:

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:

# 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:

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)

# 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().

# 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:

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:

?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:

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:

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.

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():

# 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:

# 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().

nuggets documentation built on July 22, 2026, 5:14 p.m.