For more advanced workflows — such as defining custom pattern types
or computing user-defined measures — see the section Advanced Use.
Search for Association Rules
Association rules identify conditions
(antecedents) under which a specific feature
(consequent) is present very often.
\[
A \Rightarrow C
\]
If condition A is satisfied, then the feature
C tends to be present.
For example,
university_edu & middle_age & IT_industry => high_income
can be read as:
People in middle age with university education working in IT
industry are very likely to have a high income.
In practice, the antecedent A is a set of predicates,
and the consequent C is usually a single predicate.
For a set of predicates \(I\), let
\(\text{supp}(I)\) denote the
support — the relative frequency (for logical data) or the mean
truth degree (for fuzzy data) of rows satisfying all predicates in \(I\). Using this notation, the following
rule properties and quality measures may be defined:
- Length — number of predicates in the
antecedent.
- Coverage (antecedent support) — \(\text{supp}(A)\).
- Consequent support — \(\text{supp}(C)\).
- Support — \(\text{supp}(A
\cup C)\).
- Confidence — \(\text{supp}(A \cup C) /
\text{supp}(A)\).
- Lift — \(\text{supp}(A
\cup C) / (\text{supp}(A) \text{supp}(C))\).
Rules with high support are frequent in the data. Rules with
high confidence indicate a strong association between
antecedent and consequent. Rules with high lift suggest that
the validity of antecedent increases the likelihood of the consequent
occurring.
Before searching for rules, it is recommended to create a vector
of disjoints, which specifies predicates that must not appear
together in the same condition. This vector should have the same length
as the number of dataset columns.
For example, columns representing gear=3 and
gear=4 are mutually exclusive, so their shared group label
in disj prevents meaningless conditions like
gear=3 & gear=4. You can conveniently generate this
vector with var_names():
disj <- var_names(colnames(fuzzy_mtcars))
print(disj)
#> [1] "cyl" "cyl" "cyl" "vs" "vs" "am" "am" "gear" "gear" "gear"
#> [11] "mpg" "mpg" "mpg" "disp" "disp" "disp" "hp" "hp" "hp" "drat"
#> [21] "drat" "drat" "wt" "wt" "wt" "qsec" "qsec" "qsec" "carb" "carb"
#> [31] "carb"
The dig_associations() function searches for association
rules. Its main arguments are:
x: the data matrix or data frame (logical or
numeric);antecedent, consequent: tidyselect
expressions selecting columns for each side of the rule;disjoint: a vector defining mutually exclusive
predicates;- rule filtering thresholds such as
min_support,
min_confidence, min_coverage, and limits like
min_length, max_length;
- optional parameters such as
t_norm, and
contingency_table.
In the following example, we search for fuzzy association rules in
the dataset fuzzy_mtcars, such that:
- any column except those starting with
"am" may appear
in the antecedent;
- columns starting with
"am" may appear in the
consequent;
- minimum support is
0.02, i.e., 2 % of data rows have to
contain both the antecedent and consequent of the rule;
- minimum confidence is
0.8, i.e., the conditional
probability of consequent given antecedent should be at least 80%;
- additionally to basic quality measures, the contingency table for
each rule is computed. The contingency table is a quadruplet
pp, pn, np and nn,
which contains the counts (or sums of degrees) of rows satisfying
antecedent & consequent (pp), antecedent & not
consequent (pn), not antecedent & consequent
(np), and not antecedent & not consequent
(nn). These values are important for further computation of
various additional interestingness measures.
result <- dig_associations(fuzzy_mtcars,
antecedent = !starts_with("am"),
consequent = starts_with("am"),
disjoint = disj,
min_support = 0.02,
min_confidence = 0.8,
contingency_table = TRUE)
The result is a tibble containing the discovered rules and their
quality metrics. You can arrange them, for example, by decreasing
support:
result <- arrange(result, desc(support))
print(result)
#> # A tibble: 526 × 13
#> antecedent consequent support confidence coverage
#> <chr> <chr> <dbl> <dbl> <dbl>
#> 1 {gear=3} {am=0} 0.469 1 0.469
#> 2 {gear=3,vs=0} {am=0} 0.375 1 0.375
#> 3 {cyl=eight,gear=3,vs=0} {am=0} 0.375 1 0.375
#> 4 {cyl=eight,vs=0} {am=0} 0.375 0.857 0.438
#> 5 {cyl=eight,gear=3} {am=0} 0.375 1 0.375
#> 6 {cyl=eight} {am=0} 0.375 0.857 0.438
#> 7 {mpg=(-Inf;15;20)} {am=0} 0.327 0.847 0.387
#> 8 {drat=(-Inf;2.76;3.84)} {am=0} 0.311 0.948 0.328
#> 9 {gear=3,mpg=(-Inf;15;20)} {am=0} 0.309 1 0.309
#> 10 {drat=(-Inf;2.76;3.84),gear=3} {am=0} 0.307 1 0.307
#> conseq_support lift count antecedent_length pp pn np nn
#> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl>
#> 1 0.594 1.68 15 1 15 0 4 13
#> 2 0.594 1.68 12 2 12 0 7 13
#> 3 0.594 1.68 12 3 12 0 7 13
#> 4 0.594 1.44 12 2 12 2 7 11
#> 5 0.594 1.68 12 2 12 0 7 13
#> 6 0.594 1.44 12 1 12 2 7 11
#> 7 0.594 1.43 10.5 1 10.5 1.90 8.52 11.1
#> 8 0.594 1.60 9.96 1 9.96 0.546 9.04 12.5
#> 9 0.594 1.68 9.88 2 9.88 0 9.12 13.0
#> 10 0.594 1.68 9.82 2 9.82 0 9.18 13
#> # ℹ 516 more rows
This example illustrates the typical workflow for mining association
rules with nuggets. The same structure and arguments apply
when analyzing either fuzzy or Boolean datasets.
Conditional Correlations
Conditional correlations identify strong
relationships between pairs of numeric variables under specific
conditions.
The dig_correlations() function searches for pairs of
variables that are significantly correlated within sub-data satisfying
generated conditions. This is useful for discovering context-dependent
relationships.
In the following example, we search for correlations between
different numeric variables in the original mtcars data
under conditions defined by the prepared predicates in
crisp_mtcars:
# Prepare combined dataset with both condition predicates and numeric variables
combined_mtcars <- cbind(crisp_mtcars, mtcars[, c("mpg", "disp", "hp", "wt")])
# Extend disjoint vector for the new numeric columns
disj_combined <- c(var_names(colnames(crisp_mtcars)),
c("mpg", "disp", "hp", "wt"))
# Search for conditional correlations
corr_result <- dig_correlations(combined_mtcars,
condition = colnames(crisp_mtcars),
xvars = c("mpg", "hp"),
yvars = c("wt", "disp"),
disjoint = disj_combined,
min_length = 1,
max_length = 2,
min_support = 0.2,
method = "pearson")
print(corr_result)
#> # A tibble: 536 × 10
#> condition support xvar yvar estimate p_value
#> <chr> <dbl> <chr> <chr> <dbl> <dbl>
#> 1 {carb=(-Inf;3.33]} 0.625 mpg wt -0.887 0.000000183
#> 2 {carb=(-Inf;3.33]} 0.625 mpg disp -0.816 0.0000116
#> 3 {carb=(-Inf;3.33]} 0.625 hp wt 0.791 0.0000326
#> 4 {carb=(-Inf;3.33]} 0.625 hp disp 0.877 0.000000388
#> 5 {am=0,carb=(-Inf;3.33]} 0.375 mpg wt -0.632 0.0274
#> 6 {am=0,carb=(-Inf;3.33]} 0.375 mpg disp -0.633 0.0270
#> 7 {am=0,carb=(-Inf;3.33]} 0.375 hp wt 0.755 0.00453
#> 8 {am=0,carb=(-Inf;3.33]} 0.375 hp disp 0.813 0.00131
#> 9 {carb=(-Inf;3.33],vs=0} 0.25 mpg wt -0.823 0.0121
#> 10 {carb=(-Inf;3.33],vs=0} 0.25 mpg disp -0.585 0.128
#> method alternative rows condition_length
#> <chr> <chr> <int> <int>
#> 1 Pearson's product-moment correlation two.sided 20 1
#> 2 Pearson's product-moment correlation two.sided 20 1
#> 3 Pearson's product-moment correlation two.sided 20 1
#> 4 Pearson's product-moment correlation two.sided 20 1
#> 5 Pearson's product-moment correlation two.sided 12 2
#> 6 Pearson's product-moment correlation two.sided 12 2
#> 7 Pearson's product-moment correlation two.sided 12 2
#> 8 Pearson's product-moment correlation two.sided 12 2
#> 9 Pearson's product-moment correlation two.sided 8 2
#> 10 Pearson's product-moment correlation two.sided 8 2
#> # ℹ 526 more rows
This example combines crisp predicates (from
crisp_mtcars) with numeric variables from the original
mtcars dataset. The function searches for conditions under
which pairs of numeric variables show significant Pearson correlations.
The disjoint vector is extended to include the new numeric
columns, preventing conflicts in the search algorithm.
The result shows conditions under which specific pairs of variables
exhibit strong correlations, along with correlation coefficients and
p-values.
Contrast Patterns
Contrast patterns identify conditions under which numeric variables
show statistically significant differences. The nuggets
package provides several functions for different types of contrasts.
Baseline Contrasts
Baseline contrasts identify conditions under which a
variable is significantly different from a baseline value (typically
zero) using a one-sample statistical test.
# Prepare combined dataset with predicates and numeric variables
combined_mtcars2 <- cbind(crisp_mtcars,
mtcars[, c("mpg", "hp", "wt")])
# Extend disjoint vector for the new numeric columns
disj_combined2 <- c(var_names(colnames(crisp_mtcars)),
c("mpg", "hp", "wt"))
# Search for baseline contrasts
baseline_result <- dig_baseline_contrasts(combined_mtcars2,
condition = colnames(crisp_mtcars),
vars = c("mpg", "hp", "wt"),
disjoint = disj_combined2,
min_length = 1,
max_length = 2,
min_support = 0.2,
method = "t")
head(baseline_result)
#> # A tibble: 6 × 15
#> condition support var estimate statistic df p_value n
#> <chr> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 {carb=(-Inf;3.33]} 0.625 mpg 22.5 17.1 19 5.45e-13 20
#> 2 {carb=(-Inf;3.33]} 0.625 hp 116. 11.5 19 5.16e-10 20
#> 3 {carb=(-Inf;3.33]} 0.625 wt 2.88 15.9 19 1.97e-12 20
#> 4 {am=0,carb=(-Inf;3.33]} 0.375 mpg 18.8 20.9 11 3.33e-10 12
#> 5 {am=0,carb=(-Inf;3.33]} 0.375 hp 138. 11.4 11 2.01e- 7 12
#> 6 {am=0,carb=(-Inf;3.33]} 0.375 wt 3.44 28.6 11 1.13e-11 12
#> conf_lo conf_hi stderr alternative method comment condition_length
#> <dbl> <dbl> <dbl> <chr> <chr> <chr> <int>
#> 1 19.8 25.3 1.32 two.sided One Sample t-test "" 1
#> 2 94.7 137. 10.0 two.sided One Sample t-test "" 1
#> 3 2.50 3.26 0.181 two.sided One Sample t-test "" 1
#> 4 16.8 20.8 0.900 two.sided One Sample t-test "" 2
#> 5 112. 165. 12.2 two.sided One Sample t-test "" 2
#> 6 3.18 3.71 0.120 two.sided One Sample t-test "" 2
This example tests whether the mean of numeric variables
(mpg, hp, wt) significantly
differs from zero under various conditions. The
method = "t" parameter specifies a t-test. The results show
which combinations of conditions lead to statistically significant
deviations from the baseline.
Complement Contrasts
Complement contrasts identify conditions under which a
variable differs significantly between elements that satisfy the
condition and those that don’t.
complement_result <- dig_complement_contrasts(combined_mtcars2,
condition = colnames(crisp_mtcars),
vars = c("mpg", "hp", "wt"),
disjoint = disj_combined2,
min_length = 1,
max_length = 2,
min_support = 0.15,
method = "t")
head(complement_result)
#> # A tibble: 6 × 17
#> condition support var estimate_x estimate_y statistic
#> <chr> <dbl> <chr> <dbl> <dbl> <dbl>
#> 1 {carb=(-Inf;3.33]} 0.625 mpg 22.5 16.0 3.80
#> 2 {carb=(-Inf;3.33]} 0.625 hp 116. 198. -3.60
#> 3 {carb=(-Inf;3.33]} 0.625 wt 2.88 3.78 -2.61
#> 4 {carb=(-Inf;3.33],hp=(-Inf;146]} 0.406 mpg 25.6 16.3 6.04
#> 5 {carb=(-Inf;3.33],hp=(-Inf;146]} 0.406 hp 86.5 188. -6.95
#> 6 {carb=(-Inf;3.33],hp=(-Inf;146]} 0.406 wt 2.45 3.74 -5.02
#> df p_value n_x n_y conf_lo conf_hi stderr alternative
#> <dbl> <dbl> <int> <int> <dbl> <dbl> <dbl> <chr>
#> 1 29.9 0.000662 20 12 2.99 9.94 1.70 two.sided
#> 2 16.3 0.00233 20 12 -131. -34.1 22.9 two.sided
#> 3 19.5 0.0171 20 12 -1.61 -0.178 0.343 two.sided
#> 4 18.5 0.00000929 13 19 6.06 12.5 1.54 two.sided
#> 5 24.3 0.000000318 13 19 -132. -71.3 14.6 two.sided
#> 6 28.9 0.0000244 13 19 -1.82 -0.768 0.258 two.sided
#> method comment condition_length
#> <chr> <chr> <int>
#> 1 Welch Two Sample t-test "" 1
#> 2 Welch Two Sample t-test "" 1
#> 3 Welch Two Sample t-test "" 1
#> 4 Welch Two Sample t-test "" 2
#> 5 Welch Two Sample t-test "" 2
#> 6 Welch Two Sample t-test "" 2
This example uses a two-sample t-test to compare the mean values of
numeric variables between rows that satisfy a condition and rows that
don’t. The results identify conditions where subgroups have
significantly different characteristics compared to the rest of the
data.
Paired Baseline Contrasts
Paired baseline contrasts identify conditions under which
there is a significant difference between two paired numeric
variables.
paired_result <- dig_paired_baseline_contrasts(combined_mtcars2,
condition = colnames(crisp_mtcars),
xvars = c("mpg", "hp"),
yvars = c("wt", "wt"),
disjoint = disj_combined2,
min_length = 1,
max_length = 2,
min_support = 0.2,
method = "t")
head(paired_result)
#> # A tibble: 6 × 16
#> condition support xvar yvar estimate statistic df p_value
#> <chr> <dbl> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 {carb=(-Inf;3.33]} 0.625 mpg wt 19.6 13.3 19 4.73e-11
#> 2 {carb=(-Inf;3.33]} 0.625 hp wt 113. 11.4 19 6.19e-10
#> 3 {am=0,carb=(-Inf;3.33]} 0.375 mpg wt 15.4 15.7 11 7.18e- 9
#> 4 {am=0,carb=(-Inf;3.33]} 0.375 hp wt 135. 11.2 11 2.41e- 7
#> 5 {carb=(-Inf;3.33],vs=0} 0.25 mpg wt 14.4 9.96 7 2.20e- 5
#> 6 {carb=(-Inf;3.33],vs=0} 0.25 hp wt 157. 14.7 7 1.63e- 6
#> n conf_lo conf_hi stderr alternative method comment
#> <int> <dbl> <dbl> <dbl> <chr> <chr> <chr>
#> 1 20 16.5 22.7 1.48 two.sided Paired t-test ""
#> 2 20 92.1 134. 9.90 two.sided Paired t-test ""
#> 3 12 13.2 17.5 0.980 two.sided Paired t-test ""
#> 4 12 108. 161. 12.1 two.sided Paired t-test ""
#> 5 8 11.0 17.9 1.45 two.sided Paired t-test ""
#> 6 8 131. 182. 10.7 two.sided Paired t-test ""
#> condition_length
#> <int>
#> 1 1
#> 2 1
#> 3 2
#> 4 2
#> 5 2
#> 6 2
This example performs paired t-tests to compare two variables within
the same rows under specific conditions. Here, it tests whether
mpg differs from wt (and hp from
wt) in various subgroups. This is useful for detecting
context-dependent relationships between paired measurements.
