Cubist is for numeric outcomes while RuleFit can work with numeric and categorical outcomes. For this document, we’ll focus on numeric outcomes (without loss of generality).
Both start by making rules from trees
For each model, an an initial tree-based model is created. Let’s look at a really simple example tree:
In this case, two predictors are used in splits: A and B. On this graph, the split to the left is the “<” condition. There are three terminal nodes shown as diamonds. This tree-based structure can be converted to a set of distinct rules. Each rule is a collection of if/then statements that define the path to the terminal nodes. In this example:
rule_1 <- (A < a) & (B < b)
rule_2 <- (A < a) & (B >= b)
rule_3 <- (A >= a)
As is, these paths through the tree are mutually exclusive. For deep trees, these rules can become overly complex and may contain redundancies. Cubist takes the approach of simplifying rules whenever possible. For example,
rule_x <- (A < 10) & (A < 7) & (B >= 1)
becomes
rule_y <- (A < 7) & (B >= 1)
Cubist also has an approach that can prune conditions inside of the rule that simplifies the structure without degrading performance. If B doesn’t matter much in the rule above, then Cubist could reduce it to A < 7.
Cubist and RuleFit are also ensemble methods. For RuleFit, any tree ensemble model could generate a set of rules. For the xrf package, the xgboost package creates the initial set of rules. The rules create be each tree are pooled into a broader rule set. Cubist takes a different approach and uses model committees (see APM Chapter 14). This is similar to boosting but creates a pseudo-outcome for each tree in the ensemble. This outcome adjusts the trees based on the size of the residual from the previous tree.
At the end of the rule generating process, the rules are unlikely to mutually exclusive. Any data point is likely to be fall into multiple rules.
Now let’s look at how each model uses the rules.
Cubists makes a model for each rule
Before delving into the rules, let’s start with the model tree initially created by Cubist. An initial tree is created and a regression model is created for each split in the tree. Each linear regression is created on the subset of the data covered by the current rule and uses only the predictors in the current rule. In the tree shown above, the first split produces two models with A as a predictor. The data for each are filtered either with A < a or A >= a. The entire set of models are:
split_1_low <- filter(data, A < a)
model_1_low <- lm(y ~ A, data = split_1_low)
split_1_high <- filter(data, A >= a)
model_1_high <- lm(y ~ A, data = split_1_high)
split_2_low <- filter(data, A < a & B < b)
model_2_low <- lm(y ~ A + B, data = split_2_low)
split_2_high <- filter(data, A < a & B >= b)
model_2_high <- lm(y ~ A + B, data = split_2_high)
Cubist does some feature selection on these models so they may not contain all of the possible predictors. Also, since Cubist prunes the rules, there may be not appear to be a connection between the variables used in a rule and its corresponding model.
The models associated with the rules are actually average of many models further up the tree. Since these are linear models, the models used by Cubist for each rule have coefficients that are averages:
| 1 |
A < a & B < b |
model_1_low and model_2_low |
| 2 |
A < a & B >= b |
model_1_low and model_2_high |
| 2 |
A >= a |
model_1_high |
The equations for averaging were first described in Quinlan (1992) but the updated equation for Cubist can be found in Chapter 14 of APM.
There is a model for each committee and rule within committee. When predicting, a new observation is compared to the conditions in the rules to determine which rules are active for this data point. The active linear models predict the new sample and these predictions are averaged to produce the final prediction value.
Perhaps unrelated to this document, which focuses on how rules are used, Cubist also does a nearest-neighbor correction to the predicted values.
Let’s look at an example model on the Palmer penguin data:
library(rules)
#> Loading required package: parsnip
data(penguins, package = "modeldata")
cubist_fit <-
cubist_rules(committees = 2) %>%
set_engine("Cubist") %>%
fit(body_mass_g ~ ., data = penguins)
cubist_fit
#> parsnip model object
#>
#> Fit time: 25ms
#>
#> Call:
#> cubist.default(x = x, y = y, committees = 2)
#>
#> Number of samples: 333
#> Number of predictors: 6
#>
#> Number of committees: 2
#> Number of rules per committee: 5, 1
The summary() function shows the details of the rules.
summary(cubist_fit$fit)
#>
#> Call:
#> cubist.default(x = x, y = y, committees = 2)
#>
#>
#> Cubist [Release 2.07 GPL Edition] Sat Jan 16 10:37:04 2021
#> ---------------------------------
#>
#> Target attribute `outcome'
#>
#> Read 333 cases (7 attributes) from undefined.data
#>
#> Model 1:
#>
#> Rule 1/1: [107 cases, mean 3419.2, range 2700 to 4150, est err 208.3]
#>
#> if
#> flipper_length_mm <= 202
#> sex = female
#> then
#> outcome = -1068 + 108 bill_depth_mm + 10.7 flipper_length_mm
#> + 14 bill_length_mm
#>
#> Rule 1/2: [92 cases, mean 3972.0, range 3250 to 4775, est err 275.6]
#>
#> if
#> flipper_length_mm <= 202
#> sex = male
#> then
#> outcome = 319.1 + 22.3 flipper_length_mm - 21 bill_length_mm
#> + 12 bill_depth_mm
#>
#> Rule 1/3: [58 cases, mean 4679.7, range 3950 to 5200, est err 206.6]
#>
#> if
#> flipper_length_mm > 202
#> sex = female
#> then
#> outcome = -3923.3 + 30.4 flipper_length_mm + 136 bill_depth_mm
#> + 5 bill_length_mm
#>
#> Rule 1/4: [23 cases, mean 4698.9, range 3950 to 6050, est err 275.8]
#>
#> if
#> bill_depth_mm > 16.4
#> flipper_length_mm > 202
#> then
#> outcome = -7845.8 + 58.6 flipper_length_mm
#>
#> Rule 1/5: [53 cases, mean 5475.0, range 4750 to 6300, est err 239.3]
#>
#> if
#> bill_depth_mm <= 16.4
#> sex = male
#> then
#> outcome = -138.7 + 46 bill_length_mm + 89 bill_depth_mm
#> + 8.9 flipper_length_mm
#>
#> Model 2:
#>
#> Rule 2/1: [333 cases, mean 4207.1, range 2700 to 6300, est err 315.4]
#>
#> outcome = -5815.1 + 49.7 flipper_length_mm
#>
#>
#> Evaluation on training data (333 cases):
#>
#> Average |error| 278.7
#> Relative |error| 0.41
#> Correlation coefficient 0.90
#>
#>
#> Attribute usage:
#> Conds Model
#>
#> 47% sex
#> 42% 100% flipper_length_mm
#> 11% 47% bill_depth_mm
#> 47% bill_length_mm
#>
#>
#> Time: 0.0 secs
Note that the only rule in the second committee has no conditions; all data points being predicted are affected by that rule. Also, it is possible for the linear model within a rule to only contain an intercept.
The tidy() function can extract the rule and model data:
cb_res <- tidy(cubist_fit)
cb_res
#> # A tibble: 6 x 5
#> committee rule_num rule estimate statistic
#> <int> <int> <chr> <list> <list>
#> 1 1 1 ( sex == 'female' ) & ( flipper… <tibble [4 ×… <tibble [1 …
#> 2 1 2 ( sex == 'male' ) & ( flipper_l… <tibble [4 ×… <tibble [1 …
#> 3 1 3 ( flipper_length_mm > 202 ) & (… <tibble [4 ×… <tibble [1 …
#> 4 1 4 ( flipper_length_mm > 202 ) & (… <tibble [2 ×… <tibble [1 …
#> 5 1 5 ( bill_depth_mm <= 16.4 ) & ( s… <tibble [4 ×… <tibble [1 …
#> 6 2 1 <no conditions> <tibble [2 ×… <tibble [1 …
The estimate and statistic columns contain tibbles with parameter estimates and rule statistics, respectively. They can be easily expanded using unnest():
library(tidyr)
cb_res %>%
dplyr::select(committee, rule_num, statistic) %>%
unnest(cols = c(statistic))
#> # A tibble: 6 x 8
#> committee rule_num num_conditions coverage mean min max error
#> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 1 2 107 3419. 2700 4150 208.
#> 2 1 2 2 92 3972 3250 4775 276.
#> 3 1 3 2 58 4680. 3950 5200 207.
#> 4 1 4 2 23 4699. 3950 6050 276.
#> 5 1 5 2 53 5475 4750 6300 239.
#> 6 2 1 0 333 4207. 2700 6300 315.
For the first committee, rule four contains two conditions and covers 23 data points from the original data set. We can find these data points by converting the character string for the rule to an R expression, then evaluate the expression on the data set:
library(dplyr)
library(purrr)
library(rlang)
rule_4_filter <-
cb_res %>%
dplyr::filter(rule_num == 4) %>%
pluck("rule") %>% # <- character string
parse_expr() %>% # <- R expression
eval_tidy(penguins) # <- logical vector
penguins %>%
dplyr::slice(which(rule_4_filter))
#> # A tibble: 23 x 7
#> species island bill_length_mm bill_depth_mm flipper_length_… body_mass_g
#> <fct> <fct> <dbl> <dbl> <int> <int>
#> 1 Adelie Dream 41.1 18.1 205 4300
#> 2 Adelie Dream 40.8 18.9 208 4300
#> 3 Adelie Biscoe 41 20 203 4725
#> 4 Adelie Torge… 44.1 18 210 4000
#> 5 Gentoo Biscoe 59.6 17 230 6050
#> 6 Gentoo Biscoe 44.4 17.3 219 5250
#> 7 Gentoo Biscoe 49.8 16.8 230 5700
#> 8 Gentoo Biscoe 50.8 17.3 228 5600
#> 9 Gentoo Biscoe 52.1 17 230 5550
#> 10 Gentoo Biscoe 52.2 17.1 228 5400
#> # … with 13 more rows, and 1 more variable: sex <fct>
RuleFit makes one model with rule predictors
RuleFit uses rules in a more straightforward way. It creates an initial tree (or ensemble of trees) from the data. Rules are extracted from this initial model and converted to a set of binary features. These features are then added to a regularized regression model (along with the original columns). For example:
data_with_rules <-
data %>%
mutate(
rule_1 = ifelse(A < a & B < b, 0, 1),
rule_2 = ifelse(A < a & B >= b, 0, 1),
rule_2 = ifelse(A >= a, 0, 1)
)
rule_fit_model <-
glmnet(x = data_with_rules %>% select(A, B, starts_with("rule_") %>% as.matrix(),
y = data_with_rules$y,
alpha = 1)
It is common to use a lasso model to regularize the model and eliminate non-informative features.
While the details can depend on the implementation, the rules do not appear to be pruned or simplified.
The tuning parameters for this model inherits the parameter of the tree-based model as well as the amount of regularization used in the glmnet model. The complexity of the rules is determined by the allowed depth of the tree. For example, using a depth of four means that each rule may have up to four terms that define it. The number of rules would be primarily determined by the number of boosting iterations.
RuleFit, as implemented by the xrf package, required all of the data to be complete (i.e, non-missing). In the original data set, the body mass is encoded as integer and xrf requires a double, so:
penguins <-
penguins %>%
mutate(body_mass_g = body_mass_g + 0.0) %>%
na.omit()
Fitting the model:
rule_fit_spec <-
rule_fit(trees = 10, tree_depth = 5, penalty = 0.01) %>%
set_engine("xrf") %>%
set_mode("regression")
rule_fit_fit <-
rule_fit_spec %>%
fit(body_mass_g ~ ., data = penguins)
rule_fit_fit
#> parsnip model object
#>
#> Fit time: 1.9s
#> An eXtreme RuleFit model of 112 rules.
#>
#> Original Formula:
#>
#> body_mass_g ~ species + island + bill_length_mm + bill_depth_mm + flipper_length_mm + [truncated]
To get the rules and their associated coefficients, the tidy() method can be used again:
rf_res <- tidy(rule_fit_fit)
rf_res
#> # A tibble: 110 x 3
#> rule_id rule estimate
#> <chr> <chr> <dbl>
#> 1 (Intercept) ( TRUE ) 5753.
#> 2 bill_depth_mm ( bill_depth_mm ) -15.9
#> 3 bill_length_mm ( bill_length_mm ) 1.10
#> 4 flipper_length_… ( flipper_length_mm ) -7.67
#> 5 islandDream ( island == 'Dream' ) -29.2
#> 6 islandTorgersen ( island == 'Torgersen' ) -16.8
#> 7 r0_2 ( species == 'Gentoo' ) 472.
#> 8 r0_3 ( sex != 'male' ) & ( species != 'Gentoo' ) -78.8
#> 9 r1_2 ( flipper_length_mm >= 211.5 ) 219.
#> 10 r1_3 ( flipper_length_mm < 194.5 ) & ( flipper_length_… -142.
#> # … with 100 more rows
Note that the units of each predictor have been scaled to be between zero and one.
Looking at the rules, there are examples of some rules, such as:
( bill_depth_mm >= 14.1499996 ) &
( flipper_length_mm < 227 ) &
( flipper_length_mm < 228.5 ) &
( flipper_length_mm >= 197.5 ) &
( flipper_length_mm >= 224.5 )"
which can be simplified to have fewer conditions:
( bill_depth_mm >= 14.1499996 ) &
( flipper_length_mm < 227 ) &
( flipper_length_mm >= 197.5 ) &
Also, the tidy() function can extract the same information but use the original predictors as the unit:
rf_variable_res <- tidy(rule_fit_fit, unit = "columns")
rf_variable_res
#> # A tibble: 452 x 3
#> rule_id term estimate
#> <chr> <chr> <dbl>
#> 1 r0_3 species -78.8
#> 2 r0_2 species 472.
#> 3 r0_3 sex -78.8
#> 4 r1_3 flipper_length_mm -142.
#> 5 r1_2 flipper_length_mm 219.
#> 6 r1_3 flipper_length_mm -142.
#> 7 r2_7 flipper_length_mm 9.04
#> 8 r2_7 sex 9.04
#> 9 r2_7 species 9.04
#> 10 r3_3 flipper_length_mm -101.
#> # … with 442 more rows
A single rule might be represented with multiple rows of this version of the data. These results can also be used to compute a rough estimate or variable importance using the absolute value of the coefficients:
num_rules <- sum(grepl("^r[0-9]*_", unique(rf_res$rule_id))) + 1
rf_variable_res %>%
dplyr::filter(term != "(Intercept)") %>%
group_by(term) %>%
summarize(effect = sum(abs(estimate)), .groups = "drop") %>%
ungroup() %>%
# normalize by number of possible occurrences
mutate(effect = effect / num_rules ) %>%
arrange(desc(effect))
#> # A tibble: 6 x 2
#> term effect
#> <chr> <dbl>
#> 1 flipper_length_mm 219.
#> 2 bill_depth_mm 175.
#> 3 bill_length_mm 175.
#> 4 species 69.5
#> 5 sex 52.0
#> 6 island 9.84