# Descriptive and Graphical Analysis of the Variable and Cutpoint Selection inside Random Forests

#### 2022-03-18

Random forests are a widely used ensemble learning method for classification or regression tasks. However, they are typically used as a black box prediction method that offers only little insight into their inner workings.

In this vignette, we illustrate how the stablelearner package can be used to gain insight into this black box by visualizing and summarizing the variable and cutpoint selection of the trees within a random forest.

Recall that, in simple terms, a random forest is a tree ensemble, and the forest is grown by resampling the training data and refitting trees on the resampled data. Contrary to bagging, random forests have the restriction that the number of feature variables randomly sampled as candidates at each node of a tree (in implementations, this is typically called the mtry argument) is smaller than the total number of feature variables available. Random forests were introduced by Breiman (2001).

The stablelearner package was originally designed to provide functionality for assessing the stability of tree learners and other supervised statistical learners, both visually (Philipp, Zeileis, and Strobl 2016) and by means of computing similarity measures (Philipp et al. 2018), on the basis of repeatedly resampling the training data and refitting the learner.

However, in this vignette we are interested in visualizing the variable and cutpoint selection of the trees within a random forest. Therefore, contrary to the original design of the stablelearner package, where the aim was to assess the stability of a single original tree, we are not interested in highlighting any single tree, but want all trees to be treated as equal. As a result, some functions will later require to set the argument original = FALSE. Moreover, this vignette does not cover similarity measures for random forests, which are still work in progress.

In all sections of this vignette, we are going to work with credit scoring data where applicants are rated as "good" or "bad", which will be introduced in Section 1.

In Section 2 we will cover the stablelearner package and how to fit a random forest using the stabletree() function (Section 2.1). In Section 2.2 we show how to summarize and visualize the variable and cutpoint selection of the trees in a random forest.

In the final Section 3, we will demonstrate how the same summary and visualizations can be produced when working with random forests that were already fitted via the cforest() function of the partykit package, the cforest() function of the party package, the randomForest() function of the randomForest package, or the ranger() function of the ranger package.

Note that in the following, functions will be specified with the double colon notation, indicating the package they belong to, e.g., partykit::cforest() denoting the cforest() function of the partykit package, while party::cforest() denotes the cforest() function of the party package.

## 1 Data

In all sections we are going to work with the german dataset, which is included in the rchallenge package (note that this is a transformed version of the German Credit data set with factors instead of dummy variables, and corrected as proposed by Grömping (2019).

data("german", package = "rchallenge")

The dataset consists of 1000 observations on 21 variables. For a full description of all variables, see ?rchallenge::german. The random forest we are going to fit in this vignette predicts whether a person was classified as "good" or "bad" with respect to the credit_risk variable using all other available variables as feature variables. To allow for a lower runtime we only use a subsample of the data (500 persons):

set.seed(2409)
dat <- droplevels(german[sample(seq_len(NROW(german)), size = 500), ])
str(dat)
## 'data.frame':    500 obs. of  21 variables:
##  $status : Factor w/ 4 levels "no checking account",..: 2 4 4.. ##$ duration               : int  30 21 18 24 12 15 7 9 27 12 ...
##  $credit_history : Factor w/ 5 levels "delay in paying off in the pa".. ##$ purpose                : Factor w/ 10 levels "others","car (new)",..: 1 1 4..
##  $amount : int 4249 5003 1505 5743 3331 3594 846 2753 2520 1.. ##$ savings                : Factor w/ 5 levels "unknown/no savings account",....
##  $employment_duration : Factor w/ 5 levels "unemployed","< 1 yr",..: 1 3 3.. ##$ installment_rate       : Ord.factor w/ 4 levels ">= 35"<"25 <= ... < 35"<....
##  $personal_status_sex : Factor w/ 4 levels "male : divorced/separated",..:.. ##$ other_debtors          : Factor w/ 3 levels "none","co-applicant",..: 1 1 1..
##  $present_residence : Ord.factor w/ 4 levels "< 1 yr"<"1 <= ... < 4 yrs".. ##$ property               : Factor w/ 4 levels "unknown / no property",..: 3 2..
##  $age : int 28 29 32 24 42 46 36 35 23 22 ... ##$ other_installment_plans: Factor w/ 3 levels "bank","stores",..: 3 1 3 3 2 3..
##  $housing : Factor w/ 3 levels "for free","rent",..: 2 2 3 3 2.. ##$ number_credits         : Ord.factor w/ 4 levels "1"<"2-3"<"4-5"<..: 2 2 1 2..
##  $job : Factor w/ 4 levels "unemployed/unskilled - non-re".. ##$ people_liable          : Factor w/ 2 levels "3 or more","0 to 2": 2 2 2 2 2..
##  $telephone : Factor w/ 2 levels "no","yes (under customer name".. ##$ foreign_worker         : Factor w/ 2 levels "yes","no": 2 2 2 2 2 2 2 2 2 2..
##  $credit_risk : Factor w/ 2 levels "bad","good": 1 1 2 2 2 2 2 2 1.. ## 2 stablelearner ### 2.1 Growing a random forest in stablelearner In our first approach, we want to grow a random forest directly in stablelearner. This is possible using conditional inference trees (Hothorn, Hornik, and Zeileis 2006) as base learners relying on the function ctree() of the partykit package. This procedure results in a forest equal to a random forest fitted via partykit::cforest(). To achieve this, we have to make sure that our initial ctree, that will be repeatedly refitted on the resampled data, is specified correctly with respect to the resampling method and the number of feature variables randomly sampled as candidates at each node of a tree (argument mtry). By default, partykit::cforest() uses subsampling with a fraction of 0.632 and sets mtry = ceiling(sqrt(nvar)). In our example, this would be 5, as this dataset includes 20 feature variables. Note that setting mtry equal to the number of all feature variables available would result in bagging. In a real analysis mtry should be tuned by means of, e.g., cross-validation. We now fit our initial tree, mimicking the defaults of partykit::cforest() (see ?partykit::cforest and ?partykit::ctree_control for a description of the arguments teststat, testtype, mincriterion and saveinfo). The formula credit_risk ~ . simply indicates that we use all remaining variables of dat as feature variables to predict the credit_risk of a person. set.seed(2906) ct_partykit <- partykit::ctree(credit_risk ~ ., data = dat, control = partykit::ctree_control(mtry = 5, teststat = "quadratic", testtype = "Univariate", mincriterion = 0, saveinfo = FALSE)) We can now proceed to grow our forest based on this initial tree, using stablelearner::stabletree(). We use subsampling with a fraction of v = 0.632 and grow B = 100 trees. We set savetrees = TRUE, to be able to extract the individual trees later: set.seed(2907) cf_stablelearner <- stablelearner::stabletree(ct_partykit, sampler = stablelearner::subsampling, savetrees = TRUE, B = 100, v = 0.632) Internally, stablelearner::stabletree() does the following: For each of the 100 trees to be generated, the dataset is resampled according to the resampling method specified (in our case subsampling with a fraction of v = 0.632) and the function call of our initial tree (which we labeled ct_partykit) is updated with respect to this resampled data and reevaluated, resulting in a new tree. All the 100 trees together then build the forest. ### 2.2 Gaining insight into the forest The following summary prints the variable selection frequency (freq) as well as the average number of splits in each variable (mean) over all 100 trees. As we do not want to focus on our initial tree (remember that we just grew a forest, where all trees are of equal interest), we set original = FALSE, as already mentioned in the introduction: summary(cf_stablelearner, original = FALSE) ## ## Call: ## partykit::ctree(formula = credit_risk ~ ., data = dat, control = partykit::ctree_control(mtry = 5, ## teststat = "quadratic", testtype = "Univariate", mincriterion = 0, ## saveinfo = FALSE)) ## ## Sampler: ## B = 100 ## Method = Subsampling with 63.2% data ## ## Variable selection overview: ## ## freq mean ## status 0.96 2.38 ## duration 0.88 1.72 ## credit_history 0.88 1.69 ## employment_duration 0.87 1.62 ## amount 0.83 1.41 ## age 0.80 1.34 ## property 0.75 1.19 ## purpose 0.72 1.23 ## present_residence 0.71 1.11 ## savings 0.69 1.01 ## housing 0.69 1.00 ## installment_rate 0.68 1.12 ## number_credits 0.64 0.82 ## job 0.61 0.80 ## personal_status_sex 0.60 0.87 ## telephone 0.55 0.75 ## other_installment_plans 0.53 0.65 ## other_debtors 0.40 0.41 ## people_liable 0.39 0.47 ## foreign_worker 0.19 0.19 Looking at the status variable (status of the existing checking account of a person) for example, this variable was selected in almost all 100 trees (freq = 0.99). Moreover, this variable was often selected more than once for a split because the average number of splits is at around 2.40. Plotting the variable selection frequency is achieved via the following command (note that cex.names allows us to specify the relative font size of the x-axis labels): barplot(cf_stablelearner, original = FALSE, cex.names = 0.6) To get a more detailed view, we can also inspect the variable selection patterns displayed for each tree. The following plot shows us for each variable whether it was selected (colored in darkgrey) in each of the 100 trees within the forest, where the variables are ordered on the x-axis so that top ranking ones come first: image(cf_stablelearner, original = FALSE, cex.names = 0.6) This may allow for interesting observations, e.g., we observe that in those trees where duration was not selected, both credit_history and employment_duration were almost always selected as splitting variables. Finally, the plot() function allows us to inspect the cutpoints and resulting partitions for each variable over all 100 trees. Here we focus on the variables status, present_residence, and duration: plot(cf_stablelearner, original = FALSE, select = c("status", "present_residence", "duration")) Looking at the first variable status (an unordered categorical variable), we are given a so called image plot, visualizing the partition of this variable for each of the 100 trees.. We observe that the most frequent partition is to distinguish between persons with the values no checking account and ... < 0 DM vs. persons with the values 0 <= ... < 200 DM and ... >= 200 DM / salary for at least 1 year, but also other partitions occur. The light gray color is used when a category was no more represented by the observations left for partitioning in the particular node. For ordered categorical variables such as present_residence, a barplot is given showing the frequency of all possible cutpoints sorted on the x-axis in their natural order. Here, the cutpoint between 1 <= ... < 4 years and 4 <= ... < 7 years is selected most frequently. Lastly, for numerical variables a histogram is given, showing the distribution of cutpoints. We observe that most cutpoints for the variable duration occurred between 0 and 30, but there appears to be high variance, indicating a smooth effect of this variable rather than a pattern with a distinct change-point. For a more detailed explanation of the different kinds of plots, Section 3 of Philipp, Zeileis, and Strobl (2016) is very helpful. To conclude, the summary table and different plots helped us to gain some insight into the variable and cutpoint selection of the 100 trees within our forest. Finally, in case we want to extract individual trees, e.g., the first tree, we can do this via: cf_stablelearner$tree[[1]]

It should be noted that from a technical and performance-wise perspective, there is little reason to grow a forest directly in stablelearner, as the cforest() implementations in partykit and especially in party are more efficient. Nevertheless, it should be noted that the approach of growing a forest directly in stablelearner allows us to be more flexible with respect to, e.g., the resampling method, as we could specify any method we want, e.g., bootstrap, subsampling, samplesplitting, jackknife, splithalf or even custom samplers. For a discussion why subsampling should be preferred over bootstrap sampling, see Strobl et al. (2007).

## 3 Working with random forests fitted via other packages

In this final section we cover how to work with random forests that have already been fitted via the cforest() function of the partykit package, the cforest() function of the party package, the randomForest() function of the randomForest package, or the ranger() function of the ranger package.

Essentially, we just fit the random forest and then use stablelearner::as.stabletree() to coerce the forest to a stabletree object, which allows us to get the same summary and plots as presented above.

Fitting a cforest with 100 trees using partykit is straightforward:

set.seed(2908)
cf_partykit <- partykit::cforest(credit_risk ~ ., data = dat,
ntree = 100, mtry = 5)

stablelearner::as.stabletree() then allows us to coerce this cforest and we can produce summaries and plots as above (note that for plotting, we can now omit original = FALSE, because the coerced forest has no particular initial tree):

cf_partykit_st <- stablelearner::as.stabletree(cf_partykit)
summary(cf_partykit_st, original = FALSE)
barplot(cf_partykit_st)
image(cf_partykit_st)
plot(cf_partykit_st, select = c("status", "present_residence", "duration"))

We do not observe substantial differences compared to growing the forest directly in stablelearner (of course, this is the expected behavior, because we tried to mimic the algorithm of partykit::cforest() in the previous section), therefore we will not display the results again.

Looking at the variable importance as reported by partykit::varimp() shows substantial overlap, i.e., the variable status yields a high importance:

partykit::varimp(cf_partykit)
##                  status                duration          credit_history
##             0.247738374             0.034059673             0.077873676
##                 purpose                  amount                 savings
##            -0.004825073             0.031340832             0.040444307
##     employment_duration        installment_rate     personal_status_sex
##             0.016477162            -0.022483572            -0.004295483
##           other_debtors       present_residence                property
##             0.020315372             0.001464200             0.008682788
##                     age other_installment_plans                 housing
##             0.058371493             0.046699228             0.031434510
##          number_credits                     job           people_liable
##             0.011540270            -0.022146762             0.014946051
##               telephone          foreign_worker
##            -0.011154765             0.024520948

The coercing procedure described above is analogous for forests fitted via party::cforest():

set.seed(2909)
cf_party <- party::cforest(credit_risk ~ ., data = dat,
control = party::cforest_unbiased(ntree = 100, mtry = 5))
cf_party_st <- stablelearner::as.stabletree(cf_party)
summary(cf_party_st, original = FALSE)
barplot(cf_party_st)
image(cf_party_st)
plot(cf_party_st, select = c("status", "present_residence", "duration"))

Again, we do not observe substantial differences compared to partykit::cforest(). This is the expected behavior, as partykit::cforest() is a (pure R) reimplementation of party::cforest() (implemented in C).

For forests fitted via randomForest::randomForest, we can do the same as above. However, as these forests are not using conditional inference trees as base learners, we can expect some difference with respect to the results:

set.seed(2910)
rf <- randomForest::randomForest(credit_risk ~ ., data = dat,
ntree = 100, mtry = 5)
rf_st <- stablelearner::as.stabletree(rf)
summary(rf_st, original = FALSE)
##
## Call:
## randomForest(formula = credit_risk ~ ., data = dat, ntree = 100,
##     mtry = 5)
##
## Sampler:
## B = 100
## Method = randomForest::randomForest
##
## Variable selection overview:
##
##                         freq  mean
## status                  1.00  4.79
## duration                1.00  7.82
## credit_history          1.00  4.23
## purpose                 1.00  6.27
## amount                  1.00 10.81
## savings                 1.00  4.32
## employment_duration     1.00  5.64
## installment_rate        1.00  4.66
## property                1.00  4.50
## age                     1.00  9.08
## personal_status_sex     0.99  3.47
## present_residence       0.99  4.40
## job                     0.97  3.17
## other_installment_plans 0.94  2.26
## number_credits          0.94  2.26
## housing                 0.93  2.41
## telephone               0.85  1.90
## other_debtors           0.83  1.57
## people_liable           0.73  1.09
## foreign_worker          0.43  0.48
barplot(rf_st, cex.names = 0.6)

image(rf_st, cex.names = 0.6)

plot(rf_st, select = c("status", "present_residence", "duration"))

We observe that for numerical variables the average number of splits is much higher now, i.e., amount is selected at an average of around 10 times. Their preference for variables offering many cutpoints is a known drawback of Breiman and Cutler’s original Random Forest algorithm, which random forests based on conditional inference trees do not share. Note, however, that Breiman and Cutler did not intend the variable selection frequencies to be used as a measure of the relevance of the predictor variables, but have suggested a permutation-based variable importance measure for this purpose. For more details, see Hothorn, Hornik, and Zeileis (2006), Strobl et al. (2007), and Strobl, Malley, and Tutz (2009).

Finally, for forests fitted via ranger::ranger() (that also implements Breiman and Cutler’s original algorithm), the coercing procedure is again the same:

set.seed(2911)
rf_ranger <- ranger::ranger(credit_risk ~ ., data = dat,
num.trees = 100, mtry = 5)
rf_ranger_st <- stablelearner::as.stabletree(rf_ranger)
summary(rf_ranger_st, original = FALSE)
barplot(rf_ranger_st)
image(rf_ranger_st)
plot(rf_ranger_st, select = c("status", "present_residence", "duration"))

As a final comment on computational performance, note that just as stablelearner::stabletree(), stablelearner::as.stabletree() allows for parallel computation (see the arguments applyfun and cores). This may be helpful when dealing with the coercion of large random forests.

## References

Breiman, L. 2001. “Random Forests.” Machine Learning 45 (1): 5–32. https://doi.org/10.1023/a:1010933404324.

Grömping, U. 2019. “South German Credit Data: Correcting a Widely Used Data Set.” Beuth University of Applied Sciences Berlin, Department II.

Hothorn, T., K. Hornik, and A. Zeileis. 2006. “Unbiased Recursive Partitioning: A Conditional Inference Framework.” Journal of Computational and Graphical Statistics 15 (3): 651–74. https://doi.org/10.1198/106186006x133933.

Philipp, M., T. Rusch, K. Hornik, and C. Strobl. 2018. “Measuring the Stability of Results from Supervised Statistical Learning.” Journal of Computational and Graphical Statistics 27 (4): 685–700. https://doi.org/10.1080/10618600.2018.1473779.

Philipp, M., A. Zeileis, and C. Strobl. 2016. “A Toolkit for Stability Assessment of Tree-Based Learners.” In Proceedings of COMPSTAT 2016 – 22nd International Conference on Computational Statistics, edited by A. Colubi, A. Blanco, and C. Gatu, 315–25. The International Statistical Institute/International Association for Statistical Computing.

Strobl, C., A.-L. Boulesteix, A. Zeileis, and T. Hothorn. 2007. “Bias in Random Forest Variable Importance Measures: Illustrations, Sources and a Solution.” BMC Bioinformatics 8 (25). https://doi.org/10.1186/1471-2105-8-25.

Strobl, C., J. Malley, and G. Tutz. 2009. “An Introduction to Recursive Partitioning: Rationale, Application, and Characteristics of Classification and Regression Trees, Bagging, and Random Forests.” Psychological Methods 14 (4): 323–48. https://doi.org/10.1037/a0016973.