Comparison of Decision Boundaries of Classification Learners

Visuzalizes the decision boundaries of multiple classification learners on some artificial data sets.

The visualization of decision boundaries helps to understand what the pros and cons of individual classification learners are. This posts demonstrates how to create such plots.

Artificial Data Sets

The three artificial data sets are generated by task generators (implemented in mlr3):

library("mlr3")

N = 200
tasks = list(
  tgen("xor")$generate(N),
  tgen("moons")$generate(N),
  tgen("circle")$generate(N)
)

XOR

Points are distributed on a 2-dimensional cube with corners \((\pm 1, \pm 1)\). Class is "red" if \(x\) and \(y\) have the same sign, and "black" otherwise.

plot(tgen("xor"))

Circle

Two circles with same center but different radii. Points in the smaller circle are "black", points only in the larger circle are "red".

plot(tgen("circle"))

Moons

Two interleaving half circles (“moons”).

plot(tgen("moons"))

Learners

We consider the following learners:

library("mlr3learners")

learners = list(
  # k-nearest neighbours classifier
  lrn("classif.kknn", id = "kkn", predict_type = "prob", k = 3),

  # linear svm
  lrn("classif.svm", id = "lin. svm", predict_type = "prob", kernel = "linear"),

  # radial-basis function svm
  lrn("classif.svm", id = "rbf svm", predict_type = "prob", kernel = "radial",
    gamma = 2, cost = 1, type = "C-classification"),

  # naive bayes
  lrn("classif.naive_bayes", id = "naive bayes", predict_type = "prob"),

  # single decision tree
  lrn("classif.rpart", id = "tree", predict_type = "prob", cp = 0, maxdepth = 5),

  # random forest
  lrn("classif.ranger", id = "random forest", predict_type = "prob")
)

The hyperparameters are chosen in a way that the decision boundaries look “typical” for the respective classifier. Of course, with different hyperparameters, results may look very different.

Fitting the Models

To apply each learner on each task, we first build an exhaustive grid design of experiments with benchmark_grid() and then pass it to benchmark() to do the actual work. A simple holdout resampling is used here:

design = benchmark_grid(
  tasks = tasks,
  learners = learners,
  resamplings = rsmp("holdout")
)

set.seed(123)
bmr = benchmark(design, store_models = TRUE)

A quick look into the performance values:

perf = bmr$aggregate(msr("classif.acc"))[, c("task_id", "learner_id", "classif.acc")]
knitr::kable(perf)

task_id	learner_id	classif.acc
xor_200	kkn	0.9253731
xor_200	lin. svm	0.5223881
xor_200	rbf svm	0.9701493
xor_200	naive bayes	0.5820896
xor_200	tree	0.8955224
xor_200	random forest	0.9552239
moons_200	kkn	0.9850746
moons_200	lin. svm	0.9104478
moons_200	rbf svm	0.9701493
moons_200	naive bayes	0.8955224
moons_200	tree	0.9104478
moons_200	random forest	0.9253731
circle_200	kkn	0.9104478
circle_200	lin. svm	0.4776119
circle_200	rbf svm	0.8805970
circle_200	naive bayes	0.8358209
circle_200	tree	0.8358209
circle_200	random forest	0.9402985

Plotting

To generate the plots, we iterate over the individual ResampleResult objects stored in the BenchmarkResult, and in each iteration we store the plot of the learner prediction generated by the mlr3viz package.

library("mlr3viz")

n = bmr$n_resample_results
plots = vector("list", n)
for (i in seq_len(n)) {
  rr = bmr$resample_result(i)
  plots[[i]] = autoplot(rr, type = "prediction")
}

We now have a list of plots. Each one can be printed individually:

print(plots[[1]])

Note that only observations from the test data is plotted as points.

To get a nice annotated overview, we arranged all plots together in a single PDF file. The number in the upper right is the respective accuracy on the test set.

As you can see, the decision boundaries look very different. Some are linear, others are parallel to the axis, and yet others are highly non-linear. The boundaries are partly very smooth with a slow transition of probabilities, others are very abrupt. All these properties are important during model selection, and should be considered for your problem at hand.