# Comparison of Decision Boundaries of Classification Learners

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

Author

Michel Lang

Published

August 14, 2020

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.

library("mlr3")

We initialize the random number generator with a fixed seed for reproducibility, and decrease the verbosity of the logger to keep the output clearly represented.

set.seed(7832)
lgr::get_logger("mlr3")$set_threshold("warn") ## Artificial Data Sets The three artificial data sets are generated by task generators (implemented in mlr3): N <- 200 tasks <- list( tgen("xor")$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"),

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(
learners = learners,
resamplings = rsmp("holdout")
)

bmr <- benchmark(design, store_models = TRUE)

A quick look into the performance values:

perf <- bmr$aggregate(msr("classif.acc"))[, c("task_id", "learner_id", "classif.acc")] perf  task_id learner_id classif.acc 1: xor_200 kkn 0.9402985 2: xor_200 lin. svm 0.5223881 3: xor_200 rbf svm 0.9701493 4: xor_200 naive bayes 0.4328358 5: xor_200 tree 0.9402985 6: xor_200 random forest 1.0000000 7: moons_200 kkn 1.0000000 8: moons_200 lin. svm 0.8805970 9: moons_200 rbf svm 1.0000000 10: moons_200 naive bayes 0.8955224 11: moons_200 tree 0.8955224 12: moons_200 random forest 0.9552239 13: circle_200 kkn 0.8805970 14: circle_200 lin. svm 0.4925373 15: circle_200 rbf svm 0.8955224 16: circle_200 naive bayes 0.7014925 17: circle_200 tree 0.7462687 18: circle_200 random forest 0.7761194 ## Plotting To generate the plots, we iterate over the individual ResampleResult objects stored in the BenchmarkResult, and in each iteration we store the plot of the learner prediction generated by the mlr3viz package. library("mlr3viz") n <- bmr$n_resample_results
plots <- vector("list", n)
for (i in seq_len(n)) {
rr <- bmr$resample_result(i) plots[[i]] <- autoplot(rr, type = "prediction") } We now have a list of plots. Each one can be printed individually: print(plots[[1]]) Note that only observations from the test data is plotted as points. To get a nice annotated overview, we arranged all plots together in a single pdf file. The number in the upper right is the respective accuracy on the test set. pdf(file = "plot_learner_prediction.pdf", width = 20, height = 6) ntasks <- length(tasks) nlearners <- length(learners) m <- msr("classif.acc") # for each plot for (i in seq_along(plots)) { plots[[i]] <- plots[[i]] + # remove legend ggplot2::theme(legend.position = "none") + # remove labs ggplot2::xlab("") + ggplot2::ylab("") + # add accuracy score as annotation ggplot2::annotate("text", label = sprintf("%.2f", bmr$resample_result(i)$aggregate(m)), x = Inf, y = Inf, vjust = 2, hjust = 1.5 ) } # for each plot of the first column for (i in seq_len(ntasks)) { ii <- (i - 1) * nlearners + 1L plots[[ii]] <- plots[[ii]] + ggplot2::ylab(sub("_[0-9]+$", "", tasks[[i]]$id)) } # for each plot of the first row for (i in seq_len(nlearners)) { plots[[i]] <- plots[[i]] + ggplot2::ggtitle(learners[[i]]$id)
}

gridExtra::grid.arrange(grobs = plots, nrow = length(tasks))
dev.off()

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.