perClass Documentation
version 5.1 (11-May-2017)

kb22: Note on decision tree performance and speed

Keywords: decision tree, random forest, classification speed, PRTools, C4.5

22.1. Introduction ↩

perClass 3.1 has introduced several new classifiers such as decision tree and random forest. In this article, we focus on the sdtree decision tree and compare its performance with two state-of-the-art packages, namely C4.5 and PRTools.

Decision tree is a fast discriminant based on recursive thresholding. At each step (node), the threshold is found that yield more homogeneuous solution i.e. separates one or mode classes. For the sake of practical use, the tree needs to be pruned in order to generalize well to unseen data. Therefore, we compere pruned trees for all three packages.

22.2. Experiment 1: Spherical data set ↩

In the first experiment, we used the "spherical" data set generated using the PRTools gendatc command. The data set consists of two overlapping Gaussians with different variances in 2D feature space. The Bayes (minimal) error of this problem is about 0.16.

>> a=gendatc(1000)
Spherical Set, 1000 by 2 dataset with 2 classes: [454  546]
>> sdscatter(a)

Interactive scatter plot sdscatter of spherical data set showing class distributions

22.2.1. Classification error ↩

In order to compare different decision tree implementations, we build a learning curve. For a growing number of training data samples, we train the decision tree classifier using the three packages and estimate its error on an independent test set with 10000 samples.

The following image shows the test set error:

Test error of decision tree built with perClass, C4.5 and PRTools.

In case of perClass, we trained sdtree using default settings i.e. pruning using internal 80%/20% data split to estimate error.

The PRTools treec classifier was trained using the pruning as follows: w=treec(tr,'maxcrit',-1). Because treec crashed for 2000 or more samples using the default recursion limit of Matlab, we inscreased it using set(0,'RecursionLimit',2000). However, also the increased limit was not sufficient for data sets with 8000 and 10000 trainig samples.

The C4.5 was executed using the command line on data sets exported from Matlab. We use the reported test set error based on default pruning.

$ time ./c4.5 -u -f data

We may observe that both perClass and C4.5 exhibit similar behaviour getting close to the Bayes error. PRTools treec seems to have a difficulty building a generalizing tree.

The cross-validation of perClass sdtree shows the standard deviation of mean error lower than 1% (not shown in the plot for the sake of clarity).

22.2.2. Training time ↩

Apart of classification error, the training time is a very important performance indicator. We have measured the training time using tic/toc Matlab timer for perClass and PRTools implementations and Unix time command for C4.5. The results are visualized in the following figure:

Training time for different decision tree implementations.

On the left side, we can see that the Matlab decision tree implementation in PRTools exhibits quadratic time complexity with a growing number of samples. In the right figure, we zoomed-in the time axis. It appears that both C-based implementations show linear training complexity. Training on 10000 samples takes less than 1.5 seconds with perClass.

22.3. Experiment 2: Decision tree when increasing feature size ↩

In the second experiment, we investigate how decision trees behave when increasing the data set dimensionality. We use the "difficult" data set generated using the PRTools gendatd command.

>> a=gendatd(1000)
Difficult Dataset, 1000 by 2 dataset with 2 classes: [520  480]
>> sdscatter(a)

'Difficult' data set used in experiment with perClass and C4.5 decision trees.

This data set contains two strongly correlated Gaussians. The gendatd command enables us to increase the number of features by adding non-informative dimensions. With this data set, we may investiagate how the decision tree implementations cope with stong correlation and large amount of noise.

We use 10000 samples for training a separate 10000 samples as a test set. The perClass sdtree and C4.5 are trained with the default settings. The test set errors are visualized in the following figure:

Error of decision tree as a function of dimensionality.

It is interesting to note that both decision tree implementations exhibit different trends when dealing with large number of uninformative features.

The training time of both implementations is linear when increasing the feature size. Training a decision tree on 10000 samples with 100 features takes 30 seconds for perClass and 5 seconds for C4.5.

Decision tree training times a function of dimensionality for perClass and C4.5

22.4. Note on execution speed ↩

Classifier execution speed is of crucial importance for real-time industrial deployment. We have used circular data set with 5000 samples for training and measured the execution speed on a separate set with 100000 samples:

>> a=gendatc(5000)
Spherical Set, 5000 by 2 dataset with 2 classes: [2511  2489]
>> w=treec(a)
Decision Tree, 2 to 2 trained  mapping   --> tree_map
>> ts=gendatc(100000)
Spherical Set, 100000 by 2 dataset with 2 classes: [49776  50224]

perClass sdtree classifier:

>> p=sdtree(a)
Decision tree pipeline  2x2   (sdp_tree)
>> pd=sddecide(p);
>> tic; dec=ts*pd; toc
Elapsed time is 0.044217 seconds.

PRTools treec classifier:

>> prwarning off
>> prwaitbar off
>> tic; out=ts*w*labeld; toc
Elapsed time is 100.443812 seconds.

We did not find the way to time only the C4.5 execution on new data.

22.5. Conclusions ↩

We have seen that the decision tree implementation, introduced in perClass 3.1, exhibits similar performance to the state-of-the-art C4.5 package and offers similarly high training speed.

The sdtree offers a number of additional options, described in documentation and is a part of an integrated toolbox which allows, for example, to leverage powerful feature selection (sdfeatsel), optimize the resulting tree classifier with ROC analysis (sdroc) or use a tree as a part of a multiple classifier system (sdrandforest, sdcascade).

In addition, sdtree is built for real-time industrial deployment using the perClass runtime.

22.5.1. Additional remarks ↩

Experiments were performed using the 64-bit Matlab 2010b (7.11) on a laptop with 2.8 GHz Intel Core 2 Duo and 8GB of memory running Mac OS X 10.5.

The perClass 3.1.2, PRTools 4.2.1 and C4.5 R8 were used.

PRTools was used with prwarning off option to limit textual output.

The C4.5 was compiled using gcc 4.0 with -O2 option.