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Rule Extraction from Random Forest: the RF+HC Methods

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Random forest (RF) is a tree-based learning method, which exhibits a high ability to generalize on real data sets. Nevertheless, a possible limitation of RF is that it generates a forest consisting of many trees and rules, thus it is viewed as a black box model. In this paper, the RF+HC methods for rule extraction from RF are proposed. Once the RF is built, a hill climbing algorithm is used to search for a rule set such that it reduces the number of rules dramatically, which significantly improves comprehensibility of the underlying model built by RF. The proposed methods are evaluated on eighteen UCI and four microarray data sets. Our experimental results show that the proposed methods outperform one of the state-of-the-art methods in terms of scalability and comprehensibility while preserving the same level of accuracy.
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Rule Extraction from Random Forest:
The RF+HC Methods
Morteza Mashayekhi(B
)and Robin Gras
School of Computer Science, University of Windsor, Windsor, ON, Canada
{mashaye,rgras}@uwindsor.ca
http://www.uwindsor.ca
Abstract. Random forest (RF) is a tree-based learning method, which
exhibits a high ability to generalize on real data sets. Nevertheless, a
possible limitation of RF is that it generates a forest consisting of many
trees and rules, thus it is viewed as a black box model. In this paper,
the RF+HC methods for rule extraction from RF are proposed. Once
the RF is built, a hill climbing algorithm is used to search for a rule set
such that it reduces the number of rules dramatically, which significantly
improves comprehensibility of the underlying model built by RF. The
proposed methods are evaluated on eighteen UCI and four microarray
data sets. Our experimental results show that the proposed methods
outperform one of the state-of-the-art methods in terms of scalability
and comprehensibility while preserving the same level of accuracy.
Keywords: Rule extraction ·Random forest ·Hill climbing
1 Introduction
Random forest (RF) is an ensemble learning method for both classification and
regression that constructs and integrates multiple decision trees at training step
using bootstrapping. Additionally, it aggregates the outputs of all trees via plu-
rality voting in order to classify a new input. It has few parameters to tune
and it is robust against overfitting. It runs efficiently on large data sets and
can handle thousands of input variables. Moreover, RF has an effective method
for estimating missing data, and has some mechanisms to deal with unbalanced
data sets [4]. In some applications, RF outperforms well-known classifiers such
as support vector machines (SVMs) and neural networks (NNs) [5,18]. Despite
good performance of RF in different domains, its major drawback is that, it
generates a ‘black box’model in the sense that it does not have the ability to
explain and interpret the model in an understandable form [23,27] given that
it generates a vast number of propositional if-then rules. As a result, ensemble
predictors such as RF are very rarely used in domains where making transparent
models is mandatory, such as predicting clinical outcomes [23]. In order to bear
this limitation, the hypothesis generated by RF should be transformed into a
more comprehensible representation.
c
Springer International Publishing Switzerland 2015
D. Barbosa and E. Milios (Eds.): Canadian AI 2015, LNAI 9091, pp. 223–237, 2015.
DOI: 10.1007/978-3-319-18356-5 20
224 M. Mashayekhi and R. Gras
To obtain a comprehensible model which is simpler to interpret, accuracy is
often sacrificed. This fact is normally referred to as the ‘accuracy vs. comprehens-
ibility tradeoff’. The importance of accuracy or comprehensibility is completely
related to the application. One way to obtain a transparent model is to induce rules
directly from the training set or to build a decision tree. However, another option
is to take advantage of the good performance of the existing opaque models such
as SVMs, RF, or NNs and generate rules based on them. This process is called rule
extraction (RE), which is aimed at providing explanations for the predictive mod-
els’ outputs. There are two different rule extraction methods based on an opaque
model: decompositional and pedagogical [11]. Decompositional methods extract
rules at the level of individual units of the prediction model such as neurons in
neural networks, and therefore rely on the model’s architecture. In contrast, in
pedagogical approaches, the predictive model is only used to produce predictions.
In previous years, a high number of rule extraction methods using trained
NNs and SVMs have been published (see [11] for a good survey). Nevertheless,
in the case of the RF model, few research projects have been conducted. In this
paper, the RF+HC methods for the interpretation of the RF model are proposed.
The proposed methods can be treated as a decompositional rule extraction app-
roach given that we employed all the generated rules by RF, which are dependent
on the number of trees and also the tree structures in the RF.
This paper is organized as follows: the background description including
foundation of the RF followed by a discussion of related research projects are
explained in section 2. In section 3, the RF+HC methods are introduced. Exper-
imental results of our methods applied to several data sets and comparisons are
described in section 4. Finally, we present our conclusions along with possible
future directions for our work.
2 Background
2.1 Random Forest
The RF is an ensemble learning method such that successive trees do not depend
on the previous ones and each tree is constructed independently using a boot-
strap sample of the data set. At the end, a majority voting procedure is used for
making predictions. In addition, each node is split using the best feature among
a subset of features (m) randomly chosen at that node. Parameter mis usually
equal to 0.5×sqrt(n), sqrt(n), or 2 ×sqrt(n) where nis the number of features.
Error estimations are performed on a subset of data which are not included in
the bootstrap sample at each bootstrap iteration (This subset is called out-of
bag or OOB). RF can also estimate the importance of a feature by permutation
of the values associated with a feature and comparing of the average OOB error
before and after the permutation over all trees. However, it does not consider
the dependency between features.
RF deserves to be considered as one of the important prediction methods
because it demonstrates a high prediction accuracy and it can be used for clus-
tering and feature selection applications as well [4,7,22]. Moreover, estimating
Rule Extraction from Random Forest: The RF+HC Methods 225
the out-of-bag error often eliminates the need for cross-validation. More impor-
tantly, as it generates a multitude of propositional if-then rules, which is the
most widespread rule type in RE domain, it has a very high potential to provide
clear explanations and interpretations of its underlying model.
2.2 Related Work
One of the projects focusing on this topic was conducted by Zhang et al. [27]to
search for the smallest RF. Although their method is not a rule extraction strat-
egy, it seeks out a sub-forest that can achieve the accuracy of a large RF. They
used three different measures in order to determine the importance of trees in
terms of their predictive power. The experimental results demonstrate that such
a sub-forest with performance as good as a large forest exists. Latinne et al. [13]
attempted to reduce the number of trees in RF using the McNemar test of signif-
icance on the prediction outputs of the trees. Similarly, others tried to select an
optimal sub-set of decision trees in RF [2,16,26]. These methods are not really rule
extraction methods and mostly concentrate on reducing the number of decision
trees in the RF or in a similar ensemble method such as bagging.
There are also some other methods to increase comprehensibility of an ensem-
ble or RF by compacting them into one decision tree. For example, a single
decision tree was used to approximate an ensemble of decision trees [24]. In this
method, class distributions were estimated from the ensemble in order to deter-
mine the tests to be used in the new tree. A similar method was employed to
approximate the RF with just one decision tree [12]. The aim was to generate a
weaker but transparent model using combinations of regular training data and
test data initially labeled by the RF.
Other methods with different approaches were proposed to select an optimal
set of rules generated by RF [9,17]. More specifically, Liu et al. [14,15]used
RF as an ensemble of rules and proposed a joint rule extraction and feature
selection method (CRF) based on 1-norm regularized RF, using sparse encoding
and solving an optimization problem applying linear programming method.
3RF+HCMethods
RE can be expressed as an optimization problem [8] and one solution of this
problem is to apply heuristic search methods. These methods overcome the com-
plexity of finding the best rule set, which is an NP-hard problem.
In this section, we present our algorithm (Algorithm 1) to extract compre-
hensible rules from RF as follows. The algorithm consists of four parts: In the
first part, RF is constructed and all the rules in the forest are extracted into the
Rs set. The second part of the algorithm computes the score of all rules based
on the RsCoverage, a sparse matrix that shows which rules cover each sample
and its corresponding class label. Afterwards, the scores are assigned to the rules
in order to control the rule selection process, which can be based on different
factors such as accuracy and rule coverage. We used equation (1) that has been
shown to be a promising fitness function [20]:
226 M. Mashayekhi and R. Gras
Algorithm 1. RF+HC
Input: trainSet, testSet,iniRuleNo, treeNo
Step 1: // Construct Random Forest
RF = trainRF( trainSet, treeNo );
Rs = getAllTerminalNodes (RF );
Step 2: // Compute rules coverage
m= size( trainSet );
n= size(Rs );
RsCoverage =zeros( m, n );
foreach sample in trainSet {
foreach rule in Rs
if match(rule,sample )
RsCoverage (sample,rule )=class ;
}
RScore = ruleScore (RsCoverage );
Step 3: // Repeat the HC method to obtain best rules
iniRs = getRuleSet(RScore, n, iniRuleNo );
impRs = iniRs; bestRs=iniRs;
for i=1 to MaxIteration {
impRs = HeuristicSearch (impRs, RScore );
if AccimpRs >AccbestRs
bestRs =impRs ;
impRs = getRuleSet( RScore, n, iniRuleNo );
}
Step 4: // Calculate the accuracy on test set
calcPerformance(testSet,bestRs );
ruleScore1=cc ic
cc +ic +cc
ic +k(1)
In this formula, cc (correct classification) is the number of training sam-
ples that are covered and correctly classified by the rule. Variable ic (incorrect
classification) is the number of incorrectly classified training samples that are
covered by the rule. Finally, kis a predefined positive constant value. In our case
k=4, though other values can be used as it is mostly to avoid the denominator
becomes zero and there is no significant change in the results by modifying k).
This scoring function ensures the retention of the rules with higher classification
accuracy and higher coverage and to remove the noisy rules. Obviously, other
fitness measures can be used instead. One possibility would be to employ the rule
score based on metrics such as number of features in the extracted rule set and
number of antecedents to increase the quality of rules in terms of comprehensi-
bility. In the third step of the algorithm, a fitness proportionate selection method
is used iniRuleNo times to generate an initial rule set (iniRs) with a probability
to select a rule proportional to its score. In order to search the RF rules space,
Rule Extraction from Random Forest: The RF+HC Methods 227
we used the random-restart stochastic hill climbing method, which gives a local
optimum point of the search space based on the random start locations.
Any other search methods such as simulated annealing or genetic algorithm
can be applied instead of HeuristicSearch function in the algorithm. We repeated
the search with a predefined maximum number of iterations (MaxIteration), each
time with a new initial rule set. This can compensate some of the deficiencies in
hill climbing due to the randomized and incomplete search strategy [21]. The hill
climbing algorithm, searches for the best neighbor, the one with the highest score,
of the current location based on equation (1) in the search space and by changing
(adding/removing) one rule to the current rule set. For adding/removing a rule,
we used the same fitness proportionate selection procedure that was employed
for producing the iniRs. The hill climbing score function was defined based only
on the overall accuracy because the scoring schema of the second step already
took into account both rule coverage and rule accuracy. If the new movement in
the rule set space improves the score value, that change is retained. Otherwise it
is discarded and then another neighbor in the rule space is sought. We repeat this
step for a pre-defined maximum number of iterations (MaxIteration). Finally, in
the fourth step, we apply the best extracted rule set on the test set to evaluate
the generalization ability of the extracted rules.
One of the RF characteristics is that there is no pruning while it is con-
structed. Therefore, we expect to have long rules (with a large number of anteced-
ents) in the rule set as well as in the extracted rule set using the proposed
algorithm. Having long rules damages the interpretability of the model and thus
it should be considered in the applications for which the interpretation of the
rules is important. Therefore, we proposed the second algorithm, which is basi-
cally similar to Algorithm 1 except that a modified version of the rule score
function (i.e. equation (2)) was used, where rl shows rule length or number of
antecedents. We called the new method RF+HC CMPR. In the RF+HC CMPR
method more generalized rules (shorter length rules with higher accuracy) have
higher priority than the more specialized rules (the longer rules with lower accu-
racy) based on the following equation:
ruleScore2=ruleScore1+cc
rl (2)
The inputs of the proposed methods are the training/test sets, initial num-
ber of rules (iniRuleNo) and the number of trees in the RF (see Algorithm 1).
Variable iniRuleNo adjusts the tradeoff between accuracy and comprehensibil-
ity. In cases where prediction ability is important, higher values are used and in
cases where the interpretation of the underlying model is important lower values
should be used. For the implementation, we used Matlab as the same as the
source code available for the CRF method.
4 Experiments and Discussion
To compare our proposed methods with other methods, we also applied CRF
[14,15] and RF on 22 different data sets. Different criteria have been proposed to
228 M. Mashayekhi and R. Gras
evaluate a RE algorithm [11]. For instance, accuracy is defined as the ability of
extracted rules to predict unseen test sets. Another major factor is comprehensi-
bility, which is not easy to measure due to the subjective nature of this concept.
There are different factors that are used to determine comprehensibility such as,
the number of rules and the average number of antecedents. Another desirable
characteristic of a RE method is its potential to be applicable to a wide range of
applications. If a RE algorithm is applicable to data sets with a large number of
samples, features, or classes then it is said to be scalable. This scalability notion
includes time and algorithm complexity.
In our work, we measured the average accuracy of 10 times 3-fold cross-
validation (by randomizing the data set for every repetition) for evaluating
accuracy, as it gives more accurate results in compare to one time k-fold cross-
validation. This measure demonstrates the prediction and generalization ability
of the extracted rules. Majority voting is used to classify a sample when more
than one rule covers a sample. We assumed a default rule such that the sam-
ples not covered by any of the extracted rules are simply assigned to the high
frequency class in the dataset. In the RF+HC methods, due to their stochastic
nature, we repeated the whole procedure 10 times and computed the average
results along with their standard deviations. For the CRF method, 10 differ-
ent values for the lambda parameter (which indicates the tradeoff between the
number of rules and accuracy) were used. To determine these values, we did a
few pilot runs with each data set separately. To determine the best lambda, a
cross-validation step is incorporated in the CRF method such that it selects the
lambda value, which gives the minimum error for cross-validation. In order to
show the comprehensibility of the methods, we considered the number of rules,
maximum rule length, and total number of antecedents in the extracted rule
set. For the CRF method, these values are related to the lambda parameter
value, which gives the lowest cross-validation error. On the other hand, those of
RF+HC methods are related to 10 repetitions of the process. All the values are
rounded to the closest integer value.
Scalability is one of the most important evaluation metrics often overlooked
in most of the RE methods such as the CRF method. We measured the compu-
tational time as a metric to evaluate the scalability. To have a fair comparison,
we used 10 different lambdas in the CRF method and we divided the required
time to find the best lambda by 10. This means that we only considered the
time for cross-validation using the best lambda plus the time for training and
test steps. We considered the cross-validation time because it is an important
part of the CRF method , which finds the best lambda in each iteration of the
algorithm. At each iteration, that includes cross-validation, optimization, and
feature selection, the features in the extracted rules are kept and the rest are
removed. In the case of RF+HC, we repeated each experiment 10 times and then
divided the overall time by 10. We also considered the hill climbing repetition
time (MaxIteration) in order to calculate the computation time.
As the input of the proposed algorithms, we should specify the initial number
of rules (iniRuleNo) for each data set. We used 500 decision trees to build RF
Rule Extraction from Random Forest: The RF+HC Methods 229
with m=sqrt(n), which is a default value mostly used in the literature, where
(m) is the number of features randomly chosen at that node for splitting. In the
random-restart hill climbing, we repeated hill climbing from 10 initial rule sets.
We took MaxIteration = 500 in all of our experiments. Higher values provide
hill climbing with more opportunities for likely improving the rule set score,
although it did not happen in our case. For comparing the proposed methods
with CRF in terms of performance, comprehensibility, and computation time
complexity, we used Wilcoxon and Friedman tests as suggested in [6].
4.1 Data Sets
We used 22 data sets with various characteristics in terms of the number of
features, the number of samples, and the number of classes to observe how the
performance of the proposed methods varies depending on the data set type.
Eighteen data sets were taken from UCI machine learning repository [3]and
another four data sets (Golub [10], Colon [1], Nutt [19], Veer [25]), which are
gene expression microarray data sets. The extreme cases are Veer with 24188
features, Magic with 19020 samples, and Yeast and Cardio with 10 classes (see
Table 1).
Table 1. Data sets along with their characteristics
Data set Feature# Class# Sample#
Breast Cancer 9 2 699
Magic 10 2 19020
Musk Clean1 166 2 476
Wine 13 2 178
Wine Quality 11 6 1599
Iris 4 3 150
Yeast 8 10 1485
Cardiography 20 10 1726
Balance Scale 4 3 625
Cmc 9 3 1473
Glass 9 6 214
Haberman 3 2 306
Iono 34 2 351
Segmentation 19 7 210
Tae 5 3 151
Zoo 16 7 101
Ecoli 7 8 336
Spam 57 2 4601
Golub 5147 2 72
Colon 2000 2 62
Glimo Nutt 12625 2 50
Veer 24188 2 77
230 M. Mashayekhi and R. Gras
4.2 Accuracy and Generalization Ability
On average, both the RF+HC and RF+HC CMPR methods gave almost the
same level of accuracy as the CRF method with marginal differences (Table 2).
Moreover, all three methods obtained 96% of the RF accuracy for the whole
data sets on average. For some datasets, they demonstrated higher accuracy
than RF such as for Tae, Cmc, and Golub with RF+HC or Tae and Clean with
CRF method. A similar result was observed in [28] when the authors used a NN
ensemble to extract the rules, observing higher accuracy for extracted rules than
for the underlying model.
The generalization ability of RF+HC is due to the selection of the high
score rules in RF and it is also due to some level of stochasticity, which results
in assigning odds to the rules with low scores in the training set, but they
may be important for unseen data. Comparing the accuracy of CRF method
with the proposed methods revealed that the null hypothesis with α=0.05
cannot be rejected with z=0.41 (CRF vs. FR+HC) and z=0.42 (CRF vs.
RF+HC CMPR), while the critical zvalue is -1.96 in Wilcoxon test. Therefore,
the difference is not significant, which proves that two methods are equivalent
in terms of accuracy.
Table 2. Percentage accuracy of the RF+HC, RF+HC CMPR,CRF,andRFmethods
on the selected data sets along with the standard deviations in parenthesis
Data set RF+HC RF+HC CMPR CRF RF
Cancer 96.18 (0.32) 96.23 (1.56) 95.71 (1.01) 96.65 (1.75)
Magic 85.37 (0.46) 85.6 (0.28) 83.65 (1.3) 88.12 (0.3)
Clean 81.34 (3.25) 83.17 (4.3) 88.45 (1.55) 88.68 (2.18)
Wine 92.07 (3.29) 95.93 (1.8) 91.93 (5.91) 98.99 (0.9)
Wineqlty 65.13 (1.93) 62 (1.8) 62.79 (0.57) 68.59 (3.47)
Iris 93.36 (2.4) 94.12 (3.25) 94.4 (2.61) 96.40 (1.67)
Yeast 59.98 (1.5) 61.3 (0.7) 55.02 (2.75) 62.02 (1)
Cardio 81.74 (0.82) 82 (0.6) 84.01 (0.84) 85.67 (2.19)
BalancS 84.48 (0.52) 83.75 (2.36) 82.87 (2.86) 87.24 (1.6)
Cmc 52.87 (0.99) 52.6 (2.5) 49.42 (3.65) 52.46 (2.57)
Glass 74.33 (2.7) 73.75 (7.3) 72.77 (2.15) 78.02 (7.51)
Haber 67.69 (2.1) 69.14 (1.7) 70.2 (4.42) 73.92 (4.2)
Iono 90.14 (3.53) 91.9 (3.3) 91.45 (1.6) 93.16 (1.9)
Segment 87.54 (1.86) 89.97 (2.4) 88.86 (3.7) 93.14 (2.1)
Tae 57.60 (3.46) 53.45 (4.1) 62.29 (4.8) 55.60 (1)
Zoo 91.33 (9.6) 92.96 (5.87) 93.94 (8.2) 97.02 (2)
Ecoli 84.2 (3.11) 79.9 (4) 86.67 (11.54) 86.96 (1.74)
Spam 94.04 (0.71) 94.33 (0.5) 94.2 (1.05) 95.24 (0.3)
Golub 93.00 (6.7) 87.25 (7.6) 86.11 (9.62) 92.5 (4.5)
Colon 74.76 (5.26) 76.1 (3.9) 82.46 (17.94) 75.00 (11.85)
Glimo 64.11 (4.26) 66.3 (7.36) 54.9 (8.99) 71.69 (14.47)
Veer 58.27 (7.88) 63.11 (8) 60.97 (8.99) 66.43 (13.76)
Rule Extraction from Random Forest: The RF+HC Methods 231
4.3 Comprehensibility
Although a feature selection phase was incorporated in the CRF method, our
methods were superior in the number of extracted rules in all the data sets except
the Golub data set (Table 3). The number of rules extracted by RF+HC or RF+
HC CMPR in average are 0.6% of the total number of rules in RF while that of
CRF is 11.66%, which demonstrates very good improvement compared to RF and
CRF . The proposed methods significantly reduced the number of rules in compar-
ison to CRF (z=4.06) and as a result improved the comprehensibility. However,
the difference in terms of rule numbers for the two proposed methods was not sig-
nificant (z=1.89). There is one dataset, i.e. Golub, for which CRF extracted
only one rule. In such cases, the extracted rule is related to one class and it can only
explain that class. However, there is no information and interpretation regard-
ing the other class(es). Therefore, we believe that this type of rule set is not fully
comprehensible as it cannot describe the underlying model completely. We found
an issue in the implementation of the CRF method, which will affect the results.
When the number of rules is reported, only the rules with the weights greater than
a threshold (in this case 10e-6) are considered. However, all the extracted rules are
used to do prediction for the test set which is not correct. The CRF results in Table
3 corresponds to the correct number of rules.
We used the modified version of the rule score function (i.e. equation (2)) in
order to give higher priority to the more generalized rules. Table 3 shows the com-
parison between the original algorithm and RF+HC CMPR. The results showed
that RF+HC CMPR have a stronger impact on the maximum rule length and
also on the total number of antecedents (42% and 18% decrease respectively) in
the rule set in comparison with RF+HC. In addition, we observed no significant
change in the accuracy. These results indicate that RF+HC CMPR improves
the comprehensibility significantly (z=4.16).
Comparing the CRF method with the two proposed methods using Wilcoxon
test (critical z=-1.96) indicates that RF+HC had a significant lower maximum
rule length (z=3.13) and also number of antecedents (z=4.07) in compare
to CRF. RF+HC CMPR was superior in all data sets in terms of maximum
rule length (z=4.09) and number of antecedents (z=4.07) except for the
maximum rule length for Golub.
One important aspect of comprehensibility is the number of rules extracted
from an underlying model. However, we have to consider the importance of the
tradeoff between accuracy and comprehensibility. The extracted rules should not
only be concise but also have good performance on unseen samples. This is, in
fact, the main objective of rule extraction. Therefore, a good rule extraction
method should consider two facts simultaneously: comprehensibility and gen-
eralization ability, although it should be adjustable based on the application.
For example, for the Magic dataset, RF generates 608155 rules with approxi-
mately 88% accuracy. This number of rules shows the complexity of the model
for this dataset. RF+HC methods extract only about 0.4% of the RF rules and
give about 85% accuracy for this data set. We still can generate fewer rules by
decreasing iniRuleNo, although it will reduce accuracy. Therefore, what needs to
232 M. Mashayekhi and R. Gras
be considered in order to have a fair judgment is the combination of the number
of rules and accuracy. The results we have presented in this paper correspond to
the smallest number of rules in order to achieve a level of accuracy as close as
possible to the level of accuracy for RF. We provided the samples of extracted
rules for two data sets in the Table 4.
Table 3. Each cell shows ‘Number of extracted rules / Maximum length of rule / Total
number of antecedents’in each method. The values in bold show the best results.
Data set RF+HC RF+HC CMPR CRF RF
Cancer 36/8/159 33/6/129 463/9/1940 12075/13/65869
Magic 2604/8/8186 2597/3/5697 3182/37/50668 608155/58/8514170
Clean 83/15/586 78/10/473 104/18/947 18392/20/150309
Wine 16/8/64 14/5/55 176/7/619 7590/10/26784
Wineqlty 1258/21/12301 1259/12/10526 2282/24/22256 138889/30/1757860
Iris 13/6/39 11/5/28 43/5/145 4202/9/13222
Yeast 1037/25/13460 1303/13/11621 1836/27/18430 126936/32/1469328
Cardio 1609/20/15720 1606/11/12951 2121/20/19003 126412/22/1150839
BalancS 88/9/471 83/5/339 360/11/1768 19764/13/124447
Cmc 332/16/2390 322/10/1818 2025/19/14695 74257/22/754197
Glass 88/13/398 59/8/335 10050/12/30662 16530/16/115932
Haber 28/13/165 25/8/140 410/16/2417 19697/18/142512
Iono 41/11/193 36/7/145 155/12/784 10641/14/57312
Segment 42/10/267 54/6/175 11134/13/24065 9905/12/59837
Tae 91/13/495 76/8/359 177/13/997 14437/16/93530
Zoo 16/6/66 15/4/51 185/7/608 4954/9/17615
Ecoli 138/11/762 141/7/649 8900/14/29421 16761/16/105260
Spam 476/34/5076 473/21/4228 1154/41/14852 118878/44/1455859
Golub 9/3/18 6/2/10 1/3/32322/4/4939
Colon 17/4/46 19/3/39 27/5/85 2620/6/8154
Glimo 9/3/23 12/2/20 17/4/47 1953/4/4716
Veer 18/4/45 17/3/33 39/5/128 3254/6/8513
4.4 Complexity and Scalability
We found a significant difference in terms of computational time between our
methods and CRF (z=4.07). For all data sets, the RF+HC methods were
superior to CRF with the exception of the Iris data set, which had only a one-
second difference (Table 5). More specifically, in some cases with large numbers
of classes such as Yeast, Glass, Ecoli, and Segment, our methods were 136,
310, 518, and 842 times faster than CRF respectively. We observed the same
circumstance for data sets with a large number of samples such as Magic, Spam,
and CMC such that RF+HC and RF+HC CMPR were 13, 18, and 130 times
faster than CRF. On average, the overhead time for the proposed methods and
CRF method was 1.12, and 11.8 times respectively relative to RF time.
Moreover, we observed more computational time for CRF when there was a
larger number of classes (Table 5) because the CRF method considers cclassifiers
Rule Extraction from Random Forest: The RF+HC Methods 233
Table 4. Sample of rules extracted by the RF+HC CMPR method from Iris and Golub
data sets (The features in the data sets are shown by “V” and a subscripted number.
The consequence of each rule is specified by a class label, for example, “Class 1”. The
value in the parenthesis is the score of the rule based on equation 2. Acc. is test set
accuracy).
Iris (Acc. 98%)
V40.80 : Class 1 (38.50)
V32.70: Class 1 (38.50)
V32.60: Class 1 (38.50)
V34.85 & V3>2.70: Class 2 (20.88)
V23.05 & V34.75 & V3>2.45 & V2>2.55
: Class 2 (10.50)
V4>0.80 & V2>2.95 & V41.70 & V35.15
: Class 2 (5.50)
V4>1.60: Class 3 (43.50)
Golub (Acc. 100%)
V1727 >1570.00: Class 1 (35.73)
V4572 >1116.50 & V737 526.50: Class 1 (18.25)
V3607 13177.00 & V4005 >44.50: Class 1 (18.81)
V4969 540.50: Class 2 (21.00)
V4648 >489.00: Class 2 (16.46)
V4929 >5863.00: Class 2 (12.25)
V1556 >2699.00 & V3595 2939.50: Class 2 (10.67)
V1394 63.50 & V3776 211.00: Class 2 (9.31)
V4594 530.50: Class 2 (6.67)
(cis number of classes) and finds a weight vector for each class. When there are a
relatively large number of samples and a large number of classes simultaneously,
the CRF method has an even worse performance. In addition, a large number
of features can increase the computational time as CRF has a repeating feature
selection step. However, in RF+HC methods, the overhead time on top of RF
in RF+HC method has a strong linear correlation with the number of samples
in the data sets (R2=0.994).
4.5 Overall Comparison and Major Contributions
The major contributions of the proposed methods in comparison to RF are that
they refine RF in selecting the most valuable rules, which leads to a huge decre-
ment in the number of rules i.e. 0.6% of the random forest rules, while at the
same time attaining 96% of the RF accuracy with a reasonable overhead time
on top of RF time. In addition, both methods improved the comprehensibility in
comparison with CRF while retaining the same accuracy. RF+HC decreased the
number of rules, the maximum rule length, and the total number of antecedents
by 27%, 16%, and 49% respectively in average. RF+HC CMPR also reduced
them by 25%, 50%, and 59%. The RF+HC methods decreased the computa-
tional time in 21 of the 22 data sets. Moreover, for the data sets with a large
234 M. Mashayekhi and R. Gras
Table 5. Computational time for RF+HC, RF+HC CMPR, CRF, and RF in second
Dataset RF+HC RF+HC CRF RF
CMPR
Cancer 16 16 36 5
Magic 1409 1425 19338 1050
Clean 34 34 118 26
Wine 4 5 13 1
Wineqlty 52 56 5317 17
Iris 4 9 3 1
Yeast 46 49 6276 15
Cardio 80 83 6410 31
BalancS 17 17 233 4
Cmc 36 36 4696 14
Glass 5 5 1551 1
Haber 10 10 15 2
Iono 9 9 24 3
Segment 7 7 5900 2
Tae 5 5 14 1
Zoo 3 3 14 1
Ecoli 9 9 4669 2
Spam 236 239 4479 166
Golub 230 230 253 228
Colon 56 56 62 54
Glimo 633 633 720 631
Veer 3165 3165 3558 3162
number of samples and/or a large number of classes, they were much faster (up
to about 800 times) in terms of the computational time. Table 6 summarizes the
overall comparisons of RF+HC and RF+HC CMPR with the CRF method. The
numbers in the table specify the average rank of each method for Friedman test
computed for the mentioned criteria in the table, where lower value demonstrates
the better method. The Friedman test showed significant difference between the
average ranks and the mean rank for each criterion. However, the difference was
marginal for the accuracy as it was also confirmed by the Wilcoxon test. These
results show that our proposed methods are better than the CRF in terms of
Table 6. Comparison summary for different methods. The values are the average rank
with the standard deviation in the parenthesis.
RF+HC RF+HC CMPR CRF
Accuracy 2.23 (0.81) 1.73 (0.7) 2.05 (0.9)
Rule# 1.77 (0.53) 1.32 (0.48) 2.91 (0.43)
Time 1.34 (0.24) 1.7 (0.37) 2.95 (0.21)
MaxCond 2.11 (0.26) 1.02 (0.11) 2.86 (0.35)
Cond# 2 (0) 1 (0) 3 (0)
Rule Extraction from Random Forest: The RF+HC Methods 235
number of rules, computational time, maximum rule length, and also number of
antecedents while they keep level of accuracy as the same as CRF method.
5 Conclusions and Future Works
In this paper, we introduced new rule extraction methods from RF. Experimental
results showed that these methods are superior to the CRF method in terms
of comprehensibility while keeping the same level of accuracy. In addition, our
methods are much more scalable than the state-of-the-art method, CRF and they
can be applied more generally and on data sets with various characteristics.
This work can be extended in several different directions in future research.
We plan to compare the proposed methods with other related methods, especially
the ones described in [2,17,26]. Another possible direction would be improving
the rule score and fitness function based on other metrics such as number of
features in the extracted rule set and number of antecedents to increase the
quality of rules in terms of comprehensibility. Yet another direction is to examine
other heuristic search methods such as simulated annealing, tabu search, and
genetic algorithms in order to find better sets of rules than those obtained with
hill climbing.
Acknowledgments. This research was supported by the CRC grant 950-2-3617 and
NSERC grant ORGPIN 341854. We greatly appreciate Brian MacPherson for his com-
mentsonthispaper.
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