 Proceedings
 Open Access
Building interpretable fuzzy models for high dimensional data analysis in cancer diagnosis
 Zhenyu Wang^{1} and
 Vasile Palade^{1}Email author
https://doi.org/10.1186/1471216412S2S5
© Wang and Palade; licensee BioMed Central Ltd. 2011
 Published: 27 July 2011
Abstract
Background
Analysing gene expression data from microarray technologies is a very important task in biology and medicine, and particularly in cancer diagnosis. Different from most other popular methods in high dimensional biomedical data analysis, such as microarray gene expression or proteomics mass spectroscopy data analysis, fuzzy rulebased models can not only provide good classification results, but also easily be explained and interpreted in human understandable terms, by using fuzzy rules. However, the advantages offered by fuzzybased techniques in microarray data analysis have not yet been fully explored in the literature. Although some recently developed fuzzybased modeling approaches can provide satisfactory classification results, the rule bases generated by most of the reported fuzzy models for gene expression data are still too large to be easily comprehensible.
Results
In this paper, we develop some MultiObjective Evolutionary Algorithms based Interpretable Fuzzy (MOEAIF) methods for analysing high dimensional biomedical data sets, such as microarray gene expression data and proteomics mass spectroscopy data. We mainly focus on evaluating our proposed models on microarray gene expression cancer data sets, i.e., the lung cancer data set and the colon cancer data set, but we extend our investigations to other type of cancer data set, such as the ovarian cancer data set. The experimental studies have shown that relatively simple and small fuzzy rule bases, with satisfactory classification performance, can be successfully obtained for challenging microarray gene expression datasets.
Conclusions
We believe that fuzzybased techniques, and in particular the methods proposed in this paper, can be very useful tools in dealing with high dimensional cancer data. We also argue that the potential of applying fuzzybased techniques to microarray data analysis need to be further explored.
Keywords
 Fuzzy Rule
 Cancer Data
 Candidate Rule
 Lung Cancer Data
 Colon Cancer Data
Background
Microarray techniques allow simultaneous measuring of the expression of thousands of genes under different experimental environments and conditions. They allow us to analyse the gene information very rapidly by managing them at one time. The gene expression profiles from particular microarray experiments have been widely used for cancer classification [1–3]. However, the amount of data produced by this new technology is usually too large to be manually analysed. Hence, the need to automatically analyse the microarray data offers an opportunity for Machine Learning (ML) methods to have a significant impact on cancer research.
A typical gene expression matrix X, where rows represent samples obtained under different experimental conditions and columns represent genes
Gene 1  Gene 2  …  Gene n1  Gene n  Class  

1  165.1  276.4  …  636.6  784.9  1 
2  653.6  1735.1  …  524.1  104.5  1 
…  …  …  …  …  …  … 
m1  675.0  45.1  …  841.9  782.8  1 
m  78.2  893.8  …  467.9  330.1  1 
Unsupervised methods, such as Clustering [4], and SelfOrganizing Maps (SOMs) [5] were initially used to analyse the relationships among different genes. Subsequently, supervised methods, such as Support Vector Machines (SVMs) [6], MultiLayer Perceptrons (MLPs or NNs) [7, 8], K Nearest Neighbor (KNN) methods [9, 10], etc., have been successfully applied to the classification of different tissues. But, most of the current methods in microarray data analysis are black box methods; these models can not satisfactorily reveal the hidden information in the data. This information usually plays a very important role in making a quality clinical diagnosis.
Different from blackbox methods, fuzzy rulebased models can not only provide good classification results, but also easily be explained and interpreted in human understandable terms by using fuzzy rules. This provides the researchers or clinician an insight into the developed models. At the same time, fuzzy systems adapt numerical data (input/output pairs) onto human linguistic terms and offer very good capabilities of dealing with noisy and missing data. Compared to other popular rulebased models in the area, such as C4.5 Decision Trees (DTs) [11, 12], the linguistic rules generated by our fuzzybased models are short and easy to be read.
Unfortunately, rulebased methods have suffered some wellknown limitations in dealing with high dimensional data. Very high dimensional feature vectors and lack of enough training samples are two major challenges for modeling microarray data in general, hence for the success of applying fuzzy models to this problem too. However, some recent developments in fuzzy systems provide us with some good ways to obtain good diagnosis results. For example, Vinterbo et al. [13] firstly used fuzzy rule bases to classify gene expression data, but this model only allow linear discrimination, and the classification performance is limited; an AdaptiveNetworkbased Fuzzy Inference System (ANFIS) was successfully applied for this problem in [14] too; Woolf and Wang [15] developed a fuzzy based model to analyse the relationships between genes, while Ressom et al. [16] used a clusteringbased preprocessing method to increase the efficiency of the fuzzy models. All these reported systems are either small models which perform well on small data sets, or huge models which are difficult to be understood by the human experts. Other machine learning techniques, like genetic algorithms (GA) [17] or ensemble learning [18, 19] have been adopted to allow fuzzy rulebased models to deal with a relative large number of features, but the obtained rule bases still look very complex to be easily comprehensible. Large model complexity significantly damages the main advantage of applying fuzzy models to this application, i.e., the interpretability of the models. The computational cost of constructing these models is generally very high too.
Normally, the accuracy of each fuzzy rulebased classifier is measured by the number of correctly classified training or testing patterns, while its interpretability is measured by the complexity of the model, more specifically, the number of fuzzy rules and the total number of antecedent conditions. Whereas both accuracy and interpretability were considered, multiobjective evolutionary based methods are introduced into our systems, hence, the name of MultiObjective Evolutionary Algorithms based Interpretable Fuzzy (MOEAIF) models. We evaluated our proposed model on some wellknown cancer data sets, i.e., the ovarian cancer data set, the lung cancer data set and the colon cancer data set. Experimental results are listed and discussed in the later section. Compared with most previously reported models, accurate and small fuzzy rule bases were obtained.
Methods
Gene Selection
A major goal for diagnostic research is to develop diagnostic procedures based on inexpensive microarrays that have enough probes to detect certain diseases. This requires the selection of some genes which are highly related to the particular classification problem, i.e., the informative genes. This process is called Gene Selection (GS), which corresponds to feature selection from any machine learning task in general. Two basic approaches for feature selection used in machine learning and information theory literature are the filter methods and the wrapper methods [9, 20, 20, 21]. In theory, wrapper methods should provide more accurate classification results than filter methods [21]. The main disadvantage of the wrapper approach is its computational cost when combined with more complex algorithms such as SVM for example. The wrapper approach, which is popular in many machine learning applications, is not extensively used in DNA microarray tasks, and in most cases the gene selection is performed by ranking genes on the basis of scores, correlation coefficients, mutual information and sensitivity analysis. More detailed discussions of these two approaches can be found in [9, 20, 22–24]. As suggested in [25], a Fuzzy CMean Clustering based Enhanced Gene Selection method (FCCEGS) is applied in this paper as well for gene selection.
Improved Methods for Obtaining Interpretable Fuzzy Models
An ideal design of fuzzy rulebased models for microarray data analysis is when we find fuzzy rulebased models with good interpretability but with acceptable testing accuracy too. Compared to most popular methods in cancer microarray gene expression data analysis area, rulebased fuzzy models usually have relative high computational complexity. In order to obtain good fuzzy rulebased models, we adopted the following recent techniques to reduce the complexity of fuzzy rulebased models.
Weighted Fuzzy Rules
The fuzzy rules R_{ q } used in our models are in the form of:
• R_{ q }: If x_{1} is A_{ q }_{1} and … and x_{ n } is A_{ qn }, then Class C_{ q } with CF_{ q }.
In the above rule, x = (x_{1},…,x_{ n }) is the ndimensional input vector, A_{ qi } is an antecedent fuzzy set for the i – th input variable, C_{ q } is a consequent class, and CF_{ q } is a certainty degree (i.e., rule weight). The rule weight is a real number in the unit interval [0, 1].
Multiple Fuzzy Partitions
Simple Fuzzy Reasoning
Since we have 15 antecedent fuzzy sets for each attribute of our ndimensional pattern classification problem, the total number of combinations of the antecedent fuzzy sets is 15^{ n }. Each combination is used as the antecedent part A_{ q } of the fuzzy rule R_{ q }. Its consequent class C_{ q } and rule weight CF_{ q }are specified from compatible training patterns with A_{ q } in the following heuristic manner.
First, we calculate the compatibility degree of each pattern x_{ p } with the antecedent part A_{ q } of the rule R_{ q } via a product operation like:
µ_{ A }_{ q }(x_{ p }) = _{ µ }_{Aq1}(x_{ p }_{1}) ⋅…⋅µ_{ A }_{ qn }(x_{ pn }) (1)
where m denotes the number of training patterns.
where M is the number of classes.
Let S be a subset of candidate fuzzy rules, i.e., a fuzzy rulebased classifier. Each pattern x_{ p } is classified by a single winner rule R_{ w }, which is chosen from the rule set S as follows:
µ_{ Aw }(x_{ p }) ⋅ CF_{ w } = max{µ_{ A }_{ q }(x_{ p }) ⋅ CF_{ q }R_{ q } ∈ S}.
We only generate short fuzzy rules with a few antecedent conditions, and it should be noted that the DC conditions can be omitted from fuzzy rules. This restriction is used in order to find a compact set of fuzzy rules with high interpretability. As for short fuzzy rules, we only use fuzzy rules that satisfy both the minimum confidence and support as candidate rules for the multiobjective genetic fuzzy rule selection mechanism.
Rule confidence and support can be used as prescreening criteria for finding a tractable number of candidate fuzzy rules. The generated fuzzy ifthen rules are then divided into T groups according to their consequent classes, where T is a userdefined parameter. Fuzzy ifthen rules in each group are sorted in the descending order of a prescreening criterion (i.e. confidence, support, or their product). For selecting Q candidate rules, the first Q/T rules are chosen from each of the T groups, and in this manner, we can choose an arbitrarily specified number of candidate fuzzy ifthen rules (i.e., Q candidate rules). It should be noted that the aim of the candidate rule prescreening is not to construct a fuzzy rulebased system, but to find candidate rules, from which a small number of fuzzy ifthen rules are later selected. For using a variety of candidate rules in rule selection, we choose the same number of fuzzy ifthen rules (i.e., Q/T candidate rules) for each class. By applying these new techniques, the models complexity and computational cost is significantly decreased.
The MultiObjective Evolutionary Algorithms based Interpretable Fuzzy (MOEAIF) Model
In this section, we describe how to apply multiobjective evolutionary algorithms (MOEA) to extract fuzzy rule sets considering the balance between model accuracy and model interpretability.
Our task is to select a smaller number of simple fuzzy ifthen rules with high classification performance, and this is performed by maximizing the classification accuracy, minimizing the number of selected rules, and minimizing the total rule length at the same time. Therefore, the fitness value of each string S (i.e., each rule set S) in the current population is defined by the three objectives using the following fitness function:
f(S) = w_{1}⋅NCCP(S) –w_{2}⋅NOR(S) –w_{3}⋅NOA(S),
where w = (w_{1}, w_{2},w_{3}), NCCP(S), NOR(S), and NOA(S) are the weight vector, the number of correctly classified training patterns, the number of selected fuzzy rules in S, and the total number of antecedent conditions in S, respectively. The weights w_{1},w_{2},w_{3} must satisfy the following conditions:
w_{1},w_{2},w_{3}≥0;
w_{1}+w_{2}+w_{3} = 1;
As suggested by [29, 30], a rule subset R consisting of the Q candidate rules can be represented by a binary string as:
R = r_{1}r_{2}r_{3}…r_{ Q },
where r_{ q } = 0 means that the qth candidate rule r_{ q } is not included in the rule set R, while r_{ q' } = 1 means that r_{ q' } is included in R.
where w_{1},w_{2},w_{3} are the user defined weights of the three search objectives in the fitness function, and PA_{10} and PA_{10} are two initial fixed parameters. A larger probability is normally assigned to the mutation from 1 to 0 than to that from 0 to 1, in order to efficiently decrease the number of fuzzy ifthen rules (i.e., the number of 1s) included in each string. By applying the Equations 11 and 12, the mutation rate can be automatically adjusted according to user different purposes. Our MOEAIF can be summarized as follows:
1. Step 1: Randomly generate N_{ pop } (number of individuals in the population) binary strings of length Q as an initial population. Specify the crossover probability p_{ c }, two mutation probabilities, PA_{10} and PA_{10}, and the stopping condition;
2. Step 2: Generate N_{ pop } children strings by applying crossover and mutation to the current population;
3. Step 3: Calculate the threeobjectives fitness value for each string; unnecessary rules are removed from each string;
4. Step 4: Update the next population by selecting top ranked individuals;
5. Step 5: Stop, if the stopping condition is satisfied or the maximum number of training epochs is reached, otherwise return to Step 2.
Results and Discussion
Cancer Data Sets
We evaluated our proposed MOEAIF models on three cancer data sets, namely the ovarian cancer data set, the lung cancer data set and the colon cancer data set.
• Lung Cancer Data Set
Lung Cancer Classification differentiates between malignant pleural mesothelioma (MPM) and adenocarcinoma (ADCA) of the lung. There are 181 reported samples in total, where 31 samples belong to MPM and 150 samples belong to ADCA. The training set contains 32 samples, 16 MPM and 16 ADCA, and the remaining 149 samples are used for testing. The expression levels of 12,533 features were report in each sample. Each feature represents one probe, for example, the feature 1018.at represents the probe 1018 at. The data is available at http://cilab.ujn.edu.cn/datasets.htm.
• Ovarian Cancer Data Set
The ovarian cancer data set was first reported in [31]. The aim of the experiment was to identify proteomic patterns in serum that distinguish ovarian cancer from noncancer. This study is significant to women who have a high risk of ovarian cancer due to family or personal history of cancer. The proteomic spectra were generated by mass spectroscopy and the raw data can be found at http://clinicalproteomics.steem.com. There are 253 reported samples in this data set, where 91 samples belong to normal and 162 samples belong to ovarian cancers. The normalization is done over all the 253 samples for all 15154 M/Z identities. After the normalization, each intensity value is to fall within the range of 0 to 1. The data is available at http://cilab.ujn.edu.cn/datasets.htm.
• Colon Cancer Data Set
The data set used here was firstly reported in [1]. This data set contains 62 samples, of which 40 are tumour samples, and 22 normal samples. About 6000 genes are represented in each sample in the original data set, out of which only 2000 were selected. The data is available at http://sdmc.i2r.astar.edu.sg/rp/ColonTumor/ColonTumor.html.
Accuracy of the MOEAIF Models
Classification accuracy and interpretability of models on the lung cancer data set.
w _{1}  w _{2}  w _{3}  Number of Rules  Average Rule Length  Testing Accuracy 

0.1  0.7  0.2  2  1.5  89.26 
0.5  0.1  0.4  6  1.8  90.06 
0.5  0.4  0.1  3  2  90.06 
0.5  0.2  0.2  3  2  89.93 
0.7  0.1  0.2  3  2  91.28 
1  0  0  23  2  90.06 
Classification accuracy and interpretability of models on the ovarian cancer data set.
w _{1}  w _{2}  w _{3}  Number of Rules  Average Rule Length  Testing Accuracy 

0.7  0.2  0.1  36  2.3  86.71 
0.5  0.2  0.3  16  2  78.03 
0.3  0.4  0.3  8  2  63.75 
Interpretability of the MOEAIF Models
We further explore the rule base obtained by our MOEAIF approach for lung cancer data set, where w_{1}=0.5, w_{2}=0.3, and w_{3}=0.2.
Rule Extraction
The selected rule subset for lung cancer data when testing accuracy = 0.8993; “–” denotes “don’t care” condition.
40256.at  1018.at  35792.at  33357.at  CF  Class  

Rule 1 
    
 0.9999  1 
Rule 2   
  
 0.9829  1 
Rule 3   

   0.9725  1 
• Rule 1: If the feature 40256.at is “large” and the feature 33357.at is “large”, then the sample is Cancer with CF=99.99%.
• Rule 2: If the feature 1018.at is “large” and the feature 33357.at is “medium”, then the sample belongs to Normal with CF = 98.29%.
• Rule 3: If the feature 1018.at is “large” and the feature 35792.at is “relatively small”, then the sample belongs to Normal with CF = 97.25%.
The membership functions of the feature 1018.at and the feature 35792.at in Rule 1 are “dont’t care”, which can reduce the length of Rule 1. The rules generated by our MOEAIF models are shorter than the rules from our previously built models [14, 18, 25] and some other reported rulebased models [11, 32].
The selected rule subset for ovarian cancer data when testing accuracy = 0.6375. “–” denotes “don’t care”.
MZ820.8  MZ6880.2  MZ1730.9  MZ1866.7  MZ18871.5  MZ827.3  Class  

Rule 1   

       1 
Rule 2   
      
 1 (0.9995) 
Rule 3   
    
   1 (0.9994) 
Rule 4         

 1 (0.9999) 
Rule 5       

   1 (0.9999) 
Rule 6   
    
   1 (0.9997) 
Rule 7   
    
   1 (0.9996) 
Rule 8 
      
   1 (0.9994) 
Conclusions
In this paper, small and linguistically understandable fuzzy rule bases were obtained from challenging high dimensional cancer data sets by using our proposed MultiObjective Evolutionary Algorithms based Interpretable Fuzzy (MOEAIF) method. The classification performance obtained by our models is also competitive. We also point out that an ideal design of fuzzy rulebased models for microarray gene expression data analysis includes two important tasks: designing lowcomplexity fuzzy models, and finding tradeoff points between classification accuracy and model interpretability. We believe that fuzzy techniques and, in particular, the methods proposed in this paper can be very useful tools in dealing with microarray data. There also are some important issues that need to be addressed in the future. For example, some microarray gene expression data sets were generated directly from the probes set, and in some cases several probes may correspond to the same gene, or several different genes may hybridise to the same probes (i.e.,crosshybridisation). If some of the input features (or probes) in a single rule are specific to the same gene(s), then this rule need to be deleted. Due to lack of enough training examples, satisfactory classification results were not always guaranteed in some small data sets, for example, the colon cancer data set in this paper.
Declarations
Acknowledgements
The authors would like to thank to the anonymous reviewers who have reviewed the original conference paper, whose comments were helpful in improving the paper. Zhenyu Wang would like to thank the Computing Laboratory, University of Oxford, for hosting this research while doing his PhD.
This article has been published as part of BMC Genomics Volume 12 Supplement 2, 2011: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2010. The full contents of the supplement are available online at http://www.biomedcentral.com/14712164/12?issue=S2.
Authors’ Affiliations
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