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Research article Special Issues

Towards a robust algorithm to determine topological domains from colocalization data

  • One of the most important tasks in understanding the complex spatial organization of the genome consists in extracting information about this spatial organization, the function and structure of chromatin topological domains from existing experimental data, in particular, from genome colocalization (Hi-C) matrices. Here we present an algorithm allowing to reveal the underlying hierarchical domain structure of a polymer conformation from analyzing the modularity of colocalization matrices. We also test this algorithm on several model polymer structures: equilibrium globules, random fractal globules and regular fractal (Peano) conformations. We define what we call a spectrum of cluster borders, and show that these spectra behave strikingly di erently for equilibrium and fractal conformations, allowing us to suggest an additional criterion to identify fractal polymer conformations.

    Citation: Alexander P. Moscalets, Leonid I. Nazarov, Mikhail V. Tamm. Towards a robust algorithm to determine topological domains from colocalization data[J]. AIMS Biophysics, 2015, 2(4): 503-516. doi: 10.3934/biophy.2015.4.503

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  • One of the most important tasks in understanding the complex spatial organization of the genome consists in extracting information about this spatial organization, the function and structure of chromatin topological domains from existing experimental data, in particular, from genome colocalization (Hi-C) matrices. Here we present an algorithm allowing to reveal the underlying hierarchical domain structure of a polymer conformation from analyzing the modularity of colocalization matrices. We also test this algorithm on several model polymer structures: equilibrium globules, random fractal globules and regular fractal (Peano) conformations. We define what we call a spectrum of cluster borders, and show that these spectra behave strikingly di erently for equilibrium and fractal conformations, allowing us to suggest an additional criterion to identify fractal polymer conformations.


    Recently, classification systems have a wide range in many fields such as text classification, intrusion detection, bio-informatics, and image retrieval [1]. Unfortunately, a huge amount of data which may include irrelevant or redundant features is one of the main challenges of these systems. The negative effect of these undesirable features reduces the classification performance [2]. For this reason, reducing the number of features by finding an effective subset of features is an important task in classification systems [2]. Feature reduction has two techniques: feature selection and feature extraction [3]. Both reduce a high-dimensional dataset into a representative feature subset of low-dimensional. Feature extraction is effective when the original features fail to discriminate the classes [4], but it requires extra computation. Moreover, it changes the true meaning of the original features. In contrast, the feature selection preserves the true meaning of the selected features, which is important for some classification systems [5]. Furthermore, the result of FS is more understandable for domain experts [6].

    FS tries to find the best feature subset which represents the dataset well and improves the performance of classification systems [7]. It can be classified into three approaches [6]: wrapper, embedded, and filter. According to an evaluation strategy, wrapper and embedded are called classifier-dependent approaches, while filter is called classifier-independent approach [8]. In this paper, we use the filter approach according to its advantages over wrapper or embedded approaches in terms of efficiency, simplicity, scalability, practicality, and classifier-independently [6,9]. Filter approach is a pre-processing task which finds the highly ranked features to be the input of classification systems [7,10]. There are two criteria to rank features: feature relevance, and feature redundancy [11]. Feature relevance is related to how features discriminate different classes, while feature redundancy is related to how features share the same information of each other [12]. To define these criteria, filter approach uses many weighting functions which rank features based on their significance [10] such as correlation [13], mutual information (MI) [14]. MI overcomes the weakness of correlation, whereas, correlation is suitable only for linear relationship and numerical features [1]. MI is suitable for any kind of relationship such as linear and non-linear. Moreover, MI deals with both numerical and categorical features [1].

    Although MI has been widely used in many methods to find the best feature subset that maximizes the relevancy between the candidate feature and class label, and minimizes the redundancy between the candidate feature and pre-selected features [15]. The main limitations of these methods are: (1) difficult to indicate the best candidate features with the same new classification information [16], (2) difficult to deal with continuous features without information loss [17], and (3) consider the inner-class information only [18]. In this paper, we integrate fuzzy concept with mutual information to propose a new FS method called Fuzzy Joint Mutual Information Maximization (FJMIM). The fuzzy concept helps the proposed method to exploit all possible information of data where it can deal with any numerical data and extract the inner and outer-class information. Moreover, the objective function of FJMIM can overcome the feature overestimation problem which happens when the candidate feature be completely correlated with some of pre-selected features and does not depend on the majority of the subset at the same time [8].

    The rest of this paper is organized as follows: Section 2 presents the basic measures of fuzzy information theory. Then, we present the proposed method in section 3. After that, the experiment design was presented in section 4, followed by the results and discussion in section 5. Finally, section 6 concludes the paper.

    For the purpose of measuring the significance of features, information theory introduced many information measures such as entropy, and mutual information. To enhance these measures, fuzzy concept is used to estimate new extensions of information measures based on fuzzy equivalence relations such as fuzzy entropy, and fuzzy mutual information [19,20].‎ Fuzzy entropy measures the average amount of uncertainty of fuzzy relation in order to estimate its discriminative power, while fuzzy mutual information measures the shared amount of information between two fuzzy relations. In the following, we present the basic measures of fuzzy information theory:

    Given a dataset $ D = F\cup C $, where $ F $ is a set of $ n $ features, and $ C $ is the class label. Let $ {\bar{F}} = \{a_1, a_2, ….., a_m\} $ be a feature of $ m $ samples, where $ {\bar{F}} \in F $. Let $ S $ is the feature subset with $ d $ of selected features, and the remaining set is $ \{F-S\} $, where $ \bar{F}_f \in F-S $ and $ \bar{F}_s \in S $. Based on the fuzzy equivalence relation $ R_{\bar{F}} $ on $ \bar{F} $, the feature $ \bar{F} $ can be represented by the relation matrix $ M(R_{\bar{F}}) $.

    $ M(RˉF)=(r11r12r1mr21r2mrm1rm2rmm) $ (2.1)

    where $ r_{ij} = R_{\bar{F}}(a_i, a_j) $ is the fuzzy equivalence relation between two samples $ a_i $ and $ a_j $.

    In this paper, the used fuzzy equivalence relation between two elements $ a_i $ and $ a_j $ is defined as [21]:

    $ RˉF(ai,aj)=expaiaj $ (2.2)

    Fuzzy equivalence class of sample $ a_i $ on $ R_{\bar{F}} $ ‎can be defined as:‎

    $ [ai]RˉF=[ri1]a1+[ri2]a2++[rim]am $ (2.3)

    Fuzzy entropy of feature $ \bar{F_1} $ based on fuzzy equivalence relation is defined as:

    $ H(ˉF)=1mmi=1logm|[ai]RˉF| $ (2.4)

    ‎ where $ \lvert[a_i]_{R_{\bar{F}}}\rvert = \sum_{i = 1}^{m} r_{ij} $.

    Let $ \bar{F_1} $ and $ \bar{F_2} $ be two features of $ F $, fuzzy joint entropy of $ \bar{F_1} $ and $ \bar{F_2} $ is defined as: ‎

    $ H(¯F1,¯F2)=H(R¯F1,R¯F2)=1mmi=1logm|[ai]R¯F1[ai]R¯F2| $ (2.5)

    Fuzzy conditional entropy of $ \bar{F_1} $ given $ \bar{F_2} $ is defined as

    $ H(¯F1|¯F2)=H(R¯F1|R¯F2)=1mmi=1log|[ai]R¯F2||[ai]R¯F1[ai]R¯F2| $ (2.6)

    Fuzzy Mutual information between two features $ \bar{F_1} $ and $ \bar{F_2} $ is defined as:

    $ I(¯F1;¯F2)=I(R¯F1;R¯F2)=1mmi=1logm|[ai]R¯F1[ai]R¯F2||[ai]R¯F1.[ai]R¯F2| $ (2.7)

    Fuzzy conditional mutual information between feature $ \bar{F_1} $ and $ \bar{F_2} $ given class $ C $ is defined as:

    $ I(¯F1;¯F2|C)=H(¯F1|C)+H(¯F2|C)H(¯F1,¯F2|C) $ (2.8)

    Fuzzy joint mutual information between two features $ \bar{F_1} $, $ \bar{F_2} $ and class $ C $ is defined as:

    $ I(¯F1,¯F2;C)=I(¯F1;C)+I(¯F2;C|¯F1) $ (2.9)

    Fuzzy interaction information between among $ \bar{F_1} $, $ \bar{F_2} $ and $ C $ is defined as:

    $ I(¯F1;¯F2;C)=I(¯F1;C)+I(¯F2;C)I(¯F1,¯F2;C) $ (2.10)

    In this section, we presented the general theoretical frameworks of different feature selection methods based on mutual information. Then, we studied the limitation of previous work. Finally, we introduced the proposed method.

    Brown et al. [22] studied the exist feature selection methods based on MI and analyzed the different criteria to propose the following theoretical framework of these methods.

    $ J(ˉFf)=I(ˉFf;C)βˉFsSI(ˉFf;ˉFs)+γˉFsSI(ˉFf;ˉFs|C) $ (3.1)

    This framework is a linear combination of three terms: relevance, redundancy, and conditional that measures the individual predictive power of the feature, the unconditional relation, and the class-conditional relation, respectively. The criteria of different feature selection based on MI depends on the value of $ \beta $ and $ \gamma $. MIM ($ \beta = \gamma = 0 $) [23] is the simplest FS method based on MI. It considers only the relevance relation only. However, It may suffer from the redundant features. MIFS ($ \gamma = 0 $) [24] introduced two criteria to estimate the feature relevance and redundancy. An extension of MIFS, called MIFS-U [24] is proposed to improve the redundancy term of MIFS by considering the uniform distribution of the information. However, Both MIFS and MIFS-U still require an input parameter $ \beta $. To avoid this limitation, MRMR ($ \beta = \frac{1}{\lvert S\rvert} $, $ \gamma = 0 $) [25] introduced the mean of the redundancy term as automatic value to the input parameter ($ \beta $). JMI ($ \beta = \gamma = \frac{1}{\lvert S\rvert} $) [26] extended MRMR to extract the benefit of conditional term. In addition, Brown et al. [22] introduced also a similar non-linear framework to represent some methods as CMIM method [27]. According to [22], CMIM can be written as:

    $ Jcmim=minˉFsS[I(ˉFf;C|ˉFs)]=I(ˉFf;C)maxˉFsS[I(ˉFf;ˉFs)I(ˉFf;ˉFs|C)] $ (3.2)

    The reason of the non-linear relation on CMIM returns to the using of $ max $ operation. Similar to CMIM, JMIM [8] introduces a non-linear relation as follows:

    $ Jjmim=minˉFsS[I(ˉFs;C)+I(ˉFf;C|ˉFs)]=I(ˉFf;C)maxˉFsS[I(ˉFf;ˉFs)I(ˉFf;ˉFs|C)I(ˉFs;C)] $ (3.3)

    Although MI has been widely used in many feature selection methods such as MIFS [24], JMI [26], mRMR [25], DISR [28], IGFS [29], NMIFS [30] and MIFS-ND [31]. These methods suffer from the overestimation of the feature significance problem [8]. For this reason, Bennasar et al. [8] proposed JMIM method to address the overestimation of the feature significance problem. However, it may fail to select the best candidate features if they have the same new classification information. To illustrate this problem, Figure 1 shows the FS scenario, where $ \bar{F_1} $ and $ \bar{F_2} $ are two candidate features, $ \bar{F_s} $ is the pre-selected feature subset, and $ C $ is the class label. $ \bar{F_1} $ is partially redundant with $ \bar{F_s} $, while $ \bar{F_2} $ is independent to $ \bar{F_s} $. Suppose that $ \bar{F_1} $ and $ \bar{F_2} $ have the same new classification information $ I(\bar{F_1};C|\bar{F_s}) $ = (area 3) and $ I(\bar{F_2};C|\bar{F_s}) $ = (area 5) respectively. In this case, JMIM may fail to indicate the best feature where $ I(\bar{F_1}, \bar{F_s}; C) $ and $ I(\bar{F_1}, \bar{F_s}; C) $ are equal.

    Figure 1.  Venn diagram presents the feature selection scenario where the two candidate features $\bar{F_1}$ and $\bar{F_2}$ have the same new classification information.

    Unfortunately, JMIM also shares some limitations with the previous methods. Firstly, it can not directly estimate MI between continuous features [17]. To address this limitation, there are two methods were introduced. One is to estimate MI based on Parzen window [18], but it is inefficient in high-dimensional feature spaces with spare samples [17]. Moreover, its performance depends on the used window function which requires a window width parameter [32]. The other one is to discretize continuous features before estimating MI [33], but it may cause information loss [34]. Secondly, JMIM depends only on inner-class information without considering outer-class information [18].

    Motivated by the previous limitation of JMIM, we proposed a new FS method, called Fuzzy Joint Mutual Information Maximization (FJMIM). Both of FJMIM and JMIM depends on "maximum of the minimum" approach. The main difference is that JMIM maximize the joint mutual information of the candidate feature and pre-selected feature subset with class, whereas FJMIM maximizes the joint mutual information of the candidate feature and pre-selected feature subset with class without considering the class-relevant redundancy. To illustrate the difference in Figure 1, JMIM depends on the union of areas 2, 4 and 3, while FJMIM depends on the union of areas 2 and 3. The proposed method discarded the class-relevant redundancy (area 4) because it can reduce the predictive ability of the feature subset when a candidate feature is selected [15]. On the other hand, integrating fuzzy concept with MI has many benefits. Firstly, using a fuzzy concept helps to deal directly with continuous features. Furthermore, it enables MI to take the advantages of inner and outer-class information [35]. Moreover, FS methods based on fuzzy concept are more robust toward any change of the data than methods based on probability concept [36].

    According to FJMIM, the candidate feature $ \bar{F}_f $ must satisfy the following condition

    $ ˉFf=argmaxˉFfFS(minˉFsS(I(ˉFf,ˉFs;C)I(ˉFf;ˉFs;C))) $ (3.4)

    FJMIM also can be written according to the non-linear framework as follows:

    $ ˉFf=2I(ˉFf;C)maxˉFsS[I(ˉFf;ˉFs)I(C;ˉFs)I(ˉFf;ˉFs|C)I(ˉFf;C|ˉFs)] $ (3.5)

    The proposed method can be summarized to find the best feature subset of size $ d $ as follows:

    Input: $ F $ is a set of $ n $ features, $ C $ is the class label, and $ d $ is the number of selected features.

    Step 1: Initialize the empty selected feature subset $ S $.

    Step 2: Update the selected feature set $ S $ and the feature set $ F $.

    Step 2.1: Compute $ I(\bar{F}; C) $ for all $ \bar{F} $ in the feature set $ F $.

    Step 2.2: Add the feature $ \bar{F} $ that maximizes $ I(\bar{F}; C) $ to the selected feature set $ S $.

    Step 2.3: Remove the feature $ \bar{F} $ from the feature set $ F $.

    Step 3: Repeat until $ \lvert S \lvert = d $

    Step 3.1: Add the feature $ \bar{F} $ that satisfies

    $arg \max_{\bar{F}_f \in F-S} (\min_{\bar{F}_s \in S} (I(\bar{F}_f, \bar{F}_s; C)-I(\bar{F}_f; \bar{F}_s; C))) $

    to the selected feature set $ S $.

    Step 3.2: Remove the feature $ \bar{F} $ from the feature set $ F $.

    Output: Return the selected feature set $ S $.

    Success of FS methods depends on different criteria such as classification performance, and stability [11]. Consequently, we design the experiment based on these criteria (Figure 2). To clarify our improvement, we compared our proposed method FJMIM, with four conventional methods (CMIM [27], JMI [26], QPFS [37], Relief [38]) and five state-of-the-art methods (CMIM3 [39], JMI3 [39], JMIM [8], MIGM [40], and WRFS [41]). The compared methods can be divided into two groups: FS based on fuzzy concept and FS based on probability concept. For the methods which depend on probability concept, data discretization is required as a pre-processing step prior to the FS process. So, the continuous features are transformed into ten bins using EqualWidth discretization [42]. Then, we selected feature subset from all methods based on threshold which is defined as the median position of the ranked features (or the nearest integer position when the number of ranked features is even).

    Figure 2.  Experiment design of the proposed method Fuzzy Joint Mutual Information (FJMIM): firstly, a discretization pre-processing is applied before FS methods based on probability concept. Then, the FS methods are evaluated according to classification performance, and feature stability.

    FS plays an important role to improve the classification performance. There are many measures to evaluate classification performance such as classification accuracy, precision, F-measure, area-under-ROC (AUC), and area-under-PRC (AUCPR) [43]. To clarify our improvement, popular classifiers were used in this study such as Naive Bayes (NB), Support Vector Machine (SVM), and 3-Nearest Neighbors (KNN) [44]. The average classification performance measures were computed by 10-fold cross-validation approach [45].

    Another important evaluation criterion for FS is stability. FS stability measures the impact of any change in the input data on FS result [46]. In this study, we measure the impact of noise on the selected feature subset. Firstly, we produce the noise using standard deviation and the normal distribution of each feature [47]. Then, we injected 10% of the data by adding noise. After that, we repeated this step ten times. Each time produces a different sequence of selected features. Finally, we computed the stability of each method using Kuncheva stability index [48].

    Our experiment was conducted using 13 datasets from UCI machine learning repository [49]. Table 1 presents a brief description about these datasets.

    Table 1.  Datasets Description.
    No. Datasets Instances Features Classes
    1 Acute Inflammations 120 6 2
    2 Arrhythmia 452 279 13
    3 Blogger 100 6 2
    4 Diabetic Retinopathy Debrecen (DRD) 1151 20 2
    5 Hayes-Roth 160 5 3
    6 Indian Liver Patient Dataset (ILPD) 583 10 2
    7 Lenses 24 4 3
    8 Lymphography 148 18 4
    9 Congressional Voting Records (CVR) 435 16 2
    10 Sonar 208 60 2
    11 Thoracic Surgery 470 17 2
    12 Wilt 4889 6 2
    13 Zoo 101 17 7

     | Show Table
    DownLoad: CSV

    1) Accuracy: A paired two-tailed t-test is employed between FJMIM and other compared methods. The notations ([=], [+], and [-]) indicate the statistically significant (5%) that the proposed method (equals, wins, and losses) other methods. According to NB classifier, FJMIM achieved the maximum average accuracy with score 78.02%, while Relief achieved the minimum average accuracy with score 75.34% (Table 2). The proposed method outperformed compared methods in the range from 0.06 to 2.68%. In SVM classifier, FJMIM outperformed other methods with score 80.03%, while Relief achieved the minimum average accuracy with score 77.33% (Table 3). The proposed method outperformed compared methods in the range from 0.69 to 2.7%. Similarly with KNN classifier, FJMIM kept the maximum average accuracy by 81.45%, while Relief achieved the minimum average accuracy with score 77.98% (Table 4). The proposed method outperformed compared methods in the range from 0.57 to 3.47%. Across all datasets, Figure 3(a) shows the distribution of the average accuracy values of all used classifiers. In a detailed box-plot, the box represents upper and lower quartiles, while the black circle represents the median. The box-plot confirms that FJMIM is more consistent and outperformed other compared methods. Figure 3(b) shows the average accuracy of the three used classifiers. FJMIM achieved the best accuracy, followed by QPFS, JMI3, both of JMI and CMIM, both of CMIM3 and JMIM, MIGM, WRFS, and Relief respectively. The proposed method outperformed compared methods in the range from 0.6 to 2.9%.

    Table 2.  Classification accuracy using NB Classifier, FJMIM achieved the highest average accuracy.
    Dataset CMIM CMIM3 JMI JMI3 JMIM MIGM QPFS Relief WRFS FJMIM
    Acute Inflammations 100±0[=] 100±0[=] 100±0[=] 100±0[=] 100±0[=] 99.75±2.5[=] 99.75±2.5[=] 100±0[=] 90.83±8.5[+] 100±0
    Arrhythmia 60.64±7.01[=] 59.22±6.17[=] 59.42±6.49[=] 59.58±6.58[=] 60.62±6.93[=] 58.98±6.39[=] 63.7±6.57[-] 64.94±6.33[-] 59.31±6.3[=] 59.74±6.43
    Blogger 61.9±10.7[=] 61.9±10.7[=] 61.9±10.7[=] 64.9±10.2[=] 61.9±10.7[=] 61.9±10.7[=] 61.9±10.7[=] 65.4±7.84[=] 66.1±9.09[=] 65.9±10.26
    DRD 57.59±4.95[=] 60.66±4.69[-] 60.3±5.04[-] 60.6±4.58[-] 57.59±4.95[=] 60.3±5.04[-] 61.19±4.49[-] 60.61±5.06[-] 57.47±4.98[=] 57.56±4.83
    Hayes-Roth 45.81±10.77[=] 45.81±10.77[=] 45.81±10.77[=] 45.81±10.77[=] 45.81±10.77[=] 45.81±10.77[=] 49.37±10.53[=] 46.31±11.48[=] 45.81±10.77[=] 49.37±10.53
    ILPD 68.99±5.68[=] 67.33±5.87[=] 68.99±5.68[=] 67.33±5.87[=] 68.99±5.68[=] 68.99±5.68[=] 68.73±5.64[=] 64.96±6.45[+] 70.11±6.74[=] 69.54±5.75
    Lenses 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 54.83±30.73[+] 86.83±21.88[=] 86.83±21.88
    Lymphography 79.02±10.57[=] 82.05±9.59[=] 79.02±10.57[=] 77.93±10.87[=] 78.27±10.98[=] 77.25±11.38[=] 79.02±10.57[=] 82.61±9.77[=] 76.91±11.64[+] 81.5±10.29
    CVR 94.3±3.15[=] 94.32±3.12[=] 93.75±3.57[=] 93.75±3.57[=] 93.75±3.57[=] 93.75±3.57[=] 94.32±3.26[=] 93.82±3.28[=] 93.52±3.18[=] 93.75±3.57
    Sonar 75.05±8.33[=] 75.42±9.08[=] 75.75±8.85[=] 76.23±9.99[=] 72.2±9.93[=] 72.14±9.54[+] 76.42±8.69[=] 74.25±8.33[=] 77.91±8.74[=] 77±8.98
    Thoracic Surgery 83.83±3.1[=] 83.38±3.09[=] 83.38±3.09[=] 83.85±2.24[=] 83.38±3.09[=] 84.64±1.83[=] 83.55±3.08[=] 83.06±2.77[=] 83.85±2.74[=] 83.38±3.09
    Wilt 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06[=] 94.61±0.06
    Zoo 95.15±5.74[=] 96.05±5.6[=] 96.03±5.66[=] 96.03±5.66[=] 96.03±5.66[=] 96.93±5.42[=] 94.08±6.6[=] 94.08±6.6[=] 92.96±7.05[=] 95.05±5.87
    Average 77.21 77.51 77.37 77.50 76.92 77.07 77.96 75.34 76.63 78.02

     | Show Table
    DownLoad: CSV
    Table 3.  Classification accuracy using SVM Classifier, FJMIM achieved the highest average accuracy.
    Dataset CMIM CMIM3 JMI JMI3 JMIM MIGM QPFS Relief WRFS FJMIM
    Acute Inflammations 98.33±3.35[=] 99.67±1.64[=] 98.33±3.35[=] 98.33±3.35[=] 98.33±3.35[=] 99.42±2.44[=] 99.42±2.44[=] 100±0[+] 89.92±9.5[+] 99.42±2.44
    Arrhythmia 66.09±6.81[=] 64.08±6.11[=] 65.01±5.96[=] 65.7±6.23[=] 66.14±6.61[=] 65.23±6.39[=] 66.82±6.2[=] 67.22±6.05[=] 65.23±6.15[=] 67.11±6.7
    Blogger 68±4.02[=] 67.7±4.89[=] 68±4.02[=] 67.4±5.62[=] 68±4.02[=] 68±4.02[=] 67.7±4.89[=] 67.7±4.89[=] 68±4.02[=] 67.7±4.89
    DRD 68.03±3.92[=] 67.55±4.07[=] 67.15±3.85[=] 67.03±3.95[=] 68.03±3.92[=] 67.19±3.8[=] 65.1±4.33[+] 61.91±4.51[+] 67.7±4[=] 67.96±4.23
    Hayes-Roth 49.94±12.83[=] 49.94±12.83[=] 49.94±12.83[=] 49.94±12.83[=] 49.94±12.83[=] 49.94±12.83[=] 52±11.61[=] 48.75±13.21[=] 49.94±12.83[=] 52±11.61
    ILPD 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73[=] 71.36±0.73
    Lenses 76.17±22.13[=] 76.17±22.13[=] 76.17±22.13[=] 76.17±22.13[=] 76.17±22.13[=] 76.17±22.13[=] 76.17±22.13[=] 54.5±30.42[+] 76.17±22.13[=] 76.17±22.13
    Lymphography 81.57±10.06[=] 80.56±9.09[=] 81.42±10.15[=] 84.55±8.84[=] 82.02±10.48[=] 79.06±10.44[+] 81.57±10.11[=] 80.3±9[=] 74.25±11.45[+] 85.5±8.3
    CVR 94.09±3.33[=] 94.46±3.37[=] 94.3±3.34[=] 94.32±3.32[=] 94.3±3.34[=] 94.3±3.34[=] 94±3.44[=] 94.43±3.41[=] 94.37±3.41[=] 94.3±3.34
    Sonar 79.34±8.37[=] 80.48±8.05[=] 80.18±8.33[=] 78.46±8.44[=] 80.54±8.43[=] 76.93±8.71[=] 79.91±7.73[=] 82.1±8.07[=] 77.41±8.41[=] 80.67±7.88
    Thoracic Surgery 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0[=] 85.11±0
    Wilt 97.62±0.57[+] 97.97±0.49[=] 97.97±0.49[=] 97.97±0.49[=] 97.62±0.57[+] 97.97±0.49[=] 94.61±0.06[+] 97.97±0.49[=] 97.62±0.57[+] 97.98±0.49
    Zoo 95.75±5.12[=] 93.88±6.72[=] 89.39±5.72[+] 89.49±5.64[+] 89.49±5.64[+] 94.06±6.9[=] 95.27±5.85[=] 93.89±6.96[=] 90.49±7.96[+] 95.06±6.2
    Average 79.34 79.15 78.79 78.91 79.00 78.83 79.16 77.33 77.51 80.03

     | Show Table
    DownLoad: CSV
    Table 4.  Classification accuracy using KNN Classifier, FJMIM achieved the highest average accuracy.
    Dataset CMIM CMIM3 JMI JMI3 JMIM MIGM QPFS Relief WRFS FJMIM
    Acute Inflammations 99.5±1.99[=] 100±0[=] 99.5±1.99[=] 99.5±1.99[=] 99.5±1.99[=] 100±0[=] 100±0[=] 100±0[=] 99.67±1.64[=] 100±0
    Arrhythmia 58.27±5.17[+] 57.63±5.17[+] 57.79±5.34[+] 58.32±5.08[+] 58.94±5.13[=] 57.67±5.21[+] 60.73±4.64[=] 65.13±4.43[-] 58.39±5.22[+] 61.11±5.23
    Blogger 72.1±11.31[=] 72.1±11.31[=] 72.1±11.31[=] 78.8±10.18[=] 72.1±11.31[=] 72.1±11.31[=] 72.1±11.31[=] 70.3±11.41[=] 76.1±12.05[=] 78.6±11.37
    DRD 64.3±4.19[=] 62.62±5.29[=] 63.69±4.58[=] 61.96±4.68[=] 64.3±4.19[=] 63.58±4.56[=] 60.47±4.64[+] 63.51±4.63[=] 64.06±4.4[=] 64.7±4.23
    Hayes-Roth 59.19±10.41[=] 59.19±10.41[=] 59.19±10.41[=] 59.19±10.41[=] 59.19±10.41[=] 59.19±10.41[=] 63.12±11.32[=] 57.56±10.82[=] 59.19±10.41[=] 63.12±11.32
    ILPD 67.16±5.41[=] 68.22±5.06[=] 67.54±5.44[=] 68.22±5.06[=] 67.16±5.41[=] 67.3±5.42[=] 68.74±5.01[=] 67.84±5.56[=] 66.66±5.25[=] 69.15±5.62
    Lenses 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 86.83±21.88[=] 54.17±30.46[+] 86.83±21.88[=] 86.83±21.88
    Lymphography 84.61±7.58[=] 72.61±9.14[+] 84.61±7.58[=] 84.16±9.46[=] 80.99±9.49[=] 76.62±8.87[=] 84.61±7.58[=] 81.44±9.42[=] 79.74±9.36[=] 78.87±8.35
    CVR 93.63±3.43[=] 94.09±3.33[=] 95.26±3.01[=] 95.26±3.01[=] 95.26±3.01[=] 95.26±3.01[=] 94.14±3.37[=] 93.67±3.5[=] 94.09±2.95[=] 95.26±3.01
    Sonar 85.09±7.43[=] 83.51±7.68[=] 87.74±7.51[=] 85.53±8.83[=] 82.7±9.06[=] 82.4±7.72[+] 86.15±7.24[=] 87.95±6.78[=] 84.17±7.73[=] 87.65±7.27
    Thoracic Surgery 80.96±3.45[=] 81.38±3.53[=] 81.38±3.53[=] 83.45±3.05[=] 81.38±3.53[=] 82.64±3.28[=] 83.89±2.95[=] 82.68±3.08[=] 84.85±0.97[-] 81.38±3.53
    Wilt 97.69±0.57[+] 98.12±0.55[=] 98.12±0.55[=] 98.12±0.55[=] 97.69±0.57[+] 98.12±0.55[=] 94.44±0.52[+] 98.12±0.55[=] 97.69±0.57[+] 98.12±0.55
    Zoo 91.9±7.21[=] 92.06±7.35[=] 92.05±7.24[=] 92.05±7.24[=] 92.05±7.24[=] 91.16±7.48[=] 91.42±6.94[=] 91.42±6.94[=] 90.29±7.45[+] 94.08±6.6
    Average 80.09 79.10 80.45 80.88 79.85 79.45 80.51 77.98 80.13 81.45

     | Show Table
    DownLoad: CSV
    Figure 3.  Accuracy result of all compared methods. FJMIM outperformed in all cases.

    2) Precision: Figure 4 shows the precision results of NB, SVM, KNN, and their average. FJMIM achieved the highest precision, while Relief achieved the lowest precision. The proposed method outperformed other methods in the range from 0.6 to 12.7% based on NB, from 0.3 to 3.3% based on SVM, and from 2 to 8.2% based on KNN. According to the average of all classifiers, FJMIM achieved the highest precision by 71.2%, while Relief kept the lowest precision by 63.1%. The second-best method achieved by CMIM3, followed by JMI3, QPFS, JMI, JMIM, MIGM, CMIM and WRFS. The results of precision distribution are inconsistent as shown in Figure 5. Both FJMIM and QPFS achieved the best distribution. FJMIM achieved the highest upper quartile and median, while QPFS achieved the highest lower quartile and median. In addition, FJMIM shares the highest upper quartile with JMI, JMIM and MIGM.

    Figure 4.  Precision result of NB, SVM, KNN and their average. FJMIM achieved the best result in all cases.
    Figure 5.  Precision distribution across all datasets.

    3) F-measure: Figure 6 shows the F-measure results of the three used classifiers and their average. FJMIM achieved the highest F-measure by 79.8, 71.5 and 84% on NB, SVM and KNN respectively. Relief achieved the lowest F-measure on NB and SVM, while WRFS achieved the lowest score on SVM. The proposed method outperformed other methods in the range from 0.3 to 16.6% based on NB, from 0.1 to 1.4% based on SVM, and from 0.6 to 15.2% based on KNN. According to the average of all classifiers, FJMIM achieved the highest precision by 78.4% and outperformed other methods in the range from 2.5 to 10.8%. The second-best method achieved by JMI3, followed by QPFS, CMIM3, JMI, JMIM, CMIM, MIGM, WRFS and Relief. Figure 7 shows the distribution of F-measure across all datasets. The box-plot confirms the outperformance of FJMIM compared to other methods.

    Figure 6.  F-measure result of NB, SVM, KNN and their average. FJMIM achieved the best result in all cases.
    Figure 7.  F-measure distribution across all datasets.

    4) AUC: Figure 8 shows the AUC results of the used classifiers and their average. FJMIM achieved the highest AUC on NB and KNN by 83.9 and 85.2%, while it achieved the second-best score on SVM by 74.8%. On the other hand, Relief achieved the lowest AUC on all classifiers. Although MIGM achieved the best AUC on SVM, FJMIM outperformed on the average of all classifiers. The proposed method outperformed other methods in the range from 0.4 to 4.2%. As shown in the box-plot (Figure 9), the proposed method achieved the highest lower quartile and median values. On the other hand, JMI achieved also the highest lower quartile, while CMIM achieved the highest upper quartile.

    Figure 8.  AUC result of NB, SVM, KNN and their average. FJMIM achieved the best result in all cases except SVM.
    Figure 9.  AUC distribution across all datasets.

    5) AUCPR: The highest AUCPR was achieved by both FJMIM and CMIM3 using NB, MIGM using SVM, and FJMIM using KNN (Figure 10). On the other hand, Relief achieved the lowest AUCPR using all classifiers. According to the average of all classifiers, FJMIM achieved the best AUCPR by 81.6%, while Relief kept the lowest AUCPR by 77.2%. The proposed method outperformed other methods in the range from 0.3 to 4.4%. The second-best AUCPR was achieved by CMIM3, followed by MIGM, both QPFS and JMIM, both CMIM and JMI, and both JMI3 and WRFS. Figure 11 shows the distribution of AUCPR across all datasets. It's obvious that CMIM achieved the highest median, lower, and upper quartiles, while FJMIM achieved the highest median, upper quartile.

    Figure 10.  AUCPR result of NB, SVM, KNN and their average. FJMIM achieved the best result in all cases except SVM.
    Figure 11.  AUCPR distribution across all datasets.

    Figure 12(a) shows the stability of the used FS methods on all datasets. It's obvious that FJMIM is more consistent and stable compared to all other methods. Figure 12(b) confirms the stability of the compared method. FJMIM achieved the highest average stability by 87.8%. The proposed method outperformed other methods in the range from 6.6 to 43%. JMI achieved the second-best position by 81.2%, while Relief achieved the lowest stability by 44.3%.

    Figure 12.  Stability result of all compared methods. FJMIM achieved the best result in all cases.

    More detailed results are presented in the appendix section. According to previous results, It is obvious that FJMIM achieves the best results in most measures. This is expected because our proposed method addresses the feature overestimation problem and handle the candidate feature problem well. Moreover, it avoids the discretization step. Another reason is the advantages of both inner and outer class information which FJMIM depends on it. This information helps the proposed method to be more robust toward the noise. On the other hand, the other compared methods are close to FJMIM than Relief that achieved the lowest result. This is because all compared method except Relief depends on mutual information as the proposed method to estimate the significant of features.

    In this paper, we propose a new FS method, called, Fuzzy Joint Mutual Information Maximization (FJMIM). The proposed method depends on integrating an improved JMIM objective function with fuzzy concept. The benefits of our proposed method include: 1) The ability to deal directly with discrete and continuous features. 2) The suitability to handle any kind of relations between features such as linear and non-linear relation. 3) The ability to take the advantages of inner and outer class information. 4) The robustness toward the noise. 5) The ability to select the most significant feature subset and avoiding the undesirable features.

    To confirm the effectiveness of FJMIM, 13 benchmark datasets have been used to evaluate the proposed method in the term of classification performance (accuracy, precision, F-measure, AUC, and AUCPR) and feature selection stability. According to nine conventional and state-of-the-art feature selection methods, the proposed method achieved promising improvement on feature selection in the terms of classification performance and stability.

    In future work, we plan to extend the proposed method to cover multi-label classification problem. Moreover, we plan to study the effectiveness of imbalanced data on the proposed method.

    This research has been supported by the National Natural Science Foundation (61572368).

    The authors declare no conflict of interest.

    Tables A1A6 show some numerical results of Figures 3(b), 4, 6, 8, 10 and 12(b), while Figures A1–A4 show the statistical results (mean ± standard deviation) of some datasets (DRD, Sonar, Wilt and Zoo) by precision, F-measure, AUC and AUCPR, respectively.

    Figure A1.  The precision results of four datasets (DRD, Sonar, Wilt and Zoo) on the used classifiers (NB, SVM and KNN).
    Figure A2.  The F-measure results of four datasets (DRD, Sonar, Wilt and Zoo) on the used classifiers (NB, SVM and KNN).
    Figure A3.  The AUC results of four datasets (DRD, Sonar, Wilt and Zoo) on the used classifiers (NB, SVM and KNN).
    Figure A4.  The AUCPR results of four datasets (DRD, Sonar, Wilt and Zoo) on the used classifiers (NB, SVM and KNN).
    Table A1.  Average accuracy of compared FS methods.
    FS methodAverage
    CMIM 78.9
    CMIM3 78.6
    JMI 78.9
    JMI3 79.1
    JMIM 78.6
    MIGM 78.4
    QPFS 79.2
    Relief 76.9
    WRFS 78.1
    FJMIM 79.8

     | Show Table
    DownLoad: CSV
    Table A2.  Precision results according to the used classifiers and their average.
    FS method NB SVM KNN Average
    CMIM 0.702 0.641 0.704 0.682
    CMIM3 0.742 0.641 0.705 0.696
    JMI 0.705 0.640 0.713 0.686
    JMI3 0.712 0.636 0.723 0.690
    JMIM 0.701 0.642 0.707 0.683
    MIGM 0.701 0.637 0.708 0.682
    QPFS 0.712 0.636 0.711 0.686
    Relief 0.621 0.612 0.661 0.631
    WRFS 0.698 0.632 0.708 0.679
    FJMIM 0.748 0.645 0.743 0.712

     | Show Table
    DownLoad: CSV
    Table A3.  F-measure results according to the used classifiers and their average.
    FS method NB SVM KNN Average
    CMIM 0.722 0.712 0.818 0.750
    CMIM3 0.795 0.712 0.752 0.753
    JMI 0.724 0.712 0.823 0.753
    JMI3 0.729 0.712 0.834 0.759
    JMIM 0.719 0.714 0.821 0.751
    MIGM 0.722 0.710 0.818 0.750
    QPFS 0.732 0.712 0.825 0.756
    Relief 0.632 0.708 0.688 0.676
    WRFS 0.716 0.701 0.826 0.748
    FJMIM 0.798 0.715 0.840 0.784

     | Show Table
    DownLoad: CSV
    Table A4.  AUC results according to the used classifiers and their average.
    FS method NB SVM KNN Average
    CMIM 0.835 0.733 0.812 0.793
    CMIM3 0.836 0.748 0.844 0.809
    JMI 0.837 0.734 0.815 0.795
    JMI3 0.831 0.729 0.805 0.788
    JMIM 0.836 0.742 0.812 0.796
    MIGM 0.832 0.754 0.807 0.798
    QPFS 0.838 0.710 0.807 0.785
    Relief 0.804 0.707 0.801 0.771
    WRFS 0.831 0.732 0.806 0.789
    FJMIM 0.839 0.748 0.852 0.813

     | Show Table
    DownLoad: CSV
    Table A5.  AUCPR results according to the used classifiers and their average.
    FS method NB SVM KNN Average
    CMIM 0.848 0.708 0.798 0.785
    CMIM3 0.862 0.715 0.862 0.813
    JMI 0.850 0.708 0.797 0.785
    JMI3 0.837 0.705 0.800 0.781
    JMIM 0.852 0.713 0.797 0.787
    MIGM 0.857 0.724 0.827 0.803
    QPFS 0.857 0.708 0.797 0.787
    Relief 0.842 0.685 0.791 0.772
    WRFS 0.850 0.697 0.795 0.781
    FJMIM 0.862 0.718 0.869 0.816

     | Show Table
    DownLoad: CSV
    Table A6.  Average stability of compared FS methods.
    FS method Stability
    CMIM 72.6
    CMIM3 67.3
    JMI 81.2
    JMI3 75.2
    JMIM 75.9
    MIGM 66.2
    QPFS 71.7
    Relief 44.3
    WRFS 66.6
    FJMIM 87.8

     | Show Table
    DownLoad: CSV
    [1] de Gennes PG (1979) Scaling Concepts in Polymer Physics, Cornell University Press.
    [2] Grosberg AY, Khokhlov AR (1994) Statistical Physics of Macromolecules, AIP Press.
    [3] Rubinstein M, Colby RH (2003) Polymer Physics, Oxford:OUP.
    [4] Grosberg AY, Nechaev SK, Shakhnovich EI (1988) The role of topological constraints in the kinetics of collapse of macromolecules. J Phys France 49(12):2095-2100.
    [5] Grosberg A, Rabin Y, Havlin S, et al. (1993) Crumpled globule model of the three-dimensional structure of DNA. EPL-Europhys Lett 23(5):373.
    [6] Rosa A, Everaers R (2008) Structure and dynamics of interphase chromosomes. PLoS Comput Biol 4(8):e1000153.
    [7] Lieberman-Aiden E, van Berkum NL, Williams L, et al. (2009) Comprehensive mapping of longrange interactions reveals folding principles of the human genome. Science 326(5950):289-293.
    [8] Mirny LA (2011) The fractal globule as a model of chromatin architecture in the cell. Chromosome Res 19(1):37-51.
    [9] Halverson JD, Smrek J, Kremer K, et al. (2014) From a melt of rings to chromosome territories: the role of topological constraints in genome folding. Rep Prog Phys 77(2):022601.
    [10] Grosberg AY (2014) Annealed lattice animal model and Flory theory for the melt of nonconcatenated rings: towards the physics of crumpling. Soft Matter 10:560-565. doi: 10.1039/C3SM52805G
    [11] Rosa A, Everaers R (2014) Ring polymers in the melt state: The physics of crumpling. Phys Rev Lett 112:118302. doi: 10.1103/PhysRevLett.112.118302
    [12] Nazarov LI, TammMV, Avetisov VA, et al. (2015) A statistical model of intra-chromosome contact maps. Soft Matter 11(5):1019-1025.
    [13] Tamm MV, Nazarov LI, Gavrilov AA, et al. (2015) Anomalous diffusion in fractal globules. Phys Rev Lett 114:178102. doi: 10.1103/PhysRevLett.114.178102
    [14] Bunin G, Kardar M (2015) Coalescence model for crumpled globules formed in polymer collapse. Phys Rev Lett 115:088303. doi: 10.1103/PhysRevLett.115.088303
    [15] Sachs RK, van den Engh G, Trask B, et al. (1995) A random-walk/giant-loop model for interphase chromosomes. P Natl Acad Sci 92(7):2710-2714.
    [16] Münkel C, Langowski J (1998) Chromosome structure predicted by a polymer model. Phys Rev E 57:5888-5896. doi: 10.1103/PhysRevE.57.5888
    [17] Ostashevsky J (1998) A polymer model for the structural organization of chromatin loops and minibands in interphase chromosomes. Mol Biol Cell 9(11):3031-3040.
    [18] Mateos-Langerak J, Bohn M, de Leeuw W, et al. (2009) Spatially confined folding of chromatin in the interphase nucleus. P Natl Acad Sci 106(10):3812-3817.
    [19] Iyer BVS, Arya G (2012) Lattice animal model of chromosome organization. Phys Rev E86:011911.
    [20] Barbieri M, Chotalia M, Fraser J, et al. (2012) Complexity of chromatin folding is captured by the strings and binders switch model. P Natl Acad Sci 109(40):16173-16178.
    [21] Fritsch CC, Langowski J (2010) Anomalous diffusion in the interphase cell nucleus: The effect of spatial correlations of chromatin. J Chem Phys 133(2):025101.
    [22] Fritsch CC, Langowski J (2011) Chromosome dynamics, molecular crowding, and diffusion in the interphase cell nucleus: a monte carlo lattice simulation study. Chromosome Res 19(1):63-81.
    [23] Dekker J, Rippe K, Dekker M, et al. (2002) Capturing chromosome conformation. Science 295(5558):1306-1311.
    [24] Naumova N, Imakaev M, Fudenberg G, et al. (2013) Organization of the mitotic chromosome. Science 342(6161):948-953.
    [25] Nagano T, Lubling Y, Stevens TJ, et al. (2013) Single-cell Hi-C reveals cell-to-cell variability in chromosome structure. Nature 502(7469):59-64.
    [26] Newman MEJ (2004) Analysis of weighted networks. Phys Rev E 70:056131. doi: 10.1103/PhysRevE.70.056131
    [27] Lancichinetti A, Fortunato S (2011) Limits of modularity maximization in community detection. Phys Rev E 84:066122. doi: 10.1103/PhysRevE.84.066122
    [28] Granell C, Gomez S, Arenas A (2012) Hierarchical multiresolution method to overcome the resolution limit in complex networks. Int J Bifurcat Chaos 22(07):1250171.
    [29] Arenas A, Fernandez A, Gomez S (2008) Analysis of the structure of complex networks at different resolution levels. New J Phys 10(5):053039.
    [30] Fortunato S (2010) Community detection in graphs. Phys Rep 486(3-5):75-174.
    [31] Newman MEJ (2006) Modularity and community structure in networks. P Natl Acad Sci 103(23):8577-8582.
    [32] Krzakala F, Moore C, Mossel E, et al. (2013) Spectral redemption in clustering sparse networks. P Natl Acad Sci 110(52):20935-20940.
    [33] Arenas A, Diaz-Guilera A, Kurths J, et al. (2008) Synchronization in complex networks. Phys Rep 469(3):93-153.
    [34] Sethna JP (2006) Statistical Mechanics: Entropy, Order Parameters and Complexity, Oxford:OUP.
    [35] Mezard M, Montanari A, (2009) Information, Physics, and Computation, Oxford:OUP.
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