 Research Article
 Open access
 Published:
On the Impact of Entropy Estimation on Transcriptional Regulatory Network Inference Based on Mutual Information
EURASIP Journal on Bioinformatics and Systems Biology volume 2009, Article number: 308959 (2008)
Abstract
The reverse engineering of transcription regulatory networks from expression data is gaining large interest in the bioinformatics community. An important family of inference techniques is represented by algorithms based on information theoretic measures which rely on the computation of pairwise mutual information. This paper aims to study the impact of the entropy estimator on the quality of the inferred networks. This is done by means of a comprehensive study which takes into consideration three stateoftheart mutual information algorithms: ARACNE, CLR, and MRNET. Two different setups are considered in this work. The first one considers a set of 12 synthetically generated datasets to compare 8 different entropy estimators and three network inference algorithms. The two methods emerging as the most accurate ones from the first set of experiments are the MRNET method combined with the newly applied Spearman correlation and the CLR method combined with the Pearson correlation. The validation of these two techniques is then carried out on a set of 10 public domain microarray datasets measuring the transcriptional regulatory activity in the yeast organism.
1. Introduction
The inference of regulatory networks by modeling dependencies at the transcription level aims at providing biologists with an additional insight about cell activities. This task belongs to the domain of systems biology which studies the interactions between the components of biological systems and how these interactions give rise to the function and the behavior of the whole system. This approach differs from the socalled "reductionist approach" that limits its focus to the building blocks of the system without providing a global picture of the cell behavior, as stated in [1]:
"the reductionist approach has successfully identified most of the components and many of the interactions but, unfortunately, offers no convincing concepts or methods to understand how system properties emerg the pluralism of causes and effects in biological networks is better addressed by observing, through quantitative measures, multiple components simultaneously and by rigorous data integration with mathematical models."
The reverse engineering of transcriptional regulatory networks (TRNs) from expression data is known to be a very challenging task because of the large amount of noise intrinsic to the microarray technology, the high dimensionality, and the combinatorial nature of the problem. Also, a genetogene network inferred on the basis of transcriptional measurements returns only a rough approximation of a complete biochemical regulatory network since many physical connections between macromolecules might be hidden by shortcuts. Notwithstanding, in recent years, computational techniques have been applied with success to this domain, as witnessed by successful validations of the interaction networks predicted by the algorithms [2].
Network inference consists in representing the stochastic dependencies between the variables of a dataset by means of a graph. Mutual information networks are an important category of network inference methods.
Informationtheoretic approaches typically rely on the estimation of mutual information (MI) from expression data in order to measure the statistical dependence between genes [3]. In these methods, a link between two nodes is established if it exhibits a significant score estimated by mutual information. The role of the mutual information estimator is therefore essential to guarantee a high accuracy rate. Notwithstanding, few experimental studies about the impact of the estimator on the quality of the inferred network exist [4]. To the best of our knowledge, this paper presents the first comprehensive experimental comparison of several mutual information estimation techniques and stateoftheart inference methods like MRNET [3], ARACNE [5], and CLR [6]. An additional contribution of this paper is the study of the impact of the correlation estimator (notably Spearman and Pearson) on the mutual information computation once a hypothesis of normality is done. Interestingly enough, the Spearman and the Pearsonbased information estimators emerge as the most competitive techniques once combined with the MRNET and the CLR inference strategies, respectively.
The first part of the experimental session aims at studying the sensitivity to noise and missing values of different discretization, estimation, and network inference methods. For this purpose, a synthetic benchmark is created by means of the SynTReN data generator [7].
In the second part, the techniques which appeared to be the most effective in the synthetic session are assessed by means of a biological microarray benchmark which integrates several public domain yeast microarray datasets.
The outline of the paper is as follows. Section2 reviews the most important mutual information estimators. Section 3 introduces some stateoftheart network inference algorithms. Section 4 contains the description of the synthetic data generator, the description of the real data setting, and the related discussions of the results. Section 5 concludes the paper.
2. Estimators of Information
An information theoretic network inference technique aims at identifying connections between two genes (variables) by estimating the amount of information between them. Different information measures exist in the literature [8]. In this article, we focus on the mutual information measure and the related estimation techniques. Note that, if the estimation technique has been conceived for discrete random variables, a discretization procedure has to be executed before applying the estimation procedure to expression data.
2.1. Mutual Information
Mutual information is a wellknown measure which quantifies the stochastic dependency between two random variables without making any assumption (e.g., linearity) about the nature of the relation [9].
Let be a discrete random vector whose th component takes values in the discrete set of size . The th element of the mutual information matrix (MIM) associated to is defined by
where the entropy of a discrete random variable is defined as
and is the mutual information between the random variables and .
2.2. Entropy Estimation
In practical setups, the underlying distribution of the variables is unknown. Consequently, the entropy terms in (1) cannot be computed directly but require an estimation. Many different approaches to entropy estimation have been introduced. In this paper, we restrict the choice to the following five estimators: empirical, MillerMadow, shrink, Pearson and Spearman correlation.
2.2.1. Empirical
Let be a continuous random variable taking values in the real interval . Suppose the interval is partitioned into bins, where denotes the bin index vector, denotes the number of data points in the th bin, and stands for the total number of observations.
The empirical estimator, also known as the maximum likelihood estimator, is the entropy of the empirical distribution
It has been shown in [10] that the asymptotic bias of this estimator amounts to
2.2.2. MillerMadow
The MillerMadow estimator [10] corrects the biased empirical estimator by removing the estimated bias term from it (4):
This estimator reduces the bias of (3) without increasing its variance.
2.2.3. Shrink
The shrink estimator [8] combines two different estimators, one with low variance and one with low bias, by using the weighting factor :
Let
be the value minimizing the mean square function [8]. It has been shown in [11] that the optimal is given by
2.2.4. Pearson Correlation
Correlation is a statistic measuring the strength and the direction of the linear relationship between two random variables. The Pearson correlation between two random variables and is defined as
Correlation takes values in , where denotes a linear relation between the variables and . If the variables are independent, the correlation is equal to zero while the opposite is not necessarily true (e.g., nonlinear dependency).
It can be shown that correlation and mutual information are related if the joint distribution is normal.
Let
be the density of a multivariate Gaussian variable with mean and covariance matrix . The entropy of this distribution is given by
where is the determinant of the covariance matrix [12]. The mutual information between two variables and is then given by
where is the Pearson's correlation.
Since the functional relation (13) between the mutual information and the correlation is a monotone function, it is sufficient to use when computing this value.
The Pearson correlation can be estimated from the measurements and of two genes and by the following equation:
2.2.5. Spearman Correlation
The Spearman rank correlation coefficient is a special case of the Pearson correlation in which the data are converted to rankings before calculating the coefficient.
The Spearman correlation can be calculated using (14), where the terms and are replaced by their respective ranks. Note that the Spearman rank correlation coefficient generalizes the Pearson correlation coefficient by being able to detect not only linear relationships between the variables but also any kind of monotone relation without making any assumptions about the distributions of the variables.
2.3. Discretization Methods
The mutual information estimators in Sections 2.2.1, 2.2.2, and 2.2.3 apply to discrete random variables. In order to use them for continuous random variables, a discretization step is required. The two most widely used methods for discretization are the equal width and the equal frequency methods [13].
Equal Width. This discretization method partitions the domain of into subintervals of equal size. As a consequence, the number of data points in each bin is likely to be different.
Equal Frequency. This method divides the interval into subintervals, each containing the same number of data points. It follows that subinterval sizes are typically different.
The number of subintervals should be chosen so that all bins contain a significant number of samples. In [14], the authors propose to use , where is the total number of samples.
3. Network Inference Algorithms
The network inference proceeds in two steps. In the first step, the mutual information matrix is calculated. In the second step, the chosen algorithm is applied to the mutual information matrix in order to compute a score that is used to weigh the links between network nodes.
3.1. The Mrnet Method
The MRNET method [3] is based on the maximum relevance/minimum redundancy (MRMR) feature selection technique [15]. This iterative selection technique chooses at each step, among the least redundant variables, the one having the highest mutual information with the target.
The method ranks the set of inputs according to a score which is the difference between the mutual information with the output variable (maximum relevance) and the average mutual information with the previously ranked variables (minimum redundancy). The network is inferred by deleting all edges whose score lies below a given threshold.
Direct interactions should be well ranked whereas indirect interactions should be badly ranked. In the first step, variable which has the highest mutual information to the target is selected. The second selected variable will be the one with a high information to the target and at the same time a low information to the previously selected variable.
In the next steps, given a set of selected variables, the criterion updates by choosing the variable that maximizes the score
which can be described as a relevance term minus a redundancy term.
For each pair , the algorithm returns two scores and and computes the maximum of the two. All edges with a score below a given threshold are then deleted.
3.2. The Aracne Method
The algorithm for the reconstruction of accurate cellular networks (ARACNEs) [5] is based on the data processing inequality [16]. This inequality states that if the interaction between and depends on , then
The algorithm assigns a weight to each pair of nodes which is equal to the mutual information between the variables. Then, the minimal mutual information between three variables is computed, and eventually the edge with the lowest value is interpreted as an indirect connection and removed if the difference between the two lowest weights is above a given threshold.
3.3. The Clr Method
In the context likelihood or relatedness (CLR) algorithm [6], the mutual information is calculated for each pair of variables. Then, a score related to the empirical distribution of these MI values is computed. In particular, instead of considering the information between two variables and , the algorithm takes into account the score , where
and and are, respectively, the mean and the standard deviations of the empirical distribution of the mutual information values .
4. Experiments
This section contains two parts. In the first part, several inference methods and estimators are applied to synthetic datasets with different noise and missing values configurations. The aim of this part is to identify the best combination of estimator and inference method. Once the assessment on the synthetic benchmark is done, the best performing techniques are then applied to a biological problem. The aim of this second experiment is to assess the capability of the algorithms of discovering interactions of the yeast transcriptome uniquely on the basis of expression data.
All computations were carried out using the Rpackage MINET [17] (http://cran.rproject.org/web/packages/minet). This recently introduced package allows the use of three different inference methods, namely, ARACNE [5], CLR [6], and MRNET [3].
The following entropy estimators are also made available to calculate the mutual information: empirical, MillerMadow, shrink, and Pearson correlation.
Note that, in order to apply the first three estimators to expression data, two different discretization methods are implemented: equal frequency and equal width discretization with default size .
4.1. Synthetic Data
4.1.1. Network Generation
The synthetic benchmark relies on several artificial microarray datasets generated by the SynTReN generator [7]. This simulator emulates the gene expression process by adopting topologies derived from subnetworks of E.coli and S.cerevisiae networks. Interaction kinetics are modeled by nonlinear differential equations based on MichaelisMenten and Hill kinetics.
We used the SynTReN generator to create twelve benchmark datasets whose number of samples and number of genes are detailed in Table 1.
4.1.2. Introducing Missing Values
In order to study the impact of missing values, expression values were removed from the generated datasets. The number of missing values is distributed according to the distribution with parameters and . The maximal allowed number of missing values is a third of the entire dataset. This distribution was utilized, instead of the uniform distribution, because the latter one could have favored the empirical estimator.
4.1.3. Setup
For each experiment, ten repetitions were carried out. Each dataset was analyzed using three inference methods (i.e., MRNET, ARACNE, and CLR) and the following estimators: Pearson correlation, empirical, MillerMadow, shrink, and the Spearman correlation coefficient. The empirical, the MillerMadow, and the shrink estimator were computed applying the equal width and the equal frequency discretization approaches. Furthermore, the computation was carried out with and without additive Gaussian noise (having 50% variance of the observed values). Each of these setups was also assessed with introduced missing values.
4.1.4. Validation
Network inference algorithms infer either the presence or the absence of an edge for each pair of nodes. Similarly to classification, we define the possible outcomes of inference as follows. A true positive (TP) occurs when an edge is correctly predicted as existing, a false positive (FP) occurs when a nonexisting edge is inferred, true negative (TN) occurs when a nonexisting edge is not inferred, and false negative (FN) occurs when an existing edge is not detected.
Once the numbers of TP, FP, TN, and FN are computed, we can measure precision and recall
Precision measures the fraction of real edges among the ones classified as positive while recall quantifies the fraction of real edges that are correctly inferred.
A weighted harmonic average of precision and recall is returned by the score [18]:
which attains its maximum value when the returned network is without any error.
To validate the simulation's results, the maximal score was computed for each experiment. Using a paired ttest, the maximal scores were then compared and statistically validated.
4.1.5. Discussion of Results
The results of the synthetic benchmark are collected in Table 2 which returns the score for each combination of inference method, mutual information estimator, and nature of the dataset (noisy versus not noisy, complete versus missing data). Note that the maximal score is highlighted together with the scores which are not significantly different from the best.
We analyze the results according to four different aspects: the impact of the estimator, the impact of the discretization, the impact of the inference algorithm, and the influence of sample and network size.
The section concludes with the identification of the best combination of inference algorithm and estimator.
Impact of The Estimator. In case of complete datasets with no noise, the empirical and the MillerMadow estimators with equalfrequency binning lead to the highest scores for the MRNET and the ARACNE inference methods. The Spearman correlation is not significantly different from the best, in case of ARACNE, and is close to the best in case of MRNET. The CLR method is less sensitive to the estimator, and the best result is obtained with the Pearson correlation.
In case of noisy data or missing value (NA) configurations, the Pearson correlation and the Spearman correlation lead to the highest score for all inference methods. A slight better accuracy of the Pearson correlation can be observed in presence of missing values. The Spearman correlation outperforms the other estimators in MRNET and ARACNE when complete yet noisy datasets are considered. In CLR, Pearson and Spearman correlations lead the ranking without being significantly different.
Impact of The Discretization. In case of complete datasets with no noise, the equal frequency binning approach outperforms the equal width binning approach for all discrete estimators. The gap between the two discretization methods is clearly evident in MRNET and less striking in ARACNE and CLR. In case of noisy or missing data configurations, differences are attenuated.
Impact of The Inference Algorithm. In case of complete datasets with no noise, the MRNET inference technique outperforms the other algorithms.
The situation changes in presence of noisy or missing values. Here, CLR appears to be the most robust by returning the highest scores for all combinations of noise and missing values.
Impact of Number of Sample and Network Sizes. The role of network size is illustrated in Figure 1 (first row) which shows how the score decreases as long as the network size increases. This behavior can be explained by the increasing difficulty of recovering a larger underlying network in front of an increasing dimensionality of the modeling task.
In Figure 1 (second row), the values of the score seem not to be influenced substantially by the number of samples.
Conclusion. A concise summary of the previously discussed results is displayed in Table 3 which averages the accuracy over the different data configurations.
It emerges that the most promising combinations are represented by the MRNET algorithm with the Spearman estimator and the CLR algorithm with the Pearson correlation. The former seems to be less biased because of its good performance in front of nonnoisy datasets while the latter seems to be more robust since it is less variant in front of additive noise.
4.2. Biological Data
The second part of the experimental session aims to assess the performance of the two selected techniques once applied to a real biological task.
We proceeded by (i) setting up a dataset which combines several public domain microarray datasets about the yeast transcriptome activity, (ii) carrying out the inference with the two selected techniques, and (iii) assessing the quality of the inferred network with respect to two independent sources of information: the list of interactions measured by means of an alternative genomic technology and a list of biologically known gene interactions derived from the TRANSFAC database.
4.2.1. The Dataset
The dataset was built by first normalizing and then joining ten public domain yeast microarray datasets, whose number of samples and origin is detailed in Table 4. The resulting dataset contains the expression of 6352 yeast genes in 711 experimental conditions.
4.2.2. Assessment by ChipChip Technology
The first validation of the network inference outcome is obtained by comparing the inferred interactions with the outcome of a set of ChIPchip experiments. The ChIPchip technology, detailed in [28], measures the interactions between proteins and DNA by identifying the binding sites of DNAbinding proteins. The procedure can be summarized as follows. First, the protein of interest is crosslinked with the DNA site it binds to, then doublestranded parts of DNA fragments are extracted. The ones which were crosslinked to the protein of interest are filtered out from this set and reverse crosslinked. Also, their DNA is purified. In the last step, the fragments are analyzed using a DNA microarray in order to identify genegene connections. For our purposes, it is interesting to remark that the ChIpchip technology returns for each pair of genes a probability of interaction. In particular we use, for the validation of our inference procedures, the ChIpchip measures of the yeast transcriptome provided in [29].
4.2.3. Assessment by Biological Knowledge
The second validation of the network inference outcome relies on existing biological knowledge and in particular on the list of putative interactions in Saccharomyces cerevisiae published in [30].
This list contains 1222 interactions involving 725 genes, and in the following we will refer to this as the Simonis list.
4.2.4. Results
In order to make a comparison with the Simonis list of known interactions, we limited our inference procedure to the genes contained in the list.
The quantitative assessment of the final results is displayed by means of receiver operating characteristics (ROCs) and the associated area (AUC). This curve compares the true positive rate (TPR) to the false positive rate (FPR) which are defined as follows:
Note that this assessment considers as true only the interactions contained in the Simonis list.
Figure 2 displays the ROC curves, and Table 5 reports the associated AUC for the following techniques: the ChIPchip technique, the MRNETSpearman correlation combination, the CLRGaussian combination, the CLRMillerMadow combination, the MRNETMillerMadow combination, and the random guess.
A first consideration to be made about these results is that network inference methods are able to be significantly better than a random guess also in real biological settings. Also the two combinations which appeared to be the best in synthetic datasets confirmed their supremacy over the MillerMadowbased techniques also in real data.
However, the weak, though significative, performance of the networks inferred from microarray data requires some specific considerations.

(1)
With respect to the ChIPchip technology, it is worth mentioning that the information coming from microarray datasets is known to be less informative than the one coming from the ChIPchip technology. Microarray datasets remain nowadays however more easily accessible to the experimental community, and techniques able to extract complex information from them are still essential for system biology purposes.

(2)
Both the microarray dataset we set up for our experiment and the list of known interactions we used for assessment are strongly heterogeneous and concern different functionalities in yeast. We are confident that more specific analysis on specific functionalities could increase the final accuracy.

(3)
Like in any biological validation of bioinformatics methods, the final assessment is done with respect to a list of putative interactions. It is probable that some of our false positives could be potentially true interactions or at least deserve additional investigation.
5. Conclusion
The paper presented an experimental study of the influence of the information measure and the estimator on the quality of the inferred interaction network. The study concerned both synthetic and real datasets.
The study on synthetically generated datasets allowed to identify two effective techniques with complementary properties. The MRNET method combined with the Spearman correlation appeared to be effective mainly in front of complete and accurate measures. The CLR method combined with the Pearson correlation was ranked as the best one in the case of noisy and missing values.
The experiments on real microarray data confirmed the potential of these inference methods and showed that, though in presence of noisy and heterogeneous datasets, the techniques are able to return significative results.
References
Sauer U, Heinemann M, Zamboni N: Getting closer to the whole picture. Science 2007, 316(5824):550551. 10.1126/science.1142502
Gardner TS, Faith JJ: Reverseengineering transcription control networks. Physics of Life Reviews 2005, 2(1):6588. 10.1016/j.plrev.2005.01.001
Meyer PE, Kontos K, Lafitte F, Bontempi G: Informationtheoretic inference of large transcriptional regulatory networks. EURASIP Journal on Bioinformatics and Systems Biology 2007, 2007:9.
Olsen C, Meyer PM, Bontempi G: On the impact of entropy estimator in transcriptional regulatory network inference. Proceedings of the 5th International Workshop on Computational Systems Biology (WCSB '08)Leipzig, Germany June 2008
Margolin AA, Nemenman I, Basso K, et al.: ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinformatics 2006, 7(supplement 1, article S7):115.
Faith JJ, Hayete B, Thaden JT, et al.: Largescale mapping and validation of Escherichia coli transcriptional regulation from a compendium of expression profiles. PLoS Biology 2007, 5(1, article e8):13. 10.1371/journal.pbio.0050001
Van den Bulcke T, Van Leemput K, Naudts B, et al.: SynTReN: a generator of synthetic gene expression data for design and analysis of structure learning algorithms. BMC Bioinformatics 2006, 7, article 43: 112.
Hausser J: Improving entropy estimation and the inference of genetic regulatory networks, M.S. thesis. Department of Biosciences, National Institute of Applied Sciences of Lyon, Cedex, France; 2006.
Steuer R, Kurths J, Daub CO, Weise J, Selbig J: The mutual information: detecting and evaluating dependencies between variables. Bioinformatics 2002, 18(supplement 2):S231S240.
Paninski L: Estimation of entropy and mutual information. Neural Computation 2003, 15(6):11911253. 10.1162/089976603321780272
Schäfer J, Strimmer K: A shrinkage approach to largescale covariance matrix estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology 2005., 4(1, article 32):
Haykin S: Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River, NJ, USA; 1999.
Dougherty J, Kohavi R, Sahami M: Supervised and unsupervised discretization of continuous features. Proceedings of the 20th International Conference on Machine Learning (ICML '95) 194202. Tahoe City, Calif, USAJuly 1995
Yang Y, Webb GI: On why discretization works for naïvebayes classifiers. Proceedings of the 16th Australian Joint Conference on Artificial Intelligence (AI '03) Perth, Australia December 2003 440452.
Ding C, Peng H: Minimum redundancy feature selection from microarray gene expression data. Journal of Bioinformatics and Computational Biology 2005, 3(2):185205. 10.1142/S0219720005001004
Cover TM, Thomas JA: Elements of Information Theory. John Wiley & Sons, New York, NY, USA; 1990.
Meyer PE, Lafitte F, Bontempi G: minet: Mutual Information Network Inference. R package version 1.1.3
Sokolova M, Japkowicz N, Szpakowicz S: Beyond accuracy, fscore and roc: a family of discriminant measures for performance evaluation. Proceedings of the AAAI Workshop on Evaluation Methods for Machine Learning Boston, Mass, USA July 2006s
DeRisi JL, Iyer VR, Brown PO: Exploring the metabolic and genetic control of gene expression on a genomic scale. Science 1997, 278(5338):680686. 10.1126/science.278.5338.680
Chu S, DeRisi J, Eisen M, et al.: The transcriptional program of sporulation in budding yeast. Science 1998, 282(5389):699705.
Spellman PT, Sherlock G, Zhang MQ, et al.: Comprehensive identification of cell cycleregulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization. Molecular Biology of the Cell 1998, 9(12):32733297.
Ferea TL, Botstein D, Brown PO, Rosenzweig RF: Systematic changes in gene expression patterns following adaptive evolution in yeast. Proceedings of the National Academy of Sciences of the United States of America 1999, 96(17):97219726. 10.1073/pnas.96.17.9721
Gasch AP, Spellman PT, Kao CM, et al.: Genomic expression programs in the response of yeast cells to environmental changes. Molecular Biology of the Cell 2000, 11(12):42414257.
Gasch AP, Huang M, Metzner S, Botstein D, Elledge SJ, Brown PO: Genomic expression responses to DNAdamaging agents and the regulatory role of the yeast ATR homolog Mec1p. Molecular Biology of the Cell 2001, 12(10):29873003.
Hughes TR, Marton MJ, Jones AR, et al.: Functional discovery via a compendium of expression profiles. Cell 2000, 102(1):109126. 10.1016/S00928674(00)000155
Ogawa N, DeRisi J, Brown PO: New components of a system for phosphate accumulation and polyphosphate metabolism in Saccharomyces cerevisiae revealed by genomic expression analysis. Molecular Biology of the Cell 2000, 11(12):43094321.
Godard P, Urrestarazu A, Vissers S, et al.: Effect of 21 different nitrogen sources on global gene expression in the yeast Saccharomyces cerevisiae . Molecular and Cellular Biology 2007, 27(8):30653086. 10.1128/MCB.0108406
Buck MJ, Lieb JD: ChIPchip: considerations for the design, analysis, and application of genomewide chromatin immunoprecipitation experiments. Genomics 2004, 83(3):349360. 10.1016/j.ygeno.2003.11.004
Harbison CT, Gordon DB, Lee TI, et al.: Transcriptional regulatory code of a eukaryotic genome. Nature 2004, 430(7004):99104.
Simonis N, Wodak SJ, Cohen GN, van Helden J: Combining pattern discovery and discriminant analysis to predict gene coregulation. Bioinformatics 2004, 20(15):23702379. 10.1093/bioinformatics/bth252
Acknowledgments
The authors intend to thank Professor Jacques Van Helden and Kevin Kontos for useful suggestions and comments and for providing and formatting the datasets of the biological experience.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Olsen, C., Meyer, P.E. & Bontempi, G. On the Impact of Entropy Estimation on Transcriptional Regulatory Network Inference Based on Mutual Information. J Bioinform Sys Biology 2009, 308959 (2008). https://doi.org/10.1155/2009/308959
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/308959