Causality Inference Techniques for In-Silico Gene Regulatory Network

Causality Inference Techniques for In-Silico Gene Regulatory Network Swarup Roy1 , Dipankar Das1 , Dhrubajyoti Choudhury1 , Gunenja G. Gohain1 , Ramesh Sharma1 , and Dhruba K. Bhattacharyya2 1

Dept of IT, North Eastern Hill University, Shillong 793022, Meghalaya, India [email protected], {dipankar.rinku.das,dchoudhury325,gunenja}@gmail.com, [email protected] 2 Dept of CSE, Tezpur University, Napaam 784028, Assam, India [email protected]

Abstract. Causality detection in gene regulatory networks (GRN) is a challenging problem due to the limit of available data and lack of efficiency in the existing techniques. A number of techniques proposed so far to reconstruct GRN. However, majority of them ignore drawing causality among genes which indicates regulatory relationship. In this paper, we study few techniques available for inferring causality. We select four state-of-the-art causality detection techniques namely, Bayesian network, Granger causality, Mutual information(MI) and Transfer entropy based approach for our study. Performance of the techniques are evaluated using DREAM challenge data based on associated in-silico regulatory networks. Experimental results reveal the superiority of MI based approach in terms of prediction accuracy in comparison to other techniques. Keywords: causality, gene regulatory networks, microarray, prediction, Bayesian, Granger, mutual information, transfer entropy.

1

Introduction

Biological network presents an integrated way to look into the dynamic behaviour of the cellular system through the interactions of components. Biological networks may be categorised [1] as metabolic pathways, signal transduction pathways, gene regulatory networks, protein-protein interaction(PPI) [2] networks. Advent of micro-array technology has enabled system biologist to study the dynamic behaviour of genes with respect to different conditions [36]. Due to availability of large collection of microarray data and next generation sequencing technologies, it is now possible to reconstruct or reverse engineer the cellular system in-silico. Gene Regulatory Networks (GRN) is a collection of genes in a cell which interact with each other and with other substances in the cell such as proteins or metabolites, thereby governing the rates at which genes in the network are transcribed into mRNA. Mathematically, GRN can be represented as directed graph, where node represents gene or gene products and edge represents biochemical processes like reaction, transformation, interaction, activation, inhibition. R. Prasath and T. Kathirvalavakumar (Eds.): MIKE 2013, LNAI 8284, pp. 432–443, 2013. c Springer International Publishing Switzerland 2013 

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In GRN, causal information is one of the important component in inferring regulatory relationship between the genes or gene products. Causation refers to the relation that exists between the cause and its effect, where the effect is an outcome of the cause. It makes a huge contribution in GRN and represented as directed graph. The directed edges in GRNs correspond to causal influences between gene-activities (nodes). These could include regulation of transcription by transcription factors, but also less intuitive causal effects between genes involving signal-transduction or metabolism.

Fig. 1. Possible causal relationship between five nodes in a GRN

A causal effect may be direct or indirect [3]. A gene may influence activity of other gene or gene product directly. On the other hand, a gene may influence activity of other gene or itself by coding a transcription factor (TF) that in turn regulates another gene or itself. A possible causal relationship in GRN is shown in Fig. 1. Apparently, four different types of causal relationship may be possible in a living cell. Based on the above figure we can derive following relationship. 1. A gene can enhance the activity of more than one gene (relationship between A, B, C and D). 2. A gene’s activity may be influenced by more than one gene (relationship between B, D and F) 3. Gene can also influence the activity of itself (node B). 4. A gene may inhibit activity of other gene (D inhibit E). Inhibition or negative regulation may also follow above three relationship i.e. many-to one, one-tomany and self. A number of techniques have been proposed for network construction [4–8, 37]. Many approaches use statistical [38], machine learning or soft-computing techniques [9] as discovery tools. Broadly the techniques can be classified into two categories [10, 11], (i) supervised and (ii) unsupervised approach. Supervised approach requires prior knowledge about the regulatory interactions to infer novel interactions and unsupervised approach infer networks exclusively based on data (e.g. differential gene expression). Most exiting techniques infer regulatory networks without causality information. They depict GRN as an undirected graph, where edge represents some kind of association among the genes

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(e.g. correlation or mutual information). Sometime they referred as co-expression network [12, 13]. A few computational techniques are available for reconstruction of GRN that represent GRN as directed graph with causal information. In this work, we review some of the state-of-the-art GRN construction techniques that compute casual relationship between the genes. Prediction accuracy are evaluated using in-silico regulatory networks along with associated gene expression data.

2

Causality Detection Techniques

Causality is considered to be fundamentals to natural science like physics, biology and also a topic studied from the perspectives of economics, philosophy and statistics. Inspired from different causality finding techniques applied in statistics, economics or physics, a number of computational techniques proposed so far for reconstruction of GRN with causality. In this work we consider only four benchmark techniques for analysis and comparison. Below we present a brief discussion on all the techniques. 2.1

Bayesian Network

A pioneering work by Friedman et al. [14, 15] introduced Bayesian networks as a probabilistic tool for the identification of regulatory genes using high throughput experimental data [16, 17]. A Bayesian network [18] represents the joint probability distribution of a set of random variables and captures the dependencies and conditional independencies between variables in a graphical manner. A Bayesian network is a representation of a joint probability distribution. This representation consists of two components. The first component, G, is a directed acyclic graph (DAG) whose vertices correspond to the random variables X1 , · · · , Xn . The second component, Θ, describes a conditional distribution for each variable, given its parents in G. Together, these two components specify a unique distribution on X1 , · · · , Xn . The graph G represents conditional independence assumptions that allow the joint distribution to be decomposed, economizing on the number of parameters. The graph G encodes the Markov assumption i.e. each variable Xi is independent of its non-descendants, given its parents in G. By applying the chain rule of probabilities and properties of conditional independences, any joint distribution that satisfies Markov property can be decomposed into the product form P (X1 , · · · , Xn ) =

n 

P (Xi |parent(Xi )).

(1)

i=1

Learning Bayesian Networks: The problem of learning a Bayesian network can be stated as follows. Given a training set D = {X 1 , · · · , X n } of independent instances of X, find a network B = (G, Θ) that best matches D. The common

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approach to this problem is to introduce a statistically motivated scoring function that evaluates each network with respect to the training data and to search for the optimal network according to this score. Score-based learning algorithms are general purpose heuristic optimization algorithms. It searches all the possible DAGs and uses a scoring function to evaluate each stage of the search process. One of the popular score and search based method is the Hill Climbing algorithm [19]. Initially it creates a solution and that solution is the best solution until the heuristic finds a better one. The steps involved in the Hill Climbing techniques are as follows: 1. Initially a solution to the problem is created. 2. Next a small part of the solution is changed to create a slightly different solution. The changes that can be applied to the graph are arc addition, deletion and removal operation. After performing these operations we get a new solution. Now if the new solution is better than the original then the new solution is taken as the current working solution and it is used to create new solutions. 3. If it is not better it is rejected and a different change is made to the original solution. 4. At some point none of the available changes will produce an improvement. When none of the small changes produce an improvement the solution is known as a local optimum. The search process is normally stopped when a local optimum is reached. To mitigate this problem the algorithm is restarted. Discussion: It is effective in detecting non-linear relationship and easy to recognize the dependence and independence between nodes. Bayesian network is acyclic in nature i.e. there can be no cycle or loop in it. However, in real gene regulatory network loop can not be ignored. Moreover, construction of the network is a NP-hard (nondeterministic polynomial-time hard) problem. The search space increases super-exponentially if the number of variables increases in the network. 2.2

Granger Causality

Clive J Granger introduced the Granger causality tests [20], to analyze the effect of one time series on another one. He thought out of the box and said that ‘regressions’ does not only show ‘correlations’ but if certain tests are performed on them they may reveal information about causality. It was then widely used in economics but now a days it has found its application in neuroscience, bioinformatics and some other domains. Granger causality is applied successfully to identify gene-gene interactions from mRNA experiment data to elucidate biological process in disease development. It also used to discover GRN from DNA microarray time-series data [21–24]. Granger causality is a statistical concept of causality that is based on prediction. According to Granger causality, if a signal A “Granger-causes” (or “Gcause”) a signal B, then past values of A should contain information that helps

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predict future value of B above and beyond the information contained in past values of B alone. This means that A can help in reducing the errors which were inevitable if the calculation was done using the values of B only. G-causality is normally tested in the context of linear regression models. For illustration, consider a bivariate linear autoregressive model of two variables X1 and X2 :

X1 (t) = X2 (t) =

p  j=1 p  j=1

A11,j X1 (t − j) + A21,j X1 (t − j) +

p  j=1 p 

A12,j X2 (t − j) + E1 (t) A22,j X2 (t − j) + E2 (t)

(2)

j=1

where p is the maximum number of lagged observations included in the model (the model order), the matrix A contains the coefficients of the model (i.e., the contributions of each lagged observation to the predicted values of X1 (t) and X2 (t) , and E1 and E2 are residuals (prediction errors) for each time series. If the variance of E1 (or E2 ) is reduced by the inclusion of the X2 (or X1 ) terms in the first (or second) equation, then it is said that X2 (or X1 ) Granger-(G)-causes X1 (or X2 ). In other words, X2 G-causes X1 if the coefficients in A12 are jointly significantly different from zero. This can be tested by performing an F-test of the null hypothesis that A12 = 0, given assumptions of covariance stationarity on X1 and X2 . The magnitude of a G-causality interaction can be estimated by the logarithm of the corresponding F-statistic. Discussion: Granger causality is not necessarily true causality. If both X and Y are driven by a common third process with different lags, one might still accept the alternative hypothesis of Granger causality. Yet, manipulation of one of the variables would not change the other. Indeed, the Granger test is designed to handle pairs of variables, and may produce misleading results when the true relationship involves three or more variables. 2.3

Mutual Information Based Technique

Mutual Information is an effective information theoretic measure applied in gene expression for finding biologically significant relationship among genes [25]. However, like correlation, mutual information (MI) along can’t able to give causality between two variables. Catharina Olsen et al. used conditional mutual information with MI to successfully infer causal relation in GRN [26]. Mutual information between two events is defined as the information that one event contains about the occurrence of the other event in a particular environment. Mutual information of two variables reacts as the mutual reduction in uncertainty of one by knowing the other one. On the other hand, conditional mutual information is the expected value of the mutual information of two random variables given the value of a third.

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The mutual information I(X; Y) between two random variables X and Y is then defined as: I(X; Y ) = H(X) + H(Y ) − H(X, Y ) (3) and conditional MI between random variables X and Y given Z is defined as: I(X, Y |Z) = H(X|Z) + H(Y |Z) − H(X, Y |Z)

(4)

where, H(X), H(Y ), H(X, Y ) are the measure of uncertainty in the values of the genes X, Y and (X, Y ) and conditional entropy H(X|Y ) quantifies the amount of information needed to describe the outcome of a random variable X given that the value of another random variable Y is known. Based on MI and conditional MI, they calculated a interaction score between the variables X, Y and Z as: C(X, Y, Z) = I(X, Y )I(X, Y |Z)

(5)

where, I(X, Y) is the MI between X, Y and I(X, Y |Z) is conditional MI between X, Y assuming Z has already occurred. Catharina Olsen et al. used the concept of V-structure in their work. If in a network three nodes X, Y, Z is connected as X → Y ← Z, then B is called a collider and the structure is called a v-structure. The negative value of the interaction information C(X, Y, Z), implies Z is a collider. Following are the steps involved in above approach: 1. Inference of the undirected network using a method able to cope with the usually high number of variables (e.g. ARACNE [8], CLR [27] or MRNET [28]). 2. Estimate the interaction information for all possible v-structures. 3. Use the interaction information criterion to orient the v-structures. By using the undirected graph obtained in the first step, the interaction information for all possible v-structures is measured. Colliders are detected based on negative value of interaction information. In the third step, all the v-structures are orientated. The orientation can be done by arranging the valid values calculated in the previous step in random order, in decreasing order, or in decreasing order of average interaction information values which are undetectable by the MI based technique. Discussion: The graph obtained in this approach is a mixed graph, containing both the directed and the undirected edges, which is an important component in real GRN. The MI based approach works well only with v-structure in a graph. As shown in Fig. 1, other then v shaped structure a number of other structures are also available in a GRN. 2.4

Transfer Entropy Based Technique

Schreiber [29] developed the concept of Transfer Entropy between two processes to capture the nonlinearities that are common in real-world system. The Transfer

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Entropy is developed from the concept of Kullback entropy [30]. It measures the uncertainty reduction in inferring the future state of a process by learning the current and past states of other processes. The transfer entropy is the amount of information flow from one process to the other. Originally applied in Physics, transfer entropy later applied in inferring genetic networks [31, 32]. Let Xi and Yi be two time series with discrete states xi , yi at time i. Assume that the series can be approximated by a stationary Markov process of order k. Then the dynamical structure of that process is reflected by the transition (k) probability p(xi+1 |Xi ). The true transition probability is usually not known (k) so that one has to assume a prior transition probability q(xi+1 |Xi ). Suppose that the future state xi of Xi depends on k past states of Xi but not on the l past states of Yi then the generalized Markov property holds: (k)

(l)

(k)

p(xi+1 |Xi , yi ) = p(xi+1 |Xi )

(6)

If there is a dependence of X on Y , it can be quantified by Kullback entropy (k) (l) (k) with p(xi+1 |Xi , yi ) as underlying transition probability and p(xi+1 |Xi ) as a prior transition probability. The amount of information transferred from Y to X is defined using Transfer Entropy as follows: TY →X =

 (k) (l) xi+1 ,Xi ,yi

(k)

(k)

(l)

p(xi+1 , Xi , yi ) log

(l)

p(xi+1 |Xi , yi ) (k)

p(xi+1 |Xi )

(7)

By assessing the transfer entropy between all pair of genes one can infer a causal network of genes and then apply a heuristic rule to differentiate indirect and direct causal relations. Tung et al. [32] proposed a three steps method for reconstruction of gene regulatory network from microarray time series data. The steps are as follows: 1. Quantify causality relations between all pair of genes by measuring the transfer entropy between their time series data. 2. Estimate the significant levels of all causality relations and select ones whose significant level is greater than a predefined threshold value. The selected relations are used to construct a directed graph. 3. Refine the graph by identifying and removing edges which are consider as indirect causal relations. Discussion: The transfer entropy cannot only identify the linear causality but also nonlinear causality. It also requires a low computational effort and applicable to a large-scale analysis because it is a model-free network reconstruction method which is based on pair wise statistical measure of causal relations. Along with the advantages discussed there are some disadvantages of transfer entropy. The causal relationships inferred by transfer entropy are often misleading when the underlying system contains indirect connections, dominance of neighboring dynamics, or anticipatory couplings.

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Table 1. Properties of different causality inference techniques

Nodes Edges Input Parameters Causality

Bayesian Network Random variables Joint probability distribution Multivariate

Granger Mutual Transfer Causality Information Entropy Random Random Random variables variables variables Linear Conditional Transition regression MI probability Bivariate Multivariate Bivariate

Non linear

Linear

Linear

Non linear

The overall properties of the four techniques are summarized in Table 1. Next, we evaluate all the four causality finding techniques experimentally in terms of accuracy of GRN prediction.

3

Performance Evaluation

In this section we compare the strength of the candidate causality inference techniques for GRN against prediction accuracy. We use in-silico gene regulatory networks and associated gene expression data from DREAM (Dialogue for Reverse Engineering Assessments and Methods) network inference challenge, provided by Marbach’s Dream Net Weaver [33] platform. Dream3 and Dream4 are the two challenges that are available. Dream3 involves fifteen benchmark datasets, five each of various sizes (10, 50 and 100). The structures of the benchmark networks are obtained by extracting modules from real biological networks. At each size, two of the networks are extracted from the regulatory network of E. coli and Yeast. Dream4 is very similar to Dream3 containing a total of 10 networks, five of each size, 10 and 100. The in silico datasets generated based on [33] platform for our experiments are characterized in Table 2. Table 2. In silico DREAM Challenge datasets Challenges Dataset In silico Size of network the network 1 Ecoli1 10 2 Ecoli2 10 Dream3 3 Yeast1 50 5 Yeast2 50 Dream4 4 insilico1 10 6 insilico2 10

Prediction effectiveness is compared against the actual networks generated from in slico DREAM Challenge data, using two different metrics for evaluating accuracy: AUPvR (Area under Precision vs Recall curve) and AUROC (Area under Receiver-Operator Characteristics curve) score. The ROC is also known

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as a relative operating characteristic curve, because it is a comparison of two operating characteristics (True Positive Rate and False Positive Rate) as the criterion changes [34]. ROC curves may not be the appropriate measure when a dataset contains large skews in the class distribution, which is commonly the case in transcriptional network inference. As an alternative, precision vs. recall (PvR) curves are considered for measuring prediction accuracy [35]. ROC curves are commonly used to evaluate prediction results. However, PvR curve may be more sensitive when there is a much larger negative set than positive set. Computing the area under the curve (AUC) of a ROC or PvR is a way to reduce ROC or PvR performance to a single value, representing expected performance. The effectiveness of prediction by the four techniques on all the datasets are shown in Fig. 2. From the figure (Fig. 2) it is evident that MI based technique outperforms rest three candidate techniques in terms of network prediction on two different scores. In case of dataset 1, MI based technique achieved a very high AU(PvR) score of .82 and AUROC of .74. In all cases performance of Bayesian network

(a) AU(PvR) curve of different algorithms

(b) AUROC curve showing prediction performance Fig. 2. Performance comparison of four causality inference techniques on in silico dataset

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are not effective. Transfer entropy and Granger causality performs variably for different datasets.

4

Conclusion

In this work we study and analyze four techniques such as Bayesian network, Granger causality, Mutual information and Transfer entropy for inferring causality among the genes to form a Gene Regulatory Network. Based on experimental results it is evident that the Mutual information based techniques yields significantly better results than the other techniques. Majority of the techniques for drawing causal relationship are not biologically motivated. A gene may up- or down-regulate another gene in a network which is missing in the above techniques. Moreover, they consider only few network structures from real GRN. A new causality inference technique which is biologically motivated and draw all the relationship between the genes in a network (Fig. 1) is a current issue in gene regulatory network inference research.

References 1. Tavazoie, S., Hughes, J., Campbell, M., Cho, R., Church, G., et al.: Systematic determination of genetic network architecture. Nature Genetics 22, 281–285 (1999) 2. Panchenko, A., Przytycka, T.: Protein-protein interactions and networks: identification, computer analysis, and prediction, vol. 9. Springer (2008) 3. Brazhnik, P., de la Fuente, A., Mendes, P.: Gene networks: how to put the function in genomics. TRENDS in Biotechnology 20(11), 467–472 (2002) 4. Friedman, N., Linial, M., Nachman, I., Pe’er, D.: Using bayesian networks to analyze expression data. J. of Computational Biology 7(3-4), 601–620 (2000) 5. Davidich, M., Bornholdt, S.: Boolean network model predicts cell cycle sequence of fission yeast. PLoS One 3(2), e1672 (2008) 6. Segal, E., et al.: Rich probabilistic models for gene expression. Bioinformatics 17(suppl. 1), S243–S252 (2001) 7. Kuo, W., et al.: Functional relationships between gene pairs in oral squamous cell carcinoma. In: Proc. of AMIA Symposium, pp. 371–375 (2003) 8. Margolin, A., Nemenman, I., Basso, K., Wiggins, C., Stolovitzky, G., Favera, R., Califano, A.: Aracne: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinformatics 7(suppl. 1), S7 (2006) 9. Mitra, S., Das, R., Hayashi, Y.: Genetic networks and soft computing. IEEE/ACM Transactions on Computational Biology and Bioinformatics 8(1), 94–107 (2011) 10. De Smet, R., Marchal, K.: Advantages and limitations of current network inference methods. Nature Reviews Microbiology 8(10), 717–729 (2010) 11. Ambroise, J., Robert, A., Macq, B., Gala, J.L., et al.: Transcriptional network inference from functional similarity and expression data: a global supervised approach. Statistical Applications in Genetics and Molecular Biology 11(1), 1–24 (2012) 12. Fuente, A.D.L.: What are gene regulatory networks? In: Handbook of Research on Computational Methodologies in Gene Regulatory Networks, pp. 1–27 (2010) 13. Lee, H., Hsu, A., Sajdak, J., Qin, J., Pavlidis, P.: Coexpression analysis of human genes across many microarray data sets. Genome Research 14(6), 1085–1094 (2004)

442

S. Roy et al.

14. Friedman, N.: Inferring cellular networks using probabilistic graphical models. Science 303(5659), 799–805 (2004) 15. Friedman, N., Linial, M., Nachman, I., Pe’er, D.: Using bayesian networks to analyze expression data. Journal of Computational Biology 7(3-4), 601–620 (2000) 16. Dondelinger, F., Husmeier, D., L`ebre, S.: Dynamic Bayesian networks in molecular plant science: inferring gene regulatory networks from multiple gene expression time series. Euphytica 183(3), 361–377 (2012) 17. Zou, M., Conzen, S.D.: A new dynamic Bayesian network (DBN) approach for identifying gene regulatory networks from time course microarray data. Bioinformatics 21(1), 71–79 (2005) 18. Pearl, J.: Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann (1988) 19. G´ amez, J.A., Mateo, J.L., Puerta, J.M.: Learning bayesian networks by hill climbing: efficient methods based on progressive restriction of the neighborhood. Data Mining and Knowledge Discovery 22(1-2), 106–148 (2011) 20. Granger, C.W.: Investigating causal relations by econometric models and crossspectral methods. Econometrica: Journal of the Econometric Society, 424–438 (1969) 21. Zhang, Z.G., Hung, Y.S., Chan, S.C., Xu, W.C., Hu, Y.: Modeling and identification of gene regulatory networks: a granger causality approach. In: 2010 International Conference on Machine Learning and Cybernetics (ICMLC), vol. 6, pp. 3073–3078. IEEE (2010) 22. Mukhopadhyay, N.D., Chatterjee, S.: Causality and pathway search in microarray time series experiment. Bioinformatics 23(4), 442–449 (2007) 23. Nagarajan, R., Upreti, M.: Granger causality analysis of human cell-cycle gene expression profiles. Statistical Applications in Genetics and Molecular Biology 9(1) (2010) 24. Tam, G.H.F., Chang, C., Hung, Y.S.: Gene regulatory network discovery using pairwise Granger causality. IET Systems Biology 7(5), 195–204 (2013) 25. Priness, I., Maimon, O., Ben-Gal, I.: Evaluation of gene-expression clustering via mutual information distance measure. BMC Bioinformatics 8(1), 111 (2007) 26. Olsen, C., Meyer, P.E., Bontempi, G.: Inferring causal relationships using informationtheoretic measures. In: Proceedings of the 5th Benelux Bioinformatics Conference, BBC 2009 (2009) 27. Faith, J., Hayete, B., Thaden, J., Mogno, I., Wierzbowski, J., Cottarel, G., Kasif, S., Collins, J., Gardner, T.: Large-scale mapping and validation of escherichia coli transcriptional regulation from a compendium of expression profiles. PLoS Biology 5(1), e8 (2007) 28. Meyer, P., Kontos, K., Lafitte, F., Bontempi, G.: Information-theoretic inference of large transcriptional regulatory networks. EURASIP Journal on Bioinformatics and Systems Biology 2007 (2007) 29. Schreiber, T.: Measuring information transfer. Physical Review Letters 85(2), 461 (2000) 30. Cover, T.M., Thomas, J.A.: Elements of information theory. John Wiley & Sons (2012) 31. Banerji, C.R., Severini, S., Teschendorff, A.E.: Network transfer entropy and metric space for causality inference. Physical Review E 87(5), 052814 (2013) 32. Tung, T.Q., Ryu, T., Lee, K.H., Lee, D.: Inferring gene regulatory networks from microarray time series data using transfer entropy. In: Twentieth IEEE International Symposium on Computer-Based Medical Systems, CBMS 2007, pp. 383–388. IEEE (2007)

Causality Inference Techniques

443

33. Marbach, D., Schaffter, T., Mattiussi, C., Floreano, D.: Generating realistic in silico gene networks for performance assessment of reverse engineering methods. Journal of Computational Biology 16(2), 229–239 (2009) 34. Swets, J.A.: Signal detection theory and ROC analysis in psychology and diagnostics: Collected papers. Lawrence Erlbaum Associates, Inc. (1996) 35. Craven, J.: Markov networks for detecting overlapping elements in sequence data. In: Advances in Neural Information Processing Systems 17: Proceedings of the 2004 Conference, vol. 17, p. 193. MIT Press (2005) 36. Roy, S., Bhattacharyya, D.K., Kalita, J.K.: CoBi: Pattern based co-regulated biclustering of gene expression data. Pattern Recognition Letters 34(14), 1669–1678 (2013) 37. Roy, S., Bhattacharyya, D.K.: Reconstruction of genetic networks in yeast using support based approach. In: Trendz in Information Sciences & Computing (TISC), pp. 116–121. IEEE (2010) 38. Roy, S., Bhattacharyya, D.K.: Mining strongly correlated item pairs in large transaction databases. International Journal of Data Mining, Modelling and Management 5(1), 76–96 (2013)