Engin A. Sungur

A copula method for modeling directional dependence of genes

BackgroundGenes interact with each other as basic building blocks of life, forming a complicated network. The relationship between groups of genes with different functions can be represented as gene networks. With the deposition of huge microarray data sets in public domains, study on gene networking is now possible. In recent years, there has been an increasing interest in the reconstruction of gene networks from gene expression data. Recent work includes linear models, Boolean network models, and Bayesian networks. Among them, Bayesian networks seem to be the most effective in constructing gene networks. A major problem with the Bayesian network approach is the excessive computational time. This problem is due to the interactive feature of the method that requires large search space. Since fitting a model by using the copulas does not require iterations, elicitation of the priors, and complicated calculations of posterior distributions, the need for reference to extensive search spaces can be eliminated leading to manageable computational affords. Bayesian network approach produces a discretely expression of conditional probabilities. Discreteness of the characteristics is not required in the copula approach which involves use of uniform representation of the continuous random variables. Our method is able to overcome the limitation of Bayesian network method for gene-gene interaction, i.e. information loss due to binary transformation.ResultsWe analyzed the gene interactions for two gene data sets (one group is eight histone genes and the other group is 19 genes which include DNA polymerases, DNA helicase, type B cyclin genes, DNA primases, radiation sensitive genes, repaire related genes, replication protein A encoding gene, DNA replication initiation factor, securin gene, nucleosome assembly factor, and a subunit of the cohesin complex) by adopting a measure of directional dependence based on a copula function. We have compared our results with those from other methods in the literature. Although microarray results show a transcriptional co-regulation pattern and do not imply that the gene products are physically interactive, this tight genetic connection may suggest that each gene product has either direct or indirect connections between the other gene products. Indeed, recent comprehensive analysis of a protein interaction map revealed that those histone genes are physically connected with each other, supporting the results obtained by our method.ConclusionThe results illustrate that our method can be an alternative to Bayesian networks in modeling gene interactions. One advantage of our approach is that dependence between genes is not assumed to be linear. Another advantage is that our approach can detect directional dependence. We expect that our study may help to design artificial drug candidates, which can block or activate biologically meaningful pathways. Moreover, our copula approach can be extended to investigate the effects of local environments on protein-protein interactions. The copula mutual information approach will help to propose the new variant of ARACNE (Algorithm for the Reconstruction of Accurate Cellular Networks): an algorithm for the reconstruction of gene regulatory networks.

A Note on Directional Dependence in Regression Setting

ABSTRACT In this article, we define and study the concept of directional dependence in bivariate regression setting by using copulas. We consider two cases of directional dependence; one originating from marginals and the other originating from the joint behavior of variables. We also generalize and clarify the results given by Dodge and Rousson (2000) and Muddapur (2003).

Community Service Statistics Projects

Abstract Statistics instructors know that interesting, real-world problems are crucial to motivate student learning. As an extension of our efforts to build student interest and ownership in applications, we recently incorporated service learning into our statistics courses. Service learning provides an active-learning experience associated with a community service application. In this article we describe our experiences using service learning in our statistics courses. We give examples of projects used at the University of Minnesota-Morris, a public, liberal arts college.

Diagonal copulas of archimedean class

In this paper the concept of diagonal copulas is introduced and its properties are examined. It is shown that for the Archimedean class, the diagonal copula uniquely determines the corresponding copula. This fact helps to reduce the dimension and makes it easier to understand the underlying dependence structure without losing any information. Therefore diagonal copulas can be used in various stages of dependence model building, selection, fitting and diagnostics. In this paper we consider some of the possibilities on estimation, test of hypotheses and graphical data analysis.

Partial correlation with copula modeling

We propose a new partial correlation approach using Gaussian copula. Our empirical study found that the Gaussian copula partial correlation has the same value as that which is obtained by performing a Pearsons partial correlation. With the proposed method, based on canonical vine and d-vine, we captured direct interactions among eight histone genes.

Some Observations on Copula Regression Functions

ABSTRACT The main objective of this article is to introduce an alternative way of looking at regression analysis by using copulas. To achieve our objective we work on copula regression function, study its properties, and discuss advantages that will come out from our approach.

Generalized bivariate copulas and their properties

Copulas are useful devices to explain the dependence structure among variables by eliminating the influence of marginals. In this paper, we propose a new class of bivariate copulas to quantify dependency and incorporate it into various iterated copula families. We investigate properties of the new class of bivariate copulas and derive the measure of association, such as Spearmans �, Kendalls � , and the regression function for the new class. We also provide the concept of directional dependence in bivariate regression setting by using copulas.

Some results on truncation dependence invariant class of copulas

ABSTRACT In this paper, m-dimensional distribution functions with truncation invariant dependence structure are studied. Some of the properties of generalized Archimedean class of copulas under this dependence structure are presented including some results on the conditions of compatibility. It has been shown that Archimedean copula generalized as it is described by Jouini and Clemen[1] which has the truncation invariant dependence structure has to have the form of independence or Cook-Johnson copula. We also consider a multi-parameter class of copulas derived from one-parameter Archimedean copulas. It has been shown that this class has a probabilistic meaning as a connecting copula of the truncated random pair with a right truncation region on the third variable. Multi-parameter copulas generated in this paper stays in the Archimedean class. We provide formulas to compute Kendalls tau and explore the dependence behavior of this multi-parameter class through examples.

Dependence information in parameterized copulas

Copulas are useful devices to explain the dependence structure between variables by eliminating the influence of marginals. In this paper we develop a formula for better understanding of the dependence mechanism hidden in copulas and discuss some approximations on copulas. We explore the copula of multivariate normal distribution in detail and try to reveal interesting features of the class.

Truncation invariant dependence structures

In this paper, a special class of m-dimensional distribution functions which can be uniquely determined in terms of their 2-dimensional marginals is studied. The members of the class can be characterized as having truncation invariant dependence structure. The representation given in this paper provides a physical meaning to the multivariate Cook-Johnson distribution, and introduces a systematic way of generating higher dimensional distributions by using rich 2-dimensional distributions provided that the 2-dimensional marginals are compatible. A class of 3-dimensional multivariate normal distribution has been generated and bounds in terms of lower dimensional marginals are provided.