School of Mathematics and Statistics

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    A bayesian framework for data and hypotheses driven fusion of high throughput data: Application to mouse organogenesis
    ( 2008-12-01) Bhattacharjee, Madhuchhanda ; Pritchard, Colin ; Nelson, Peter
    In this paper we present a framework for integrating diverse data sets under a coherent probabilistic setup. The necessity of a probabilistic modeling arises from the fact that data integration does not restrict to compiling information from data bases with data that are typically thought to be non-random. Currently wide range of experimental data is also available however rarely these data sets can be summarized in simple output data, e.g. in categorical form. Moreover it may not even be appropriate to do so. The proposed setup allows modeling not only the observed data and parameters of interest but most importantly to incorporate prior knowledge. Additionally the setup easily extends to facilitate more popular data-driven analysis. © 2008 World Scientific Publishing Co. Pte. Ltd.
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    A bayesian mixed regression based prediction of quantitative traits from molecular marker and gene expression data
    ( 2011-11-07) Bhattacharjee, Madhuchhanda ; Sillanpää, Mikko J.
    Both molecular marker and gene expression data were considered alone as well as jointly to serve as additive predictors for two pathogen-activity-phenotypes in real recombinant inbred lines of soybean. For unobserved phenotype prediction, we used a Bayesian hierarchical regression modeling, where the number of possible predictors in the model was controlled by different selection strategies tested. Our initial findings were submitted for DREAM5 (the 5th Dialogue on Reverse Engineering Assessment and Methods challenge) and were judged to be the best in sub-challenge B3 wherein both functional genomic and genetic data were used to predict the phenotypes. In this work we further improve upon this previous work by considering various predictor selection strategies and cross-validation was used to measure accuracy of in-data and out-data predictions. The results from various model choices indicate that for this data use of both data types (namely functional genomic and genetic) simultaneously improves out-data prediction accuracy. Adequate goodness-of-fit can be easily achieved with more complex models for both phenotypes, since the number of potential predictors is large and the sample size is not small. We also further studied gene-set enrichment (for continuous phenotype) in the biological process in question and chromosomal enrichment of the gene set. The methodological contribution of this paper is in exploration of variable selection techniques to alleviate the problem of over-fitting. Different strategies based on the nature of covariates were explored and all methods were implemented under the Bayesian hierarchical modeling framework with indicator-based covariate selection. All the models based in careful variable selection procedure were found to produce significant results based on permutation test. © 2011 Bhattacharjee, Sillanpää. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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    A class of general solutions of the unsteady Oseen equations
    ( 2019-06-01) Tumuluri, Suman Kumar ; Padmavati, B. Sri
    In this paper, we present a class of general solutions of the unsteady Oseen equations in terms of two scalar functions. We also give a condition for a divergence-free vector to be a possible solution of Oseen equations.
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    A community effort to assess and improve drug sensitivity prediction algorithms
    ( 2014-12-01) Costello, James C. ; Heiser, Laura M. ; Georgii, Elisabeth ; Gönen, Mehmet ; Menden, Michael P. ; Wang, Nicholas J. ; Bansal, Mukesh ; Ammad-Ud-Din, Muhammad ; Hintsanen, Petteri ; Khan, Suleiman A. ; Mpindi, John Patrick ; Kallioniemi, Olli ; Honkela, Antti ; Aittokallio, Tero ; Wennerberg, Krister ; Collins, James J. ; Gallahan, Dan ; Singer, Dinah ; Saez-Rodriguez, Julio ; Kaski, Samuel ; Gray, Joe W. ; Stolovitzky, Gustavo ; Abbuehl, Jean Paul ; Allen, Jeffrey ; Altman, Russ B. ; Balcome, Shawn ; Battle, Alexis ; Bender, Andreas ; Berger, Bonnie ; Bernard, Jonathan ; Bhattacharjee, Madhuchhanda ; Bhuvaneshwar, Krithika ; Bieberich, Andrew A. ; Boehm, Fred ; Califano, Andrea ; Chan, Christina ; Chen, Beibei ; Chen, Ting Huei ; Choi, Jaejoon ; Coelho, Luis Pedro ; Cokelaer, Thomas ; Collins, James C. ; Creighton, Chad J. ; Cui, Jike ; Dampier, Will ; Davisson, V. Jo ; De Baets, Bernard ; Deshpande, Raamesh ; DiCamillo, Barbara ; Dundar, Murat ; Duren, Zhana ; Ertel, Adam ; Fan, Haoyang ; Fang, Hongbin ; Gauba, Robinder ; Gottlieb, Assaf ; Grau, Michael ; Gusev, Yuriy ; Ha, Min Jin ; Han, Leng ; Harris, Michael ; Henderson, Nicholas ; Hejase, Hussein A. ; Homicsko, Krisztian ; Hou, Jack P. ; Hwang, Woochang ; IJzerman, Adriaan P. ; Karacali, Bilge ; Keles, Sunduz ; Kendziorski, Christina ; Kim, Junho ; Kim, Min ; Kim, Youngchul ; Knowles, David A. ; Koller, Daphne ; Lee, Junehawk ; Lee, Jae K. ; Lenselink, Eelke B. ; Li, Biao ; Li, Bin ; Li, Jun ; Liang, Han ; Ma, Jian ; Madhavan, Subha ; Mooney, Sean ; Myers, Chad L. ; Newton, Michael A. ; Overington, John P. ; Pal, Ranadip ; Peng, Jian ; Pestell, Richard ; Prill, Robert J. ; Qiu, Peng ; Rajwa, Bartek ; Sadanandam, Anguraj ; Sambo, Francesco ; Shin, Hyunjin ; Song, Jiuzhou ; Song, Lei ; Sridhar, Arvind
    Predicting the best treatment strategy from genomic information is a core goal of precision medicine. Here we focus on predicting drug response based on a cohort of genomic, epigenomic and proteomic profiling data sets measured in human breast cancer cell lines. Through a collaborative effort between the National Cancer Institute (NCI) and the Dialogue on Reverse Engineering Assessment and Methods (DREAM) project, we analyzed a total of 44 drug sensitivity prediction algorithms. The top-performing approaches modeled nonlinear relationships and incorporated biological pathway information. We found that gene expression microarrays consistently provided the best predictive power of the individual profiling data sets; however, performance was increased by including multiple, independent data sets. We discuss the innovations underlying the top-performing methodology, Bayesian multitask MKL, and we provide detailed descriptions of all methods. This study establishes benchmarks for drug sensitivity prediction and identifies approaches that can be leveraged for the development of new methods.
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    A complete general solution of the unsteady Brinkman equations
    ( 2018-05-15) Tumuluri, Suman Kumar ; Amaranath, T.
    In this paper, we present a complete general solution of the unsteady Brinkman equations. To this end, we introduce a representation for velocity and pressure in terms of two scalar functions. One of these scalar functions satisfies a second order partial differential equation (PDE) while the other satisfies a fourth order PDE which can be factorized into a pair of second order PDEs. We show that the solution of this fourth order PDE is indeed the sum of the solutions of the two second order PDEs. We also use these solutions to obtain a complete general solution of the unsteady Brinkman equations.
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    A factorization theorem for operators occurring in the Stokes, Brinkman and Oseen equations
    ( 2016-03-05) Tumuluri, Suman Kumar ; Amaranath, T.
    In many physical problems one is faced with solving partial differential equations of the form L1(L1+L2)u=0, where L1 and L2 are linear operators. It is found in many cases that the solution u is of the form u1+u2 where L1u1=0 and (L1+L2)u2=0. In this paper we present sufficient conditions under which such a splitting is possible. Moreover, we give explicit formulae for u1 and u2 for a given u. We also show in some examples where the operators satisfy the sufficient conditions and such a splitting is used extensively. In particular, we find a class of solutions for the unsteady Brinkman and unsteady Oseen equations using the splitting that we propose.
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    A new approximate analytical solution for arbitrary Stokes flow past rigid bodies
    ( 2012-12-01) Radha, R. ; Sri Padmavati, B. ; Amaranath, T.
    A method of computing general Stokes flows in the presence of rigid boundaries of arbitrary shape is proposed. The solution satisfies the governing field equations exactly and the boundary conditions approximately. The method has been illustrated with three examples. The advantage of the method lies in the ease of implementation for rigid bodies of arbitrary shape, providing an approximate but analytical solution throughout the domain. © 2012 Springer Basel AG.
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    A new numerical method based on Daftardar-Gejji and Jafari technique for solving differential equations
    ( 2015-01-01) Patade, Jayvant ; Bhalekar, Sachin
    In the present work we introduce a new numerical method (NNM) for solving differential equations. We apply Daftardar-Gejji and Jafari technique on implicit trapezium formula to derive new method. We discuss error, stability and convergence analysis of the proposed method and also provide some software packages based on this method. We solve various types of equations using this method and show that the results are matching with exact solutions.
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    A new predictor-corrector method for fractional differential equations
    ( 2014-10-01) Daftardar-Gejji, Varsha ; Sukale, Yogita ; Bhalekar, Sachin
    We present a new predictor-corrector method to solve non-linear fractional differential equations involving Caputo derivative. The proposed method is compared with the fractional Adams method. Numerous illustrative examples discussed here demonstrate that the new method is more accurate and time efficient. A detailed error analysis given points to the higher accuracy of the new method. Furthermore the proposed method when applied to fractional analog of chaotic system introduced by Bhalekar and Daftardar-Gejji, unravels the underlying rich dynamics of the system. © 2014 Elsevier Inc. All rights reserved.
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    A nonlinear hyperbolic system modeling currency hoarding
    ( 2019-07-01) Tumuluri, Suman Kumar ; Murthy, A. S.Vasudeva
    We present a nonlinear model for replacement, regulation of currency in circulation and hoarding currency. The nonlinearity enters the model in the regulatory term which depends on the total currency in the circulation. We provide an existence and uniqueness result for the model as well as its steady state. Local and global dynamics of the solution is studied for large time. In fact, convergence of the solution to the nontrivial steady state is obtained in both the linearized and nonlinear cases. Furthermore, we have analyzed the dynamics of the total currency in circulation and hoarding for large time by constructing a Lyapunov function. Convergence of the total currency in circulation and hoarding to the steady state is established using this Lyapunov function.
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    A note on a neuron network model with diffusion
    ( 2020-09-01) Michel, Philippe ; Tumuluri, Suman Kumar
    We study the dynamics of inhomogeneous neuronal networks parametrized by a real number σ and structured by the time elapsed since the last discharge. The dynamics are governed by the parabolic PDE which describes the probability density of neurons with elapsed time s after its last discharge. We prove existence and uniqueness of a solution to the model. Moreover, we show that under some conditions on the connectivity and the firing rate, the networks exhibit total desynchronization.
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    A note on Gaussian distributions in ℝn
    ( 2012-01-01) Manjunath, B. G. ; Parthasarathy, K. R.
    Given any finite set F of (n - 1)-dimensional subspaces of Rn we give examples of nonGaussian probability measures in R{double-struck}n whose marginal distribution in each subspace from F is Gaussian. However, if F is an infinite family of such (n - 1)- dimensional subspaces then such a nonGaussian probability measure in R{double-struck}n does not exist. ©Indian Academy of Sciences.
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    A Note on the Linear Stability of the Steady State of a Nonlinear Renewal Equation with a Parameter
    ( 2020-09-01) Tumuluri, Suman Kumar
    In this article we consider a variant of age-structured nonlinear Lebowitz-Rubinow equation. We study the linear stability of this equation near the nontrivial steady state by analyzing the corresponding characteristic equation. In particular, we provide some sufficient conditions under which the nonzero steady state is linearly stable.
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    A Novel Numerical Method for Solving Volterra Integro-Differential Equations
    ( 2020-02-01) Patade, Jayvant ; Bhalekar, Sachin
    In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.
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    A numerical scheme to the McKendrick–von Foerster equation with diffusion in age
    ( 2018-11-01) Kakumani, Bhargav Kumar ; Tumuluri, Suman Kumar
    In this paper a numerical scheme for McKendrick–von Foerster equation with diffusion in age (MV-D) is proposed. First, we discretize the time variable to get a second-order ordinary differential equation (ODE). At each time level, well-posedness of this ODE is established using classical methods. Stability estimates for this semidiscrete scheme are derived. Later we construct piecewise linear (in time) functions using the solutions of the semidiscrete problems to approximate the solution to MV-D and establish the convergence result. Numerical results are presented in some cases and compared with the corresponding analytic solutions where the latter is known explicitly.
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    Adaptive PORT-MVRB estimation: an empirical comparison of two heuristic algorithms
    ( 2013-06-01) Gomes, M. Ivette ; Henriques-Rodrigues, Lígia ; Fraga Alves, M. Isabel ; Manjunath, B. G.
    In this article, we deal with an empirical comparison of two data-driven heuristic procedures of estimation of a positive extreme value index (EVI), working thus with heavy right tails. The semi-parametric EVI-estimators under consideration, the so-called peaks over random threshold (PORT)-minimum-variance reduced-bias (MVRB) EVI-estimators, are location and scale-invariant estimators, based on the PORT methodology applied to second-order MVRB EVI-estimators. Trivial adaptations of these algorithms make them work for a similar estimation of other parameters of extreme events, such as the Value-at-Risk at a level p, the expected shortfall and the probability of exceedance of a high level x, among others. Applications to simulated data sets and to real data sets in the field of finance are provided. © 2013 Copyright Taylor and Francis Group, LLC.
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    Additive problems with smooth integers
    ( 2016-01-01) Ki, H. ; Maier, H. ; Sankaranarayanan, A.
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    Admissibility over function fields of P-Adic curves
    (University of Hyderabad, 2011-07-20) Surendranath Reddy, B ; Suresh, V.
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    All Conditional Distributions for Y Given X that are Compatible with a Given Conditional Distribution for X Given Y
    ( 2020-01-01) Arnold, Barry C. ; Manjunath, B. G.
    For a given conditional distribution for X given Y, it is important to identify the class of all conditional distributions for Y given X such that there exists at least one bivariate distribution with the given particular conditional densities. Such problems are addressed as dealing with “compatibility” of two conditional distributions. In the present note our goal is to identify all possible conditional densities for Y given X that are compatible with the given family of distributions of X given Y.
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    Analysing the stability of a delay differential equation involving two delays
    ( 2019-08-01) Bhalekar, Sachin
    Analysis of systems involving delay is a popular topic among the applied scientists. In the present work, we analyse the generalised equation Dαx(t) = g(x(t- τ1) , x(t- τ2)) involving two delays, viz. τ1≥ 0 and τ2≥ 0. We use stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.