Gene Sets Net Correlations Analysis (GSNCA): a multivariate differential coexpression test for gene sets

Motivation: To date, Gene Set Analysis (GSA) approaches primarily focus on identifying differentially expressed gene sets (pathways). Methods for identifying differentially coexpressed pathways also exist but are mostly based on aggregated pairwise correlations, or other pairwise measures of coexpression. Instead, we propose Gene Sets Net Correlations Analysis (GSNCA), a multivariate differential coexpression test that accounts for the complete correlation structure between genes.

Results: In GSNCA, weight factors are assigned to genes in proportion to the genes' cross-correlations (intergene correlations). The problem of finding the weight vectors is formulated as an eigenvector problem with a unique solution. GSNCA tests the null hypothesis that for a gene set there is no difference in the weight vectors of the genes between two conditions. In simulation studies and the analyses of experimental data, we demonstrate that GSNCA, indeed, captures changes in the structure of genes' cross-correlations rather than differences in the averaged pairwise correlations. Thus, GSNCA infers differences in coexpression networks, however, bypassing method-dependent steps of network inference. As an additional result from GSNCA, we define hub genes as genes with the largest weights and show that these genes correspond frequently to major and specific pathway regulators, as well as to genes that are most affected by the biological difference between two conditions. In summary, GSNCA is a new approach for the analysis of differentially coexpressed pathways that also evaluates the importance of the genes in the pathways, thus providing unique information that may result in the generation of novel biological hypotheses.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Kuznetsov independence for interval-valued expectations and sets of probability distributions: Properties and algorithms

Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y), E[f(X)g(Y)]=E[f(X)] *times* E[g(Y)], where E[.] denotes interval-valued expectation and *times* denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.

Progress on core outcomes sets for critical care research

Purpose of review: Appropriate selection and definition of outcome measures are essential for clinical trials to be maximally informative. Core outcome sets (an agreed, standardized collection of outcomes measured and reported in all trials for a specific clinical area) were developed due to established inconsistencies in trial outcome selection. This review discusses the rationale for, and methods of, core outcome set development, as well as current initiatives in critical care.

Recent findings: Recent systematic reviews of reported outcomes and measurement instruments relevant to the critically ill highlight inconsistencies in outcome selection, definition, and measurement, thus establishing the need for core outcome sets. Current critical care initiatives include development of core outcome sets for trials aimed at reducing mechanical ventilation duration; rehabilitation following critical illness; long-term outcomes in acute respiratory failure; and epidemic and pandemic studies of severe acute respiratory infection.

Summary: Development and utilization of core outcome sets for studies relevant to the critically ill is in its infancy compared to other specialties. Notwithstanding, core outcome set development frameworks and guidelines are available, several sets are in various stages of development, and there is strong support from international investigator-led collaborations including the International Forum for Acute Care Trialists.

Sets of p-multiplicity in locally compact groups

We initiate the study of sets of p-multiplicity in locally compact groups and their operator versions. We show that a closed subset E of a second countable locally compact group G is a set of p-multiplicity if and only if E∗={(s,t):ts−1∈E} is a set of operator p-multiplicity. We exhibit examples of sets of p-multiplicity, establish preservation properties for unions and direct products, and prove a p-version of the Stone–von Neumann Theorem.

COS-STAR: a reporting guideline for studies developing core outcome sets (protocol)

BACKGROUND: Core outcome sets can increase the efficiency and value of research and, as a result, there are an increasing number of studies looking to develop core outcome sets (COS). However, the credibility of a COS depends on both the use of sound methodology in its development and clear and transparent reporting of the processes adopted. To date there is no reporting guideline for reporting COS studies. The aim of this programme of research is to develop a reporting guideline for studies developing COS and to highlight some of the important methodological considerations in the process.

METHODS/DESIGN: The study will include a reporting guideline item generation stage which will then be used in a Delphi study. The Delphi study is anticipated to include two rounds. The first round will ask stakeholders to score the items listed and to add any new items they think are relevant. In the second round of the process, participants will be shown the distribution of scores for all stakeholder groups separately and asked to re-score. A final consensus meeting will be held with an expert panel and stakeholder representatives to review the guideline item list. Following the consensus meeting, a reporting guideline will be drafted and review and testing will be undertaken until the guideline is finalised. The final outcome will be the COS-STAR (Core Outcome Set-STAndards for Reporting) guideline for studies developing COS and a supporting explanatory document.

DISCUSSION: To assess the credibility and usefulness of a COS, readers of a COS development report need complete, clear and transparent information on its methodology and proposed core set of outcomes. The COS-STAR guideline will potentially benefit all stakeholders in COS development: COS developers, COS users, e.g. trialists and systematic reviewers, journal editors, policy-makers and patient groups.

Core outcome sets and trial registries

Some reasons for registering trials might be considered as self-serving, such as satisfying the requirements of a journal in which the researchers wish to publish their eventual findings or publicising the trial to boost recruitment. Registry entries also help others, including systematic reviewers, to know about ongoing or unpublished studies and contribute to reducing research waste by making it clear what studies are ongoing. Other sources of research waste include inconsistency in outcome measurement across trials in the same area, missing data on important outcomes from some trials, and selective reporting of outcomes. One way to reduce this waste is through the use of core outcome sets: standardised sets of outcomes for research in specific areas of health and social care. These do not restrict the outcomes that will be measured, but provide the minimum to include if a trial is to be of the most use to potential users. We propose that trial registries, such as ISRCTN, encourage researchers to note their use of a core outcome set in their entry. This will help people searching for trials and those worried about selective reporting in closed trials. Trial registries can facilitate these efforts to make new trials as useful as possible and reduce waste. The outcomes section in the entry could prompt the researcher to consider using a core outcome set and facilitate the specification of that core outcome set and its component outcomes through linking to the original core outcome set. In doing this, registries will contribute to the global effort to ensure that trials answer important uncertainties, can be brought together in systematic reviews, and better serve their ultimate aim of improving health and well-being through improving health and social care.

Inference in Polytrees with Sets of Probabilities

Inferences in directed acyclic graphs associated with probability intervals and sets of probabilities are NP-hard, even for polytrees. We propose: 1) an improvement on Tessem’s A/R algorithm for inferences on polytrees associated with probability intervals; 2) a new algorithm for approximate inferences based on local search; 3) branch-and-bound algorithms that combine the previous techniques. The first two algorithms produce complementary approximate solutions, while branch-and-bound procedures can generate either exact or approximate solutions. We report improvements on existing techniques for inference with probability sets and intervals, in some cases reducing computational effort by several orders of magnitude.

Besov spaces and the Laplacian on fractal H-sets

Nesta tese são estudados espaços de Besov de suavidade generalizada em espaços euclidianos, numa classe de fractais designados conjuntos-h e em estruturas abstractas designadas por espaços-h. Foram obtidas caracterizações e propriedades para estes espaços de funções. Em particular, no caso de espaços de Besov em espaços euclidianos, foram obtidas caracterizações por diferenças e por decomposições em átomos não suaves, foi provada uma propriedade de homogeneidade e foram estudados multiplicadores pontuais. Para espaços de Besov em conjuntos-h foi obtida uma caracterização por decomposições em átomos não suaves e foi construído um operador extensão. Com o recurso a cartas, os resultados obtidos para estes espaços de funções em fractais foram aplicados para definir e trabalhar com espaços de Besov de suavidade generalizada em estruturas abstractas. Nesta tese foi também estudado o laplaciano fractal, considerado a actuar em espaços de Besov de suavidade generalizada em domínios que contêm um conjunto-h fractal. Foram obtidos resultados no contexto de teoria espectral para este operador e foi estudado, à custa deste operador, um problema de Dirichlet fractal no contexto de conjuntos-h.

Main eigenvalues and (κ, τ)-regular sets

A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.

Polyhedral study of mixed integer sets arising from inventory problems

“Branch-and-cut” algorithm is one of the most efficient exact approaches to solve mixed integer programs. This algorithm combines the advantages of a pure branch-and-bound approach and cutting planes scheme. Branch-and-cut algorithm computes the linear programming relaxation of the problem at each node of the search tree which is improved by the use of cuts, i.e. by the inclusion of valid inequalities. It should be taken into account that selection of strongest cuts is crucial for their effective use in branch-and-cut algorithm. In this thesis, we focus on the derivation and use of cutting planes to solve general mixed integer problems, and in particular inventory problems combined with other problems such as distribution, supplier selection, vehicle routing, etc. In order to achieve this goal, we first consider substructures (relaxations) of such problems which are obtained by the coherent loss of information. The polyhedral structure of those simpler mixed integer sets is studied to derive strong valid inequalities. Finally those strong inequalities are included in the cutting plane algorithms to solve the general mixed integer problems. We study three mixed integer sets in this dissertation. The first two mixed integer sets arise as a subproblem of the lot-sizing with supplier selection, the network design and the vendor-managed inventory routing problems. These sets are variants of the well-known single node fixed-charge network set where a binary or integer variable is associated with the node. The third set occurs as a subproblem of mixed integer sets where incompatibility between binary variables is considered. We generate families of valid inequalities for those sets, identify classes of facet-defining inequalities, and discuss the separation problems associated with the inequalities. Then cutting plane frameworks are implemented to solve some mixed integer programs. Preliminary computational experiments are presented in this direction.

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.