Unknown Subset Research Articles

Reconstructing Chain Functions in Genetic Networks

The following problems arise in the analysis of biological networks: We have a boolean function of n variables, each of which has some default value. An experiment fixes the values of any subset of the variables, the remaining variables assume their default values, and the function value is the result of the experiment. How many experiments are needed to determine (reconstruct) the function? How many experiments that involve fixing at most q values are needed? What are the answers to these questions when an unknown subset of the variables are actually involved in the function? In the biological context, the variables are genes and the values are gene expression intensities. An experiment measures the gene levels under conditions that perturb the values of a subset of the genes. The goal is to reconstruct the particular logic (regulation function) by which a subset of the genes together regulate one target gene, using few experiments that involve minor perturbations. We study these questions under the assumption that all functions belong to a biologically motivated set of so‐called chain functions. We give optimal reconstruction schemes for several scenarios and show their application in reconstructing the regulation of galactose utilization in yeast.

ALFA-9000 Trial - An Analysis of Cytogenetic Subsets: Significant Long-Term Benefit of Early Induction Reinforcement Is Observed in CBF-AML Only.

We have recently reported the results of the ALFA-9000 randomized trial in which younger adults with AML had a significant better long-term relapse-free interval when receiving a timed-sequentially reinforced induction (arm C) as compared to either standard 3+7 (arm A) or double (arm B) induction (Castaigne et al, Blood 2004). Both arms B and C included the same induction reinforcement with mitoxantrone and intermediate-dose cytarabine (ID-AraC) systematically administered at day 20 in arm B or at day 8 in arm C. Although patients aged between 15 and 65 years were included in this study, the gain in relapse incidence associated with arm C was observed in those less than 50 years of age only. We report here the outcomes of cytogenetic subsets, according to the arm C versus arm A/B stratification. Overall, 405 karyotypes were available (68%) and classified according to the SWOG classification as 56 favorable (CBF-AML, 46/56 under the age of 50 years), 215 intermediate (including 194 normal), 91 unfavorable, and 43 of unknown significance. Outcome of unfavorable and unknown subsets did not differ significantly in C versus A/B. Patients with complex karyotype or chromosome 7, 5q, or 3q abnormalities, as well as those with 11q23 abnormalities, had a similar outcome whatever their induction arm. Similarly, no benefit of early reinforcement was observed in the intermediate group, nor in patients with a normal karyotype even after adjustment on WBC or when considering only those aged less than 50 years. In the favorable subset, patients with either t(8;21) (N=30) or inv(16) (N=26) had identical long-term outcome. Contrary to other subsets, this outcome differed strikingly among induction arms with better disease-free survival in arm C (64% versus 28% at 5 years, P=.05 after adjustment on WBC) with a similar trend observed for event-free survival. The value of higher than standard doses of cytarabine in CBF-AML patients, either during induction or as post-remission therapy, has been demonstrated by several cooperative groups. Our results do suggest that early introduction of ID-AraC is also associated with long-term benefit in this favorable subgroup of patients, even greater when made at day 8 instead of day 20. We will evaluate prospectively this induction approach in the next French Intergroup CBF-AML study, in which all patients will then receive high-dose cytarabine during post-remission therapy.

Optimal group testing algorithms with interval queries and their application to splice site detection

Given an ordered set of n items and an unknown subset P of up to p positive elements, we want to identify P by asking the least number of queries 'does Q intersect P?' where Q must consist of consecutive elements. This Interval Group Testing problem arises in the context of splice site detection in genes. We study algorithms that operate in a few stages where queries chosen depending on previous answers, are performed in parallel. We obtain tight bounds for two-stage strategies. Finally, we get results for any number of stages and positives.

Optimal Two-Stage Algorithms for Group Testing Problems

Group testing refers to the situation in which one is given a set of objects ${\cal O}$, an unknown subset ${\cal P}\subseteq {\cal O}$, and the task of determining ${\cal P}$ by asking queries of the type does ${\cal P}$ intersect $\cal Q$?, where $\cal Q$ is a subset of ${\cal O}$. Group testing is a basic search paradigm that occurs in a variety of situations such as quality control testing, searching in storage systems, multiple access communications, and data compression, among others. Group testing procedures have been recently applied in computational molecular biology, where they are used for screening libraries of clones with hybridization probes and sequencing by hybridization. Motivated by particular features of group testing algorithms used in biological screening, we study the efficiency of two-stage group testing procedures. Our main result is the first optimal two-stage algorithm that uses a number of tests of the same order as the information-theoretic lower bound on the problem. We also provide efficient algorithms for the case in which there is a Bernoulli probability distribution on the possible sets ${\cal P}$, and an optimal algorithm for the case in which the outcome of tests may be unreliable because of the presence of inhibitory items in ${\cal O}$. Our results depend on a combinatorial structure introduced in this paper. We believe that it will prove useful in other contexts, too.

Guessing Secrets with Inner Product Questions

We suppose we are given some fixed (but unknown) subset _X_ of a set Ω = 𝔽^<em>n</em>₂ , where 𝔽₂ denotes the field of two elements. Our goal is to learn as much as possible about the elements of _X_ by asking certain binary questions. Each "question" _Q_ is just some element of Ω, and the "answer" to _Q_ is just the inner product _Q_ . _x_ ∈ 𝔽₂ for some _x_ ∈ _X_. However, the choice of _x_ is made by a truthful (but possibly malevolent) adversary A, whom we may assume is trying to choose answers so as to yield as little information as possible about _X_. In this note, we investigate several aspects of this problem. In particular, we are interested in extracting as much information as possible about _X_ from A's answers. Although Acan prevent us from learning the identity of any particular element of _X_, with appropriate questions we can still learn quite a bit about _X_. We determine the maximum amount of information that can be recovered under these assumptions and describe explicit sets of questions for achieving this goal. For the case that |_X_| = 2, we give an _O_(_n_³) algorithm for recovering the desired information. On the other hand, when |_X_| ≥ 3, we show that no polynomial-time algorithm can exist for producing a secret set consistent with the answers given, unless _P = NP_.

Biodiversity curves for the Ordovician of Baltoscandia

Biodiversity curves for the Ordovician of Baltoscandia show a substantial increase in taxonomical diversity through the period, as seen also in global data sets. A database of 10,340 records of first and last appearances of species at different localities in the region has been analysed using simple species counts, and partly validated with resampling methods. While the biodiversity curve for all fossil groups combined is probably reasonably accurate except for an unknown scaling constant, taxonomical or geographical subsets may not be sufficiently well sampled to allow precise measurement of their species counts through time. The analysis shows a major diversification event commencing in the middle Arenig (Ibex-Whiterock boundary), and more limited diversification events in the Llanvirn, Caradoc and Ashgill. The Scandinavian (Norwegian and Swedish) biodiversity curves are broadly correlated with major changes in sea level, with low biodiversity at highstands and high biodiversity at lowstands, although this pattern is not clear for all fossil groups. In the Arenig, graptolites and trilobites appear to have higher diversities at high sea levels, while the brachiopods and ostracodes show higher diversities at low sea level. As a consequence, the Arenig diversification is delayed for the latter two groups until the upper end of the interval.

Algorithms for phylogenetic footprinting.

Phylogenetic footprinting is a technique that identifies regulatory elements by finding unusually well conserved regions in a set of orthologous noncoding DNA sequences from multiple species. We introduce a new motif-finding problem, the Substring Parsimony Problem, which is a formalization of the ideas behind phylogenetic footprinting, and we present an exact dynamic programming algorithm to solve it. We then present a number of algorithmic optimizations that allow our program to run quickly on most biologically interesting datasets. We show how to handle data sets in which only an unknown subset of the sequences contains the regulatory element. Finally, we describe how to empirically assess the statistical significance of the motifs found. Each technique is implemented and successfully identifies a number of known binding sites, as well as several highly conserved but uncharacterized regions. The program is available at http://bio.cs.washington.edu/software.html.

Guessing Secrets

Suppose we are given some fixed (but unknown) subset $X$ of a set $\Omega$, and our object is to learn as much as possible about the elements of $X$ by asking binary questions. Specifically, each question is just a function $F: \Omega \rightarrow \{0,1\}$, and the answer to $F$ is just the value $F(X_i)$ for some $X_i \in X$, (determined, for example, by a potentially malevolent but truthful, adversary). In this paper, we describe various algorithms for solving this problem, and establish upper and lower bounds on the efficiency of such algorithms.

Endogenous targets of transcriptional gene silencing in Arabidopsis.

Transcriptional gene silencing (TGS) frequently inactivates foreign genes integrated into plant genomes but very likely also suppresses an unknown subset of chromosomal information. Accordingly, RNA analysis of mutants impaired in silencing should uncover endogenous targets of this epigenetic regulation. We compared transcripts from wild-type Arabidopsis carrying a silent transgene with RNA from an isogenic transgene-expressing TGS mutant. Two cDNA clones were identified representing endogenous RNA expressed only in the mutant. The synthesis of these RNAs was found to be released in several mutants affected in TGS, implying that TGS in general and not a particular mutation controls the transcriptional activity of their templates. Detailed analysis revealed that the two clones are part of longer transcripts termed TSI (for transcriptionally silent information). Two major classes of related TSI transcripts were found in a mutant cDNA library. They are synthesized from repeats present in heterochromatic pericentromeric regions of Arabidopsis chromosomes. These repeats share sequence homology with the 3' terminal part of the putative retrotransposon Athila. However, the transcriptional activation does not include the transposon itself and does not promote its movement. There is no evidence for a general release of silencing from retroelements. Thus, foreign genes in plants encounter the epigenetic control normally directed, at least in part, toward a subset of pericentromeric repeats.

An efficient implementation of the backward greedy algorithm for sparse signal reconstruction

Recent work in sparse signal reconstruction has shown that the backward greedy algorithm can select the optimal subset of unknowns if the perturbation of the data is sufficiently small. We propose an efficient implementation of the backward greedy algorithm that yields a significant improvement in computational efficiency over the standard implementation. Furthermore, we propose an efficient algorithm for the case in which the transform matrix is too large to be stored. We analyze the computational complexity and compare the algorithms, and we illustrate the improved efficiency with examples.

Hotelling's competition with demand location uncertainty

Firms' product variety choices are examined in a differentiated products model where the range of consumers' preferences is an unknown (strict) subset of the range of potential innovations. It is found that the firms, locating sequentially, will follow the basic pattern of locations found by Neven ( International Journal of Industrial Organization, 1987, 5, 419–434). The uncertainty, however, generally causes a greater, more symmetric, amount of differentiation among varieties as firms gamble on finding a market niche with less fierce competition. For a sufficiently large pool of potential entrants, some firms will lose this gamble and will fail.

Testing the equivalence of multiple regression models with additional data

The problem of testing the equality of multiple regression parameter sets among several models when additional data from some unknown (or unidentified) subset of these models are available is considered. The standard test (T 0) for this problem discards the additional data and tests the equivalence of regression models using only the data from the identified models. Two test procedures are proposed based upon all the available data. The additional data may contain valuable information about the parameters that can be used to improve the power of the test and make discrimination among the regression models easier. A power comparison among the tests is also presented. The test procedure T 1 is shown to have larger power than To.For certain parameter values, T2 has the largest power.

Multivariate analysis of variance with additional data from an unknown subset of the populations

In this paper we consider the problem of testing the means of k multivariate normal populations with additional data from an unknown subset of the k populations. The purpose of this research is to offer test procedures utilizing all the available data for the multivariate analysis of variance problem because the additional data may contain valuable information about the parameters of the k populations. The standard procedure uses only the data from identified populations. We provide a test using all available data based upon Hotelling' s generalized T2statistic. The power of this test is computed using Betz's approximation of Hotelling' s generalized T2statistic by an F-distribution. A comparison of the power of the test and the standard test procedure is also given.

On Sets that are Uniquely Determined by a Restricted Set of Integrals

In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset $S$ of a measure space $(X,\lambda )$ such as ${{\mathbf {R}}^n}$, or an unknown function $0 \leqslant \phi \leqslant 1$ on $X$, having known moments (integrals) relative to a specified class $F$ of functions $f:X \to {\mathbf {R}}$. Usually, these $F$-moments do not fully determine the object $S$ or function $\phi$. We will say that $S$ is a set of uniqueness if no other function $0 \leqslant \psi \leqslant 1$ has the same $F$-moments as $S$ in so far as the latter moments exist. Here, $S$ is identified with its indicator function. Within this general setting and with no further assumptions, we develop a powerful sufficient condition for uniqueness, called generalized additivity. When $F$ is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each $\phi$, which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same $F$-moments as $\phi$, provided $(X,\lambda ,F)$ is nonatomic or regular and, moreover, ‘strongly rich’, a condition which is satisfied in many applications. Using such general results, we also study the uniqueness problem for measures with given marginals relative to a given system of projections ${\pi _j}:X \to {Y_j}(j \in J)$. Here, one likes to know, for instance, what subsets $S$ of $X$ are uniquely determined by the corresponding set of projections (X-ray pictures). It is allowed that $\lambda (S) = \infty$. Our results are also relevant to a wide class of optimization problems.

The expected value of some functions of the convex hull of a random set of points sampled inR d

This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton (d−1)/(d+1), which generalizes Renyi and Sulanke’s asymptotic raten (1/3) ford=2 and agrees with Raynaud’s asymptotic raten (d−1)/(d+1) for the expected number of facets, as it should be, by Barany’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset ofR d .

Role of Maggi's equations in computational methods for constrained multibody systems

This paper presents a unified theoretical basis for a class of methods that generate the governing equations of constrained dynamical systems by eliminating the constraints. By using Maggi's equations in conjunction with a common projective theory from numerical analysis, it is shown that members of the class are precisely characterized by the basis they choose for the null-space of the variational form of the constraints. For each method considered, the specific basis chosen for the null-space of the variational constraints is derived, as well as a dual basis for the orthogonal complement. The latter basis is of particular interest since it is shown that its knowledge theoretically enables one to generalize certain methods of the class to calculate constraint forces and torques. Practical approaches based on orthogonal transformations to effect this strategy are also outlined. In addition, since the theory presented herein stresses a common, fundamental structure to the various methods, it is especially useful as a means of comparing and evaluating individual numerical algorithms. The theory presented makes clear the relationship between certain numerical instabilities that have been noted in some methods that eliminate a priori constraint contributions to the governing equations by selecting an independent subset of unknowns. It is also briefly indicated how this formalism can be extended, in principle, to the wider class of nonlinear nonholonomic constraints.

Locating a robot with angle measurements

We consider the problem posed by Sugihara of locating a robot in a planar environment containing n indistinguishable mark points. The robot takes k angular measurements between an unknown subset of mark points. When the robot has a compass, we show that all valid placements of the robot consistent with the measurements may be determined in O ( kn 2 ) time and O ( kn ) space. When the robot does not have a compass, we show how to determine all valid placements in O ( kn 3 ) time and O ( kn ) space. We show that with polynomial-time preprocessing both types of location queries can be answered in O (log n ) time. For both cases we give bounds on the maximum number of valid placements. We also consider the case where there are image processing errors involved in the angle measurements.

Equivalent Sparse Matrix Reordering by Elimination Tree Rotations

In this paper, we introduce a class of fill-preserving equivalent reorderings for sparse Cholesky factorization. This class is based on performing a special form of tree rotation on the elimination tree of the given sparse matrix. Some applications on the use of such elimination tree rotation are described: core storage reduction in a sparse out-of-core scheme, working storage reduction in the multifrontal method, and solution of a subset of unknowns. Some experimental results are also provided to demonstrate its effectiveness in some of these applications.

On two random search problems

The paper is concerned with static search on a finite set. An unknown subset of cardinality k of the finite set is to be found by testing its subsets. We investigate two problems: in the first, the number of common elements of the tested and the unknown subset is given; in the second, only the information whether the tested and the unknown subset are disjoint or not is given. Both problems correspond to problems on false coins. If the unknown subset is taken from the family of k-element sets with uniform distribution, we determine the minimum of the lengths of the strategies that find the unknown element with small error probability. The strategies are constructed by probabilistic means.

Reaching Agreement in the Presence of Faults

The problem addressed here concerns a set of isolated processors, some unknown subset of which may be faulty, that communicate only by means of two-party messages. Each nonfaulty processor has a private value of information that must be communicated to each other nonfaulty processor. Nonfaulty processors always communicate honestly, whereas faulty processors may lie. The problem is to devise an algorithm in which processors communicate their own values and relay values received from others that allows each nonfaulty processor to infer a value for each other processor. The value inferred for a nonfaulty processor must be that processor's private value, and the value inferred for a faulty one must be consistent with the corresponding value inferred by each other nonfaulty processor. It is shown that the problem is solvable for, and only for, n ≥ 3 m + 1, where m is the number of faulty processors and n is the total number. It is also shown that if faulty processors can refuse to pass on information but cannot falsely relay information, the problem is solvable for arbitrary n ≥ m ≥ 0. This weaker assumption can be approximated in practice using cryptographic methods.