Long-term dynamics of chromosomal instability in cancer: A transition probability model

A stochastic model of chromosomal instability has been previously developed which has included one adjustable parameter—the probability of a segregation error. Using computer simulations, we have previously analyzed this model and were able to reproduce a short-term dynamics of chromosome copy number distributions in clones of cancer cells. In a short run, segregation errors provide a continuous production of deviant cells with increasing variation of cell karyotypes, which depends upon the rate of segregation errors. In the long-term observations, many tumors and cancer cell lines have been observed to maintain a stable, although abnormal, distribution of chromosome number for hundreds of cell generations. This phenomenon of “stability within instability” presents an interesting paradox, which could be addressed mathematically. However, this would require modeling of long term growth of tumor cell clones for hundreds of generations, which has far exceeded capabilities of modern computer systems. In this study, we have analyzed asymptotic behavior of our model using a semianalytical approach. A transition probability matrix was derived analytically and implemented in a recursive algorithm for computational experiments. Using this transition probability model, the expected frequencies of chromosome copy number have been calculated under various initial and boundary conditions. We have also tested several alternative models, which describe various mechanisms of errors in segregation of chromosomes, and found conditions for stabilization of distribution of chromosomes copy numbers over a large number of cell generations. Stable clonal frequencies were estimated which are independent of initial conditions, i.e., chromosome copy numbers in the initiator cells. These stable distributions were, however, dependent on the model assumptions regarding particular mechanism of errors in segregation of chromosomes. Thus, our modeling results have suggested a possible connection between the form of stable distribution of chromosome numbers in tumors and the underlying mechanism of errors in segregation of chromosomes. This new analytical approach allows us to overcome technical impairments and limitations of computer simulation, and, for the first time, provides mathematical insight into long-term evolution of chromosome numerical changes in human tumors.     

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.     

A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K

A reliability system subject to shocks producing damage and failure is considered. The source of shocks producing failures is governed by a Markovian arrival process. All the shocks produce deterioration and some of them failures, which can be repairable or non-repairable. Repair times are governed by a phase-type distribution. The number of deteriorating shocks that the system can stand is fixed. After a fatal failure the system is replaced by another identical one. For this model the availability, the reliability, and the rate of occurrence of the different types of failures are calculated. It is shown that this model extends other previously published in the literature.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

Decision rules for efficient classification of biological data

Cancer classification using genomic data is one of the major research areas in the medical field. Therefore, a number of binary classification methods have been proposed in recent years. Top Scoring Pair (TSP) method is one of the most promising techniques that classify genomic data in a lower dimensional subspace using a simple decision rule. In the present paper, we propose a supervised classification technique that utilizes incremental generalized eigenvalue and top scoring pair classifiers to obtain higher classification accuracy with a small training set. We validate our method by applying it to well known microarray data sets.     

Equivalence classes of Niho bent functions

Equivalence classes of Niho bent functions are in one-to-one correspondence with equivalence classes of ovals in a projective plane. Since a hyperoval can produce several ovals, each hyperoval is associated with several inequivalent Niho bent functions. For all known types of hyperovals we described the equivalence classes of the corresponding Niho bent functions. For some types of hyperovals the number of equivalence classes of the associated Niho bent functions are at most 4. In general, the number of equivalence classes of associated Niho bent functions increases exponentially as the dimension of the underlying vector space grows. In small dimensions the equivalence classes were considered in detail.

     

Multi-state Systems with Graduate Failure and Equal Transition Intensities

We consider unrecoverable homogeneous multi-state systems with graduate failures, where each component can work at M + 1 linearly ordered levels of performance. The underlying process of failure for each component is a homogeneous Markov process such that the level of performance of one component can change only for one level lower than the observed one, and the failures are independent for different components. We derive the probability distribution of the random vector X, representing the state of the system at the moment of failure and use it for testing the hypothesis of equal transition intensities. Under the assumption that these intensities are equal, we derive the method of moments estimators for probabilities of failure in a given state vector and the intensity of failure. At the end we calculate the reliability function for such systems. Received: May 18, 2007., Revised: July 8, 2008., Accepted: September 29, 2008.     

A study of biases of DNA copy number estimation based on PICR model

Affymetrix single-nucleotide polymorphism (SNP) arrays have been widely used for SNP genotype calling and copy number variation (CNV) studies, both of which are dependent on accurate DNA copy number estimation significantly. However, the methods for copy number estimation may suffer from kinds of difficulties: probe dependent binding affinity, crosshybridization of probes, and the whole genome amplification (WGA) of DNA sequences. The probe intensity composite representation (PICR) model, one former established approach, can cope with most complexities and achieve high accuracy in SNP genotyping. Nevertheless, the copy numbers estimated by PICR model still show array and site dependent biases for CNV studies. In this paper, we propose a procedure to adjust the biases and then make CNV inference based on both PICR model and our method. The comparison indicates that our correction of copy numbers is necessary for CNV studies.     

Entropy embedding and fluctuation analysis in genomic manifolds

The complexity of typically high-dimensional genomic data requires computational work prone to integrate different biological information sources through efficient model solutions. Usually, one step involves dimensionality reduction (DR), which requires projecting the input data onto a low dimensional subspace, and often leads to an embedding. Thus, DR should be able to filter out the uninformative dimensions and recover the original variables. This step is of crucial relevance for any reverse engineering and statistical inference attempt to reconstruct the dynamics underlying the biological systems under study, i.e. the interactions between its genes or proteins. DR has become almost a standard practice just following the pre-processing steps typically applied to the experimental measurements (mining, normalization, etc.). In this work, the data for the analysis reflect expression values of genes whose dynamics are affected by perturbation experiments. In particular, the aims are to monitor the response of genes involved in a certain pathway, and then to isolate their biological variability from any possible external influence. Last, it is of interest to control the stability of the system; with this regard, we look at dynamical aspects of data through embedding theory and entropy fluctuation analysis. We demonstrate that a redundant biological system can in principle be reduced to a minimal number of almost independent components. In particular, such structures detect the higher-order statistical dependencies in the training data in addition to the correlations. Two popular DR techniques are compared in relation to their ability to extract the most salient features, allow gene selection, and minimize the various interferences due to algorithmic approximation errors and variable noise covers.     

云计算模式下SLA中的补偿策略及风险

本文基于云计算的特性,首次提出了服务水平协议(Service Level Agreement,SLA)中的三种补偿策略(分别为基于服务失败数量、基于长服务中断数量和基于累积中断时间的补偿策略),通过在SLA中约定客户容忍阀值,针对服务失败等级对客户进行分级补偿。针对三种策略的补偿成本,本文对云服务提供商造成的损失风险进行了测量和分析,得到了风险价值(Value-at-Risk,VaR)变化的特征和规律。在平均损失相同情况下,基于累积中断时间补偿策略的VaR值大于基于服务失败数量补偿策略的VaR值,因此采取基于中断时间的补偿策略风险更大。实例数据分析表明,本文提出的补偿策略是合理可行的。本文结论对于云服务提供商的补偿策略制定具有指导意义。     

Estimation of the number of failures in the Weibull model using the ordinary differential equation

In estimating the number of failures using right truncated grouped data, we often encounter cases that the estimate is smaller than the true one when we use the likelihood principle to conditional probability. In infectious disease spread predictions, the SIR model described by simultaneous ordinary differential equations is commonly used, and it can predict reasonably well the number of infected patients even when the size of observed data is small. We have investigated whether the ordinary differential equation model can estimate the number of failures more accurately than does the likelihood principle under the condition of right truncated grouped data. The positive results are obtained in the Weibull model, similarly to the cases of the SARS, A(H1N1), and FMD.     

Mixture representation for the residual lifetime of a repairable system

In this paper, we consider a repairable system in which two types of failures can occur on each failure. One is a minor failure that can be corrected with minimal repair, whereas the other type is a catastrophic failure that destroys the system. The total number of failures until the catastrophic failure is a positive random variable with a given probability vector. It is assumed that there is some partial information about the failure status of the system, and then various properties of the conditional probability of the system failure are studied. Mixture representations of the reliability function for the system in terms of the reliability function of the residual lifetimes of record values are obtained. Some stochastic properties of the conditional probabilities and the residual lifetimes of two systems are finally discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

估计的和理论的保证以及启动Ⅲ期试验的概率

预期的Ⅱ期试验通常会导致Ⅲ期试验失败.对于随机分为两个治疗组的随机对照的Ⅱ期和Ⅲ期试验,在假设正态分布响应的方差已知的情况下,我们解析性地获得了这三种情况下的估计的和理论的保证(无、加性和乘性偏差调整).在一些较小的假设下,我们证明了对这三种情况下的估计保证分别是Ⅱ期试验的每组患者数和Ⅱ期试验观察到的治疗效果的增加函数...     

父代种群参与竞争遗传算法几乎必然收敛

熟知,标准遗传算法如不采用“杰出者记录策略”则必不收敛。本文发现：允许父代种群参与竞争是标准遗传算法几乎必然收敛的条件。特别地,我们运用鞅收敛定理证明：允许父代种群参与竞争型遗传算法能以概率1确保在限步内达到全局最优解,且收敛与种群规模无关。所获结果对该类遗传算法的应用奠定了可靠基础。     

Application of auxiliary Markov chains to start-up demonstration tests

We use auxiliary Markov chains to derive probabilistic results for five types of start-up demonstration tests, with start-ups that are Markovian of a general order. Four of the tests are based on consecutive (or total) successful start-ups and consecutive (or total) failures; the fifth has two rejection criteria. For each test type, we obtain the probability of the test ending with acceptance of the unit, the probability distribution and moments of the number of start-ups in the test, the probability of acceptance (or rejection) of the equipment in a specified number of trials, and the conditional distribution of the number of start-ups in the test given that the unit is accepted or rejected. Numerical examples are given. Though the results are for these specific types of start-up demonstration tests, the method of derivation may be used for tests with other stopping criteria, and in other situations as well.     

A Sparse Structured Shrinkage Estimator for Nonparametric Varying-Coefficient Model with an Application in Genomics

Many problems in genomics are related to variable selection where high-dimensional genomic data are treated as covariates. Such genomic covariates often have certain structures and can be represented as vertices of an undirected graph. Biological processes also vary as functions depending upon some biological state, such as time. High-dimensional variable selection where covariates are graph-structured and underlying model is nonparametric presents an important but largely unaddressed statistical challenge. Motivated by the problem of regression-based motif discovery, we consider the problem of variable selection for high-dimensional nonparametric varying-coefficient models and introduce a sparse structured shrinkage (SSS) estimator based on basis function expansions and a novel smoothed penalty function. We present an efficient algorithm for computing the SSS estimator. Results on model selection consistency and estimation bounds are derived. Moreover, finite-sample performances are studied via simulations, and the effects of high-dimensionality and structural information of the covariates are especially highlighted. We apply our method to motif finding problem using a yeast cell-cycle gene expression dataset and word counts in genes' promoter sequences. Our results demonstrate that the proposed method can result in better variable selection and prediction for high-dimensional regression when the underlying model is nonparametric and covariates are structured. Supplemental materials for the article are available online.     

A Sparse Structured Shrinkage Estimator for Nonparametric Varying-Coefficient Model With an Application in Genomics

Many problems in genomics are related to variable selection where high-dimensional genomic data are treated as covariates. Such genomic covariates often have certain structures and can be represented as vertices of an undirected graph. Biological processes also vary as functions depending upon some biological state, such as time. High-dimensional variable selection where covariates are graph-structured and underlying model is nonparametric presents an important but largely unaddressed statistical challenge. Motivated by the problem of regression-based motif discovery, we consider the problem of variable selection for high-dimensional nonparametric varying-coefficient models and introduce a sparse structured shrinkage (SSS) estimator based on basis function expansions and a novel smoothed penalty function. We present an efficient algorithm for computing the SSS estimator. Results on model selection consistency and estimation bounds are derived. Moreover, finite-sample performances are studied via simulations, and the effects of high-dimensionality and structural information of the covariates are especially highlighted. We apply our method to motif finding problem using a yeast cell-cycle gene expression dataset and word counts in genes’ promoter sequences. Our results demonstrate that the proposed method can result in better variable selection and prediction for high-dimensional regression when the underlying model is nonparametric and covariates are structured. Supplemental materials for the article are available online.     

M/M/1排队系统中的维修策略

以平均报酬率为目标函数的维修策略问题引入可修排队系统.在M/M/1/模型下,利用几何过程描述服务台随机退化过程,考虑了基于服务台失效次数N的策略,即当失效次数到达N次时,对服务台进行替换.根据更新报酬定理,获得了基于维修次数N的平均报酬率的表达式.     

Max Horn SAT and the minimum cut problem in directed hypergraphs

In this paper we consider the Maximum Horn Satisfiability problem, which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph. For the latter problem, we propose different IP formulations, related to three different definitions of hyperpath weight. We investigate the properties of their linear relaxations, showing that they define a hierarchy. The weakest relaxation is shown to be equivalent to the relaxation of a well known IP formulation of Max Horn SAT, and to a max-flow problem on hypergraphs. The tightest relaxation, which is a disjunctive programming problem, is shown to have integer optimum. The intermediate relaxation consists in a set covering problem with a possible exponential number of constraints. This latter relaxation provides an approximation of the convex hull of the integer solutions which, as proven by the experimental results given, is much tighter than the one known in the literature. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.