Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves

SUMMARY: In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback-Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors.     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

Accurate genetic and environmental covariance estimation with composite likelihood in genome-wide association studies

Genetic and environmental covariances between pairs of complex traits are important quantitative measurements that characterize their shared genetic and environmental architectures. Accurate estimation of genetic and environmental covariances in genome-wide association studies (GWASs) can help us identify common genetic and environmental factors associated with both traits and facilitate the investigation of their causal relationship. Genetic and environmental covariances are often modeled through multivariate linear mixed models. Existing algorithms for covariance estimation include the traditional restricted maximum likelihood (REML) method and the recent method of moments (MoM). Compared to REML, MoM approaches are computationally efficient and require only GWAS summary statistics. However, MoM approaches can be statistically inefficient, often yielding inaccurate covariance estimates. In addition, existing MoM approaches have so far focused on estimating genetic covariance and have largely ignored environmental covariance estimation. Here we introduce a new computational method, GECKO, for estimating both genetic and environmental covariances, that improves the estimation accuracy of MoM while keeping computation in check. GECKO is based on composite likelihood, relies on only summary statistics for scalable computation, provides accurate genetic and environmental covariance estimates across a range of scenarios, and can accommodate SNP annotation stratified covariance estimation. We illustrate the benefits of GECKO through simulations and applications on analyzing 22 traits from five large-scale GWASs. In the real data applications, GECKO identified 50 significant genetic covariances among analyzed trait pairs, resulting in a twofold power gain compared to the previous MoM method LDSC. In addition, GECKO identified 20 significant environmental covariances. The ability of GECKO to estimate environmental covariance in addition to genetic covariance helps us reveal strong positive correlation between the genetic and environmental covariance estimates across trait pairs, suggesting that common pathways may underlie the shared genetic and environmental architectures between traits.     

THE COMPARISON OF QUANTITATIVE GENETIC PARAMETERS BETWEEN POPULATIONS

A statistical method for comparing matrices of genetic variation and covariation between groups (e.g., species, populations, a single population grown in distinct environments) is proposed. This maximum-likelihood method provides a test of the overall null hypothesis that two covariance component matrices are identical. Moreover, when the overall null hypothesis is rejected, the method provides a framework for isolating the particular components that differ significantly between the groups. Simulation studies reveal that discouragingly large experiments are necessary to obtain acceptable power for comparing genetic covariance component matrices. For example, even in cases of a single trait measured on 900 individuals in a nested design of 100 sires and three dams per sire in each population, the power was only about 0.5 when additive genetic variance differed by a factor of 2.5. Nevertheless, this flexible method makes valid comparison of covariance component matrices possible.     

基于堆积协变信息与最小自由能预测含伪结的RNA二级结构

RNA伪结预测是RNA研究的一个难点问题。文中提出一种基于堆积协变信息与最小自由能的RNA伪结预测方法。该方法使用已知结构的RNA比对序列(ClustalW比对和结构比对)测试此方法, 侧重考虑相邻碱基对之间相互作用形成的堆积协变信息, 并结合最小自由能方法对碱基配对综合评分, 通过逐步迭代求得含伪结的RNA二级结构。结果表明, 此方法能正确预测伪结, 其平均敏感性和特异性优于参考算法, 并且结构比对的预测性能比ClustalW比对的预测性能更加稳定。文中同时讨论了不同协变信息权重因子对预测性能的影响, 发现权重因子比值在l1: l2=5:1时, 预测性能达到最优。     

Quantitative covariance NMR by regularization

The square root of a covariance spectrum, which offers high spectral resolution along both dimensions requiring only few t ₁ increments, yields in good approximation the idealized 2D FT spectrum provided that the amount of magnetization exchanged between spins is relatively small. When this condition is violated, 2D FT and covariance peak volumes may differ. A regularization method is presented that produces a modified covariance spectrum with cross-peak volumes that closely match their 2D FT analogues. The method is demonstrated for TOCSY spectra with variable mixing times.     

Dynamic and structural analysis of isotropically distributed molecular ensembles.

An efficient new method is presented for the characterization of motional correlations derived from a set of protein structures without requiring the separation of overall and internal motion. In this method, termed isotropically distributed ensemble (IDE) analysis, each structure is represented by an ensemble of isotropically distributed replicas corresponding to the situation found in an isotropic protein solution. This leads to a covariance matrix of the cartesian atomic positions with elements proportional to the ensemble average of scalar products of the position vectors with respect to the center of mass. Diagonalization of the covariance matrix yields eigenmodes and amplitudes that describe concerted motions of atoms, including overall rotational and intramolecular dynamics. It is demonstrated that this covariance matrix naturally distinguishes between "rigid" and "mobile" parts without necessitating a priori selection of a reference structure and an atom set for the orientational alignment process. The method was applied to the analysis of a 5-ns molecular dynamics trajectory of native ubiquitin and a 40-ns trajectory of a partially folded state of ubiquitin. The results were compared with essential dynamics analysis. By taking advantage of the spherical symmetry of the IDE covariance matrix, more than a 10-fold speed up is achieved for the computation of eigenmodes and mode amplitudes. IDE analysis is particularly suitable for studying the correlated dynamics of flexible and large molecules.     

A compound decision approach to covariance matrix estimation

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is suboptimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example, sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals.     

Improved covariance matrix estimators for weighted analysis of microarray data.

Empirical Bayes models have been shown to be powerful tools for identifying differentially expressed genes from gene expression microarray data. An example is the WAME model, where a global covariance matrix accounts for array-to-array correlations as well as differing variances between arrays. However, the existing method for estimating the covariance matrix is very computationally intensive and the estimator is biased when data contains many regulated genes. In this paper, two new methods for estimating the covariance matrix are proposed. The first method is a direct application of the EM algorithm for fitting the multivariate t-distribution of the WAME model. In the second method, a prior distribution for the log fold-change is added to the WAME model, and a discrete approximation is used for this prior. Both methods are evaluated using simulated and real data. The first method shows equal performance compared to the existing method in terms of bias and variability, but is superior in terms of computer time. For large data sets (>15 arrays), the second method also shows superior computer run time. Moreover, for simulated data with regulated genes the second method greatly reduces the bias. With the proposed methods it is possible to apply the WAME model to large data sets with reasonable computer run times. The second method shows a small bias for simulated data, but appears to have a larger bias for real data with many regulated genes.     

Prediction of Protein Structural Classes by Modified Mahalanobis Discriminant Algorithm

We first discuss quantitative rules for determining the protein structural classes based on their secondary structures. Then we propose a modification of the least Mahalanobis distance method for prediction of protein classes. It is a generalization of a quadratic discriminant function to the case of degenerate covariance matrices. The resubstitution tests and leave-one-out tests are carried out to compare several methods. When the class sample sizes or the covariance matrices of different classes are significantly different, the modified method should be used to replace the least Mahalanobis distance method. Two lemmas for the derivation of our new algorithm are proved in an appendix.     

Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains

The specific growth rate for P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY-mutM is estimated by a suggested Maximum Likelihood, ML, method which takes the autocorrelation of the observation into account. For each bacteria strain, six wells of optical density, OD, measurements are used for parameter estimation. The data is log-transformed such that a linear model can be applied. The transformation changes the variance structure, and hence an OD-dependent variance is implemented in the model. The autocorrelation in the data is demonstrated, and a correlation model with an exponentially decaying function of the time between observations is suggested. A model with a full covariance structure containing OD-dependent variance and an autocorrelation structure is compared to a model with variance only and with no variance or correlation implemented. It is shown that the model that best describes data is a model taking into account the full covariance structure. An inference study is made in order to determine whether the growth rate of the five bacteria strains is the same. After applying a likelihood-ratio test to models with a full covariance structure, it is concluded that the specific growth rate is the same for all bacteria strains. This study highlights the importance of carrying out an explorative examination of residuals in order to make a correct parametrization of a model including the covariance structure. The ML method is shown to be a strong tool as it enables estimation of covariance parameters along with the other model parameters and it makes way for strong statistical tools for inference studies.     

Robust frequency recognition for SSVEP-based BCI with temporally local multivariate synchronization index

Multivariate synchronization index (MSI) has been proved to be an efficient method for frequency recognition in SSVEP-BCI systems. It measures the correlation according to the entropy of the normalized eigenvalues of the covariance matrix of multichannel signals. In the MSI method, the estimation of covariance matrix omits the temporally local structure of samples. In this study, a new spatio-temporal method, termed temporally local MSI (TMSI), was presented. This new method explicitly exploits temporally local information in modelling the covariance matrix. In order to evaluate the performance of the TMSI, we performs a comparison between the two methods on the real SSVEP datasets from eleven subjects. The results show that the TMSI outperforms the standard MSI. TMSI benefits from exploiting the temporally local structure of EEG signals, and could be a potential method for robust performance of SSVEP-based BCI.     

Predicting genetic values: a kernel-based best linear unbiased prediction with genomic data

Genomic data provide a valuable source of information for modeling covariance structures, allowing a more accurate prediction of total genetic values (GVs). We apply the kriging concept, originally developed in the geostatistical context for predictions in the low-dimensional space, to the high-dimensional space spanned by genomic single nucleotide polymorphism (SNP) vectors and study its properties in different gene-action scenarios. Two different kriging methods ["universal kriging" (UK) and "simple kriging" (SK)] are presented. As a novelty, we suggest use of the family of Matérn covariance functions to model the covariance structure of SNP vectors. A genomic best linear unbiased prediction (GBLUP) is applied as a reference method. The three approaches are compared in a whole-genome simulation study considering additive, additive-dominance, and epistatic gene-action models. Predictive performance is measured in terms of correlation between true and predicted GVs and average true GVs of the individuals ranked best by prediction. We show that UK outperforms GBLUP in the presence of dominance and epistatic effects. In a limiting case, it is shown that the genomic covariance structure proposed by VanRaden (2008) can be considered as a covariance function with corresponding quadratic variogram. We also prove theoretically that if a specific linear relationship exists between covariance matrices for two linear mixed models, the GVs resulting from BLUP are linked by a scaling factor. Finally, the relation of kriging to other models is discussed and further options for modeling the covariance structure, which might be more appropriate in the genomic context, are suggested.     

Directional Variance Adjustment: Bias Reduction in Covariance Matrices Based on Factor Analysis with an Application to Portfolio Optimization

Robust and reliable covariance estimates play a decisive role in financial and many other applications. An important class of estimators is based on factor models. Here, we show by extensive Monte Carlo simulations that covariance matrices derived from the statistical Factor Analysis model exhibit a systematic error, which is similar to the well-known systematic error of the spectrum of the sample covariance matrix. Moreover, we introduce the Directional Variance Adjustment (DVA) algorithm, which diminishes the systematic error. In a thorough empirical study for the US, European, and Hong Kong stock market we show that our proposed method leads to improved portfolio allocation.     

Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains

The specific growth rate for P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY–mutM is estimated by a suggested Maximum Likelihood, ML, method which takes the autocorrelation of the observation into account. For each bacteria strain, six wells of optical density, OD, measurements are used for parameter estimation. The data is log-transformed such that a linear model can be applied. The transformation changes the variance structure, and hence an OD-dependent variance is implemented in the model. The autocorrelation in the data is demonstrated, and a correlation model with an exponentially decaying function of the time between observations is suggested. A model with a full covariance structure containing OD-dependent variance and an autocorrelation structure is compared to a model with variance only and with no variance or correlation implemented. It is shown that the model that best describes data is a model taking into account the full covariance structure. An inference study is made in order to determine whether the growth rate of the five bacteria strains is the same. After applying a likelihood-ratio test to models with a full covariance structure, it is concluded that the specific growth rate is the same for all bacteria strains. This study highlights the importance of carrying out an explorative examination of residuals in order to make a correct parametrization of a model including the covariance structure. The ML method is shown to be a strong tool as it enables estimation of covariance parameters along with the other model parameters and it makes way for strong statistical tools for inference studies.     

Analysis of cytoplasmic and maternal effects I. A genetic model for diploid plant seeds and animals

A genetic model for modified diallel crosses is proposed for estimating variance and covariance components of cytoplasmic, maternal additive and dominance effects, as well as direct additive and dominance effects. Monte Carlo simulations were conducted to compare the efficiencies of minimum norm quadratic unbiased estimation (MINQUE) methods. For both balanced and unbalanced mating designs, MINQUE (0/1), which has 0 for all the prior covariances and 1 for all the prior variances, has similar efficiency to MINQUE(), which has parameter values for the prior values. Unbiased estimates of variance and covariance components and their sampling variances could be obtained with MINQUE(0/1) and jackknifing. A t-test following jackknifing is applicable to test hypotheses for zero variance and covariance components. The genetic model is robust for estimating variance and covariance components under several situations of no specific effects. A MINQUE(0/1) procedure is suggested for unbiased estimation of covariance components between two traits with equal design matrices. Methods of unbiased prediction for random genetic effects are discussed. A linear unbiased prediction (LUP) method is shown to be efficient for the genetic model. An example is given for a demonstration of estimating variance and covariance components and predicting genetic effects.     

Generalized least squares for the synthesis of correlated information

This paper deals with the synthesis of information from different studies when there is lack of independence in some of the contrasts to be combined. This problem can arise in several different situations in both case-control studies and clinical trials. For efficient estimation we appeal to the method of generalized least squares to estimate the summary effect and its standard error. This method requires estimates of the covariances between those contrasts that are not independent. Although it is not possible to estimate the covariance between effects that have been adjusted for confounding factors we present a method for finding upper and lower bounds for this covariance. In the simplest discussion homogeneity of the relative risks is assumed but the method is then extended to allow for heterogeneity in an overall estimate. We then illustrate the method with several examples from an analysis in which case-control studies of cervical cancer and oral contraceptive use are synthesized.     

Ordered genotypes: an extended ITO method and a general formula for genetic covariance

Traditionally, the stochastic ITO transition matrices provide a simple general method for obtaining the joint genotype distribution and genotypic correlations between any specified pair of noninbred relatives. The ITO method has been widely used in modern genetic analysis; however, since it was originally derived for unordered genotypes, it is not very useful in some new applications -- for example, when one is modeling genomic imprinting and must keep track of the parental origin of alleles. To address these new, emerging problems, here we extend the ITO method to handle ordered genotypes. Our extended method is applied to calculate the covariance in unilineal and bilineal relatives under genomic imprinting, and some generalized linear functions of the transition matrices are given. Since the ITO method is limited to biallelic loci and to unilineal and bilineal relatives, we derive a general formula for calculating the genetic covariance using ordered genotypes for any type of relative pair.     

Covariance analyses with heterogeneity of slopes in fixed models

Techniques that describe the use of covariance when heterogeneity of slopes exists are severely limited. Although a few procedures for model selection have been recommended, none, except the hierarchical approach, is straightforward and usable with present computer programs. The hierarchical subset selection procedure presented in this paper is based on the proposition that heterogeneity may be present only for certain terms in the model. After hierarchical selection, those terms which do not involve heterogeneity are interpreted as in the usual analysis for covariance. The interpretations of those terms which do involve heterogeneity are modified with respect to significance tests performed at various values of the covariate. The hierarchical subset selection method allows one to investigate heterogeneity of slopes in covariance models as functions of the classification variables present in the design.