Multiple Testing and Error Control in Gaussian Graphical Model Selection

Loss function, unbiasedness, and optimality of Gaussian graphical model selection

A novel framework of compressed sensing, namely statistical compressed\nsensing (SCS), that aims at efficiently sampling a collection of signals that\nfollow a statistical distribution, and achieving accurate reconstruction on\naverage, is introduced. SCS based on Gaussian models is investigated in depth.\nFor signals that follow a single Gaussian model, with Gaussian or Bernoulli\nsensing matrices of O(k) measurements, considerably smaller than the O(k\nlog(N/k)) required by conventional CS based on sparse models, where N is the\nsignal dimension, and with an optimal decoder implemented via linear filtering,\nsignificantly faster than the pursuit decoders applied in conventional CS, the\nerror of SCS is shown tightly upper bounded by a constant times the best k-term\napproximation error, with overwhelming probability. The failure probability is\nalso significantly smaller than that of conventional sparsity-oriented CS.\nStronger yet simpler results further show that for any sensing matrix, the\nerror of Gaussian SCS is upper bounded by a constant times the best k-term\napproximation with probability one, and the bound constant can be efficiently\ncalculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian\ndistributions and that each signal follows one of them with an unknown index, a\npiecewise linear estimator is introduced to decode SCS. The accuracy of model\nselection, at the heart of the piecewise linear decoder, is analyzed in terms\nof the properties of the Gaussian distributions and the number of sensing\nmeasurements. A maximum a posteriori expectation-maximization algorithm that\niteratively estimates the Gaussian models parameters, the signals model\nselection, and decodes the signals, is presented for GMM-based SCS. In real\nimage sensing applications, GMM-based SCS is shown to lead to improved results\ncompared to conventional CS, at a considerably lower computational cost.\n

A major challenge in contemporary neuroscience is to analyze data from large numbers of neurons recorded simultaneously across many experimental replications (trials), where the data are counts of neural firing events, and one of the basic problems is to characterize the dependence structure among such multivariate counts. Methods of estimating high-dimensional covariation based on ℓ 1-regularization are most appropriate when there are a small number of relatively large partial correlations, but in neural data there are often large numbers of relatively small partial correlations. Furthermore, the variation across trials is often confounded by Poisson-like variation within trials. To overcome these problems we introduce a comprehensive methodology that imbeds a Gaussian graphical model into a hierarchical structure: the counts are assumed Poisson, conditionally on latent variables that follow a Gaussian graphical model, and the graphical model parameters, in turn, are assumed to depend on physiologically-motivated covariates, which can greatly improve correct detection of interactions (non-zero partial correlations). We develop a Bayesian approach to fitting this covariate-adjusted generalized graphical model and we demonstrate its success in simulation studies. We then apply it to data from an experiment on visual attention, where we assess functional interactions between neurons recorded from two brain areas.

The multivariate adaptive regression spline (MARS) is one of the alternative nonparametric approaches of the Gaussian graphical model (GGM) in order to construct the undirected networks of complex biological systems. In GGM, it is assumed that the binary form of the estimated precision matrix, computed from the systems' elements under their conditional independencies, represents the undirected interactions of these elements by using the lasso regression. From previous studies, it has been shown that if MARS is estimated by discarding all interactions and nonlinear components, resulting in a sole inference of the main effects, it can be a strong alternative of GGM. We call this type of the summarized MARS model the lasso-based MARS model. Furthermore, in the construction of the networks via the optimal MARS model, it is assumed that the estimated regression coefficients, i.e., the important terms of the estimated model, can be the indication of the pairwise links between the species of the systems. Hereby, in this study, we investigate whether the inference of the networks via the Pearson, Spearman, and Kendall's tau correlation coefficients has better performance than the networks constructed by the important regression coefficients in the lasso-based MARS model. In the analyses, we use both simulated and real datasets and compare the estimated networks by various accuracy measures from specificity and F-measure to Matthews correlation coefficient. From the analyses, we find that the suggested approaches improve all the listed criteria significantly without any computational cost.

In machine learning, graphical models like Bayesian networks are one of important visualization tools that can be learned from data to represent pictorially a complex system. In order to compare two complex systems (or one complex system functioning in two different contexts), one usually compares directly their representative graphs. However, with small sample size data, it is hard to learn the graph that represents precisely the system. That's why ensemble methods (e.g. Bootstrapping, evolutionary algorithm, etc...) are proposed to learn from data of each system a set of graphs that represents more precisely this system. Then, for comparing two systems, one needs a mechanism to compare two sets of graphs. We propose in this work an approach based on multiple hypothesis testing and quasi essential graph (QEG) to compare two sets of Bayesian networks.

Multiantenna measurements were performed in three small aircraft (Beech Baron 58-P, Rockwell T-39 Sabreliner, and the Diamond DA-42 Twin Star) and compared to those of six statistical multiantenna channel models, being the Gaussian, Nakagami-m, Weibull, hyperRayleigh, virtual, and Weichselberger models. The results were also compared to that of a site-specific3Dray-tracing-basedmodel.Theeffectsofthemeasurementsystem,likeantenna polarization and radiation pattern, were added to all of the models. It was found that the 3D ray-tracing model provides the most accurate predictions, closely followed by the Nakagamim and Gaussian models. Differences between measured and modeled data were typically less than 1bit/s/Hz for the three aircraft for the 3D ray-tracing model. For the Nakagami-m model, the differences were typically in the range of 0.5‐1.5bits/s/Hz whereas the Gaussian model had differences typically in the range of 1‐2bit/s/Hz. The simpler statistical hyperRayleigh model gives differences of the order of 2‐3bit/s/Hz for the Beech Baron 58-P and 3‐5bits/s/Hz for the Rockwell T-39 Sabreliner. The virtual and Weichselberger models were in the range of 2‐4bits/s/Hz.

This doctoral thesis contributes to the advancement of statistical analyses of gene expression profiles using Gaussian graphical models (GGMs). The main focus has been on studying the properties of GGMs obtained with the LW shrinkage. The premise is that, despite its advantages, the shrinkage approach introduces some biases that have not been studied in sufficient detail. These biases have the potential of obscuring the interpretation of the network structure and impeding the validation of earlier analyses. In this sense, this thesis may be considered a continuation of the works of Schäfer and Strimmer, where the LW-shrinkage was originally used to model GRNs with GGMs. The thesis is organized as follows; Chapter 1 presents a test of statistical significance for GGMs based on the LW-shrinkage. Here the probability density of the ‘shrunk’ partial correlation is derived by means of geometric arguments. In Chapter 2 a network analysis of (matched) nasal and bronchial expression profiles is presented. The method developed in the previous chapter is employed to explore whether expression profiles from nasal epithelial cells can be used as a proxy for bronchial epithelial cells. Chapter 3 shows the existence of a non-linear bias on the partial correlations obtained with the LW-shrinkage. This bias is removed via ‘un-shrinking’; a new concept that de-regularizes the partial correlation. In Chapter 4 the LW-shrinkage is revisited from a data-level perspective. The goal is to explore the correspondence between the shrinkage-based covariance matrix and the dataset.

A new methodology for model determination in decomposable graphical Gaussian models (Dawid and Lauritzen in Ann. Stat. 21(3), 1272---1317, 1993) is developed. The Bayesian paradigm is used and, for each given graph, a hyper-inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models's posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (Stat. Comput. 19(3), 303---316, 2009). We illustrate the efficiency of this new scheme on simulated and real datasets.

Mapping of the Ionospheric Total Electron Content over the East African Low–Latitude Region

Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests. In this work, we introduce a modified version of the FDR called the “positive false discovery rate” (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the “q-value” is introduced and investigated, which is a natural “Bayesian posterior p-value,” or rather the pFDR analogue of the p-value. 1. Introduction. When testing a single hypothesis, one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists. In statistical terms, we maximize the power conditional on the Type I error rate being at or below some level. The field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously. One must define an appropriate compound error measure according to the rate of false positives one is willing to encounter. Then a procedure is developed that allows one to control the error rate at a desired level, while maintaining the power of each test as much as possible. The most commonly controlled quantity when testing multiple hypotheses is the family wise error rate (FWER), which is the probability of yielding one or more false positives out of all hypotheses tested. The most familiar example of this is the Bonferroni method. If there are m hypothesis tests, each test is controlled so that the probability of a false positive is less than or equal to α/m for some chosen value of α. It then follows that the overall FWER is less than or equal to α .M any

There is uncertainty in the neuromusculoskeletal system, and deterministic models cannot describe this significant presence of uncertainty, affecting the accuracy of model predictions. In this paper, a knee joint angle prediction model based on surface electromyography (sEMG) signals is proposed. To address the instability of EMG signals and the uncertainty of the neuromusculoskeletal system, a non-parametric probabilistic model is developed using a Gaussian process model combined with the physiological properties of muscle activation. Since the neuromusculoskeletal system is a dynamic system, the Gaussian process model is further combined with a non-linear autoregressive with eXogenous inputs (NARX) model to create a Gaussian process autoregression model. In this paper, the normalized root mean square error (NRMSE) and the correlation coefficient (CC) are compared between the joint angle prediction results of the Gaussian process autoregressive model prediction and the actual joint angle under three test scenarios: speed-dependent, multi-speed and speed-independent. The mean of NRMSE and the mean of CC for all test scenarios in the healthy subjects dataset and the hemiplegic patients dataset outperform the results of the Gaussian process model, with significant differences (p < 0.05 and p < 0.05, p < 0.05 and p < 0.05). From the perspective of uncertainty, a non-parametric probabilistic model for joint angle prediction is established by using Gaussian process autoregressive model to achieve accurate prediction of human movement.

Research background: Every real estate may be described with a multitude of attributes. In the process of real estate appraisal only those properties are taken into account that significantly affect its value. Mass appraisal involves a simultaneous valuation of many similar real properties, carried out in the same manner and at the same time. The algorithm applied to mass appraisal ought to ensure a uniform approach to the valuation of all real estate of the same type in an objective fashion. Purpose: The purpose of the paper is to define the weights of attributes in the process of real estate mass appraisal on the basis of relationships between unit property value and the values of attributes. Research methodology: The weights were defined on the grounds of partial correlation coefficients for the qualitative properties (Spearman rank correlation coefficient and τB Kendall correlation coefficient). Results: The signs of certain correlation coefficients were discordant with the actual direction of the relations between the analysed properties. The problem was avoided by employing partial correlation coefficients. On the basis of the calculated partial correlation coefficients the weights of individual attributes were calculated. Of all analysed coefficients the partial τB Kendall correlation coefficient is methodologically the most suitable one. Novelty: The use of partial correlation coefficients for determining attribute weights is an innovative approach and is applied in the article.

This protocol describes how to build a gene network based on the graphical Gaussian model (GGM) from large scale microarray data. GGM uses partial correlation coefficient (pcor) to infer co-expression relationship between genes. Compared to the traditional Pearson’ correlation coefficient, partial correlation is a better measurement of direct dependency between genes. However, to calculate pcor requires a large number of observations (microarray slides) greatly exceeding the number of variables (genes). This protocol uses a regularized method to circumvent this obstacle, and is capable of building a network for ~20,000 genes from ~2,000 microarray slides. For more details, see Ma et al. (2007). For help regarding the script, please contact the author.

This paper introduces a MATLAB toolbox, MGraph, which applies graphical models as a natural environment to formulate and solve problems in microarray data analysis. MGraph with its graphical interface allows the user to predict genetic regulatory networks by a graphical gaussian model (GGM), and to quantify the effects of different experimental treatment conditions on gene expression profiles by a graphical log-linear model (GLM). The power of graphical models was explored and illustrated through two example applications. First, four MAPK pathways in yeast were meaningfully reconstructed through GGM. Second, GLM was used to quantify the contributions of sex, genotype and age to transcriptional variance in Drosophila melanogaster. This application may provide a valuable aid in the prediction of genetic regulatory networks, as well as in investigations of various experimental conditions that affect global gene expression profiles. The MATLAB program MGraph is freely available at http://www.uio.no/~junbaiw/mgraph/mgraph.html for academics.

Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.