Analysis of Large Data Sets

For general covariance matrix, we derive the asymptotic risk of LASSO in the limit of n and p going to infinity with fixed ratio n/p. A phase boundary is precisely established in the phase space. Above this boundary, LASSO perfectly recovers the signals with high probability. Below this boundary, LASSO fails to recover the signals with high probability. While the values of the non-zero elements of the signals do not have any effect on the phase transition curve, our analysis shows that the curve does depend on the signed pattern of the nonzero values of the signal for non-i.i.d. covariance matrix. Underlying our formalism is a recently developed efficient algorithm called approximate message passing (AMP) algorithm. We generalize the state evolution of AMP from i.i.d. case to general case. Extensive computational experiments confirm that our theoretical predictions are consistent with simulation results on moderate size system.

In nonparametric Bayesian mixture models we assume that the data is an iid sample from an infinite mixture of component distributions; the mixture weights and the parameters of the components are random. This approach can be applied to cluster analysis; in this context it is convenient to think that we have a prior distribution on the space of all possible partitions of data (e.g. the Chineses Restaurant Process) and the likelihood is obtained by marginalising the parameter out of component distributions. This can be transformed into the posterior on the space of possible clusterings; as is rather common in the Bayesian setting, this posterior is known up to intractable normalising constant. During the talk we are going to present some properties of the MAP partition - the one that maximises the posterior probability. Firstly we will describe the results of (1) that concern the Chinese Restaurant prior on partitions and the component distributions being multivariate Gaussian with unknown mean but known (and the same for all components) covariance. Those findings concern the geometry of the MAP partition (we prove that in this case the MAP clusters are linearly separated from each other) and its asymptotic behaviour. Then we will show generalisations of some of these results to general infinite mixture models, when the component measures and the prior on their parameters belong to conjugate exponential families. Finally we present a ‘score function’ for clusterings, related to our analysis. This score function can be used to choose among clustering propositions suggested by more computationally efficient algorithms, like K-means.

We shall review the notion of conic intrinsic volumes and provide some explicit examples. In particular, we will show examples of random polytopes exhibiting phase transitions when the dimension goes to infinity.

The use of multiple sources of information in regression modeling has recently received a lot of attention in the statistical and brain imaging literature. This talk introduces a novel, fully-automatic statistical procedure that addresses the problem of linear regression coefficients' estimation in the situation when the additional information about connectivities between variables is given. Our method, “Adaptive Information Merging Estimator for Regression” (AIMER) enables for the incorporation of multiple sources of such information as well as for the division of one source into pieces and determining their impact on the estimates. We performed extensive simulations to visualize the desired adjusting properties of our method and show its advantages over the existing approaches. We also applied AIMER to analyze structural brain imaging data and to reveal the association between cortical thickness and HIV-related outcomes.

The amount of medical data collected has been growing exponentially over the past few decades. This growth in data acquisition has not been, unfortunately, paralleled by the same growth rate of statistical learning methods’ development. In my talk, I will give a brief overview of the analytical methods developed by my group and their applications in the medical and public health areas. Specifically, (1) regularization methods applied to the structural brain imaging data and (2) signal processing techniques utilized in extracting physical activity information from the raw accelerometry data will be emphasized.

In the talk, we discuss existing approaches, known from the literature, to detection of stochastic ordering of the two survival curves as well as pose and solve the novel testing problem on it. Specifically, the null hypothesis asserts the lack of the ordering, while the alternative expresses its existence. An introduced test statistic is a functional of the standardized two-sample Kaplan-Meier process sampling in a randomly selected number of the random points being the observed survival times in the pooled sample and exploits the information contained in a specially defined one-sided weighted log-rank statistic. It automatically weighs the magnitude and sign of their components becoming a sensible procedure in the considered testing problem. As a result, the corresponding test asymptoticly controls the errors of both kinds at the specified significance level α. The conducted simulation study shows that the errors are also satisfactorily controlled when sample sizes are finite. Furthermore, in the comparison to the best and most popular tests, the new solution turns out to be a promising procedure which improves them upon. A real data analysis confirms that findings.

Classical scalar-response regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. In medical applications, however, regressors are often in a form of multi-dimensional arrays. For example, one may be interested in using MRI imaging to identify which brain regions are associated with a health outcome. Vectorizing the two-dimensional image arrays is an unsatisfactory approach since it destroys the inherent spatial structure of the images and can be computationally challenging. We present an alternative approach - regularized matrix regression - where the matrix of regression coefficients is defined as a solution to the specific optimization problem. The method, called SParsity Inducing Nuclear Norm EstimatoR (SpINNEr), simultaneously imposes two penalty types on the regression coefficient matrix - the nuclear norm and the lasso norm - to encourage a low rank matrix solution that also has entry-wise sparsity. A novel implementation of the alternating direction method of multipliers (ADMM) is used to build a fast and efficient numerical solver. Our simulations show that SpINNEr outperforms others methods in estimation accuracy when the response-related entries (representing the brain's functional connectivity) are arranged in well-connected communities. SpINNEr is applied to investigate associations between HIV disease-related outcomes and functional connectivity in the human brain.

We investigate the model selection properties of the Lasso estimator in finite samples with no conditions on the regressor matrix X. We show that which covariates the Lasso estimator may potentially choose in high dimensions (where the number of explanatory variables p exceeds sample size n) depends only on X and the given penalization weights. This set of potential covariates can be determined through a geometric condition on X and may be small enough (less than or equal to n in cardinality). Related to the geometric conditions in our considerations, we also provide a necessary and sufficient condition for uniqueness of the Lasso solutions. Finally, we discuss how these results carry over to other model selection procedures such as the SLOPE

Hydrogen-deuterium exchange monitored by mass spectrometry (HDX-MS) has recently become a staple tool in studies of protein structure. The main application of this technique is to compare the structure of a protein altered by several factors (so-called states). Introduced statistical frameworks address the screening part of the analysis, i.e., search for significant differences between states, but miss the post-screening phase of analysis. We critically evaluate existing models and point their strengths and weaknesses. Additionally, we provide a novel solution to a multi-state comparison problem where the region of the interest inside the protein structure is already well-defined.

Missing data exist in almost all areas of empirical research. There are various reasons why missing data may occur, including survey non-response, unavailability of measurements, and lost data. In this presentation, I will share my experience on how to do parametric estimation with missing covariates, based on likelihood methods and Expectation-Maximization algorithm. Then I will focus on recent results in a supervised learning setting, for performing logistic regression with missing values. We illustrate the method on a dataset of severely traumatized patients from Paris hospitals to predict the occurrence of hemorrhagic shock, a leading cause of early preventable death in severe trauma cases. The methodology is implemented in the R package misaem.