This article presents a novel method for causal discovery with generalized structural equation models suited for analyzing diverse types of outcomes, including discrete, continuous, and mixed data. Causal discovery often faces challenges due to unmeasured confounders that hinder the identification of causal relationships. The proposed approach addresses this issue by developing two peeling algorithms (bottom-up and top-down) to ascertain causal relationships and valid instruments. This approach first reconstructs a super-graph to represent ancestral relationships between variables, using a peeling algorithm based on nodewise GLM regressions that exploit relationships between primary and instrumental variables. Then, it estimates parent-child effects from the ancestral relationships using another peeling algorithm while deconfounding a child's model with information borrowed from its parents' models. The article offers a theoretical analysis of the proposed approach, which establishes conditions for model identifiability and provides statistical guarantees for accurately discovering parent-child relationships via the peeling algorithms. Furthermore, the article presents numerical experiments showcasing the effectiveness of our approach in comparison to state-of-the-art structure learning methods without confounders. Lastly, it demonstrates an application to Alzheimer's disease (AD), highlighting the utility of the method in constructing gene-to-gene and gene-to-disease regulatory networks involving Single Nucleotide Polymorphisms (SNPs) for healthy and AD subjects.

The article attempts to find an algebraic formula describing the correlation coefficients between random variables and the principal components representing them. As a result of the analysis, starting from selected statistics relating to individual random variables, the equivalents of these statistics relating to a set of random variables were presented in the language of linear algebra, using the concepts of vector and matrix. This made it possible, in subsequent steps, to derive the expected formula. The formula found is identical to the formula used in Factor Analysis to calculate factor loadings. The discussion showed that it is possible to apply this formula to optimize the number of principal components in Principal Component Analysis, as well as to optimize the number of factors in Factor Analysis.

In this work, the possibility of clustering correlated random variables was examined, both because of their mutual similarity and because of their similarity to the principal components. The k-means algorithm and spectral algorithms were used for clustering. For spectral methods, the similarity matrix was both the matrix of relation established on the level of correlation and the matrix of coefficients of determination. For four different sets of data, different ways of measuring the disimilarity of variables were analyzed, and the impact of the diversity of initial points on the efficiency of the k-means algorithm was analyzed.

Statistical inference is considered for variables of interest, called primary variables, when auxiliary variables are observed along with the primary variables. We consider the setting of incomplete data analysis, where some primary variables are not observed. Utilizing a parametric model of joint distribution of primary and auxiliary variables, it is possible to improve the estimation of parametric model for the primary variables when the auxiliary variables are closely related to the primary variables. However, the estimation accuracy reduces when the auxiliary variables are irrelevant to the primary variables. For selecting useful auxiliary variables, we formulate the problem as model selection, and propose an information criterion for predicting primary variables by leveraging auxiliary variables. The proposed information criterion is an asymptotically unbiased estimator of the Kullback-Leibler divergence for complete data of primary variables under some reasonable conditions. We also clarify an asymptotic equivalence between the proposed information criterion and a variant of leave-one-out cross validation. Performance of our method is demonstrated via a simulation study and a real data example.

We tackle the problem of multi-task learning with copula process. Multivariable prediction in spatial and spatial-temporal processes such as natural resource estimation and pollution monitoring have been typically addressed using techniques based on Gaussian processes and co-Kriging. While the Gaussian prior assumption is convenient from analytical and computational perspectives, nature is dominated by non-Gaussian likelihoods. Copula processes are an elegant and flexible solution to handle various non-Gaussian likelihoods by capturing the dependence structure of random variables with cumulative distribution functions rather than their marginals. We show how multi-task learning for copula processes can be used to improve multivari-able prediction for problems where the simple Gaussianity prior assumption does not hold. Then, we present a trans-ductive approximation for multi-task learning and derive analytical expressions for the copula process model. The approach is evaluated and compared to other techniques in one artificial dataset and two publicly available datasets for natural resource estimation and concrete slump prediction.