In systems biology, it is common to measure biochemical entities at different levels of the same biological system. One of the central problems for the data fusion of such data sets is the heterogeneity of the data. This thesis discusses two types of heterogeneity. The first one is the type of data, such as metabolomics, proteomics and RNAseq data in genomics. These different omics data reflect the properties of the studied biological system from different perspectives. The second one is the type of scale, which indicates the measurements obtained at different scales, such as binary, ordinal, interval and ratio-scaled variables. In this thesis, we developed several statistical methods capable to fuse data sets of these two types of heterogeneity. The advantages of the proposed methods in comparison with other approaches are assessed using comprehensive simulations as well as the analysis of real biological data sets.

Multivariate binary data is becoming abundant in current biological research. Logistic principal component analysis (PCA) is one of the commonly used tools to explore the relationships inside a multivariate binary data set by exploiting the underlying low rank structure. We re-expressed the logistic PCA model based on the latent variable interpretation of the generalized linear model on binary data. The multivariate binary data set is assumed to be the sign observation of an unobserved quantitative data set, on which a low rank structure is assumed to exist. However, the standard logistic PCA model (using exact low rank constraint) is prone to overfitting, which could lead to divergence of some estimated parameters towards infinity. We propose to fit a logistic PCA model through non-convex singular value thresholding to alleviate the overfitting issue. An efficient Majorization-Minimization algorithm is implemented to fit the model and a missing value based cross validation (CV) procedure is introduced for the model selection. Our experiments on realistic simulations of imbalanced binary data and low signal to noise ratio show that the CV error based model selection procedure is successful in selecting the proposed model. Furthermore, the selected model demonstrates superior performance in recovering the underlying low rank structure compared to models with convex nuclear norm penalty and exact low rank constraint. A binary copy number aberration data set is used to illustrate the proposed methodology in practice.