Reference frame transformation plays an important role in multi-phase AC system's control. While most existing ref-erence frame transformations are only applicable to some fixed-phase AC systems, there is a lack of a generalized transformation developing method for complex AC systems whose phase number is very large. In this paper we discuss the reference frame trans-formation for such complex AC systems from the data-driven perspective. Our key idea is based on a physical intuition, saying that a well-designed system is essentially low-rank and sparse. By leveraging this intuition, we perform low-rank decomposition and sparse representation on some high-dimensional AC system data, effectively simplifying them into a low-dimensional DC data. Comparing our transformed output with traditional methods, we find them to be remarkably similar, differing only in their initial angle. This suggests that a reference frame transformation can be developed solely using experimental data without requiring any prior model knowledge, thus providing a model-free solution for complex systems where model knowledge is not available.

MOTIVATION: Mendelian randomization (MR) is a valuable tool to examine the causal relationships between health risk factors and outcomes from observational studies. Along with the proliferation of genome-wide association studies, a variety of two-sample MR methods for summary data have been developed to account for horizontal pleiotropy (HP), primarily based on the assumption that the effects of variants on exposure (γ) and HP (α) are independent. In practice, this assumption is too strict and can be easily violated because of the correlated HP. RESULTS: To account for this correlated HP, we propose a Bayesian approach, MR-Corr2, that uses the orthogonal projection to reparameterize the bivariate normal distribution for γ and α, and a spike-slab prior to mitigate the impact of correlated HP. We have also developed an efficient algorithm with paralleled Gibbs sampling. To demonstrate the advantages of MR-Corr2 over existing methods, we conducted comprehensive simulation studies to compare for both type-I error control and point estimates in various scenarios. By applying MR-Corr2 to study the relationships between exposure-outcome pairs in complex traits, we did not identify the contradictory causal relationship between HDL-c and CAD. Moreover, the results provide a new perspective of the causal network among complex traits. AVAILABILITY AND IMPLEMENTATION: The developed R package and code to reproduce all the results are available at https://github.com/QingCheng0218/MR.Corr2. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.