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 (GWASs), 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 horizontal pleiotropy (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 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.

Considering the complexity pattern of the gold price, this paper adopts the decomposition-reconstruction-forecast-mergence scheme to perform the international gold price forecast. The original gold price data are decomposed into 12 intrinsic mode functions and a residual by the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method, and then the 13 sequences are reconstructed into a high-frequency subsequence (IMFH), a low-frequency subsequence (IMFL), and the residual (Res). According to the different characteristics of the subsequences, the IMFL and Res are forecasted by the support vector regression (SVR) model. Besides, in order to further improve the prediction accuracy of IMFH, we have developed a novel hybrid method based on the support vector regression (SVR) model and the grey wolf optimizer (GWO) algorithm with SVR for predicting the IMFH of gold prices, i.e., the CEEMDAN-GWO-SVR model. This hybrid model combines the methodology of complex systems with machine learning techniques, making it more appropriate for analyzing relationships such as high-frequency dependences and solving complex nonlinear problems. Finally, the final result is obtained by combining the forecasting results of the three subsequences. The empirical results show that the proposed model demonstrates the highest prediction ability among all of the investigated models in a comparison of prediction errors with other individual models.

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