This paper focuses on learning rate analysis of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. Using a recently developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an integral operator approach based on second-order decomposition, we succeed in deriving almost optimal learning rates of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. A series of numerical experiments are carried out to verify our theoretical results, showing the excellent learning performance of Nystrom regularization with sequential sub-sampling in learning massive time series data. All these results extend the applicable range of Nyström regularization from i.i.d. samples to non-i.i.d. sequences.

MOTIVATION: A large number of recent genome-wide association studies (GWASs) for complex phenotypes confirm the early conjecture for polygenicity, suggesting the presence of large number of variants with only tiny or moderate effects. However, due to the limited sample size of a single GWAS, many associated genetic variants are too weak to achieve the genome-wide significance. These undiscovered variants further limit the prediction capability of GWAS. Restricted access to the individual-level data and the increasing availability of the published GWAS results motivate the development of methods integrating both the individual-level and summary-level data. How to build the connection between the individual-level and summary-level data determines the efficiency of using the existing abundant summary-level resources with limited individual-level data, and this issue inspires more efforts in the existing area. RESULTS: In this study, we propose a novel statistical approach, LEP, which provides a novel way of modeling the connection between the individual-level data and summary-level data. LEP integrates both types of data by LEveraging Pleiotropy to increase the statistical power of risk variants identification and the accuracy of risk prediction. The algorithm for parameter estimation is developed to handle genome-wide-scale data. Through comprehensive simulation studies, we demonstrated the advantages of LEP over the existing methods. We further applied LEP to perform integrative analysis of Crohn's disease from WTCCC and summary statistics from GWAS of some other diseases, such as Type 1 diabetes, Ulcerative colitis and Primary biliary cirrhosis. LEP was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.39% (±0.58%) to 68.33% (±0.32%) using about 195 000 variants. AVAILABILITY AND IMPLEMENTATION: The LEP software is available at https://github.com/daviddaigithub/LEP. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Motivation: Thousands of risk variants underlying complex phenotypes (quantitative traits and diseases) have been identified in genome-wide association studies (GWAS). However, there are still two major challenges towards deepening our understanding of the genetic architectures of complex phenotypes. First, the majority of GWAS hits are in non-coding region and their biological interpretation is still unclear. Second, accumulating evidence from GWAS suggests the polygenicity of complex traits, i.e. a complex trait is often affected by many variants with small or moderate effects, whereas a large proportion of risk variants with small effects remain unknown. Results: The availability of functional annotation data enables us to address the above challenges. In this study, we propose a latent sparse mixed model (LSMM) to integrate functional annotations with GWAS data. Not only does it increase the statistical power of identifying risk variants, but also offers more biological insights by detecting relevant functional annotations. To allow LSMM scalable to millions of variants and hundreds of functional annotations, we developed an efficient variational expectation-maximization algorithm for model parameter estimation and statistical inference. We first conducted comprehensive simulation studies to evaluate the performance of LSMM. Then we applied it to analyze 30 GWAS of complex phenotypes integrated with nine genic category annotations and 127 cell-type specific functional annotations from the Roadmap project. The results demonstrate that our method possesses more statistical power than conventional methods, and can help researchers achieve deeper understanding of genetic architecture of these complex phenotypes. Availability and implementation: The LSMM software is available at https://github.com/mingjingsi/LSMM. Supplementary information: Supplementary data are available at Bioinformatics online.

MOTIVATION: Results from genome-wide association studies (GWAS) suggest that a complex phenotype is often affected by many variants with small effects, known as 'polygenicity'. Tens of thousands of samples are often required to ensure statistical power of identifying these variants with small effects. However, it is often the case that a research group can only get approval for the access to individual-level genotype data with a limited sample size (e.g. a few hundreds or thousands). Meanwhile, summary statistics generated using single-variant-based analysis are becoming publicly available. The sample sizes associated with the summary statistics datasets are usually quite large. How to make the most efficient use of existing abundant data resources largely remains an open question. RESULTS: In this study, we propose a statistical approach, IGESS, to increasing statistical power of identifying risk variants and improving accuracy of risk prediction by i ntegrating individual level ge notype data and s ummary s tatistics. An efficient algorithm based on variational inference is developed to handle the genome-wide analysis. Through comprehensive simulation studies, we demonstrated the advantages of IGESS over the methods which take either individual-level data or summary statistics data as input. We applied IGESS to perform integrative analysis of Crohns Disease from WTCCC and summary statistics from other studies. IGESS was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.2% ( ±0.4% ) to 69.4% ( ±0.1% ) using about 240 000 variants. AVAILABILITY AND IMPLEMENTATION: The IGESS software is available at https://github.com/daviddaigithub/IGESS . CONTACT: [email protected] or [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.