Multivariate methods for the treatment of high dimensional complex neuroimaging genetics data

随着神经影像学和遗传学研究的发展，关于人类大脑遗传结构的非凡信息可以被发现。事实上，现代神经成像技术提供了一个前所未有的机会，可以在个体高分辨率大脑图像的数千个体素中寻找遗传和环境影响。在这种情况下，收集双胞胎群体的大脑和遗传学数据可以为研究遗传（即可遗传）和环境影响对大脑形态和功能变异性的相对贡献提供有力的方法。扩散核磁共振成像测量大脑白质微观结构。使用这种技术，我们可以重建方向分布函数，给出大脑每个体素内定向水扩散的概率。这样就有可能检查基因对轴突纤维通路几何形状的影响。同样值得研究的是，通过功能性磁共振成像测量的神经活动的个体差异在多大程度上受到遗传或环境因素的影响。后一种技术涉及一系列磁共振图像，每个图像由大约10万个均匀间隔的体素组成，这些体素将大脑划分为大小相同的盒子。需要分析的数据总量惊人，呈现复杂的时空噪声结构，信号相对较弱。统计学在理解数据的本质和获得遗传学家/神经科学家可以使用和解释的相关结果方面起着至关重要的作用。相反，新研究中提出的问题和庞大的数据规模为统计学家的理论研究发展提供了新的和非常令人兴奋的机会。在本提案中，将开发新的强大的创新统计方法来处理这些具有挑战性的数据，主要是为了将遗传变异与大脑的结构和/或功能联系起来，但也验证数据的预先假设。这项工作将涉及几个统计领域：时间序列分析、拟合优度检验、检验的多重性、多变量分析（例如，独立成分分析或偏最小二乘回归）、R包的开发、稀疏性、复值模型、依赖度量、自举。例如，复杂正态分布的新的拟合优度检验，这是一种经常对fMRI数据中的噪声做出的假设，将被开发出来。这些结果，依赖于经验特征过程，然后将用于开发复杂值随机变量的学生t检验的模拟，这也将扩展到回归模型。同样，可以考虑将我们已经为ARMA模型开发的正态性检验的复值向量自回归（VAR）模型扩展到该模型。这些模型通常用于拟合功能磁共振成像数据。偏最小二乘（PLS）回归似乎是寻找两个数据块（脑图像和遗传矩阵）之间关联的良好候选者，因为它提取了成对的相关潜在变量（每个块的变量的一个线性组合）。另一种称为并行独立成分分析的方法最近也被开发出来，将功能性MRI数据和候选区域的snp结合起来。然而，由于数据的维度非常高，所有这些多变量方法都会遇到严重的过拟合问题。在本提案中，我们将使用稀疏性和更一般的依赖度量对这些方法进行扩展，并且我们将使用一些正则化策略来提出PLS和ICA的稀疏泛化。

As research encompassing neuroimaging and genetics gains momentum, extraordinary information can be uncovered on the genetic architecture of the human brain. Indeed, modern neuroimaging technology offers an unprecedented opportunity to search for genetic and environmental influences among thousands of voxels in individual high resolution brain images. In this context, collecting brain and genetics data on a population of twins can lead to a powerful approach for studying the relative contributions of genetic (i.e., heritable) and environmental influences on variability in brain morphometry and function. Diffusion MRI measures white matter microstructure in the brain. Using this technique, we can enable reconstruction of orientation distribution functions giving the probability of directional water diffusion within each voxel of the brain. It then becomes possible to examine genetic influences on the geometries of axon fiber pathways. It is also desirable to study the extent to which individual differences in neural activity, as measured by functional Magnetic Resonance Imaging, are influenced by genetic or environmental factors. This latter technique involves a sequence of magnetic resonance images, each consisting of roughly 100,000 uniformly spaced voxels that partition the brain into equally sized boxes. The total amount of data that needs to be analyzed is staggering and exhibit a complicated temporal and spatial noise structure with a relatively weak signal. Statistics plays a crucial role in understanding the nature of the data, and in obtaining relevant results that can be used and interpreted by geneticists/neuroscientists. Inversely, problems raised in new studies and the huge size of the data open new and very exciting opportunities for theoretical research developments for statisticians. In this proposal, new powerful innovative statistical methods will be developed to treat these challenging data, mainly in order to relate genetic variants to the structure and/or function of the brain, but also to validate pre-suppositions of the data. Several statistical areas will be involved in this work: time series analysis, goodness-of-fit tests, multiplicity of tests, multivariate analysis (e.g., Independent Component Analysis or Partial Least Square regression), development of R packages, sparsity, complex-valued models, dependence measures, bootstrap. For example, new goodness-of-fit tests of the complex normal distribution, an assumption often made for the noise in fMRI data, will be developed. These results, relying on the empirical characteristic process, will then be used to develop an analogue of the Student t-test for complex-valued random variables, which will also be extended to regression models. In the same vein, an extension to complex-valued Vector Auto Regressive (VAR) models of tests of normality which we have already developed for ARMA models could be considered. These models are often used to fit fMRI data. Partial Least Squares (PLS) regression appears to be a good candidate to look for associations between two blocks of data (brain images and genetic matrices), as it extracts pairs of correlated latent variables (one linear combination of the variables for each block). Another approach called parallel Independent Component Analysis has also been recently developed to combine functional MRI data and SNPs from candidate regions. Nevertheless, all of these multivariate methods encounter critical over-fitting issues due to very high dimensionality of the data. In this proposal, several extensions of these methods will be proposed, using sparsity and more general dependence measures, and we will use some strategies of regularization to propose a sparse generalization of PLS and ICA.