Function estimation for biased sampling and fMRI data

该提案有三个项目，涉及使用多项式样条来建模(1)点过程和时空时间序列，(2)选择偏倚（长度偏倚），以及(3)随机截断数据。项目(1)的第一部分考虑了一个线性回归模型，其中响应是平稳时间序列，解释序列是点过程和未知平滑函数的卷积。目的是估计未知光滑函数。(1)的第二部分通过使用对数样条方法估计盲源和混合矩阵的边际分布，讨论了一种基于非参数似然的独立分量分析（ICA）方法。所建议的程序的优点之一是可以灵活地纳入源的时间或空间相关性。项目(2)的目的是通过特别注意有偏抽样密度在一定速率下趋于零的情况，使用选择偏抽样估计潜在密度或条件密度函数。这将在多大程度上影响基于样条估计器的性能是一个重要的问题，研究者建议通过检查估计器的收敛速度来解决这个问题。最后一个项目涉及随机截断数据的密度和条件密度函数的估计。提出了一种基于多项式样条的新方法，并将开发该方法的软件。这种方法对于检查大范围的选择偏差或检测限制不足问题很重要。具体地说，可以仔细研究截断模式，特别是当可以获得有关某些类型的截断可能如何发生的信息时。在此拨款申请中提出的基于样条估计器的抽样特性将通过扩展研究者及其同事先前建立的渐近结果来研究。具体来说，将研究所提出估计的最优收敛率和局部渐近性，并讨论与高维解释变量有关的问题。这一建议对生命科学研究的应用有几个更广泛的影响。首先，时间序列回归模型和时空数据方法对涉及fMRI数据的大脑研究有重大影响，所提出的程序(a)偏差较小，(b)能够有效地显示时空信息，(c)在统计抽样特性方面具有数学上的可处理性。其次，对选择偏倚或随机截断的数据提出的统一方法为科学界提供了与数据获取相关问题的重要见解。在从大脑和基因组数据中寻求重要信息时，可靠的统计技术是必不可少的。第三，这里提出的方法将有助于为研究生开设统计功能磁共振成像分析课程。最后，这个项目更广泛的健康意义在于它有助于更好地理解影响人类生命的疾病。

The proposal has three projects that involve the use of polynomial splines to model (1) point process and spatial-temporal time series, (2) selection biased (length-biased), and (3) randomly truncated data. The first part of Project (1) considers a linear regression model in which the response is a stationary time series and the explanatory series is a convolution of a point process and an unknown smooth function. The aim is to estimate the unknown smooth function. The second part of (1) deals with a non-parametric likelihood based approach to independent component analysis (ICA) by estimating the marginal distributions of the blind sources and the mixing matrix using the log-spline methodology. One of the advantages of the proposed procedure is the flexibility for incorporating the temporal or spatial correlation of the sources. The aim of Project (2) is to estimate the underlying density or conditional density function using selection-biased samples, by paying special attention to the situation in which the biased sampling density tends to zero at certain rates. To what extent this will affect the performance of the spline-based estimator is an important question and the investigator proposes to address it by examining the rates of convergence of the estimator. The last project deals with the estimation of density and conditional density functions involving randomly truncated data. A new methodology based on polynomial splines is proposed and software for the methodology will be developed. This methodology is important for examining a wide range of selection-bias or under-detection limits problems. Specifically, one can study the truncation pattern closely, especially when the information about how certain type of truncation might have occurred is available. Sampling properties of the spline-based estimators proposed in this grant application will be studied by extending the asymptotic results established previously by the investigator and his colleagues. Specifically, optimal rates of convergence and local asymptotics of the proposed estimates will be investigated and issues related to high-dimensional explanatory variables will also be addressed.This proposal has several broader impacts on applications to research in life sciences. First, the time series regression models and the approach to spatial-temporal data have a significant impact on brain research involving fMRI data, the proposed procedures are (a) less-biased, (b) capable to display the spatial-temporal information effectively, and (c) mathematically tractable in terms of statistical sampling properties. Second, the proposed unified approach to selection-biased or randomly truncated data provides major insights to the scientific community into the issues related to data acquisition. Robust statistical techniques are essential in the quest for important information from brain and genomic data. Third, methods proposed here will be useful for developing a course in statistical fMRI analysis to graduate students. Finally, a much broader health significance of this project will be its contribution to the better understanding of diseases that affect human lives.