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)选择偏向(长度偏向)，和(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光滑函数的卷积。其目的是估计未知的光滑函数。(1)的第二部分讨论了一种基于非参数似然的独立分量分析(ICA)方法，该方法利用对数样条法估计盲源和混合矩阵的边缘分布。拟议程序的优点之一是可以灵活地纳入震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差的抽样密度在某些比率下趋于零的情况，使用有选择偏差样本来估计基础密度或条件密度函数。这将在多大程度上影响基于样条法的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。最后一个项目涉及包含随机截断数据的密度和条件密度函数的估计。提出了一种新的基于多项式样条函数的方法，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不足限度问题很重要。具体地说，人们可以仔细研究截断模式，特别是当关于某些类型的截断可能如何发生的信息可用时。在这项拨款申请中提出的基于样条法的估计量的抽样性质将通过扩展研究者和他的同事先前建立的渐近结果来研究。具体地说，将研究所建议估计的最优收敛速度和局部渐近性，以及与高维解释变量相关的问题。这一建议将对生命科学研究的应用产生几个更广泛的影响。首先，时间序列回归模型和时空数据的处理方法对涉及fMRI数据的脑研究有重要影响，所提出的方法具有以下特点：(A)无偏，(B)能够有效地显示时空信息，以及(C)在统计抽样性质方面易于数学处理。第二，拟议的偏向选择或随机截断数据的统一方法为科学界提供了对与数据获取有关的问题的重要见解。在从大脑和基因组数据中寻求重要信息时，强大的统计技术至关重要。第三，这里提出的方法将有助于为研究生开发一门统计功能磁共振成像分析课程。最后，该项目的更广泛的健康意义将是它有助于更好地了解影响人类生活的疾病。

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.