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摘要翻译：
设$V$是有界开集$G\子集{\MathBB{R}}^D$中的向量场。假设$v$在随机点$x_i处用随机噪声观察，I=1,..,n,$是独立且均匀分布在$G$中的问题是估计微分方程[frac{dx(t)}{dt}=v(x(t)),qquad t\geq0,x(0)=x_0\in G,\]的积分曲线，并对积分曲线到达给定集$x(0)=x_0\in G的假设进行统计检验。我们发展了一个基于Nadaraya-Watson型核回归估计量的估计过程，给出了估计积分曲线的渐近正态性，并导出了极限高斯过程的均值和协方差函数的微分和积分方程。这提供了一种不仅跟踪积分曲线，而且跟踪其估计的协方差矩阵的方法。我们还研究了积分曲线到足够光滑曲面γ-子集G$的平方极小距离的渐近分布。在此基础上，我们为积分曲线达到$\γ$的假设开发了测试过程。这种性质的问题在扩散张量成像中引起了人们的兴趣，扩散张量成像是一种基于测量脑白质中离散位置的扩散张量的脑成像技术，在那里水分子的扩散通常是各向异性的。扩散张量数据被用来估计扩散的主要方向，并跟踪白质纤维从初始位置跟随这些方向。我们的方法为这个问题的分析带来了更严格的统计工具，特别是提供了在研究白质轴突连接性时可能有用的假设检验程序。
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英文标题：
《Integral curves of noisy vector fields and statistical problems in
diffusion tensor imaging: nonparametric kernel estimation and hypotheses
testing》
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作者：
Vladimir Koltchinskii, Lyudmila Sakhanenko, Songhe Cai
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
Let $v$ be a vector field in a bounded open set $G\subset {\mathbb {R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i, i=1,...,n,$ that are independent and uniformly distributed in $G.$ The problem is to estimate the integral curve of the differential equation \[\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_0\in G,\] starting at a given point $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $\Gamma\subset G.$ We develop an estimation procedure based on a Nadaraya--Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface $\Gamma\subset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $\Gamma$. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.
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PDF链接：
https://arxiv.org/pdf/710.3509
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