The Annals of Statistics

Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: Nonparametric kernel estimation and hypotheses testing

Vladimir Koltchinskii, Lyudmila Sakhanenko, and Songhe Cai

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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 $X_i,\ i=1,\dots, n$, 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.

Article information

Ann. Statist., Volume 35, Number 4 (2007), 1576-1607.

First available in Project Euclid: 29 August 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G10: Hypothesis testing 62G20: Asymptotic properties 62P10: Applications to biology and medical sciences

diffusion tensor imaging kernel regression estimator vector fields integral curves functional central limit theorem


Koltchinskii, Vladimir; Sakhanenko, Lyudmila; Cai, Songhe. Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: Nonparametric kernel estimation and hypotheses testing. Ann. Statist. 35 (2007), no. 4, 1576--1607. doi:10.1214/009053607000000073.

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