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

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 problemis 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 givenpoint $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis thatthe integral curve reaches a specified set $\Gamma\subset G.$ We develop anestimation procedure based on a Nadaraya--Watson type kernel regressionestimator, show the asymptotic normality of the estimated integral curve andderive differential and integral equations for the mean and covariance functionof the limit Gaussian process. This provides a method of tracking not only theintegral curve, but also the covariance matrix of its estimate. We also studythe asymptotic distribution of the squared minimal distance from the integralcurve to a smooth enough surface $\Gamma\subset G$. Building upon this, wedevelop testing procedures for the hypothesis that the integral curve reaches$\Gamma$. The problems of this nature are of interest in diffusion tensorimaging, a brain imaging technique based on measuring the diffusion tensor atdiscrete locations in the cerebral white matter, where the diffusion of watermolecules is typically anisotropic. The diffusion tensor data is used toestimate the dominant orientations of the diffusion and to track white matterfibers from the initial location following these orientations. Our approachbrings more rigorous statistical tools to the analysis of this problemproviding, in particular, hypothesis testing procedures that might be useful inthe study of axonal connectivity of the white matter.