Lévy‐based Modelling in Brain Imaging

Abstract

ABSTRACT. A substantive problem in neuroscience is the lack of valid statistical methods for nonGaussian random fields. In the present study, we develop a flexible, yet tractable model for a random field based on kernel smoothing of a so-called Lévy basis. The resulting field may be Gaussian, but there are many other possibilities, including random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. It is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of test statistics commonly used in neuroscience. We give a concrete example of magnetic resonance imaging scans that are non-Gaussian. For these data, simulations under the fitted models show that traditional methods based on Gaussian random field theory may leave small, but significant changes in signal level undetected, while these changes are detectable under a non-Gaussian Lévy model.

Estimation of Sample Spacing in Stochastic Processes

“…15 ofĥ is plotted as a function of h for a concrete choice of parameter values. The chosen values of the parameters give a covariance structure similar to that obtained in a concrete analysis presented in Jónsdóttir et al (2013b). In Fig.…”

“…Note that a strictly increasing γ is equivalent to a strictly decreasing covariance function. A wide range of covariance models has this property, including the spherical, Gaussian, exponential and 3rd order autoregressive covariance functions (Jónsdóttir et al, 2013b).…”

“…For an introduction to Lévy-based modelling, see, e.g., Jónsdóttir et al (2013b). In relation to variance estimation, Lévy-based models have earlier been used in circular systematic sampling (Jónsdóttir et al, 2013a).…”

Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure

“…In [18], a model (1.1) with M following a NIG distribution was suitable for modelling the neuroscience data under consideration. For such data it is typically of interest to detect for which t ∈ B a given field obtains values that are significantly large.…”

Lévy-Based Error Prediction in Circular Systematic Sampling

“…in the modelling of turbulent flows, spatio-temporal growth, spatial point processes and random fields (Barndorff-Nielsen and Schmiegel, 2004;Jónsdóttir et al, 2008;Hellmund et al, 2008;Jónsdóttir et al, 2013). More specifically, we will consider stochastic processes of the form…”