Time Series Analysis of fMRI Data: Spatial Modelling and Bayesian Computation

Time series analysis of fMRI data is an important area of medical statistics for neuroimaging data. The neuroimaging community has embraced mean-field variational Bayes (VB) approximations, which are implemented in Statistical Parametric Mapping (SPM) software. While computationally efficient, the quality of VB approximations remains unclear even though they are commonly used in the analysis of neuroimaging data. For reliable statistical inference, it is important that these approximations be accurate and that users understand the scenarios under which they may not be accurate. We consider this issue for a particular model that includes spatially-varying coefficients. To examine the accuracy of the VB approximation we derive Hamiltonian Monte Carlo (HMC) for this model and conduct simulation studies to compare its performance with VB. As expected we find that the computation time required for VB is considerably less than that for HMC. In settings involving a high or moderate signal-to-noise ratio (SNR) we find that the two approaches produce very similar results suggesting that the VB approximation is useful in this setting. On the other hand, when one considers a low SNR, substantial differences are found, suggesting that the approximation may not be accurate in such cases and we demonstrate that VB produces Bayes estimators with larger mean squared error (MSE). A real application related to face perception is also carried out. Overall, our work clarifies the usefulness of VB for the spatiotemporal analysis of fMRI data, while also pointing out the limitation of VB when the SNR is low and the utility of HMC in this case

Necessity for quantum coherence of nondegeneracy in energy flow

In this work, we show that the quantum coherence among non-degenerate energy subspaces (CANES) is essential for the energy flow in any quantum system. CANES satisfies almost all of the requirements as a coherence measure, except that the coherence within degenerate subspaces is explicitly eliminated.We show that the energy of a system becomes frozen if and only if the corresponding CANES vanishes, which is true regardless of the form of interaction with the environment. However, CANES can remain zero even if the entanglement changes over time. Furthermore, we show how the power of energy flow is bounded by the value of CANES. An explicit relation connecting the variation of energy and CANES is also presented. These results allow us to bound the generation of system-environment correlation through the local measurement of the system's energy flow