Time-dependent spatially varying graphical models, with application to brain fMRI data analysis
In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.
This paper studied the graphical structure in spatiotemporal data through transferring the time series data into an additive model, by assuming stationary temporal correlation structure and time-varying undirected Gaussian graphical model for spatial correlation structure. With the assumption that the spatial correlations change smoothly with time, they proposed estimators for both spatial and temporal structures based on kernel method and GLasso approach. The statistical convergence property of the estimators was provided under certain assumptions. The approach presented good performance in both simulation and fMRI data application studies.
This paper is overall clearly written, with solid theoretical support and interesting application. However, some assumptions, such as a common temporal correlation matrix for different spatial locations, may be not valid in application. It may worth further discussion. There are also some other points may need further improvement.
1.	The authors didn’t provide correct supplementary, but another version of the main text.
2.	In the estimation section, tr(A) is supposed to be known. This is not so obviously satisfied in real data application. It is better to provide more discussions on this issue.
3.	Line 178 is unclear. Which lemma? It should be the inverse of B, instead of B itself, exploiting sparsity.
4.	As m / n increases, the bound in Theorem 2 / Theorem 4 converges. However, the regularization parameter \lambda_m / \lambda_n also converges to 0 at the same time, which is a sacrifice of sparsity. In simulation, the authors used a variety of regularization parameters, instead of the proposed one in the theorem. More discussions are expected on this issue.
5.	In simulation, it is recommended to compare with other methods, such as Qiu (2016). Also the authors mentioned that some previous methods were limited to AR(1) model, while their method in this paper can handle more complex models. But they only provide simulation results for AR(1) case, as shown in Fig. 2 & 3. So it is expected to see the simulation results on more general models.
6.	In simulation, it would be better to use the inverse of B(t) instead of B(t) for evaluation, as the focus is the structure of the underlying undirected graphs instead of the edge weights / strength of the undirected graphs.
7.	The assumption that all spatial points follow the same temporal structure is a bit strong. In application, its validity should be discussed. For example, in the fMRI data, different brain regions may present different developing pattern. The visual cortex may develop fast in childhood, while the dorsal prefrontal cortex related to decision and attention usually has a development peak in a later age. It would be better to check the validity of these assumptions before application.
8.	Assuming a time series follows a joint normal distribution (equation (2)) is also not very common. In the fMRI data application, its validity should also be discussed. Each of the 90 regions of interest generally has hundreds to thousands voxels, which should enable the test for normality.
9.	In Line 118, it assumes that spatial structure is sparse and m much larger than n. However, the brain connectivity is usually not sparse. In the meantime, in this resting-state fMRI dataset, some subjects has less time points (m larger than or equal to 76) than the number of ROIs (n = 90).
10.	In the application, the dataset include both healthy group and ADHD group. Did the authors use them all together for analysis? It might be better to apply the method to each group separately and compare the results. Different developing trends are expected. And if such a result is obtained, it should be more powerful to illustrate the effectiveness of this method.
11.	In consistency statements for ‘signal’ and ‘noise’, in sense of their correspondence with ‘temporal’ and ‘spatial’ structure. In abstract (line 3-4), they mentioned ‘temporally correlated signal’ and ‘spatially correlated noise’. However, in 2.1 estimator section (line 133 – 134), they mentioned ‘spatial signal variance’ and ‘noise variance … corresponding to tr(A)’, while A corresponds to temporal structure as indicated in equation (2).
12.	In the paragraph from line 44 to 51, it is better to add comparison with Qiu (2016) and Monti (2014), in sense of methodology. 
13.	Some minor points: (i) It is better to provide the definition of e_i after equation (4), as it is provided after equation (5). (ii) The subscripts in equation (8) are not consistent, where ‘t’ should be ‘i’. (iii) In line 170, B to B(t). (iv) In line196, m to n.
Update: I am satisfied with the authors' replies to my comments. Hence I would like to raise my grade.