Functional magnetic resonance imaging (fMRI) is a non-invasive technique for studying brain activity. It uses the amount of blood flowing through a brain, referred to as the blood oxygenation level dependent (BOLD) signal. However analyzing the fMRI signals is challenging because of its complicated spatio-temporal correlation structure and its massive amount of data.

There are several brain atlases available but researchers observe that fMRI signals are not coherent even within the same area in a brain atlas. Therefore providing parcellation of a brain, especially based on its functional connectivity, is necessary to understand brain activities.

One of the techniques that are used for a brain parcellation is spectral clustering. It is a well-used technique in many areas of studies, such as physics and engineering. However, its asymptotic behavior, whether spectral clustering will produce consistent clustering as samples grow large, is not fully clarified. In addition, there has previously been no available mathematical justification of the large-sample properties of spectral clustering when the data are dependent.

Von Luxburg et al. (2008) showed the consistency of eigenfunctions of spectral clustering under the assumption that data are independent and identically distributed. Because fMRI signals are spatially dependent, applying her results to fMRI data analysis is not appropriate. In this thesis, we extend von Luxburg's work to 3-dimensional spatially dependent data satisfying strong mixing conditions, which will be the case for fMRI data.

We applied the spectral clustering algorithm to simulated data to see how the algorithm can be affected by perturbation in a similarity matrix. There are two simulated data experiments. The first type of simulated data is similar to the stochastic block model, and the second is sampled independently from a Gaussian random field distribution with correlation.

We applied spectral clustering to various regions of interest (ROIs) both for a single subject and for multiple subjects. We also provided methods to analyze data from multiple subjects using spectral clustering and compared these methods using several criteria.