Functional magnetic resonance imaging or functional MRI (fMRI) is a very popular tool used for differing brain regions by measuring brain activity. It is affected by physiological noise, such as head and brain movement in the scanner from breathing, heart beats, or the subject fidgeting. The purpose of this paper is to propose a novel approach to handling fMRI data for infants with high volatility caused by sudden head movements. Another purpose is to evaluate the volatility modelling performance of multiple dependent fMRI time series data. The models examined in this paper are AR and GARCH and the modelling performance is evaluated by several statistical performance measures. The conclusions of this paper are that multiple dependent fMRI series data can be fitted with AR + GARCH model if the multiple fMRI data have many sudden head movements. The GARCH model can capture the shared volatility clustering caused by head movements across brain regions. However, the multiple fMRI data without many head movements have fitted AR + GARCH model with different performance. The conclusions are supported by statistical tests and measures. This paper highlights the difference between the proposed approach from traditional approaches when estimating model parameters and modelling conditional variances on multiple dependent time series. In the future, the proposed approach can be applied to other research fields, such as financial economics, and signal processing. Code is available at \url{https://github.com/13204942/STAT40710}.

翻译：功能磁共振成像（fMRI）是一种通过测量脑活动来区分不同脑区的常用工具。它易受生理噪声影响，例如扫描过程中因呼吸、心跳或受试者躁动引起的头部和大脑运动。本文旨在提出一种处理婴幼儿fMRI数据的新方法，该类数据因突发头部运动而呈现高波动性。另一目的是评估多个相依fMRI时间序列数据的波动率建模性能。本文检验的模型包括AR和GARCH，并通过多种统计性能指标评估建模效果。研究结论表明，若多个fMRI数据存在大量突发头部运动，可采用AR+GARCH模型进行拟合，其中GARCH模型能够捕捉由头部运动引起的跨脑区共享波动聚集效应。然而，对于无明显头部运动的多个fMRI数据，AR+GARCH模型的拟合表现存在差异。统计检验与指标进一步支持了上述结论。本文突出了所提方法与传统方法在估计模型参数及对多个相依时间序列进行条件方差建模时的差异。未来，该新方法可应用于金融经济学、信号处理等其他研究领域。代码详见\url{https://github.com/13204942/STAT40710}。