Clustering functional data is mostly based on the projection of the curves onto an adequate basis and building random effects models of the basis coefficients. The parameters can be fitted with an EM algorithm. Alternatively, distance models based on the coefficients are used in the literature. Similar to the case of clustering multidimensional data, a variety of derivations of different models has been published. Although their calculation procedure is similar, their implementations are very different including distinct hyperparameters and data formats as input. This makes it difficult for the user to apply and particularly to compare them. Furthermore, they are mostly limited to specific basis functions. This paper aims to show the common elements between existing models in highly cited articles, first on a theoretical basis. Later their implementation is analyzed and it is illustrated how they could be improved and extended to a more general level. A special consideration is given to those models designed for sparse measurements. The work resulted in the R package funcy which was built to integrate the modified and extended algorithms into a unique framework.
Functional magnetic resonance imaging (fMRI) activation detection within stimulus-based experimental paradigms is conventionally based on the assumption that activation effects remain constant over time. This assumption neglects the fact that the strength of activation may vary, for example, due to habituation processes or changing attention. Neither the functional form of time variation can be retrieved nor short-lasting effects can be detected by conventional methods. In this work, a new dynamic approach is proposed that allows to estimate time-varying effect profiles and hemodynamic response functions in event-related fMRI paradigms. To this end, we incorporate the time-varying coefficient methodology into the fMRI general regression framework. Inference is based on a voxelwise penalized least squares procedure. We assess the strength of activation and corresponding time variation on the basis of pointwise confidence intervals on a voxel level. Additionally, spatial clusters of effect curves are presented. Results of the analysis of an active oddball experiment show that activation effects deviating from a constant trend coexist with time-varying effects that exhibit different types of shapes, such as linear, (inversely) U-shaped or fluctuating forms. In a comparison to conventional approaches, like classical SPM, we observe that time-constant methods are rather insensitive to detect temporary effects, because these do not emerge when aggregated across the entire experiment. Hence, it is recommended to base activation detection analyses not merely on time-constant procedures but to include flexible time-varying effects that harbour valuable information on individual response patterns.
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