Hierarchical models for assessing variability among functions

In many applications of functional data analysis, summarising functional variation based on fits, without taking account of the estimation process, runs the risk of attributing the estimation variation to the functional variation, thereby overstating the latter. For example, the first eigenvalue of a sample covariance matrix computed from estimated functions may be biased upwards. We display a set of estimated neuronal Poisson-process intensity functions where this bias is substantial, and we discuss two methods for accounting for estimation variation. One method uses a random-coefficient model, which requires all functions to be fitted with the same basis functions. An alternative method removes the same-basis restriction by means of a hierarchical Gaussian process model. In a small simulation study the hierarchical Gaussian process model outperformed the random-coefficient model and greatly reduced the bias in the estimated first eigenvalue that would result from ignoring estimation variability. For the neuronal data the hierarchical Gaussian process estimate of the first eigenvalue was much smaller than the naive estimate that ignored variability due to function estimation. The neuronal setting also illustrates the benefit of incorporating alignment parameters into the hierarchical scheme.