  We present a new statistical modelling approach where the response is a
function of high frequency count data. Our application is about investigating
the relationship between the health outcome fat mass and physical activity (PA)
measured by accelerometer. The accelerometer quantifies the intensity of
physical activity as counts per epoch over a given period of time. We use data
from the Avon longitudinal study of parents and children (ALSPAC) where
accelerometer data is available as a time series of accelerometer counts per
minute over seven days for a subset of children. In order to compare
accelerometer profiles between individuals and to reduce the high dimension a
functional summary of the profiles is used. We use the histogram as a
functional summary due to its simplicity, suitability and ease of
interpretation. Our model is an extension of generalised regression of scalars
on functions or signal regression. It allows also multi-dimensional functional
predictors and additive non-linear predictors for metric covariates. The
additive multidimensional functional predictors allow investigating specific
questions about whether the effect of PA varies over its intensity, by gender,
by time of day or by day of the week. The key feature of the model is that it
utilises the full profile of measured PA without requiring cut-points defining
intensity levels for light, moderate and vigorous activity. We show that the
(not necessarily causal) effect of PA is not linear and not constant over the
activity intensity. Also, there is little evidence to suggest that the effect
of PA intensity varies by gender or whether it happens on weekdays or on
weekends.
