Functional density synchronization

Estimating an overall density function from repeated observations on each of a sample of independent subjects or experimental units is of interest. An example is provided by biodemographic studies, where one observes age-at-death for several cohorts of flies. Cohorts are kept in separate cages, which form the experimental units. Time variation then is likely to exist between the cohort densities and hazard rates due to cage effects on aging. Given the densities of age-at-death for the individual cohorts, one aims to obtain an estimate for the underlying overall density and hazard rate. In microarray gene expression experiments, similar problems arise when addressing the need for normalization of probe-level data from different arrays. Conventional methods, such as the cross-sectional average density, ignore time variation and hence are often not representative for such data. We view densities as functional data and model individual densities as warped versions of an underlying overall density, where the observed densities are assumed to be realizations of an underlying stochastic process. Quantile-synchronized distribution functions are obtained from an inverse warping mapping, based on quantile synchronization, leading to quantile-synchronized density and hazard functions. Kernel type smoothing methods with plug-in bandwidth selection can be used for estimating the components of the model. Asymptotic properties of the synchronized density estimates are derived. Simulation results show that functional density synchronization is often advantageous when compared to conventional density averaging or simple time-shift warping. Our approach complements previous quantile normalization methods used for microarray expression data and is illustrated with both longevity data obtained for 54 cohorts of mexflies (Mexican fruit flies) and gene expression data of the Ts1Cje mouse study for Down syndrome.