Parametric deconvolution of positive spike trains
This paper describes a parametric deconvolution method (PDPS) appropriate for a particular class of signals which we call spike-convolution models. These models arise when a sparse spike train---Dirac deltas according to our mathematical treatment---is convolved with a fixed point-spread function, and additive noise or measurement error is superimposed. We view deconvolution as an estimation problem, regarding the locations and heights of the underlying spikes, as well as the baseline and the measurement error variance as unknown parameters. Our estimation scheme consists of two parts: model fitting and model selection. To fit a spike-convolution model of a specific order, we estimate peak locations by trigonometric moments, and heights and the baseline by least squares. The model selection procedure has two stages. Its first stage is so designed that we expect a model of a somewhat larger order than the truth to be selected. In the second stage, the final model is obtained using backwards deletion. This results in not only an estimate of the model order, but also an estimate of peak locations and heights with much smaller bias and variation than that found in a direct trigonometric moment estimate. A more efficient maximum likelihood estimate can be calculated from these estimates using a Gauss-Newton algorithm. We also present some relevant results concerning the spectral structure of Toeplitz matrices which play a key role in the estimation. Finally, we illustrate the behavior of these estimates using simulated and real DNA sequencing data.