Piecewise polynomial regression with fractional residuals for the analysis of calcium imaging data

Abstract

In this work we deal with the mathematical analysis and application of piecewise (or segmented) polynomial regression. Motivated by an application in neurobiology we allow the residual processes of our model to exhibit long memory, short memory or antipersistence.As a solid biological background is essential for understanding the application in this work, we start with an introduction to neurobiology and the related experimental techniques. We conclude this introduction with a sample of data sets by means of which we illustrate piecewise polynomial regression.Thereafter, we discuss least squares estimation with piecewise polynomials when the residuals exhibit antipersistence, short memory or long memory. This purely mathematical discussion is completely detached from the initial biological application. We start with an introduction to the related mathematical foundations and then discuss consistency and the asymptotic properties of the least squares estimator. In addition to the usual least squares estimator we treat the weighted least squares estimator as well. The asymptotic distribution is represented as a stochastic integral with respect to a fractional Brownian motion (in the case of antipersistence, short memory or long memory) or as a stochastic integral with respect to a Hermite process (in the case of long memory). We derive our results by means of fractional calculus which allows us to state a unifying formula of the asymptotic covariance matrix which covers all three correlation structures. In the case of an unknown number of segments we show that an information criterion can be used to estimated this unknown number. However, as the precise normalisation of the information criterion depends on the underlying correlation structure of the residuals, the latter results is only of theoretical interest.We conclude this work by applying the derived methods on a large biological data sets. In this analysis, we apply piecewise polynomials to estimate the trend function of temporal response patterns. These estimates serve then as an input for an errors-in-variables regression model.