Reconstructing time series and their uncertainty from observations with universal noise

Abstract

A new approach is presented for the reconstruction of time series and other (y,x) functions from observables with any type of stochastic noise. In particular, noise may exist in both dependent and independent variables, i.e., y and x, or t, and may even be correlated between these variables. This situation occurs in many areas of the geosciences when the independent time variable is itself the result of a measurement process, such as in paleo– sea level estimation. Uncertainty in the recovered time series is quantified in probabilistic terms using Bayesian changepoint modeling. The main contribution of the paper is the derivation of a new form of integrated likelihood function which can measure the data fit for a curve to (y,t) observables contaminated by any type of random noise. Closed form expressions are found for the special case of correlated Gaussian data noise and curves built from the sum of piecewise linear polynomials. The technique is illustrated by estimating relative sea level variations, over the last five glacial cycles, from a data set of 1928 δ18O measurements. Comparisons are also made with other techniques including those that assume an error free “independent” variable. Experiments illustrate several benefits of accounting for timing errors. These include allowing rigorous uncertainty information of both time-dependent signals and their gradients. Derivatives of the integrated likelihood function are also given, which allow implementation of likelihood maximization. The new likelihood function better reflects real errors in data and can improve recovery of the estimated time series.