Ramp function regression: a tool for quantifying climate transitions

A method is proposed for fitting a 'ramp' to measured data. This is a continuous function, segmented in three parts: x(fit)(t) = x1 for t less than or equal to t1, x2 for t greater than or equal to t2, and linearly connected between t1 and t2. Its purpose is to measure transitions in the mean of time series as they occur, for example, in paleoclimatic records. The unknowns x1 and x2 are estimated by weighted least-squares regression, t1 and t2 by a brute-force search. Computing costs are reduced by several methods. The presented Fortran 77 program, RAMPFIT, includes analysis of weighted ordinary residuals for checking the validity of the ramp form and other assumptions. It fits an AR(1) model to the residuals to measure serial dependency; uneven time spacing is thereby allowed. Three bootstrap resampling schemes (nonparametric stationary, parametric, and wild) provide uncertainties for the estimated parameters. RAMPFIT works interactively (calculation/visualization). Example time series (one artificial, three measured) demonstrate that this approach is useful for practical applications in geosciences (n less than a few hundred, noise, unevenly spaced times), and that the ramp function may serve well to model climate transitions. (C) 2000 Elsevier Science Ltd. All rights reserved.