A Frequency-Domain Median Time Series

Abstract

This article discusses the median of a batch of time series. We focus on the frequency-domain median (FDM) series (Zeger 1985), which is calculated from the “mediancenters” (Gower 1974) of the discrete Fourier transforms. Like the univariate median, it has a breakdown value of 50%. Like the mean, the FDM time series of independent realizations from a stationary Gaussian process has a spectrum proportional to that of the component series. Analogs of the univariate median have been described for multivariate (e.g., Brown 1983), survival (Brookmeyer and Crowley 1982), and spherical (Fisher 1985) data. As in the multivariate case, there is not a unique definition of a median time series. One possibility is the median value at each time or time-domain median (TDM). A drawback of the TDM is that it can be qualitatively dissimilar in appearance to its component series because of discontinuities when the median value switches from one realization to another. This is particularly disturbing for smooth time series. Our alternative, the FDM, avoids this problem for nearly Gaussian processes. We show that the breakdown value of the mediancenter of spherically symmetric random variables is 50%. The result for the FDM then follows. We consider a set of time series in which each is composed of a common signal and an iid stationary error series. Two asymptotic scenarios are discussed—the number of iid series of finite length getting large and the length increasing for a finite number of realizations. In the former case, the FDM is asymptotically a Gaussian process with a spectrum similar to that of the components. In the latter, we show that the FDM of stationary Gaussian processes is unbiased and has a spectrum proportional to the spectrum of a single realization. An example illustrates the FDM.

Key Words:

• Mediancenter
• Spectrum
• Fourier transform
• Resistance