Cadzow’s basic algorithm, alternating projections and singular spectrum analysis

Abstract

After observing a noisy time series or signal, it is com-mon practice to try to separate the noise from the observed measurements. Singular value decomposition based methods are often used to split the observed signal into a number of components. Components associated with noise may be re-moved from the signal, and subsequent analyses may be un-dertaken. This paper will describe two methods commonly used to remove noise from a signal; the so-called singular spectrum analysis, and Cadzow's basic algorithm. Connec-tions between both methods will be drawn, and both will be related to the method of alternating projections, and struc-tured low rank approximation (finding a lower rank approx-imation of a given matrix with specified structure). A sim-ulation study and example based on real data will highlight and explain the differences between both methods.