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

After observing a noisy time series or signal, it is common 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 removed from the signal, and subsequent analyses may be undertaken. 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. Connections between both methods will be drawn, and both will be related to the method of alternating projections, and structured low rank approximation (finding a lower rank approximation of a given matrix with specified structure). A simulation study and example based on real data will highlight and explain the differences between both methods. AMS 2000 subject classifications: Primary 62M10, 62M15; secondary 62P99.