Basic Singular Spectrum Analysis and Forecasting with R

Singular Spectrum Analysis (SSA) as a tool for analysis and forecasting of time series is considered. The main features of the Rssa package, which implements the SSA algorithms and methodology in R, are described and examples of its use are presented. Analysis, forecasting and parameter estimation are demonstrated by means of case study with an accompanying code in R

Shaped extensions of singular spectrum analysis

Extensions of singular spectrum analysis (SSA) for processing of non-rectangular images and time series with gaps are considered. A circular version is suggested, which allows application of the method to the data given on a circle or on a cylinder, e.g. cylindrical projection of a 3D ellipsoid. The constructed Shaped SSA method with planar or circular topology is able to produce low-rank approximations for images of complex shapes. Together with Shaped SSA, a shaped version of the subspace-based ESPRIT method for frequency estimation is developed. Examples of 2D circular SSA and 2D Shaped ESPRIT are presented

Monte Carlo solution for the Poisson equation on the base of spherical processes with shifted centres

We consider a class of spherical processes rapidly converging to the boundary (so called Decentred Random Walks on Spheres or spherical processes with shifted centres) in comparison with the standard walk on spheres. The aim is to compare costs of the corresponding Monte Carlo estimates for the Poisson equation. Generally, these costs depend on the cost of simulation of one trajectory and on the variance of the estimate. It can be proved that for the Laplace equation the limit variance of the estimation doesn\u27t depend on the kind of spherical processes. Thus we have very effective estimator based on the decentred random walk on spheres. As for the Poisson equation, it can be shown that the variance is bounded by a constant independent of the kind of spherical processes (in standard form or with shifted centres). We use simulation for a simple model example to investigate variance behavior in more details

Statistical approach to detection of signals by Monte Carlo singular spectrum analysis: Multiple testing

The statistical approach to detection of a signal in noisy series is considered in the framework of Monte Carlo singular spectrum analysis. This approach contains a technique to control both type I and type II errors and also compare criteria. For simultaneous testing of multiple frequencies, a multiple version of MC-SSA is suggested to control the family-wise error rate

Weighted norms in subspace-based methods for time series analysis

Many modern approaches of time series analysis belong to the class of methods based on approximating high-dimensional spaces by low-dimensional subspaces. A typical method would embed a given time series into a structured matrix and find a low-dimensional approximation to this structured matrix. The purpose of this paper is twofold: (i) to establish a correspondence between a class of SVD-compatible matrix norms on the space of Hankel matrices and weighted vector norms (and provide methods to construct this correspondence) and (ii) to motivate the importance of this for problems in time series analysis. Examples are provided to demonstrate the merits of judiciously selecting weights on imputing missing data and forecasting in time series. Copyright © 2016 John Wiley & Sons, Ltd