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

Blind deconvolution of covariance matrix inverses for autoregressive processes

Matrix $\mathbf{C}$ can be blindly deconvoluted if there exist matrices $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$, where $\ast$ denotes the operation of matrix convolution. We study the problem of matrix deconvolution in the case where matrix $\mathbf{C}$ is proportional to the inverse of the autocovariance matrix of an autoregressive process. We show that the deconvolution of such matrices is important in problems of Hankel structured low-rank approximation (HSLRA). In the cases of autoregressive models of orders one and two, we fully characterize the range of parameters where such deconvolution can be performed and provide construction schemes for performing deconvolutions. We also consider general autoregressive models of order $p$, where we prove that the deconvolution $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$ does not exist if the matrix $\mathbf{B}$ is diagonal and its size is larger than $p$