By James B. Elsner

The time period singular spectrum comes from the spectral (eigenvalue) decomposition of a matrix A into its set (spectrum) of eigenvalues. those eigenvalues, A, are the numbers that make the matrix A -AI singular. The time period singular spectrum research· is unlucky because the conventional eigenvalue decomposition regarding multivariate info can be an research of the singular spectrum. extra competently, singular spectrum research (SSA) will be referred to as the research of time sequence utilizing the singular spectrum. Spectral decomposition of matrices is key to a lot the ory of linear algebra and it has many purposes to difficulties within the traditional and similar sciences. Its frequent use as a device for time sequence research within reason contemporary, although, rising to a wide quantity from functions of dynamical platforms conception (sometimes known as chaos theory). SSA was once brought into chaos thought through Fraedrich (1986) and Broomhead and King (l986a). sooner than this, SSA used to be utilized in organic oceanography via Colebrook (1978). within the digi tal sign processing neighborhood, the procedure is usually referred to as the Karhunen-Loeve (K-L) growth (Pike et aI., 1984). Like different ideas according to spectral decomposition, SSA is appealing in that it holds a promise for a discount within the dimen- • Singular spectrum research is typically referred to as singular platforms research or singular spectrum method. vii viii Preface sionality. This relief in dimensionality is frequently followed by means of a less complicated rationalization of the underlying physics.

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519 2 with the singular vectors being identical to the eigenvectors e l , e 2 , and e3 . 4. Recovering the Time Series The eigenvectors can be used to compute the principal components (ak's) by projecting the original time record onto them as follows: for i = 1, 2, ... , N, where ej represents the jth component of the kth eigenvector. 2. 473 increasing k. As a result of orthogonality, each principal component can be isolated and probed independently from the remainder of the time record. , Broomhead and King, 1986a).

Here we are interested in the eigenvalues and eigenvectors as a way to express the variability of a set of data from, for example, observations or output from numerical models. For this, A is usually symmetric with real coefficients and the process of decomposition is simplified. Before proceeding to the special case of the eigenvalues and eigenvectors of real, symmetric matrices we mention a few properties of eigenvalues and eigenvectors of an arbitrary matrix A. For an n by n matrix A, the trace of A is given by the sum of the n diagonal entries of A.

Y. Now it is an interesting mathematical observation that this equation can be solved for those special directions y and for the ).. associated with each of these directions. 2. , finding the eigenvalues and eigenvectors or the eigenmodes) is, for what values of ).. is the matrix A - ),,1 singular? For matrices, the question of singularity can be answered with determinants. Using determinants the fundamental question reduces to; for what values of ).. is det(A - ),,1) = o? This is called the characteristic equation for the matrix A.