Analysis of Time Series Structure: SSA and Related by Nina Golyandina

By Nina Golyandina

Over the past 15 years, singular spectrum research (SSA) has confirmed very winning. It has already develop into a customary software in climatic and meteorological time sequence research and popular in nonlinear physics and sign processing. although, regardless of the promise it holds for time sequence functions in different disciplines, SSA isn't well known between statisticians and econometrists, and even though the fundamental SSA set of rules appears to be like easy, knowing what it does and the place its pitfalls lay is in no way simple.Analysis of Time sequence constitution: SSA and comparable options presents a cautious, lucid description of its common conception and technique. half I introduces the fundamental options, and units forth the most findings and effects, then offers a close therapy of the technique. After introducing the elemental SSA set of rules, the authors discover forecasting and practice SSA principles to change-point detection algorithms. half II is dedicated to the idea of SSA. right here the authors formulate and end up the statements of half I. They handle the singular price decomposition (SVD) of actual matrices, time sequence of finite rank, and SVD of trajectory matrices.Based at the authors' unique paintings and full of purposes illustrated with actual info units, this publication bargains a very good chance to acquire a operating wisdom of why, whilst, and the way SSA works. It builds a powerful origin for effectively utilizing the procedure in purposes starting from arithmetic and nonlinear physics to economics, biology, oceanology, social technology, engineering, monetary econometrics, and industry study.

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Additional info for Analysis of Time Series Structure: SSA and Related Techniques (Chapman & Hall CRC Monographs on Statistics & Applied Probability)

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2), if there exists a collection of indices I ⊂ {1, . . , d} such that X(1) = i∈I Xi and consequently X(2) = i∈I X . i / In the case of separability, the contribution of X(1) , the first component in the expansion X = X(1) + X(2) , is naturally to measure by the share of the corresL ponding eigenvalues: i∈I λi i=1 λi . 2) can be looked at from different perspectives. Let us fix the set of indices I = I1 and consider the corresponding resultant matrix XI1 . 2). Moreover, if the matrices XI1 and XI2 are close to some Hankel matrices, then there exist series F (1) and F (2) such that F = F (1) + F (2) and the trajectory matrices of these series are close to XI1 and XI2 , respectively (the problem of finding these series is discussed below).

14) k 2 2 k (ak + bk ) converges. Then the measure mf is concentrated at the points ±ωk , ωk ∈ (0, 1/2), with the weights (a2k + b2k )/4. The weight of the point 1/2 equals a2 (1/2). 2 for the precise definition). Periodic series correspond to a spectral measure mf concentrated at the points ±j/T (j = 1, . . , [T /2]) for some integer T . 14), this means that the number of terms in this representation is finite and all the frequencies ωk are rational. Almost periodic series that are not periodic are called quasi-periodic.

22) wi =  N − i for K ∗ ≤ i ≤ N − 1. 23) i=0 and call the series F (1) and F (2) w-orthogonal if F (1) , F (2) w = 0. 2) that separability implies w-orthogonality. Therefore, if the series F is split into a sum of separable series F (1) , . . , F (m) , then this sum can be interpreted as an expansion of the series F with respect to a certain w-orthonormal basis, generated by the original series itself. Expansions of this kind are typical in linear algebra and analysis. 22). 23) have the form of a trapezium.

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