The Hodge Decomposition, Edition: version 27 Jan 2010 by Nikolai Nowaczyk

By Nikolai Nowaczyk

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Extra resources for The Hodge Decomposition, Edition: version 27 Jan 2010

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Ck + k Xt . 26) Considering the fact that many slowly-changing functions can be well approximated by a low-degree polynomial on an interval of finite length, the order k of differencing required in practice is quite small, being often one or two (Brockwell and Davis 2002, p. 30). 27) d Yt = Yt − Yt−d = (1 − r )Yt can be used to eliminate the seasonality. 28) represents a decomposition of d Yt into a trend (mt − mt−d ) and a noise term (Xt − Xt−d ). The remaining trend mt − mt−d can be eliminated using a power of the differencing operator (Brockwell and Davis 2002, p.

34) = Cor(Xt+h , Xt ). γX (0) The ACF is symmetrical about the origin where it attains its maximum value of one. Most physical processes have an ACF decreasing in absolute value with an increasing lag. This means that the relation between Xt at a short temporal distance is stronger than that over a longer distance. Rapidly decaying ACF values as |h| increases indicate short-term dependency, while slowly decaying ACF values suggest the presence of long-term dependency. In practical problems, one may not start directly with a model, but with observed time series data {x1 , x2 , .

The WLSE possessing the smallest MSE of all LUE is referred to as the best linear unbiased estimator (BLUE). , W = Q−1 ll . 17) where the orthogonal projectors P A and P ⊥ A are P A = A AT Q−1 ll A −1 ⊥ AT Q−1 ll , P A = I n − P A . 18) Substituting H from Eq. 17) as well as P A and P ⊥ A from Eq. 18) into Eq. 14), the resulting covariance matrices of xˆ , ˆl and eˆ are Cxˆ xˆ = σ02 AT Q−1 ll A −1 , Cˆlˆl = P A Cll , Ceˆ eˆ = P ⊥ A Cll . 19) The minimum tr(Cxˆ xˆ ) indicates that the BLUE is a minimum variance linear unbiased estimator.

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