By Dr. T. Subba Rao, Dr. M. M. Gabr (auth.)
The concept of time sequence versions has been good built during the last thirt,y years. either the frequenc.y area and time area techniques were universal within the research of linear time sequence versions. in spite of the fact that. many actual phenomena can't be accurately represented through linear types; therefore the need of nonlinear versions and better order spectra. lately a couple of nonlinear types were proposed. during this monograph we limit realization to at least one specific nonlinear version. referred to as the "bilinear model". the main fascinating function of the sort of version is that its moment order covariance research is ve~ just like that for a linear version. This demonstrates the significance of upper order covariance research for nonlinear types. For bilinear types it's also attainable to acquire analytic expressions for covariances. spectra. and so forth. that are frequently tricky to acquire for different proposed nonlinear versions. Estimation of bispectrum and its use within the development of checks for linearit,y and symmetry also are mentioned. all of the tools are illustrated with simulated and genuine information. the 1st writer want to recognize the convenience he acquired within the instruction of this monograph from offering a sequence of lectures related to bilinear types on the collage of Bielefeld. Ecole Normale Superieure. college of Paris (South) and the Mathematisch Cen trum. Ams terdam.
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Additional resources for An Introduction to Bispectral Analysis and Bilinear Time Series Models
We now obtain an expression for the bias of the bispectral estimate. the derivation of which is essentially similar to the one given by Brillinger and Rosenblatt (1967a) for the k-th order case. 8). Ftftther let Ko(8 1 ,82 ) ecK(2). 12) The mean square error is a function of V2 and BK only when the second order spectral density functions f(Wl), f(W2) and f(Wl+W2) are given. We assume that the spectral windows which are non-negative and belong to c/ 2 ) class give rise effectively to the same bias when used for estimating the bispectral density function.
C;~l window" (see Parzen, 1957; Priestley. 1981). 2) 31 MKO(Me). with Ko(e) satisfying the following conditions CD f CD (i) f Ko( e) d e = 1. Ko 2 (e) de < CD. 5) de, _CD where M = M(N) is an integer chosen so that, as N + CD. M+ CD but MIN + O. (If A(s) = 0, lsI> 1, M corresponds to the "truncation point"). The function A(s) is known as a lag window. Parzen defi ned the "characteri s ti c exponent" of the functi on Ko (e) as the largest integer r, such that exists, is finite and non-zero. Parzen (1957) has shown that the most useful class of windows to consider is that for which r = 2.
Ftftther let Ko(8 1 ,82 ) ecK(2). 12) The mean square error is a function of V2 and BK only when the second order spectral density functions f(Wl), f(W2) and f(Wl+W2) are given. We assume that the spectral windows which are non-negative and belong to c/ 2 ) class give rise effectively to the same bias when used for estimating the bispectral density function. With this assumption and for fixed f(Wl), f(W2) and f(Wl+W2), the minimization of M(Wl,W2) is the same as the minimization of Vl • In the following section we find an optimum weight function for which V2 is minimum (we note that the assumption that the non-negative windows belong to CK(2) class implies that the windows are invariant under scale transformation).
An Introduction to Bispectral Analysis and Bilinear Time Series Models by Dr. T. Subba Rao, Dr. M. M. Gabr (auth.)