加载中…Time Series Model in the view of Hilbert Spaces
(2009-03-31 08:43:56)
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数学 |
The
Nine Chapters of Mathematical Art, an ancient Chinese mathematical
work, already shed some light on the characteristics on the Hilbert
Space. The Gougu Theorem mentioned in this book actually coincides
with the Parseval Formula held in the two dimensional Euclidean
space. Time series analysis deals a lot with
distances between random variables, and such distances are defined
in a special type of Hilbert space: the square integrable functions
on a probability space. Although in linear time series regressions
the algorithm simulates that of the least square linear regression,
in linear regression the variables are defined directly on the
Euclidean space, while in time series calculation the application
of distances in Euclidean space only approximates distances in the
previous square integrable function space. The spectral analysis is
based upon an isomorphism between two types of Hilbert spaces,
using an Ito integration of an orthogonal increment process, which
transfers our perspective from the time domain to the frequency
domain.
URL: http://cos.name/2009/03/hilbert/.
URL: http://cos.name/2009/03/hilbert/.
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