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DIFFERENTIAL GEOMETRICAL STRUCTURES RELATED TO

FORECASTING ERROR VARIANCE RATIOS

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DAMING XU

*Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, U.S.A.*
(Received February 6, 1989; revised July 16, 1990)

**Abstract.**
Differential geometrical structures
(Riemannian metrics, pairs of dual
affine connections, divergences and yokes) related to multi-step
forecasting error
variance ratios are introduced to a manifold of stochastic linear
systems. They are
generalized to nonstationary cases. The problem of approximating a
given time series by a specific model is discussed. As examples, we
use the established scheme to discuss the AR (1) approximations and
the exponential smoothing
of ARMA series for multi-step forecasting purpose. In the process,
some interesting results about spectral density functions are derived
and applied.

*Key words and phrases*:
Riemannian metric, affine connection,
divergence, spectral density, forecasting error variance ratio, yoke.

**Source**
( TeX ,
DVI ,
PS )