DIFFERENTIAL GEOMETRICAL STRUCTURES RELATED TO FORECASTING ERROR VARIANCE RATIOS

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.