Generalized Dynamic principal components are presented and it is shown how to define one side inear combinations of the present and past values of the series that minimize the reconstruction mean squared error (ODPC). It is shown that the ODPC introduced in this paper can be successfully used for forecasting high-dimensional multiple time series.

Non-stationary spatial models are widely applicable in diverse disciplines, ranging from bio-medical sciences to geophysical studies. In many of theses applications, testing for structural changes in the trend and testing the specific form of the trend are highly relevant. A novel statistics based on a discrepancy measure over small regions is proposed in this paper. Such a measure can be used to construct tests for structural trends and to identify change boundaries of the trends. By virtue of the m-dependence approximation of a stationary random eld, asymptotic properties and limit distributions of these tests are established. The method is illustrated by simulations and data analysis.