Title: "Local State Space Prediction Models for Noisy Time Series"

Author: Dimitris Kugiumtzis
Department of Mathematical and Physical Sciences
Faculty of Engineering
Aristotle University of Thessaloniki, GREECE

Abstract:

State space local models, and in patricular local linear models, 
is one of several model classes that have been applied to predict 
real time series. The difference of this class from global linear,
 or in general polynomial, prediction is that, for every single point 
prediction, a different autoregressive (AR) model is estimated based 
only on a number of selected past scalar data segments. Geometrically, 
these data segments correspond to points close to the target point when 
the time series is viewed in a pseudo-state space with dimension equal 
to the order of the
local AR model.

The parameters of the local model are typically estimated using ordinary
least squares (OLS). Apart from potential linearization errors, 
a drawback of this approach is the high variance of the predictions 
under certain conditions, such as noise in the data and high dimensions
 of the pseudo-state space. It has been shown that a different set of 
so-called regularization techniques, originally derived to solve 
ill-posed regression problems, gives more stable solutions (and
thus better predictions) than OLS on noisy random-like time series. Such
regularisation techniques are the principal component regression (PCR),
the partial least squares (PLS) and the ridge regression (RR). 
These methods reduce the variance compared to OLS, but introduce more 
bias. The truncated total least squares, which is designed to solve 
''error-in-variables'' problems and thus would be expected to be more 
appropriate for noisy time series, turns out to give the worst predictions.


We will discuss the general features of local (linear) prediction and 
particularly the OLS solution and the regularisations. Some statistical
 properties of the methods will be highlighted and explained in the setting 
of local state space prediction. The superiority of the regularization 
techniques over OLS in the use of local models in high dimensional state 
space will be demonstrated with
some real world time series.