GDOP, Ridge Regression and the Kalman Filter

Abstract

Multicollinearity and its effect on parameter estimators such as the Kalman filter is analysed using the navigation application as a special example. All position-fix navigation systems suffer loss of accuracy when their navigation landmarks are nearly collinear. Nearly collinear measurement geometry is termed the geometric dilution of position (GDOP). Its presence causes the errors of the position estimates to be highly inflated. In 1970 Hoerl and Kennard developed ridge regression to combat near collinearity when it arises in the predictor matrix of a linear regression model. Since GDOP is mathematically equivalent to a nearly collinear predictor matrix, Kelly suggested using ridge regression techniques in navigation signal processors to reduce the effects of GDOP. The original programme intended to use ridge regression not only to reduce variance inflation but also to reduce bias inflation. Reducing bias inflation is an extension of Hoerl's ridge concept by Kelly. Preliminary results show that ridge regression will reduce the effects of variance inflation caused by GDOP. However, recent results (Kelly) conclude it will not reduce bias inflation as it arises in the navigation problem, GDOP is not a mismatched estimator/model problem. Even with an estimator matched to the model, GDOP may inflate the MSE of the ordinary Kalman filter while the ridge recursive filter chooses a suitable biased estimator that will reduce the MSE. The main goal is obtaining a smaller MSE for the estimator, rather than minimizing the residual sum of squares. This is a different operation than tuning the Kalman filter's dynamic process noise covariance Q, in order to compensate for unmodelled errors. Although ridge regression has not yielded a satisfactory solution to the general GDOP problem, it has provided insight into exactly what causes multicollinearity in navigation signal processors such as the Kalman filter and under what conditions an estimator's performance can be improved.