A linear-circular regression estimate for data fusion: Application to GNSS carrier-phase signal processing

Abstract

This article is dedicated to the estimation of the parameters of a linear-circular regression model. For this model, the response is circular and defined between −π and π, the predictor is linear and several sensors provide noisy observations of the response. In our approach, the noise is assumed to be distributed according to a von Mises distribution with a concentration parameter that models the accuracy of the sensors. We propose a maximum likelihood circular fusion operator for the estimation of the intercept, the slope of the regression line and the concentration parameter associated with each sensor. The proposed estimate is not direct as in the linear case and requires an iterative algorithm to maximize a periodic contrast function. In order to characterize the accuracy of our fusion operator, the theoretical expression of the variance of the proposed estimator slope is first derived. For this derivation, we approximate the von Mises distribution by a Wrapped normal distribution and we consider unwrapped observations. Then, we derive an iterative procedure to maximize the contrast function. We show, using synthetic data, that the variance of the slope of the regression line derived using the proposed estimate is in good agreement with that obtained using the theoretical expression of the variance. The proposed estimator is also used to process the carrier-phase difference between GNSS signals provided by two antennas. The objective in terms of signal processing is to estimate the linear parameters of this difference in order to derive the height between the two antennas. We show that fusing the observations provided by several satellite signals improves the accuracy of the estimated height. We also show, using real data, that the theoretical study of the proposed estimator can be used to predict the length of integration of the signal necessary for obtaining an estimate of the height with a given accuracy.