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.

本文致力于线性-循环回归模型的参数估计。对于这个模型，响应是圆形的，定义在 − π和π之间，预测器是线性的，并且几个传感器提供了对响应的嘈杂观察。在我们的方法中，假设噪声是根据 von Mises 分布分布的，该分布具有模拟传感器精度的浓度参数。我们提出了一个最大似然圆形融合算子，用于估计与每个传感器相关的截距、回归线的斜率和浓度参数。所提出的估计不像在线性情况下那样直接，并且需要迭代算法来最大化周期性对比函数。为了表征我们的融合算子的准确性，首先推导出所提出的估计器斜率方差的理论表达式。对于这个推导，我们通过 Wrapped 正态分布来近似 von Mises 分布，并考虑未包裹的观察结果。然后，我们推导出一个迭代过程来最大化对比度函数。我们使用合成数据表明，使用建议的估计得出的回归线斜率的方差与使用方差的理论表达式获得的一致。建议的估计器还用于处理由两个天线提供的 GNSS 信号之间的载波相位差。信号处理方面的目标是估计这种差异的线性参数，以便推导出两个天线之间的高度。我们表明，融合多个卫星信号提供的观察结果可以提高估计高度的准确性。我们还显示，使用真实数据，