The mean shift algorithm and its relation to kernel regression

We investigate the connection between the asymptotic bias of the well-known Nadaraya–Watson kernel regression and the mean shift (MS) vector with the Gaussian kernel. We first show that the asymptotic bias for the univariate Nadaraya–Watson kernel regression can be estimated using the MS scalar. Then, we investigate the general D-dimensional case (D > 1) and derive a formula for the asymptotic bias as a function of the MS vector. We show that when the regression function is a linear function of its entries, then the asymptotic bias can be represented as a linear function of the MS vector. The MS algorithm for the univariate and the linear cases can be used to find points where the Nadaraya–Watson kernel regression gives an unbiased estimate. Furthermore, this connection suggests that exploiting the theoretical properties of the asymptotic bias of the Nadaraya–Watson kernel regression may be helpful to show the convergence of the MS algorithm. Through the simulations, we show that how the given theoretical results can be used to estimate the bias of the estimated clean signal by just observing the noisy signal. Having access to the bias of the estimator, enables us to evaluate the accuracy of the estimated clean data, which later will be used to design a quantizer.