A Fast Approximation of the Nadaraya-Watson Regression with the k-Nearest Neighbor Crossover Kernel

For a system with inputs and outputs, a nonparametric regression has been proposed to clarify the relationship between inputs and outputs from a large amount of data. To improve estimation accuracy for the Nadaraya-Watson regression which is one of the nonparametric regressions, the regression with k-nearest neighbor crossover kernel, in which the kernel function by using neighborhood for each sample point in a sample set is made, is an effective method. However, there is a problem that the calculation time for estimation of this regression is very long, because it is needed to use all kernel functions made for all sample points. In this paper, we propose an estimation method with a fast approximation by using a few selected kernel functions instead of all kernel functions. These kernel functions are those made for only sample points included in neighborhood with the point that we want to estimate for. By this estimation method with a fast approximation, we show that the calculation time for estimation is short, and the estimation accuracy for the proposed method does not degrade, compared to that for a conventional estimation method without approximation. Moreover, we show theoretical aspects of the proposed approximation method for the regression with k-nearest neighbor crossover kernel.