Scaling of a length scale for regression and prediction

We analyze the prediction from noised data, based on a regression formulation of the problem. For the regression, we construct a model with a length scale to smooth the data, which is determined by the variance of noise and the speed of the variation of original signals. The model is found to be effective also for prediction. This is because it decreases an uncertain region near a boundary as the speed of the variation of original signals increases, which is a crucial property for accurate prediction.