The Delta Method, from the field of nonlinear regression. The Bayesian Method, from Bayesian modeling and statistics. The Mean-Variance Estimation Method, using estimated statistics. The Bootstrap Method, using data resampling and developing an ensemble of models.

A novel method for estimating prediction uncertainty using machine learning techniques is presented. Uncertainty is expressed in the form of the two quantiles (constituting the prediction interval) of the underlying distribution of prediction errors. The idea is to partition the input space into different zones or clusters having similar model errors using fuzzy c-means clustering. The prediction interval is constructed for each cluster on the basis of empirical distributions of the errors associated with all instances belonging to the cluster under consideration and propagated from each cluster to the examples according to their membership grades in each cluster. Then a regression model is built for in-sample data using computed prediction limits as targets, and finally, this model is applied to estimate the prediction intervals (limits) for out-of-sample data. The method was tested on artificial and real hydrologic data sets using various machine learning techniques. Preliminary results show that the method is superior to other methods estimating the prediction interval. A new method for evaluating performance for estimating prediction interval is proposed as well.