Backpropagation of pseudo-errors: neural networks that are adaptive to heterogeneous noise

Neural networks are used for prediction model in many applications. The backpropagation algorithm used in most cases corresponds to a statistical nonlinear regression model assuming the constant noise level. Many proposed prediction intervals in the literature so far also assume the constant noise level. There are no prediction intervals in the literature that are accurate under varying noise level and skewed noises. We propose prediction intervals that can automatically adjust to varying noise levels by applying the regression transformation model of Carroll and Rupert (1988). The parameter estimation under the transformation model with power transformations is shown to be equivalent to the backpropagation of pseudo-errors. This new backpropagation algorithm preserves the ability of online training for neural networks.