A Distribution Adaptive Framework for Prediction Interval Estimation Using Nominal Variables

Abstract: Proposed methods for prediction interval estimation so far focus on cases where input variables are numerical. In datasets with solely nominal input variables, we observe records with the exact same input $x^u$, but different real valued outputs due to the inherent noise in the system. Existing prediction interval estimation methods do not use representations that can accurately model such inherent noise in the case of nominal inputs. We propose a new prediction interval estimation method tailored for this type of data, which is prevalent in biology and medicine. We call this method Distribution Adaptive Prediction Interval Estimation given Nominal inputs (DAPIEN) and has four main phases. First, we select a distribution function that can best represent the inherent noise of the system for all unique inputs. Then we infer the parameters $\theta_i$ (e.g. $\theta_i=[mean_i, variance_i]$) of the selected distribution function for all unique input vectors $x^u_i$ and generate a new corresponding training set using pairs of $x^u_i, \theta_i$. III). Then, we train a model to predict $\theta$ given a new $x_u$. Finally, we calculate the prediction interval for a new sample using the inverse of the cumulative distribution function once the parameters $\theta$ is predicted by the trained model. We compared DAPIEN to the commonly used Bootstrap method on three synthetic datasets. Our results show that DAPIEN provides tighter prediction intervals while preserving the requested coverage when compared to Bootstrap. This work can facilitate broader usage of regression methods in medicine and biology where it is necessary to provide tight prediction intervals while preserving coverage when input variables are nominal.