Fuzzy-statistical prediction intervals from crisp regression models

Abstract

Most prediction outputs from regression models are either point estimates or interval estimates. Point estimates from a model are useful for making conclusions about model accuracy. Interval estimates on the other-hand are used to evaluate the uncertainty in the model predictions. These two approaches only produce either point or a single interval and thus do not fully represent the uncertainties in the model prediction. In this paper, previous works on constructing fuzzy numbers from arbitrary statistical intervals are extended by first constructing fuzzy-statistical prediction intervals which combines point and prediction interval estimates into a single fuzzy number which fully represents the uncertainties in the model. Then two simple metrics are introduced that can evaluate the quality of the proposed fuzzy-statistical prediction intervals. The proposed metrics are simple to calculate and use same ideas from the well-known metrics for evaluating interval estimates. To test the applicability of the proposed method, two types of scenarios are adopted. In the first scenario, the models are calibrated and then the proposed method is used to get the fuzzy-statistical prediction interval. In the second scenario, the point estimate and prediction intervals are given as output from a model by another researcher, then the proposed approach is used to get the fuzzy-statistical prediction intervals without prior knowledge of the model calibration process. The first scenario is tested by calibrating linear regression and neural network models using a well-known data set of automobile fuel consumption (auto-MPG). The second scenario is tested using outputs from point and interval estimates of two time series models (ARIMA, Kalman Filter) calibrated from a real traffic flow data set.