Uncertainty quantification plays a critical role in the process of decision making and optimization in many fields of science and engineering. The field has gained an overwhelming attention among researchers in recent years resulting in an arsenal of different methods. Probabilistic forecasting and in particular prediction intervals (PIs) are one of the techniques most widely used in the literature for uncertainty quantification. Researchers have reported studies of uncertainty quantification in critical applications such as medical diagnostics, bioinformatics, renewable energies, and power grids. The purpose of this survey paper is to comprehensively study neural network-based methods for construction of prediction intervals. It will cover how PIs are constructed, optimized, and applied for decision-making in presence of uncertainties. Also, different criteria for unbiased PI evaluation are investigated. The paper also provides some guidelines for further research in the field of neural network-based uncertainty quantification.

INTRODUCTION

Many engineering and scientific problems consisted of partly deterministic and partly random situations. The traditional point prediction is unable to predict the level of randomness or uncertainty. Prediction Intervals (PIs) have been extensively used in a range of applications for over 50 years [1]–[۳] to quantify that uncertainty. Mostly to overcome the limitations of the point prediction, an interval prediction is widely accepted in many fields of study including economics [4], food industry [5], tourism [6], medical statistics [2], power consumption [7], even in compression algorithms [8]. Moreover, the recent installation of the largescale renewable energy [9], [10], the rapid growth of the online auction systems [11], [12], and the design of different types of autonomous robots [13] are increasing the uncertainty and therefore, increasing the essence of the probabilistic forecasting. Fig. 1 presents the importance of PI with a rough sketch. The point prediction gives a value close to the median or the mean of the probable values of targets. Two green lines in Fig. 1 presents the PI. The width of the interval changes based on the probable values of the target. Observing the point forecast, the user is not provided by any complement information about the uncertainty of the system [14]–[۱۶]. Therefore, interval forecasts are a popular method of uncertainty quantification. The uncertainty in risk analysis processes is traditionally classified as follows: 1) Aleatory uncertainty (inherent randomness): The output of a system may vary slightly from time to time even for the same set of inputs. There might be day to day or year to year variation while the inputs are the same. Also, there might be no trend of such changes. Therefore these uncertainties are the inherent randomness or the aleatory uncertainty [17]. An example of such randomness is the electricity demand for a certain time. Although the temperature, the time in a day, the day in a week, humidity, wind speed is the same, the electricity demand can be slightly different [18], [19]. 2) Epistemic (subjective) uncertainty: The uncertainty may also happen due to the secondary or tertiary effects, not considered during the modeling. It can be an effect from a phenomenon, which is unknown to the research community or can be the effect of an internal parameter of an object, not readable from outside. In such scenario, the uncertainty can be reduced by enhancing knowledge or by performing measurements. An example of such uncertainty is the strength of solids such as steel and concrete. The internal defects of a solid material are difficult to measure from the outer appearance and even from the formation process. However, the provider can provide a range of the strength parameters. With additional modeling efforts, the uncertainty parameter can be transformed from a random variable to a bounded random or pseudorandom variable using probability intervals or percentile ranges [19]–[۲۳].