# An All-Batch Loss for Constructing Prediction Intervals
## 1 Introduction  
The prediction interval (PI) is widely used to evaluate uncertainty. Different from the confidence interval (CI) which only relies on statistical analyses of the observed data, the PI describes the uncertainty by means of predictions, which cover estimates of both model uncertainty and data uncertainty [1]. The PI is thus more reliable and effective than the CI. Compared with the regular point prediction, the PI offers more robust results for follow-up operations with full consideration of prediction uncertainty. The risk of over-fitting problems can also be reduced by the PI, which is beneficial for real-world tasks. PIs have found many successful applications such as wind power generation [2; 3; 4], water consumption forecasts [5], winter ore burn-through point forecasts [6], parking demand predictions [7], isolated microgrid systems [8], etc. In recent years, as statistical learning theory has evolved, feedforward neural networks (FNNs) have become universal approximators which are well suited for approximating nonlinear mappings [9]. However, commonly used point predictions generated by FNNs suffer from the inability to estimate uncertainty. Although plenty of models have been proposed to construct PIs through FNNs and have achieved remarkable results, many of them may be unsuitable for real-world tasks due to high computational costs and complex assumptions. Consequently, new techniques are necessary to efficiently generate high-quality PIs. Existing methods can be classified into two categories: (1) Those that construct PIs indirectly by point estimates based on statistical theory; and (2) those that directly estimate the lower and upper boundaries of PIs. On the basis of CIs, traditional PI construction methods tend to gain PIs through statistical relationships between point estimates and bias estimates. [PERSON] et al. [10] and [PERSON] [11] developed the Delta method for PI construction based on nonlinear regression by applying asymptotic theories and employing the NN as a nonlinear regression model. [PERSON] [12] and [PERSON] [13] utilized the Bayesian technique to construct PIs. Due to the introduction of the Hessian matrix, this method suffers from the burden of high computational complexity. [PERSON] [14], [PERSON] et al. [15] and [PERSON] et al. [16] adopted the bootstrap method to construct PIs with FNNs. Bootstrap procedures areuseful for uncertainty estimates, and are widely used in many fields, as they provide reliable solutions to obtain the predictive distribution of the output variables in NNs [17]. It has been shown that while the bootstrap method generates more reliable PIs than other methods [18], it suffers from high computational costs under large datasets. A special case of a traditional PI construction method is the mean-variance estimation (MVE) method, proposed by [PERSON] and [PERSON] [19]. The MVE method has a low computational cost and is easy to implement because of the dual-FNN structure. However, due to the assumption that the error term obeys a normal distribution with the expectation of the true target, this method may yield bad results in real-world datasets. In order to reduce computational complexity, [PERSON] et al. [20] developed the lower upper bound estimation (LUBE) method to estimate the lower and upper boundaries of PI directly by FNNs. The LUBE method is a distribution-free PI construction method which uses heuristic search algorithms such as Simulated Annealing (SA) [20], Genetic Algorithm (GA) [21] and Particle Swarm Optimization (PSO) [22] to adjust model parameters. Unlike traditional PI construction methods, the LUBE method attempts to improve the quality of PIs instead of focusing on prediction errors. Due to the flexibility of LUBE, it is gaining popularity in several communities, such as solar power prediction [23]. Nevertheless, in the case of large-scale models, the heuristic search scheme adopted by the LUBE method makes it difficult to obtain optimal results.