Algorithm

Statistics

Distribution (mathematics)

Particle swarm optimization

Computer science

Mathematical optimization

Probability distribution

Operations research

Kernel (algebra)

Mathematics

Estimator

Kernel density estimation

Mathematical analysis

Artificial intelligence

Interval (graph theory)

Combinatorics

Probability density function

Demand forecasting

Artificial neural network

Prediction interval

Machine learning

Applied Soft Computing

The current literature on water demand forecasting mostly focuses on giving accurate point predictions of water demand. However, the water demand point forecasting will encounter uninformative and unreliable problems when the uncertainty level of data increases. To solve the above problem, a hybrid model (KDE-PSO-LSTM), which combines long short-term memory networks (LSTM) to kernel density estimation (KDE) optimized by using the particle swarm optimization (PSO) algorithm, is proposed to acquire the water demand prediction interval (PI) to quantify the likely uncertainties in the predictions. At first, the prediction errors are obtained by the difference between the real values of water demand and the predictive values based on the LSTM model. Then, a novel splitting strategy is proposed to divided point predictions into different levels to deal with the problem that it is difficult to fit the prediction errors of the whole water demand using a single probability density function (PDF). Next, the PSO is used to optimize the hyper-parameter of the KDE method for fitting the PDF curves of different levels prediction errors. Moreover, due to the irregular distribution of prediction errors, a search method called confidence-window shifting is presented to determine the optimal prediction error interval from the fitted PDF curves. After that, the upper bounds and the lower bounds of the best intervals of prediction errors are added to the point predictions to attain the final PI of urban water demand. Finally, to demonstrate the superiorities of the proposed model, the proposed KDE-PSO distribution is compared to other well-known distributions, i.e, the KDE distribution, the Beta-PSO distribution and the normal distribution. The experimental results show that the comprehensive performances of the PIs generated from the proposed KDE-PSO-LSTM model are better than that of KDE-PSO-BP, KDE-PSO-RNN, ND-LSTM, KDE-LSTM, Beta-PSO-LSTM and KDE-GA-LSTM. Therefore, it can be demonstrated that the KDE-PSO-LSTM model can provide reliable decision support to policy-makers for making the optimal water supplying management.

Interval forecasting for urban water demand using PSO optimized KDE distribution and LSTM neural networks