Addressing the incorrect usage of wavelet-based hydrological and water resources forecasting models for real-world applications with best practices and a new forecasting framework

Many recent studies propose wavelet-based hydrological and water resources forecasting models that have been incorrectly developed and that cannot properly be used for real-world forecasting problems. The incorrect development of these wavelet-based forecasting models occurs during wavelet decomposition (the process of extracting high- and low-frequency information into different sub-time series known as wavelet and scaling coefficients, respectively) and as a result introduces error into the forecast model inputs. The source of this error is due to the boundary condition that is associated with wavelet decomposition (and the wavelet and scaling coefficients) and is linked to three main issues: 1) using 'future data' (i.e., data from the future that is not available); 2) inappropriately selecting decomposition levels and wavelet filters; and 3) not carefully partitioning calibration and validation data. By not addressing these boundary conditions during wavelet decomposition, incorrectly developed wavelet-based forecasting models often result in much better performance than what is realistically achievable. We demonstrate that the discrete wavelet transform (DWT) multiresolution analysis (DWT-MRA) and maximal overlap discrete wavelet transform (MODWT) multiresolution analysis (MODWT-MRA), two commonly adopted wavelet decomposition methods used in the development of hydrological and water resources wavelet-based forecasting models, suffer from these boundary conditions and cannot be used properly for real-world forecasting. However, by following a proposed set of best (correct) practices, we show that the MODWT and à trous algorithm (AT) can be used to correctly forecast target (e.g., hydrological and water resources) processes in real-world scenarios. In this vein, we contribute a set of best practices, which focusses on deriving “boundary-corrected” wavelet and scaling coefficients from time series data, overcoming the boundary condition issues and providing hydrological and water resources modellers with a justified and coherent strategy for developing wavelet-based forecasting models that may be used for real-world forecasting problems. We coalesce these best practices into a new forecasting framework named Wavelet Data-Driven Forecasting Framework (WDDFF) that uses a combination of input variable selection and data-driven models to convert “boundary-corrected” wavelet and scaling coefficients into forecasts of a target process. Through a real-world urban water demand forecasting experiment in Montreal, Canada, we demonstrate the superiority of WDDFF against benchmark forecasting models such as (non-wavelet-based) random walk, multiple linear regression, extreme learning machine, and second-order Volterra series models. For the same case study, we also show how the WDDFF provides realistic and accurate forecasts while a recently proposed wavelet-based forecasting model that adopts the (invalid) MODWT-MRA for wavelet decomposition provides incorrect and unrealistic forecasts. We conclude that WDDFF is a useful tool for forecasting real-world hydrological and water resources processes that overcomes the limitations of many earlier wavelet-based forecasting methods and should be explored further for forecasting different processes such as streamflow, rainfall, evaporation, etc.