Wavelet gated multiformer for groundwater time series forecasting.

Developing accurate models for groundwater control is paramount for planning and managing life-sustaining resources (water) from aquifer reservoirs. Significant progress has been made toward designing and employing deep-forecasting models to tackle the challenge of multivariate time-series forecasting. However, most models were initially taught only to optimize natural language processing and computer vision tasks. We propose the Wavelet Gated Multiformer, which combines the strength of a vanilla Transformer with the Wavelet Crossformer that employs inner wavelet cross-correlation blocks. The self-attention mechanism (Transformer) computes the relationship between inner time-series points, while the cross-correlation finds trending periodicity patterns. The multi-headed encoder is channeled through a mixing gate (linear combination) of sub-encoders (Transformer and Wavelet Crossformer) that output trending signatures to the decoder. This process improved the model's predictive capabilities, reducing Mean Absolute Error by 31.26 % compared to the second-best performing transformer-like models evaluated. We have also used the Multifractal Detrended Cross-Correlation Heatmaps (MF-DCCHM) to extract cyclical trends from pairs of stations across multifractal regimes by denoising the pair of signals with Daubechies wavelets. Our dataset was obtained from a network of eight wells for groundwater monitoring in Brazilian aquifers, six rainfall stations, eleven river flow stations, and three weather stations with atmospheric pressure, temperature, and humidity sensors.

Multi-fractal detrended cross-correlation heatmaps for time series analysis.

Complex systems in biology, climatology, medicine, and economy hold emergent properties such as non-linearity, adaptation, and self-organization. These emergent attributes can derive from large-scale relationships, connections, and interactive behavior despite not being apparent from their isolated components. It is possible to better comprehend complex systems by analyzing cross-correlations between time series. However, the accumulation of non-linear processes induces multiscale structures, therefore, a spectrum of power-law exponents (the fractal dimension) and distinct cyclical patterns. We propose the Multifractal detrended cross-correlation heatmaps (MF-DCCHM) based on the DCCA cross-correlation coefficients with sliding boxes, a systematic approach capable of mapping the relationships between fluctuations of signals on different scales and regimes. The MF-DCCHM uses the integrated series of magnitudes, sliding boxes with sizes of up to 5% of the entire series, and an average of DCCA coefficients on top of the heatmaps for the local analysis. The heatmaps have shown the same cyclical frequencies from the spectral analysis across different multifractal regimes. Our dataset is composed of sales and inventory from the Brazilian automotive sector and macroeconomic descriptors, namely the Gross Domestic Product (GDP) per capita, Nominal Exchange Rate (NER), and the Nominal Interest Rate (NIR) from the Central Bank of Brazil. Our results indicate cross-correlated patterns that can be directly compared with the power-law spectra for multiple regimes. We have also identified cyclical patterns of high intensities that coincide with the Brazilian presidential elections. The MF-DCCHM uncovers non-explicit cyclic patterns, quantifies the relations of two non-stationary signals (noise effect removed), and has outstanding potential for mapping cross-regime patterns in multiple domains.