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

Abstract

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.

Wavelet gated multiformer for groundwater time series forecasting

“…The Multifractal Detrended Cross-Correlation Heatmaps were strictly computed according to Paulo et al 62 . The procedure is to first obtain the long-range correlation through the Detrended Fluctuation Analysis (DFA) 69 , 73 .…”

“…Multifractal cross-correlation analysis has been used successfully in a range of studies involving time series pattern investigations, including economic trends 59 , 60 and climate 61 . This work also includes the multifractal analysis of cyclical patterns across multifractal regimes between pairs of time-series (stations) through the Multifractal Detrended Cross-Correlation Heatmaps (MF-DCCHM) 62 with Daubechies 4 Wavelets for high-frequency filtering.…”

Scale-Fractal Detrended Fluctuation Analysis for Fault Diagnosis of a Centrifugal Pump and a Reciprocating Compressor

“…In contrast, a multi-fractal time series represents a fractal system generalization that requires other descriptors apart from the fractal dimension to describe its dynamics. Consequently, their description can be performed using the singularity spectrum and the singularity exponent or Hurst exponent [ 43 ].…”

Wavelet Gated Multiformer for Groundwater Time Series Forecasting

“…The Multifractal Detrended Cross-Correlation Heatmaps were strictly computed according to Paulo et al 59 . The procedure is to first obtain the long-range correlation through the Detrended Fluctuation Analysis (DFA) 63,67 .…”

“…A multi-headed encoder with gate mixing sub-encoders (Transformer and Wavelet Crossformer) can give the decoder a more concise signature of trending signals and improve the model's predictive capabilities. This work also includes the multifractal analysis of cyclical patterns across multifractal regimes between pairs of time-series (stations) through the Multifractal Detrended Cross-Correlation Heatmaps (MF-DCCHM) 59 with Daubechies 4 Wavelets for high-frequency filtering.…”