Multi-fractal detrended cross-correlation heatmaps for time series analysis
- Paulo Roberto de Melo Barros Junior, (Co-Founder Oxaala)
- Kianny Lopes Bunge
- Vitor Hugo Serravalle Reis Rodrigues,
- Michell Thompson Ferreira Santiago, (Associate Engineer Oxaala)
- Dr. Euler Bentes dos Santos Marinho (Data Scientist Oxaala)
- Dr. Jose Luis Lima de Jesus Silva (Co-Founder Oxaala)
Scientific Reports, volume 12, Article number: 21655 (2022) Download
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.
It’s More Than Patterns
The time series analysis of pairs of signals and fluctuations can assess possible data-driven persistences (the tendency of a system to remain in the same trending state), anti-persistences (the tendency of a system to remain in opposing trending states), general trends and its cyclical patterns. Institutions can use these tools to improve efficiency and productivity throughout their supply chain by planning and executing short, medium, and long-term strategies to avoid disruptions.
For instance, measuring the cyclical fluctuations of sales and inventory could help prevent the underproduction of a particular product during a period. However, the common bottleneck is how to effectively track seasonal (up to 1 year) trends and cyclical patterns from time-evolving fluctuations.
However, the common bottleneck is how to effectively track seasonal (up to 1 year) trends and cyclical patterns from time-evolving fluctuations. Previous studies have shown the potential of cross-correlation analysis for decision-making across multiple fields. The detrended cross-correlation analysis (DCCA) has been used to investigate possible power laws over prices and volume changes in the stock market. The methods section clarifies that the DCCA generalizes the standard covariance to consider the long-range memories of two non-stationary signals. Besides, the DCCA has also been used in climatology to track the influence of seasonal patterns.
We have explored the analysis of the cross-correlation between sales and inventory from the Brazilian Automotive sector and growth descriptors such as the gross domestic product (GDP) per capita, the nominal exchange rate (NER), and the nominal interest rate (NIR) equivalent to the Special Settlement and Custody System (Selic). The following approaches were used for the analysis: (i) the detrended fluctuation analysis (DFA) for auto-correlation and detrended cross-correlation analysis (DCCA) to estimate possible trends and concurrent events that might affect decision-making processes, (ii) the cross-correlation coefficients (CCC) to verify the level of correlation for different periods, (iii) Discrete Fourier analysis to identify, distinguish, and characterize the various cycles, and (iv) MF-DCCHM to evaluate cyclic patterns from a pair of time-evolving signals. Our global analysis has shown anti-correlated patterns from fluctuations in sales and inventory, which can help identify scenarios for the automotive sector.
For more information, click here to read our article published at Scientific Reports, Nature Portfolio.
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