Probabilistic Visibility Graphs Capture Hidden Patterns In Time Series Data

The analysis of complex time series data, crucial in fields ranging from neuroscience to financial modelling, often requires discerning patterns across multiple temporal scales. Researchers are continually developing methods to capture both short-term fluctuations and long-range dependencies within these signals. Now, Roberto C. Sotero, from the University of Calgary, and Jose M. Sanchez-Bornot, from Ulster University, alongside their colleagues, present a novel approach to this challenge in their article, “Tunnelling Through Time Series: A Probabilistic Visibility Graph for Local and Global Pattern Discovery”. Their work introduces the Probabilistic Visibility Graph (PVG), a technique inspired by the quantum mechanical phenomenon of tunnelling, which allows for connections between data points even when direct visibility is obstructed, potentially revealing hidden relationships within complex datasets. The PVG’s application to electrocorticography (ECoG) data, recordings of electrical activity in the brain, demonstrates distinct network characteristics between resting and anaesthetised states, suggesting its utility in understanding neural dynamics.

Modern data analysis increasingly demands innovative techniques to extract meaningful patterns from complex, high-resolution time series data, particularly concerning the capture of both local and global dependencies. Researchers have developed the Probabilistic Visibility Graph (PVG), a novel approach inspired by the quantum mechanical phenomenon of tunnelling, designed to reveal hidden relationships within these datasets. The PVG extends the classical Visibility Graph (VG) by incorporating probabilistic connections between time points that are obstructed in the standard VG due to intervening values, offering a more comprehensive representation of the underlying dynamics.

Traditional visibility graph methods often struggle to capture long-range dependencies and subtle relationships obscured by intervening data points, limiting their applicability to complex systems. The PVG addresses this limitation by drawing inspiration from quantum tunnelling, where particles can traverse potential barriers even lacking the classical energy to do so. By analogy, the PVG allows connections between time points considered disconnected in a standard VG, effectively “tunnelling” through intervening data points to reveal hidden relationships.

The core principle of the PVG lies in its ability to probabilistically connect obstructed time points, assigning a connection probability based on the magnitude of intervening values. This is implemented by calculating a probability score for each potential connection, considering the height and width of the intervening “barrier” formed by the data points. Higher barriers result in lower connection probabilities, while lower barriers lead to higher probabilities, reflecting the intuitive notion that tunnelling through smaller obstacles is easier. This probabilistic approach captures both strong and weak relationships, providing a more complete picture of the underlying dynamics.

To demonstrate the PVG’s effectiveness, analyses were conducted using both synthetic and real-world data, focusing on amplitude-modulated signals and electrocorticography (ECoG) data recorded during rest and anesthesia. Amplitude-modulated signals served as a benchmark for evaluating the PVG’s ability to capture complex temporal patterns, while ECoG data, representing neural activity recorded directly from the brain, provided a challenging real-world application.

Key findings revealed distinct network properties between rest and anesthesia conditions, providing valuable insights into the neural mechanisms underlying brain function and dysfunction. During rest, the brain exhibited stronger “small-worldness” and “scale-free” behavior when analyzed with the PVG. Small-worldness describes a network characterised by high clustering and short path lengths, while scale-free networks exhibit a power-law degree distribution, indicating a few highly connected nodes and many sparsely connected ones.

In contrast, anesthesia resulted in a more randomized and less organized network structure, aligning with the known effects of anesthesia, which suppresses neural activity and disrupts normal communication patterns. A decrease in the number of hub nodes and a reduction in overall network connectivity were observed, suggesting that anesthesia effectively “disconnects” different brain regions.

The successful application of the PVG to ECoG data demonstrates its potential for unraveling the complexities of neural dynamics and offers new insights into the mechanisms underlying brain function and dysfunction. The PVG can serve as a powerful tool for investigating a wide range of neurological conditions, including epilepsy, Alzheimer’s disease, and stroke, providing valuable insights into underlying mechanisms and potential therapeutic targets.

The PVG’s versatility extends to various fields, including financial markets, climate patterns, and engineering systems, offering a powerful tool for analyzing complex time series data and uncovering hidden relationships. In financial markets, the PVG can identify patterns in stock prices and predict future market trends, while in climate science, it can help understand complex interactions between climate variables and predict future climate change scenarios. In engineering systems, the PVG can monitor the health of critical infrastructure and detect potential failures before they occur.

Researchers are actively exploring extensions to the PVG, including the development of adaptive algorithms that automatically adjust the connection probability based on data characteristics. The use of machine learning techniques to train the PVG to recognize specific patterns and predict future events is also under investigation.

The development of the PVG represents a step forward in time series analysis, providing a powerful tool for uncovering hidden relationships and gaining new insights into complex systems. The PVG has the potential to revolutionise a wide range of fields and its ability to capture long-range dependencies and subtle relationships makes it a valuable tool for researchers and practitioners alike, providing a new perspective on complex data and enabling more informed decision-making.