Uncovering Hidden Drivers in Dynamic Systems

Author: Denis Avetisyan

New research details a framework for mapping causal relationships in data that changes over time, even when key influencing factors are unknown.

This paper introduces a modular approach to causal discovery that handles non-stationarity, regime shifts, and spatio-temporal patterns via local independence testing and direct causal graph reconstruction.

Real-world data often violates the stationarity assumptions underlying many causal discovery algorithms, hindering reliable inference of dynamic relationships. This paper, ‘Context-Specific Causal Graph Discovery with Unobserved Contexts: Non-Stationarity, Regimes and Spatio-Temporal Patterns’, addresses this challenge by introducing a modular framework for discovering causal graphs in non-stationary data, focusing on localized independence testing and direct graph reconstruction. The approach systematically decomposes the problem, enabling the integration of existing constraint-based algorithms and facilitating improvements via related fields like change-point detection. Will this framework unlock more robust and scalable causal inference in complex, evolving systems, and how can its modularity be leveraged for future advancements?

The Shifting Sands of Causality

Many established techniques for discerning cause-and-effect relationships rely on the premise that these connections remain constant over time. However, real-world systems are rarely static; instead, they frequently exhibit dynamic behaviors where relationships between variables shift and evolve. Consequently, applying traditional causal discovery methods to non-stationary data can yield misleading or inaccurate results, as the algorithms incorrectly assume a fixed structure. For instance, a correlation observed between two variables at one point in time might not hold true later, leading to the identification of spurious causal links or the failure to detect genuine ones. This limitation is particularly problematic in fields dealing with complex temporal data, such as climate science, financial modeling, and neuroscience, where understanding evolving dependencies is crucial for effective prediction and intervention.

Assessing whether two variables are independent of each other, given a set of other variables, forms the bedrock of causal discovery. However, this seemingly straightforward task becomes profoundly difficult when dealing with non-stationary data – systems where relationships change over time. Traditional methods rely on statistical tests calibrated for stable conditions, and these tests frequently yield spurious independence or dependence claims when applied to dynamic systems. The core issue isn’t a flaw in the underlying theory, but rather the assumption of consistent data distributions; as relationships drift, the conditional independence tests become unreliable, producing inaccurate estimations of the true underlying causal structure. This inability to correctly identify independence – or dependence – fundamentally limits the accuracy of any subsequent causal inference, hindering predictive modeling and effective intervention strategies in complex, evolving environments.

Accurate causal structure discovery is paramount for both predicting future states and effectively intervening in dynamic systems, yet limitations in current methodologies significantly impede this process. When relationships between variables shift over time – a common occurrence in biological, economic, and climate systems – traditional methods struggle to differentiate spurious correlations from genuine causal links. This inability to correctly identify these relationships leads to flawed predictive models and ineffective interventions, as actions based on inaccurate causal maps may yield unintended or even detrimental consequences. Consequently, progress in fields reliant on understanding and controlling complex dynamic systems demands innovative approaches that can robustly infer causal structures even amidst non-stationarity, allowing for more reliable predictions and targeted, effective interventions.

Adapting to the Flow of Information

The proposed causal discovery framework addresses the limitations of traditional constraint-based methods when applied to non-stationary time series data. These methods typically assume a fixed underlying causal structure and stable data distribution, which is often violated in real-world scenarios. This framework moves beyond these assumptions by explicitly modeling the dynamic nature of causal relationships. It operates by iteratively learning a Partially Oriented Adjacency Graph (PAG) from data, but incorporates mechanisms to detect and adapt to shifts in conditional independence relationships. This adaptation is achieved through a novel integration of state-space reconstruction and online independence testing, enabling the framework to identify and account for changes in the underlying data-generating process. The result is a more robust and accurate causal model in non-stationary environments compared to static approaches.

The framework utilizes statistical independence testing – specifically, assessing whether two variables are uncorrelated given a set of other variables – as its primary method for identifying potential causal relationships. However, recognizing that real-world data often exhibits non-stationarity, the framework moves beyond traditional, static independence tests. It incorporates adaptive mechanisms that continuously monitor data distributions and dynamically adjust the sensitivity of the independence tests. This adaptation is achieved through techniques such as windowing and weighting of recent data, allowing the framework to prioritize information from the current data regime and reduce the impact of outdated or irrelevant data points, thus improving the accuracy of causal discovery in dynamic environments.

State-space reconstruction, implemented via techniques such as time-delay embedding, allows the framework to capture the underlying dynamics of non-stationary time series data by creating a higher-dimensional representation of the system’s phase space. This reconstruction involves creating delayed copies of the single time series, effectively generating multiple variables that reflect the system’s history and potentially reveal distinct regimes or states. By performing independence testing on this reconstructed state space, rather than the original single time series, the framework can identify conditional dependencies that vary across these regimes. Specifically, the method utilizes Takens’ embedding theorem to determine an appropriate embedding dimension and delay time, ensuring sufficient information is retained to accurately represent the system’s attractor and differentiate between its operational states. This allows for a more robust and accurate causal discovery process in the presence of non-stationarity, as the independence tests are performed within the context of each identified regime.

Building on Established Foundations

The framework leverages established constraint-based causal discovery algorithms – specifically, the Peter-Clark Momentary Conditional Independence (PCMCI) method, the Fast Causal Inference (FCI) algorithm, and the PC-Stable algorithm – to infer causal relationships from observational data. These algorithms traditionally assume stationary data; however, this implementation extends their functionality to accommodate non-stationary time series. This is achieved through modifications to the conditional independence tests performed by each algorithm, allowing for the detection of causal links even when data distributions change over time. The integration of these well-validated methods provides a robust foundation for causal inference in dynamic systems, while the adaptations ensure applicability to a wider range of real-world datasets.

The PCMCI algorithm functions as a foundational element within the framework by leveraging conditional independence testing to analyze time series data. This process involves assessing the statistical independence between variables, given a set of conditioning variables, across multiple time lags. Specifically, PCMCI employs a sliding window approach and performs tests to determine if a direct causal relationship exists between two variables at a given lag, considering the influence of potential confounders. The algorithm iteratively assesses conditional independence, adjusting for increasingly larger sets of conditioning variables to mitigate spurious correlations and identify potential causal links in non-stationary time series data. The resulting output is a partially oriented directed acyclic graph (PDAG) representing the inferred causal relationships.

To improve performance and reliability in varying data conditions, the framework integrates PCMCI variants including PCMCI_Plus and LPCMCI. PCMCI_Plus introduces a sliding window approach to the time series data, allowing for adaptation to changes in conditional independence relationships over time and mitigating the impact of non-stationarity. LPCMCI, or Local PCMCI, further optimizes the process by performing conditional independence tests only on a subset of potentially relevant variables, reducing computational cost, particularly in high-dimensional datasets. These adaptations maintain the core principles of PCMCI while enhancing its ability to accurately infer causal structures in dynamic systems.

The Language of Conditional Independence

Conditional independence, in the context of causal discovery, refers to the absence of a statistical relationship between two variables given knowledge of a third variable. Formally, variables $X$ and $Y$ are conditionally independent given $Z$ if $P(X,Y|Z) = P(X|Z)P(Y|Z)$. Accurate assessment of these relationships is fundamental because a true causal link between two variables necessitates their dependence even after conditioning on all common causes and effects; conversely, the absence of such dependence suggests the absence of a direct causal connection. Misidentifying conditional independence can lead to the construction of incorrect causal graphs and flawed inferences about the underlying system, thus emphasizing the need for robust evaluation metrics and algorithms.

Partial correlation measures the linear dependence between two variables, given a set of control variables. Specifically, it is calculated as the correlation between two variables after removing the effects of the control variables via linear regression. A partial correlation of zero indicates conditional independence – meaning the two variables are uncorrelated given the control set. Within our framework, partial correlation serves as the primary statistical test for assessing conditional independence between nodes in a graphical model, enabling the identification of potential causal links and the structure learning process. The computation relies on the inverse of the covariance matrix, allowing for efficient evaluation across multiple variable sets.

Empirical evaluation of the proposed method on generated datasets indicates superior statistical convergence characteristics as sample size increases, relative to several benchmark algorithms. Specifically, the method achieves a statistically significant reduction in error rates with larger $n$, suggesting improved reliability in estimating conditional independence. Furthermore, the method demonstrates enhanced scalability with an increasing number of nodes, exhibiting a lower computational cost and faster processing times compared to baseline approaches as the dimensionality of the data increases. These results indicate the method’s potential for application to high-dimensional datasets commonly encountered in causal discovery tasks.

Mapping the Web of Causality

The core of this analytical framework lies in its generation of a graphical representation of inferred causal links between variables. This output isn’t merely a list of correlations, but a directed graph where nodes represent variables and edges signify hypothesized causal influences – demonstrating, for example, that a change in variable A is predicted to directly impact variable B. The resulting visual map allows researchers to move beyond simple associations and explore the underlying mechanisms driving observed phenomena, offering an intuitive way to understand complex systems. Each edge is assigned a confidence score, reflecting the strength of the evidence supporting that particular causal connection, and providing a nuanced understanding of the relationships within the data. This graphical approach streamlines the process of hypothesis generation and validation, transforming raw data into a readily interpretable model of cause and effect.

The resulting graphical representation of inferred causal links provides an intuitive means of understanding complex relationships between variables. This visualization transcends the limitations of statistical tables or complex equations, allowing researchers to readily identify direct and indirect dependencies. By mapping variables as nodes and causal effects as directed edges, the framework offers a clear depiction of the system’s structure, fostering easier interpretation of findings. This accessibility extends beyond specialist audiences; the visual format greatly simplifies communication of research outcomes to stakeholders, policymakers, and the general public, promoting a broader understanding of the underlying causal mechanisms at play.

Evaluations reveal this framework surpasses several established baseline methods in reconstructing accurate causal relationships, as demonstrated by a higher quality recovered union-graph. This improved performance doesn’t come at the cost of computational efficiency; the framework achieves a notably low runtime due to its strategic implementation of local and direct testing. By focusing analyses on immediate variable dependencies, rather than exhaustive comparisons, the approach significantly reduces processing demands, enabling quicker insights into complex systems and facilitating its application to larger datasets. The combination of accuracy and speed positions this framework as a valuable tool for researchers seeking to efficiently map and understand causal structures.

The presented framework prioritizes a parsimonious approach to causal discovery, directly reconstructing the graph from local independence tests. This aligns with a fundamental principle of efficient modeling; unnecessary complexity obscures understanding. As G. H. Hardy stated, “There is no excellence in mathematics without elegance.” The pursuit of clarity in discerning causal relationships, particularly within non-stationary systems exhibiting regimes and spatio-temporal patterns, demands a rejection of superfluous parameters. The modularity of the framework facilitates precisely this-a focus on essential connections, minimizing the ‘violence’ against attention inherent in overly complex models. The emphasis on local tests mirrors a decomposition into manageable components, echoing the beauty found in mathematical simplicity.

What Lies Ahead?

The presented framework, while an advance in managing non-stationarity through localized independence testing, merely shifts the inherent difficulty. The problem isn’t simply detecting change, but acknowledging the limits of any static representation of causality. To assume a graph, even a dynamically reconstructed one, fully captures a system is a convenient fiction. Future work must confront the possibility that the very notion of a stable ‘causal graph’ is inappropriate for genuinely complex systems.

Current approaches to regime detection remain largely heuristic. A more rigorous treatment would necessitate integrating principles from dynamical systems theory – not simply as a means of smoothing data, but as a foundational element of the causal model itself. The pursuit of ‘scalability’ often comes at the cost of theoretical depth; a modular framework is useful, but not if its components are themselves built on shaky assumptions.

Ultimately, the field must move beyond the question of how to discover causal graphs in non-stationary data, and address the more fundamental question of whether such discovery is meaningfully possible. Clarity, after all, is not about finding the right answer, but about precisely defining the question. Emotion is a side effect of structure, and a persistent belief in complete causal knowledge is, perhaps, the most persistent illusion of all.

Original article: https://arxiv.org/pdf/2511.21537.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/