[Paper Review] A new method for the robust characterisation of pairwise statistical dependency between point processes

Antoine Messager, Nicos Georgiou+1 more|arXiv (Cornell University)|Apr 9, 2019

Functional Brain Connectivity StudiesNeuroscience37 references3 citations

TL;DR

This paper introduces a robust Z-score method for detecting pairwise statistical dependencies between point processes by analytically deriving the expected number and standard deviation of coincident events under independence. The approach uses a cumulative, lag-dependent statistic that converges to a normal distribution, enabling reliable Z-scores even with sparse or short-duration data, outperforming traditional correlation and coherence methods in challenging conditions.

ABSTRACT

The robust detection of statistical dependencies between the components of a complex system is a key step in gaining a network-based understanding of the system. Because of their simplicity and low computation cost, pairwise statistics are commonly used in a variety of fields. Those approaches, however, typically suffer from one or more limitations such as lack of confidence intervals requiring reliance on surrogate data, sensitivity to binning, sparsity of the signals, or short duration of the records. In this paper we develop a method for assessing pairwise dependencies in point processes that overcomes these challenges. Given two point processes $X$ and $Y$ each emitting a given number of events $m$ and $n$ in a fixed period of time $T$, we derive exact analytical expressions for the expected value and standard deviation of the number of pairs events $X_i,Y_j$ separated by a delay of less than $τ$ one should expect to observe if $X$ and $Y$ were i.i.d. uniform random variables. We prove that this statistic is normally distributed in the limit of large $T$, which enables the definition of a Z-score characterising the likelihood of the observed number of coincident events happening by chance. We numerically confirm the analytical results and show that the property of normality is robust in a wide range of experimental conditions. We then experimentally demonstrate the predictive power of the method using a noisy version of the common shock model. Our results show that our approach has excellent behaviour even in scenarios with low event density and/or when the recordings are short.

Motivation & Objective

• To address limitations in existing pairwise dependency detection methods, such as bin dependency, lack of confidence intervals, and sensitivity to short or sparse recordings.
• To develop a statistically rigorous method for assessing functional connectivity between point processes without relying on surrogate data or frequency-domain assumptions.
• To provide an analytically derived Z-score with exact expected value and standard deviation for coincident event counts under independence.
• To ensure robustness across diverse experimental conditions, including low event density and non-stationary processes.
• To enable reliable inference of statistical dependence even when traditional methods fail due to data sparsity or short duration.

Proposed method

• The method computes the number of event pairs from two point processes separated by less than a time lag τ, assuming independence.
• It derives exact analytical expressions for the expected value and standard deviation of this count under the null hypothesis of independent, uniformly distributed processes.
• A central limit theorem is proven, showing the statistic converges to a normal distribution with known parameters as T increases.
• The resulting Z-score quantifies the likelihood of observing the actual number of coincident events by chance.
• The method is validated numerically across a range of conditions, including sparse data and auto-correlated processes.
• The cumulative nature of the Z-score is leveraged to detect dependencies over multiple lags, with differencing used to infer lag-specific interaction patterns.

Experimental results

Research questions

• RQ1 Can a robust, analytical Z-score be derived for pairwise dependency detection between point processes without relying on surrogate data or binning?
• RQ2 How does the proposed method perform under conditions of low event density, short recording durations, or non-stationarity?
• RQ3 Is the distribution of the coincidence count statistic approximately normal under realistic conditions, enabling reliable Z-score interpretation?
• RQ4 To what extent does the cumulative nature of the Z-score affect the interpretation of dependency at specific lags?
• RQ5 Can the method detect delayed interactions and distinguish them from instantaneous dependencies using differencing of the Z-score?

Key findings

• The analytical derivation of the expected number of coincident events and its standard deviation under independence is exact and valid for all parameter values, including extreme cases like one event per process.
• The Z-score statistic converges to a normal distribution under the null hypothesis, enabling rigorous statistical inference without surrogate data.
• The method remains robust to sampling effects and moderate auto-correlation in the point processes, maintaining accurate Type I error rates.
• The cumulative nature of the Z-score means significant values at large lags reflect prior interactions at shorter lags, necessitating careful interpretation and potential differencing for lag-specific inference.
• The method successfully detects delayed interactions in a noisy common shock model, demonstrating predictive power even with sparse or short recordings.
• The approach outperforms traditional cross-correlation and coherence methods in scenarios with low event density or limited data duration, offering a reliable alternative for functional connectivity inference.