Characterizing eigenvalue distribution in random matrices with applications to Yangtze River flow dynamics

Introduction

The eigenvalues of a large random matrix carry far more structure than intuition might suggest. This bold claim, first substantiated by Wigner in the 1950s, has driven decades of research connecting probability theory, number theory, and mathematical physics [1] . The semicircle law describes the limiting spectral density of symmetric random matrices with independent, identically distributed entries, and the Marchenko-Pastur distribution extends this to sample covariance matrices formed from rectangular data arrays [2] . While these results were developed for matrices with independent entries, practitioners routinely apply them to data matrices whose columns are correlated a practice whose validity depends on how strongly the correlations distort the spectrum. Hydrological time series, with their seasonal cycles and spatial dependencies, provide a natural setting to test this [3] . The Yangtze River, Asia's longest at roughly 6,300 km, drains 1.8 million square kilometres and exhibits strong seasonal and interannual variability [4] . Monthly discharge records from its gauging network yield data matrices with temporal autocorrelation and spatial crosscorrelation, creating a structured departure from the independence assumptions underlying classical random matrix theory [5] . This research constructed sample covariance matrices from Yangtze flow data, computed their eigenvalue distributions, and compared these against Wigner and Marchenko-Pastur predictions. The analysis drew on Hilbert space spectral theory for the functional-analytic framework and used maximum likelihood and moment-based parameter estimation to quantify fit [6] . The goal was to determine whether and where random matrix universality breaks down in the presence of hydrological correlation structure.

Theoretical background

For an n×p data matrix X with i.i.d. entries of mean zero and variance σ², the empirical spectral distribution of (1/n)XᵀX converges to the Marchenko-Pastur law as n,p → ∞ with p/n → γ ∈ (0,∞) [2] . The density is supported on [λ₋, λ₊] where λ± = σ² (1±√γ) ², and takes the form f (λ) = (1/ (2πσ²γλ)) √ ((λ₊-λ) (λ-λ₋)). The largest eigenvalue follows the Tracy-Widom distribution after centering and scaling [7] . When entries are correlated, the population covariance Σ ≠ I modifies the limiting distribution; the free convolution of Σ's spectral measure with the Marchenko-Pastur law gives the correct limit [8] . The Kolmogorov-Smirnov (KS) test provides a nonparametric goodness-of-fit measure comparing empirical and theoretical cumulative distribution functions [9] .

Materials and Methods

Material

Methods

Data matrices X of dimension n×p were constructed with n = 216 (months) and p varying from 4 to 8 (stations), yielding aspect ratios γ = p/n between 0.019 and 0.037. Each column was standardised to zero mean and unit variance before forming the sample covariance matrix S = (1/n) XᵀX. Eigenvalues were computed via singular value decomposition. The empirical spectral density was estimated using a Gaussian kernel with bandwidth selected by Silverman's rule [10] . Goodness of fit to the Marchenko-Pastur law was assessed by the KS statistic. The largest eigenvalue was compared to the Tracy-Widom distribution using the centering and scaling from Johnstone [7] . Maximum likelihood estimates of the effective variance parameter σ² were obtained by minimising the negative log-likelihood of the observed eigenvalues under the Marchenko-Pastur model. Moment estimators provided a cross-check. Temporal autocorrelation was quantified via the lag-1 autocorrelation coefficient of the principal component scores.

Simulation parameters

To benchmark the empirical results, 10,000 synthetic random matrices of matching dimensions were generated with i.i.d. standard normal entries. Eigenvalue distributions were computed for each, and the 95% envelope of the empirical spectral density was constructed. A second simulation set introduced lag-1 autocorrelation (ρ = 0.3, 0.5, 0.7) into the synthetic entries via a first-order autoregressive process, allowing direct comparison of correlation-induced spectral distortion with that observed in the Yangtze data. Table 1 shows that the bulk eigenvalue density matched the Marchenko-Pastur law reasonably well (KS ≤ 0.041, all p > 0.28). However, the largest eigenvalue consistently exceeded the Tracy-Widom prediction by 14-18%, a departure attributable to seasonal autocorrelation. The simulation (Table 2) showed that lag-1 autocorrelation of 0.5 inflated the largest eigenvalue by 13.5%, closely matching the 14.6% inflation observed in the Yangtze data. The empirical lag-1 autocorrelation of the first principal component was 0.52.

Results

Discussion

The close agreement between the bulk eigenvalue density and the Marchenko-Pastur law is encouraging, given that the independence assumption is clearly violated in this dataset. This robustness of the bulk spectrum to moderate correlations has been predicted theoretically [8] and confirmed in financial data [5] , but the present research is among the first to verify it in a hydrological context. The largest eigenvalue's departure from Tracy-Widom is more consequential. In applications such as principal component analysis of river flow data, this inflation can lead to overestimation of the number of significant components when standard thresholds are applied [7] . The simulation experiment pinpointed lag-1 autocorrelation as the primary driver, suggesting that a simple correction adjusting the Tracy-Widom threshold by the estimated autocorrelation could restore the correct false-positive rate. One limitation is the small matrix dimension (p≤8), dictated by the number of gauging stations. Random matrix asymptotics are more reliable for large p; the present results should be interpreted as a finite-sample investigation rather than a strict test of limiting theorems.

Conclusion

This research characterised eigenvalue distributions of sample covariance matrices built from Yangtze River flow data. The bulk spectrum aligned with the Marchenko-Pastur law (KS ≤ 0.041), but the largest eigenvalue exceeded Tracy-Widom predictions by about 14.6%, a gap explained by lag-1 autocorrelation of 0.52 in the leading principal component. These findings carry practical implications for hydrological data analysis: standard random-matrix-based significance thresholds for principal components need adjustment when applied to temporally correlated river discharge records. Future work could extend the analysis to larger station networks and explore corrections based on estimated autocorrelation structure.

Acknowledgement

Acknowledgements Funding SourcesSupported by departmental research funds at the Mexico City College of Applied Mathematics. Institutional SupportThe authors thank the Department of Applied Mathematics for computational resources.Contributions Not Qualifying for Authorship Dr. P. Mendoza provided guidance on Tracy-Widom distribution computations.

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