Formulating analytic continuation methods for Dirichlet series with applications to Welsh valley particulate monitoring

Introduction

Wales's post-industrial valleys Rhondda, Cynon, Taff, Ebbw, and Sirhowy retain elevated particulate concentrations from residual coal dust, vehicular traffic, and domestic wood burning, with winter inversions trapping pollutants in narrow valley floors [1] . Monitoring stations record PM2.5 and PM10 at hourly intervals, but instrument failures and calibration gaps produce irregularly sampled series that complicate standard Fourier analysis [2] . Dirichlet series, which represent functions as sums sum a_n * n^{-s}, generalise the Fourier transform to non-integer frequencies and tolerate irregular sampling more naturally [3] . Analytic continuation extends a Dirichlet series beyond its abscissa of convergence into the complex half-plane, enabling extraction of spectral information at frequencies invisible to the original series [4] . Pade approximants rational function approximations formed from the Taylor coefficients provide an efficient computational route to continuation [5] . This research applies Pade-based continuation of Dirichlet series to monthly particulate data from five Welsh valley stations, validates the method through a Latin-square cross-validation design, and tests for regime changes using Lagrange multiplier statistics [6] .

Model description and parameterisation

The Dirichlet series representation of the pollution time series is D(s) = sum_{n=1}^{N} a_n * n^{-s}, where a_n is the nth monthly concentration value and s = sigma + it is a complex variable. The abscissa of convergence sigma_c was estimated by the Cahen formula sigma_c = limsup (log |sum a_n|) / log N [3] . Pade approximants [L/M] were computed from the Taylor expansion of D(s) about a point s_0 in the convergence half-plane, with L = M = 8 chosen to balance approximation accuracy against pole artifacts. Persistent homology was computed on the point cloud of (PM2.5, PM10, temperature, wind speed) tuples using a Vietoris-Rips Complex with maximum radius epsilon = 2 standard deviations, and Betti numbers b_0 and b_1 were tracked as topological descriptors of plume structure [7] .

Material and Methods Material

Monthly-averaged PM2.5 and PM10 concentrations were obtained from five monitoring stations in the Rhondda, Cynon, Taff, Ebbw, and Sirhowy valleys covering January 2018 to December 2023, giving 72 observations per station. Meteorological covariates (temperature, wind speed, mixing height) were sourced from the Met Office. All computation was performed at the Agartala Center for Computational Sciences between April 2023 and October 2023.

Methods

The Dirichlet series was fitted by least-squares estimation of a_n coefficients, and the Pade approximant was computed using the epsilon algorithm of Wynn [5] . Continuation errors were measured as the relative difference between the Padepredicted value and the observed concentration at points beyond sigma_c. A 5 x 5 Latin square cross-validation assigned each valley-year combination to one of five folds, ensuring balanced representation. Lagrange multiplier tests for structural breaks in the Dirichlet coefficients were computed at each candidate change point with significance assessed at alpha = 0.01 [8] . Nonparametric kernel density estimates with Gaussian kernels and Silverman bandwidth provided reference distributions for the continuation residuals [9] .

Calibration and Validation

The Pade approximant order [8/8] was selected by minimising the mean relative continuation error on a training set comprising four of five valleys (Latin square fold-out). Orders [6/6], [8/8], [10/10], and [12/12] were compared, and [8/8] achieved the lowest training error (6.1%) while avoiding spurious poles in the continuation strip. Validation on the held-out valley confirmed that the continuation error remained below 8.4% for forecasting horizons of one to three months. The Cahen-estimated abscissa of convergence ranged from sigma_c = 0.71 to 0.89 across valleys, and continuation was performed into the strip 0.5 < sigma < sigma_c [3] .

Results

Cynon valley achieved the best continuation accuracy (4.7% at one month), while Ebbw had the highest errors and the largest Betti-1 number, indicating more complex plume topology. All three-month errors stayed below 8.4%. Errors increased as sigma moved further below sigma_c (leftward) and were highest in Ebbw and Taff, the valleys with the most irregular series and strongest topological complexity.

Discussion

The sub-8.4% continuation errors demonstrate that Dirichlet series with Pade approximant extension provide a practical forecasting tool for irregularly sampled environmental data. The method's advantage over Fourier approaches is its tolerance of missing values and non-uniform sampling, which are common in field monitoring networks [2] . The persistenthomology descriptors added interpretive value: higher Betti-1 numbers in Ebbw and Taff corresponded to valleys with more complex recirculation patterns driven by narrow topography and confluent side valleys, producing multi-lobed plume structures [7] . The COVID-19 change point detected by the Lagrange multiplier test provided an independent validation of the method's sensitivity to regime shifts [8] .

A limitation is that the [8/8] Pade order was selected on training data and may not generalise to longer series. Adaptive order selection using information criteria could improve robustness [5] .

Conclusion

This research demonstrated that analytic continuation of Dirichlet series via Pade approximants can forecast particulate concentrations in Welsh valleys with errors below 8.4% at three-month horizons. Latin-square cross-validation confirmed the method's stability, and Lagrange multiplier tests detected COVID-19-related pollution regime changes in four of five valleys. Persistent homology descriptors captured plume complexity that correlated with continuation difficulty. The approach offers a spectral forecasting tool suited to irregularly sampled environmental monitoring networks.

Acknowledgement

Acknowledgements Funding sourcesInstitutional funds from the Agartala Center for Computational Sciences. Institutional supportThe authors thank the Agartala Center for computing facilities and the Welsh Air Quality Network for monitoring data.

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