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AeroMig: Multi-modal Inverse-Gamma modeling for aerosol particle number size distributions

Abdur Rahman, Juha Kangasluoma, Santtu Mikkonen, Tareq Hussein, Tuukka Petäjä, Sasu Tarkoma, Martha Arbayani Zaidan

arXiv:2610.06380v1cs.CE

TL;DR

Measured PNSDs are noisy, skewed, heavy-tailed, and multi-modal, making reliable fitting difficult. The paper models them with mixtures of Inverse-Gamma components and reports improved fitting accuracy, lower error, and greater computational efficiency than AeroMG across three monitoring stations.

  • Problem

    Measured PNSDs are noisy, highly skewed, and shaped by overlapping physical processes, while nonlinear fitting can produce local minima and parameter-identifiability problems.

    Source: §I · p. 1

  • Method

    AeroMiG represents PNSDs as mixtures of Inverse-Gamma components to model skewed, heavy-tailed aerosol size structures.

    Source: §I · p. 2, §V · p. 12

  • Results

    Across three monitoring stations, AeroMiG consistently improves fitting accuracy, reduces error, and enhances computational efficiency relative to AeroMG.

    Source: 1) Predictive Accuracies · p. 7, 1) Predictive Accuracies across NPF… · p. 8, §V · p. 12

  • Takeaways & Limitations

    AeroMiG provides a flexible parametric representation of complex aerosol size distributions that is reported to be more faithful than conventional lognormal-based approaches.

    Source: §V · p. 12

Abstract

from arXiv · show

Aerosol particles play a critical role in air quality, human health, and climate processes, making their accurate characterization essential. One common way to represent aerosol populations is through particle number size distributions (PNSDs), which describe the concentration of particles across different mobility diameters. However, these distributions are often highly skewed, heavy-tailed, and multi-modal, posing significant challenges for conventional fitting approaches. In this study, we propose a robust parametric framework (AeroMiG) for modeling PNSDs using Inverse-Gamma mixture distributions. The method represents observed size distributions as a superposition of multiple Inverse-Gamma components, enabling flexible modeling of asymmetric and heavy-tailed structures that are not well captured by widely-used Log-Gaussian based approaches. The performance of the proposed method is evaluated using real-world data from three measurement stations. Results are assessed using multiple criteria, including predictive accuracies, computational efficiency, and a composite score summarizing overall model quality. Comparative analysis against a widely used Aerosol Multi-mode Log-Gaussian (AeroMG) demonstrates that the AeroMiG framework consistently achieves improved fitting accuracy, lower error, and faster computation. The proposed approach provides a reliable and computationally efficient solution for modeling complex aerosol size distributions, with potential applications in real-time air quality monitoring and data-driven environmental analysis.

I. INTRODUCTION

Measured PNSDs are noisy, skewed, and shaped by overlapping aerosol processes, making them difficult to interpret and fit reliably. AeroMiG is introduced as a flexible alternative, evaluated using station measurements and across NPF event groups.

I. INTRODUCTION

  • Noisy PNSDs shaped by overlapping nucleation, condensation, and coagulation processes are difficult to interpret directly and require parametric fitting to extract information about aerosol populations.Fitting is also challenging because its nonlinear, multiparameter structure can produce local minima and parameter-identifiability problems.Source: §I · p. 1
  • AeroMiG targets skewed, heavy-tailed, and asymmetric PNSDs, which conventional lognormal approaches may not capture reliably.The introduction identifies challenges with stable parameter estimation and physical interpretability in existing approaches.Source: §I · pp. 1–2
  • The study evaluates AeroMiG against AeroMG using real-world measurements across multiple stations and distinct NPF event groups.The stated configurations include a 3-mode setup spanning 15 years and a 5-mode setup covering full seasonal cycles.Source: §I · p. 2, A. Aerosol Measurement data · p. 2

A. Aerosol Measurement data

  • PNSD measurements in the study are obtained with a DMPS, which measures size-resolved aerosol number concentrations based on electrical mobility diameter.The system classifies particle sizes with a differential mobility analyser and detects them with a condensation particle counter.Source: A. Aerosol Measurement data · p. 2
  • NPF days are grouped as Ia, Ib, II, or nonevent according to nucleation-mode clarity and the persistence of subsequent particle growth.Class I includes Ia and Ib; nonevent days have no detectable nucleation.Source: A. Aerosol Measurement data · p. 2

B. SMEAR Stations

The paper describes the SMEAR measurement sites and the AeroMG baseline, which models PNSDs as mixtures of lognormal modes and reduces the initial mode count.

B. SMEAR Stations

  • SMEAR I is a subarctic research station near the timberline, while SMEAR II is a non-urbanized boreal forest site northwest of Helsinki.SMEAR I is about 120 km north of the Arctic Circle; SMEAR II is about 190 km northwest of Helsinki.Source: 1) SMEAR I · p. 2, 2) SMEAR II · p. 3
  • SMEAR III is an urban-background site in Kumpula, Helsinki, complementing the subarctic and non-urbanized boreal measurement settings.Its surroundings include infrastructure, traffic corridors, and vegetated surfaces.Source: 1) SMEAR I · p. 2, 3) SMEAR III · p. 3, 2) SMEAR II · p. 3

A. AeroMG - Aerosol Multi-mode Log-Gaussian

  • AeroMG represents a measured PNSD as a superposition of K Log-Gaussian modes, each parameterized by geometric mean diameter, geometric standard deviation, and number concentration.Its fitting parameters are obtained by nonlinear optimization subject to physical constraints.Source: A. AeroMG - Aerosol Multi-mode… · p. 3
  • AeroMG fits the measured and modeled distributions by minimizing mean squared error across measurement bins.The fitting procedure begins with a maximum number of modes and then automatically reduces that number.Source: A. AeroMG - Aerosol Multi-mode… · p. 3

1) Elimination of Invalid Modes:

The mode-reduction procedure removes invalid modes and favors a simpler model when removing an overlapping mode changes fit error by less than a set tolerance.

1) Elimination of Invalid Modes:

  • Modes with negligible number concentration or physically invalid parameters are removed.The passage also identifies a small numerical threshold, ε.Source: 1) Elimination of Invalid Modes · p. 3

2) Mode Overlap Reduction:

  • Adjacent modes are tested for overlap, and a simpler model is preferred when the fit-error reduction from removing a mode is below TOLK.The reduction is applied iteratively until further mode removal would exceed the error tolerance.Source: 2) Mode Overlap Reduction · p. 3

B. AeroMiG- Aerosol Multi-mode Inverse-Gamma

AeroMiG uses an Inverse-Gamma distribution to represent positive, right-skewed particle diameters with a heavy tail. Its fitting procedure estimates parameters from measured size distributions.

B. AeroMiG- Aerosol Multi-mode Inverse-Gamma

  • DMPS diameters are positive and often right-skewed with a heavy tail, motivating an Inverse-Gamma model for particle sizes.The passage describes large particles as having low but non-negligible probability in many atmospheric and combustion environments.Source: B. AeroMiG- Aerosol Multi-mode… · p. 4
  • The Inverse-Gamma density is supported on positive diameters and has a heavy right tail.The paper derives it by taking the reciprocal of a Gamma-distributed variable.Source: B. AeroMiG- Aerosol Multi-mode… · p. 4
  • For measured size distributions, maximizing log-likelihood gives a closed-form estimate of β conditional on α, while α requires numerical solution.The α equation involves the digamma function.Source: B. AeroMiG- Aerosol Multi-mode… · p. 4

Interpretability of AeroMiG:

AeroMiG’s Inverse-Gamma parameters provide an interpretable description of aerosol size structure: α characterizes tail behavior, while β sets the characteristic diameter. Their combination summarizes DMPS aerosol structure, with the mean diameter finite only when α > 1.

Interpretability of AeroMiG:

  • β sets the characteristic diameter; larger β shifts the mode toward larger diameters, while smaller β is associated with freshly nucleated or combustion-derived ultrafine particles.The passage notes that a modal shift may reflect combined growth, secondary formation, and source influences.Source: Interpretability of AeroMiG · p. 4
  • α describes distribution sharpness and tail behavior, with smaller values associated with broader size variability and a greater probability of large particles.The passage associates α ≲ 2 with broad variability and more large, aged, or coagulated particles, while larger α indicates a more size-homogeneous population dominated by ultrafine particles.Source: Interpretability of AeroMiG · pp. 4–5
  • For α > 1, the expected mobility diameter is β α−1; for α ≤ 1, the mean diverges.Together, α and β offer a compact parameterization for comparing aerosol size structure across sources, air masses, and atmospheric processes.Source: Interpretability of AeroMiG · pp. 4–5

C. Performance Metrics

The paper evaluates fits using complementary measures of error, model complexity, and explanatory power. It transforms metrics to a common scale and combines them into a score that balances error minimization with robustness and explanatory power.

C. Performance Metrics

  • MSE summarizes average squared deviation across particle-diameter bins, but alone can favor over-parameterized fits and may not identify the optimal number of modes.The paper therefore pairs it with criteria that account for model complexity.Source: C. Performance Metrics · p. 5
  • AIC penalizes additional fitted parameters, with its likelihood term approximated as proportional to MSE under approximately Gaussian residuals.The approximation is used when MSE serves as a fitting proxy.Source: C. Performance Metrics · p. 5
  • BIC applies a stronger complexity penalty than AIC and tends to favor simpler models when data points are limited.The passage presents it as a safeguard against overfitting amid noise or measurement uncertainty in atmospheric data.Source: C. Performance Metrics · p. 5
  • R2 summarizes goodness-of-fit across diameter bins, while adjusted R2 corrects for fitted-parameter count and discourages artificial inflation as modes are added.Adjusted R2 favors parsimonious models with high explanatory power.Source: C. Performance Metrics · p. 5
  • The composite score transforms five metrics to a common scale and assigns equal total weight to error minimization and statistical robustness and explanatory power.MSE, AIC, and BIC receive 50% total weight; R2 and adjusted R2 receive the other 50%. The highest-scoring model is selected as the best-fit representation.Source: C. Performance Metrics · pp. 5–6

IV. RESULTS AND DISCUSSION

This section compares the proposed AeroMig model with AeroMG and discusses AeroMig’s practical advantages in real-world contexts.

IV. RESULTS AND DISCUSSION

  • The section presents a comparative analysis of AeroMig and AeroMG,.The passage identifies AeroMG, as the comparison model.Source: §IV · p. 6
  • The comparison concerns the proposed AeroMig model and AeroMG,.The passage does not specify the comparison outcomes here.Source: §IV · p. 6
  • The section also discusses AeroMig’s practical advantages within various real-world contexts.No specific context or advantage is detailed in this passage.Source: §IV · p. 6

A. Results

The results subsection evaluates AeroMig and AeroMG using predictive accuracy and computational efficiency across PNSD data from all SMEAR stations.

A. Results

  • The evaluation uses PNSD data collected across all SMEAR stations.The passage gives no station-specific results.Source: A. Results · p. 6
  • Predictive accuracy is assessed using MSE, R2, and the composite score.The passage names these as evaluation metrics but provides no values here.Source: A. Results · p. 6
  • Computational efficiency is included among the evaluation criteria.The passage does not state the comparative outcome.Source: A. Results · p. 6

1) Predictive Accuracies:

Across SMEAR stations, AeroMiG outperforms AeroMG on MSE, R², and the normalized multi-metric profile; representative fits also track local fluctuations and peak amplitudes more closely.

1) Predictive Accuracies:

  • MSE and R² quantify fitting accuracy, while adjusted R², AIC, and BIC are included in the composite score to support generalization beyond pointwise fit.The results are presented in Figs. 4, 5, and 6, respectively.Source: 1) Predictive Accuracies · p. 6
  • In a representative three-mode fit, AeroMG overestimates the trough near 5 × 10^-8 m and underestimates the peak near 1.5 × 10^-7 m, whereas AeroMiG tracks local fluctuations and peak amplitudes.The AeroMiG total fit follows the measured data across the submicron spectrum.Source: 1) Predictive Accuracies · pp. 6–7, Figure 3 · p. 6
  • AeroMiG has a lower median MSE and a more localized error distribution across all three stations, while AeroMG’s high-error tail is most pronounced at SMEAR III.AeroMiG’s data-point density remains concentrated near the lower error bound at each station.Source: 1) Predictive Accuracies · p. 7, Figure 4 · p. 7
  • AeroMiG shows a higher median R² and narrower IQR at all three stations, while AeroMG has a longer low-R² tail reaching approximately 0.20.AeroMiG maintains a concentrated upper-bound density across stations.Source: 1) Predictive Accuracies · p. 7, Figure 5 · p. 7
  • AeroMiG encloses a larger normalized multi-metric surface area at all three stations, despite nearly identical peak R² and adjusted R² values.The models diverge across the other dimensions, including Score, sAIC, sBIC, and sMSE.Source: 1) Predictive Accuracies · p. 7

2) Computational Efficiency:

AeroMiG maintains stronger multi-criteria performance and lower error across stations and NPF event classes, while also reducing computation time. Its fitting retains fine-scale distribution structure, including under noisy conditions.

2) Computational Efficiency:

  • AeroMiG reduces median execution time by approximately 40% to 50% across SMEAR I, II, and III, with medians below 200 seconds.AeroMG medians are approximately 330 to 350 seconds, and AeroMiG’s computation-cost distribution is narrower across stations.Source: 2) Computational Efficiency · pp. 7–8, Figure 7 · p. 8

B. Discussion

  • AeroMiG better captures localized peaks and fine fluctuations, whereas AeroMG can over-smooth modes and struggle with polluted, noisy measurements.The AeroMG fitting difficulty is especially noted at SMEAR III; AeroMiG maintains lower error variance across diverse concentrations.Source: 1) Predictive Accuracies across NPF… · p. 8

1) Predictive Accuracies across NPF classes:

  • Across Ia, Ib, II, and nonevent cases at SMEAR I and II, AeroMiG maintains MSE ≈101 while AeroMG has broader, event-dependent error distributions.AeroMG’s distributions extend to MSE 108 for classes Ib, II, and nonevents.Source: 1) Predictive Accuracies across NPF… · p. 8, Figure 8 · p. 9
  • AeroMiG’s median R2 exceeds 0.95 across event classes at SMEAR I and II, while AeroMG has a downward tail reaching R2 = 0.0.The lower interquartile bound for AeroMG drops particularly in Class Ia at SMEAR II.Source: 1) Predictive Accuracies across NPF… · p. 9, Figure 9 · p. 9
  • AeroMiG balances fit quality, error, and parsimony better than AeroMG across the multi-metric profile, despite comparable peak R2 and adjusted R2.The profile includes scaled information criteria and error metrics, with AeroMiG showing a larger composite area.Source: Figure 6 · p. 8, 1) Predictive Accuracies across NPF… · p. 9
  • The speed reduction accompanies stronger fitting accuracy and structural stability, supporting high-throughput atmospheric data processing.The authors attribute the compressed execution-time envelope to a more efficient convergence pathway, while describing that attribution as likely.Source: 1) Predictive Accuracies across NPF… · p. 9

2) Interpretability:

AeroMiG’s inverse-gamma parameters can be translated into distribution moments and interpreted across nucleation event classes. Its fitted profiles reproduce class-specific shapes and variance patterns.

2) Interpretability:

  • AeroMiG maps its inverse-gamma shape and scale parameters to each mode’s mean and standard deviation, making the fitted representation physically interpretable.The paper presents these moments as proxies for atmospheric dynamics.Source: 2) Interpretability · p. 10
  • For Class Ia, the first mode has mean 1.17 × 10^-7 and standard deviation 5.50 × 10^-8, associated with strong nucleation profiles.The paper characterizes these low-mean, low-variance profiles as constrained atmospheric conditions.Source: 2) Interpretability · p. 10, Table I · p. 11
  • On nonevent days, the first mode’s mean rises to 2.64 × 10^-7 and its standard deviation to 9.36 × 10^-8, tracking a broader, more variable regime.The paper describes increases in mean and standard deviation across all three modes from stronger NPF signatures toward nonevents.Source: 2) Interpretability · p. 10, Table I · p. 11
  • Across NPF classes, fitted distributions reproduce distinct modal shapes and shifting peak diameters while preserving the observed 1σ variance envelopes.Class Ia profiles have sharp nucleation/Aitken peaks, while nonevent profiles are accumulation-mode biased.Source: Figure 10 · p. 10, 2) Interpretability · p. 11

3) Extended Capability:

At SMEAR II, AeroMiG’s five-mode evaluation shows a larger multi-objective performance profile and lower median computation time than AeroMG. The paper presents the approach as useful for real-time monitoring and large observational datasets, while identifying future work on adaptive component selection and broader environmental data.

3) Extended Capability:

  • The five-mode evaluation uses a synchronized year-long SMEAR II dataset assembled from PSM, DMPS, and APS measurements.The measurements span sub-3 nm nucleation tracking to an APS-derived maximum diameter of 15,074 nm.Source: 3) Extended Capability · pp. 11–12
  • In the five-mode SMEAR II evaluation, AeroMiG outperforms or matches AeroMG across six criteria and reduces median processing time by approximately 35%.AeroMiG’s median is approximately 1000 s, versus approximately 1550 s for AeroMG; Fig. 11 presents the performance and execution-time comparisons.Source: Figure 11 · p. 11, 3) Extended Capability · p. 12

C. Practical uses of the proposed method

  • The authors identify real-time air-quality monitoring and data-driven atmospheric modeling as potential applications of AeroMiG’s modal descriptors.They also describe possible use in automated aerosol classification and source attribution.Source: C. Practical uses of the proposed method · p. 12

V. CONCLUSION

  • The conclusion presents AeroMiG as a scalable approach for analyzing large observational datasets, with future work targeting adaptive component selection and dynamic, high-dimensional environmental data.The paper also describes transfer to complex heavy-tailed distributions in other scientific and engineering domains as a broader possibility.Source: §V · p. 12
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