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Gaussian / Student-t Copula Generator (Chen et al. 2015; Pereira et al. 2017)
Type Parametric
Resolution Monthly
Sites Multisite
Class GaussianCopulaGenerator

Overview

Separates the marginal distributions from the dependence structure using Sklar's theorem. Per-(month, site) marginals are fitted parametrically (gamma or log-normal, selected by BIC) or empirically (Hazen plotting position). Temporal dependence is captured by a Periodic AR(1) model in normal-score space (Pereira et al. 2017), and spatial dependence among the PAR residuals is modelled by a Gaussian or Student-t copula.

The t-copula adds symmetric tail dependence via a degrees-of-freedom parameter, addressing the Gaussian copula's known zero tail dependence limitation (Tootoonchi et al. 2022).

Algorithm

Preprocessing

  1. Validate input data via _store_obs_data(); resample daily/weekly to monthly if needed.
  2. Optionally apply log(Q + offset) transform.

Fitting

  1. Marginal fitting (per calendar month, per site):
  2. Parametric: fit gamma and log-normal via MLE; select winner by BIC (Tootoonchi et al. 2022). Store shape/scale parameters.

  3. Empirical: fit NormalScoreTransform (Hazen plotting position).

  4. Probability Integral Transform (PIT): 

    u[t,s] = F_{m,s}(Q[t,s])      # CDF of fitted marginal
z[t,s] = Phi^{-1}(u[t,s])     # normal scores

  5. PAR(1) temporal model (Pereira et al. 2017, Section 3.1):

  6. For each monthly transition m-1 -> m and each site s, estimate lag-1 autocorrelation: 
    rho_{m,s} = corr(z[month=m, s], z[month=m-1, s])

  7. Compute PAR residuals (approximately i.i.d.): 

    e[t,s] = (z[t,s] - rho_{m,s} * z[t-1,s]) / sqrt(1 - rho_{m,s}^2)

  8. Copula correlation matrices (per month):

  9. Compute n_sites x n_sites Pearson correlation of PAR residuals e.
  10. Repair via spectral method to ensure positive definiteness.

  11. Cholesky decompose: R_m = L_m L_m^T.

  12. t-copula df estimation (if copula_type="t"):

  13. Grid search over df in [2, 50] maximising the multivariate-t log-likelihood on PAR residuals.

Generation

For each timestep t (calendar month m):

  1. Draw independent innovations:
  2. Gaussian: eps ~ N(0, I_{n_sites})

  3. t-copula: eps ~ t(df, I_{n_sites})

  4. Impose spatial correlation: 

    e[t] = L_m @ eps

  5. PAR(1) temporal recursion: 

    z[t,s] = rho_{m,s} * z[t-1,s] + sqrt(1 - rho_{m,s}^2) * e[t,s]

  6. Map to uniform:

  7. Gaussian: u[t,s] = Phi(z[t,s])

  8. t-copula: u[t,s] = T_df(z[t,s])

  9. Inverse marginal CDF: 

    Q[t,s] = F_{m,s}^{-1}(u[t,s])

  10. Enforce non-negativity.

Parameters

Parameter Type Default Description
copula_type str "gaussian" "gaussian" or "t". t-copula adds tail dependence.
marginal_method str "parametric" "parametric" (BIC-selected gamma/log-normal) or "empirical" (Hazen CDF).
log_transform bool False Apply log(Q + offset) before fitting.
offset float 1.0 Additive offset for log transform.
matrix_repair_method str "spectral" Method for repairing non-PSD correlation matrices.

Properties Preserved

  • Marginal distributions per (month, site) via parametric or empirical CDF
  • Spatial cross-site correlation via copula correlation matrix
  • Lag-1 temporal autocorrelation via PAR(1) in normal-score space
  • Seasonal cycle (implicit in per-month marginals and PAR parameters)

Not preserved:

  • Higher-order temporal dependence (only lag-1 via PAR(1))
  • Asymmetric tail dependence (Gaussian copula only; t-copula provides symmetric tail dependence)
  • Long-range dependence / Hurst exponent (no fractional differencing)

Limitations

  • The Gaussian copula has zero upper and lower tail dependence (Renard and Lang, 2007). Use copula_type="t" to model symmetric tail dependence.
  • PAR(1) captures only lag-1 persistence. Higher-order temporal structure (e.g., multi-month drought persistence) is not explicitly modelled.
  • Vine copulas (Czado and Nagler, 2022) offer more flexible dependence structures but are beyond the scope of this implementation.
  • With short records (< 20 years per month), parametric marginal fitting and copula parameter estimation may be unreliable.

References

Primary: Genest, C., and Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12(4), 347-368. https://doi.org/10.1061/(ASCE)1084-0699(2007)12:4(347)

Chen, L., Singh, V.P., Guo, S., Zhou, J., and Zhang, J. (2015). Copula-based method for multisite monthly and daily streamflow simulation. Journal of Hydrology, 526, 360-381. https://doi.org/10.1016/j.jhydrol.2015.05.018

Pereira, G.A.A., Veiga, A., Erhardt, T., and Czado, C. (2017). A periodic spatial vine copula model for multi-site streamflow simulation. Electric Power Systems Research, 152, 9-17. https://doi.org/10.1016/j.epsr.2017.06.017

See also: - Tootoonchi, F. et al. (2022). Copulas for hydroclimatic analysis: A practice-oriented overview. WIREs Water, 9(2), e1579. - Nelsen, R.B. (2006). An Introduction to Copulas. 2nd ed. Springer. - Sklar, A. (1959). Fonctions de repartition a n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229-231.

Implementation: src/synhydro/methods/generation/parametric/gaussian_copula.py Tests: tests/test_gaussian_copula_generator.py