SPARTA -- Stochastic Periodic AutoRegressive To Anything (Tsoukalas et al., 2018)¶


Type	Parametric
Resolution	Monthly
Sites	Multisite

Overview¶

SPARTA generates cyclostationary (seasonally varying) synthetic timeseries with per-month parametric marginal distributions and month-to-month temporal persistence. The core idea parallels SMARTA: simulate an auxiliary Gaussian process, then map to the target domain via inverse CDFs. However, SPARTA uses a Periodic AutoRegressive PAR(1) model as the auxiliary Gaussian process instead of SMA, which naturally handles seasonal nonstationarity by allowing different autoregressive coefficients, marginal distributions, and cross-correlation structures at each month. Equivalent correlations are identified through the Nataf joint distribution model.

The Tsoukalas et al. (2018) framework supports any continuous marginal distribution. The current SynHydro implementation restricts marginal selection to gamma and lognormal families, chosen per (month, site) by BIC. Extending to additional families (e.g., kappa, Weibull, GEV) is a planned enhancement.

Notation¶

Symbol	Description
\(x_{s,t}^i\)	Observed process \(i\) at season (month) \(s\), year \(t\)
\(z_{s,t}^i\)	Auxiliary standard Gaussian process for site \(i\) at season \(s\), year \(t\)
\(F_{x_s^i}\)	Target marginal CDF of process \(i\) at season \(s\)
\(F_{x_s^i}^{-1}\)	Target marginal ICDF at season \(s\)
\(\Phi(\cdot)\)	Standard normal CDF
\(\rho_s^i\)	Target lag-1 season-to-season autocorrelation of process \(i\) at season \(s\)
\(\tilde{\rho}_s^i\)	Equivalent autocorrelation in the Gaussian domain
\(\rho_s^{i,j}\)	Target lag-0 cross-correlation between processes \(i\) and \(j\) at season \(s\)
\(\tilde{\rho}_s^{i,j}\)	Equivalent lag-0 cross-correlation
\(\tilde{A}_s\)	Diagonal autoregressive coefficient matrix at season \(s\)
\(\tilde{B}_s\)	Lower Cholesky factor of innovation covariance at season \(s\)
\(\tilde{C}_s\)	Equivalent lag-0 cross-correlation matrix at season \(s\)
\(\tilde{G}_s\)	Innovation covariance matrix at season \(s\)
\(m\)	Number of sites
\(S\)	Number of seasons (12 for monthly)
\(w_{s,t}\)	Standard normal i.i.d. innovation vector

Formulation¶

Nataf Mapping¶

Identical to SMARTA. Each auxiliary Gaussian variate is mapped to the target domain by:

\[ x_{s,t}^i = F_{x_s^i}^{-1}\!\left(\Phi(z_{s,t}^i)\right) \]

The key difference from SMARTA is that the ICDF varies by season \(s\) and site \(i\).

PAR(1)-N Auxiliary Model¶

The multivariate contemporaneous PAR(1) model in the Gaussian domain is:

\[ \mathbf{z}_s = \tilde{A}_s \mathbf{z}_{s-1} + \tilde{B}_s \mathbf{w}_s, \qquad \mathbf{w}_s \sim \mathcal{N}(\mathbf{0}, \mathbf{I}) \]

where: - \(\tilde{A}_s = \text{diag}(\tilde{\rho}_s^1, \ldots, \tilde{\rho}_s^m)\) is the diagonal matrix of equivalent autoregressive coefficients - \(\tilde{G}_s = \tilde{C}_s - \tilde{A}_s \tilde{C}_{s-1} \tilde{A}_s^\top\) is the innovation covariance - \(\tilde{B}_s \tilde{B}_s^\top = \tilde{G}_s\) (Cholesky decomposition)

For the univariate case (\(m = 1\)), this simplifies to:

\[ z_s = \tilde{\rho}_s \, z_{s-1} + \sqrt{1 - \tilde{\rho}_s^2} \, w_s \]

Equivalent Correlation Identification¶

The Nataf procedure is applied pairwise. For autocorrelations, the pair involves different marginals (month \(s\) and month \(s-1\)):

\[ \tilde{\rho}_s^i = \mathcal{F}^{-1}\!\left(\rho_s^i \mid F_{x_s^i}, F_{x_{s-1}^i}\right) \]

This requires \(S \times m\) Nataf inversions for autocorrelations, plus \(S \times m(m-1)/2\) for cross-correlations.

Parameter Estimation¶

Marginal fitting: Fit a distribution \(F_{x_s^i}\) for each (month, site) pair -- \(S \times m\) distributions.
Target autocorrelations: Compute empirical lag-1 season-to-season correlations for each site.
Target cross-correlations: Compute empirical lag-0 cross-correlations per month for each site pair.
Equivalent autocorrelations: Nataf inversion for each (site, month) pair.
Equivalent cross-correlations: Nataf inversion for each (site pair, month).
PAR(1) matrices: Build \(\tilde{A}_s\), compute \(\tilde{G}_s\), Cholesky decompose to get \(\tilde{B}_s\).

Synthesis Procedure¶

For each year \(t = 1, \ldots, T\) and month \(s = 1, \ldots, 12\), draw \(\mathbf{w}_{s,t} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_m)\).
Compute the auxiliary Gaussian series via PAR(1) recursion:

\[ \mathbf{z}_{s,t} = \tilde{A}_s \, \mathbf{z}_{s-1,t} + \tilde{B}_s \, \mathbf{w}_{s,t} \]

with wrapping: \(\mathbf{z}_{0,t} = \mathbf{z}_{12,t-1}\).

Map to the target domain using per-(month, site) ICDFs:

\[ x_{s,t}^i = F_{x_s^i}^{-1}\!\left(\Phi(z_{s,t}^i)\right) \]

Statistical Properties¶

SPARTA exactly preserves the target marginal distribution at each (month, site) by construction. Lag-1 season-to-season autocorrelations are preserved through the PAR(1) structure and Nataf inversion. Lag-0 cross-correlations across sites are preserved per month. Higher-order autocorrelations (lag > 1) are not explicitly modeled but emerge from the PAR(1) chain.

Limitations¶

Only lag-1 autocorrelation is explicitly modeled. Long-range dependence requires SMARTA or higher-order PAR.
Per-season Nataf inversion is computationally intensive: \(S \times (m + m(m-1)/2)\) inversions.
Innovation covariance \(\tilde{G}_s\) may not be positive-definite when sites have very different distributions or autocorrelation strengths across seasons.
The model assumes cyclostationarity (same seasonal pattern every year). Trends or regime shifts are not modeled.
Only lag-0 cross-correlations are explicitly preserved; lagged cross-correlations emerge implicitly.

References¶

Primary: Tsoukalas, I., Efstratiadis, A., & Makropoulos, C. (2018). Stochastic periodic autoregressive to anything (SPARTA): Modeling and simulation of cyclostationary processes with arbitrary marginal distributions. Water Resources Research, 54(1), 161-185. https://doi.org/10.1002/2017WR021394

See also: - Tsoukalas, I., Makropoulos, C., & Koutsoyiannis, D. (2018). Simulation of stochastic processes exhibiting any-range dependence and arbitrary marginal distributions. Water Resources Research, 54(11), 9484-9513. https://doi.org/10.1029/2017WR022462 - Cario, M. C., & Nelson, B. L. (1996). Autoregressive to anything: Time-series input processes for simulation. Operations Research Letters, 19(2), 51-58. - Tsoukalas, I., Kossieris, P., & Makropoulos, C. (2020). Simulation of non-Gaussian correlated random variables, stochastic processes and random fields: Introducing the anySim R-package for environmental applications and beyond. Water, 12(6), 1645. https://doi.org/10.3390/w12061645

Implementation: src/synhydro/methods/generation/parametric/sparta.py