A Matsuoka-Based GARMA Model for Hydrological Forecasting: Theory, Estimation, and Applications

TL;DR

The paper tackles the challenge of modeling time series valued in the unit interval $(0,1)$, where Gaussian ARMA models are inappropriate. It introduces the MARMA model, combining a Matsuoka distribution for the random component with an ARMA-like systematic structure and exogenous covariates, and develops partial maximum likelihood estimation with asymptotic theory. A novel bootstrap-based method is proposed to construct prediction intervals that account for dependence-structure uncertainty. Through Monte Carlo simulations and a real-data application to the Guarapiranga Reservoir UV, MARMA demonstrates competitive forecasting performance with fewer parameters and lower computational cost, providing a practical tool for hydrological forecasting with bounded data.

Abstract

Time series in natural sciences, such as hydrology and climatology, and other environmental applications, often consist of continuous observations constrained to the unit interval (0,1). Traditional Gaussian-based models fail to capture these bounds, requiring more flexible approaches. This paper introduces the Matsuoka Autoregressive Moving Average (MARMA) model, extending the GARMA framework by assuming a Matsuoka-distributed random component taking values in (0,1) and an ARMA-like systematic structure allowing for random time-dependent covariates. Parameter estimation is performed via partial maximum likelihood (PMLE), for which we present the asymptotic theory. It enables statistical inference, including confidence intervals and model selection. To construct prediction intervals, we propose a novel bootstrap-based method that accounts for dependence structure uncertainty. A comprehensive Monte Carlo simulation study assesses the finite sample performance of the proposed methodologies, while an application to forecasting the useful water volume of the Guarapiranga Reservoir in Brazil showcases their practical usefulness.

Figures (6)

• Figure 1: A typical example of a time series considered in the simulation study. The plot was generated considering $n=500$, $\alpha = 0.5$, $\beta=-0.5$, $\phi=0.2$, $\theta=-0.4$.
• Figure 2: Boxplots of the simulation results for all parameter for $\alpha=0.5$ (top) and $\alpha=1$ (bottom), with $\beta=-0.5$ fixed. Parameter $(\phi,\theta)$ are presented as follows. Cases 1 and 5: $(0.2,-0.4)$, cases 2 and 6: $(-0.8,0.2)$, cases 3 and 7: $(-0.4,-0.2)$, and cases 4 and 8: $(0.4,0.2)$. Vertical blue lines indicate the true parameter value
• Figure 3: Pairwise joint and marginal behavior of the estimated values for $\alpha=1$, $\beta=-0.5$, $\phi=-0.4$, $\theta=-0.2$. Solid lines in the scatter plot represent the true values.
• Figure 4: Simulation results for the prediction confidence interval exercise. Each block presents the coverage of the bootstrap prediction confidence interval for a given $n$ (column) and $\delta$ (row) as a function of the forecasting horizon $h$. Model 1 refers to parameter $\phi=-0.8$ and $\theta=0.2$; Model 2: $\phi=-0.4$ and $\theta=-0.2$; Model 3: $\phi=0.2$ and $\theta=-0.4$; Model 4: $\phi=0.2$ and $\theta=0.4$. The nominal level is indicated by the blue lines.
• Figure 5: Guarapiranga UV data, Brazil: (a) the observed time series and the corresponding (b) autocorrelation function (ACF), (c) partial autocorrelation (PACF) function and (d) seasonal plot.