Beta Autoregressive Moving Average Models

Fits Beta Autoregressive Moving Average (BARMA) models for time series data distributed in the standard unit interval (0, 1). The estimation is performed via the conditional maximum likelihood method using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm. A ridge penalization scheme is available to improve numerical stability of the estimation, as proposed by Cribari-Neto, Costa and Fonseca (2025) . The package includes tools for model fitting, diagnostic checking, and forecasting, along with two hydro-environmental datasets from Brazil. Based on the work of Rocha and Cribari-Neto (2009) and the associated erratum Rocha and Cribari-Neto (2017) . The original code was developed by Fabio M. Bayer.

betaARMA: Beta Autoregressive Moving Average Models

The betaARMA package provides a comprehensive suite of tools for fitting, diagnosing, and forecasting Beta Autoregressive Moving Average models. These models are specifically tailored for time series data that assume values in the standard unit interval (0, 1), such as rates, proportions, and relative indices.

Features

• Estimation: Supports both standard Conditional Maximum Likelihood Estimation (CMLE) and Penalized Conditional Maximum Likelihood Estimation (PCMLE) via ridge penalization to handle numerical instabilities and flat likelihood surfaces (Cribari-Neto, Costa, and Fonseca, 2025).
• Diagnostics: Highly optimized, fully vectorized implementations of the Ljung-Box and Monti portmanteau tests, correctly adjusted for the effective sample size and the number of estimated parameters.
• Forecasting: Built-in methods for out-of-sample predictions. True to the theoretical properties of the betaARMA model, all forecasts naturally remain strictly within the (0, 1) bounds.

What’s New in v1.2.0

• Ridge Penalization: New penalty argument in barma() enables Penalized CMLE (PCMLE) as proposed by Cribari-Neto, Costa, and Fonseca (2025). Addresses numerical instability caused by flat log-likelihood surfaces in challenging data structures.
• Advanced Optimization Backends: Users can now specify the optimization algorithm via the optimization argument in barma(). Supported methods include stats::optim (BFGS and L-BFGS-B with bounds) and lbfgs::lbfgs.
• Enhanced Model Summaries: The summary() method has been significantly upgraded. It now outputs significance levels (stars) for coefficients, explicitly details the optimization procedure and convergence status used, and reports the $\lambda$ penalty parameter when ridge penalization is applied.
• Diagnostic Plots: New plot() method for barma objects displays a comprehensive diagnostic grid.
• New Vignette: A new, comprehensive applied vignette is now included. It demonstrates the handling of numerical instability, the application of ridge penalization (PCMLE), and a full modeling workflow using monthly relative humidity data from Brasília, Brazil.

Table of Contents

• Project Motivation
• Foundational Literature
• Key Features
• Repository Structure
• Installation
• Getting Started
• Citation
• Contributing
• License
• Contact

Project Motivation

Modeling time series data bounded within the unit interval $(0, 1)$ presents unique challenges. Standard Gaussian methods (like ARIMA) are often inappropriate because they do not respect the natural boundaries of the data, potentially leading to fitted values or forecasts outside the admissible range.

The $\beta\text{ARMA}$ model addresses this by assuming the conditional distribution of the variable follows a Beta law. While the theoretical foundations exist, there has been a need for a modern, robust R package that:

1. Unifies the workflow for $\beta$AR, $\beta$MA, and $\beta$ARMA specifications.
2. Ensures stability through numerical optimization with analytic gradients, including ridge penalization for challenging data structures.
3. Provides a standard API consistent with popular time series tools (like forecast).

This project aims to fill that gap, serving as a go-to resource for hydrologists, economists, and data scientists working with bounded data.

Foundational Literature

The foundational methodology implemented in this package was published in TEST, journal in the fields of Statistics and Probability.

Key Publications:

• Rocha, A. V., & Cribari-Neto, F. (2009). “Beta autoregressive moving average models.” TEST, 18(3), 529–545. doi:10.1007/s11749-008-0112-z
• Rocha, A. V., & Cribari-Neto, F. (2017). “Erratum to: Beta autoregressive moving average models.” TEST, 26(2), 451–459. doi:10.1007/s11749-017-0528-4
• Cribari-Neto, F., Costa, E., & Fonseca, R. V. (2025). “Numerical stability enhancements in beta autoregressive moving average model estimation.” Brazilian Journal of Probability and Statistics, 39(4), 410–437. doi:10.1214/25-BJPS645 — Basis for the penalty argument in barma().
• Costa, E., Cribari-Neto, F., & Scher, V. T. (2024). “Test inferences and link function selection in dynamic beta modeling of seasonal hydro-environmental time series with temporary abnormal regimes.” Journal of Hydrology, 638, 131489.

Key Features

This package utilizes modern R development standards (S3 classes, roxygen2 documentation) to provide a seamless user experience.

• Unified Model Fitting: A single core function, barma(), handles any combination of AR and MA lags, exogenous regressors (xreg), and ridge penalization (penalty = TRUE).
• Estimation: Implements Conditional Maximum Likelihood Estimation (CMLE) and Penalized CMLE (PCMLE). Users can select the numerical optimization backend (BFGS, L-BFGS-B, or lbfgs) utilizing analytical gradients (score vectors) for improved convergence and speed.
• Correctness: Fully implements the corrections to the Information Matrix and Score Vector detailed in the 2017 Erratum, for all supported link functions (logit, cloglog, loglog, probit).
• Standard S3 Methods: Works with standard R functions:
  • summary(): Detailed coefficients, significance levels (stars), optimization tracking, and penalization metrics.
  • plot(): Six-panel diagnostic grid (observed vs. fitted, residuals over time, ACF, PACF, Ljung-Box p-values, Monti p-values) via which = ‘all’, or a four-panel default view.
  • forecast(): Multi-step-ahead point forecasts.
  • residuals(): Extraction of Pearson, quantile, raw, or link-scale residuals.
  • fitted(): Extraction of fitted values.
• Advanced Access: Core mathematical functions (loglik_barma(), score_vector_barma(), fim_barma()) are exported for researchers who need direct access to gradients or model internals.

Repository Structure

The repository is structured as a standard R package for clarity and reproducibility.

.
├── R/                  # Source code for all R functions.
├── data/               # Processed data objects (e.g., brasilia_ts).
├── data-raw/           # Scripts and raw CSV files used to generate the datasets.
├── doc/                # Compiled HTML vignettes and extracted R code.
├── inst/               # Additional package files (CITATION, REFERENCES.bib).
├── man/                # R package documentation files (generated by roxygen2).
├── vignettes/          # R Markdown source files for vignettes.
├── DESCRIPTION         # Package metadata and dependencies.
├── NAMESPACE           # Manages the package's namespace (generated by roxygen2).
├── NEWS.md             # Package changelog.
├── LICENSE             # MIT License file.
├── betaARMA.Rproj      # RStudio project file.
└── README.md           # This file.

Installation

CRAN (Recommended)

You can install the stable version of betaARMA directly from CRAN:

install.packages("betaARMA")

Development Version

You can install the development version from GitHub using the remotes package:

# install.packages("remotes")
remotes::install_github("Everton-da-Costa/betaARMA", dependencies = TRUE)

Usage and Vignettes

The package includes a detailed vignette demonstrating a complete modeling workflow, including how to handle convergence failures using the penalized estimator.

To explore the application of betaARMA modeling on empirical data, check out the main vignette:

vignette("humidity_brasilia", package = "betaARMA")

Application: Modeling Relative Humidity in Brasília

This vignette walks through a full analysis of monthly relative humidity in Brasília, Brazil. It highlights a specific lag specification where standard CMLE fails to converge. It demonstrates how applying the PCMLE (ridge-penalized) estimator achieves convergence, allows for valid model reduction, and ultimately produces a stable, well-specified model that passes all residual diagnostic tests.

Getting Started

Here is a quick example of how to simulate data, fit a model, and generate forecasts.

library(betaARMA)

# Example 1: Fit a BAR(1) model

# 1. Simulate data from a BAR(1) process
set.seed(2025)
y_sim_bar <- simu_barma(
  n = 250,
  alpha = 0.0,
  varphi = 0.6,
  phi = 25.0,
  link = "logit",
  freq = 12
)

# 2. Fit the model
fit_bar <- barma(y_sim_bar, ar = 1, link = "logit")

# 3. Inspect results
summary(fit_bar)
coef(fit_bar)

# 4. Diagnostic plots
plot(fit_bar)

# 5. Forecast the next 6 steps
pred_bar <- forecast(fit_bar, h = 6)
print(pred_bar)

# Example 2: Fit a BARMA(1,1) model using L-BFGS-B optimization

# 1. Simulate data from a BARMA(1,1) process
set.seed(2025)
y_sim_barma <- simu_barma(
    n = 250,
    alpha = 0.0,
    varphi = 0.6, 
    theta = 0.3, 
    phi = 25,
    link = "logit",
    freq = 12
)

# 2. Fit the model explicitly using L-BFGS-B
fit_barma <- barma(
    y_sim_barma, 
    ar = 1, 
    ma = 1, 
    link = "logit",
    optimization = list(method = "L-BFGS-B")
)

# 3. Inspect results
summary(fit_barma)
coef(fit_barma)

# 4. Forecast the next 6 steps
pred_barma <- forecast(fit_barma, h = 6)
print(pred_barma)

# Example 3: Ridge-penalized estimation (PCMLE)
# Useful when standard CMLE produces implausible estimates.

fit_penalized <- barma(y_sim_barma, ar = 1, ma = 1, link = "logit", penalty = TRUE)
summary(fit_penalized)

Citation

If you use this package in your research, please cite it as follows:

@Manual{,
  title = {betaARMA: Beta Autoregressive Moving Average Models},
  author = {Everton da Costa, Francisco Cribari-Neto and Vinícius T. Scher},
  year = {2026},
  note = {R package version 1.2.0},
  doi = {10.32614/CRAN.package.betaARMA},
  url = {https://CRAN.R-project.org/package=betaARMA}
}

If you use the ridge penalization (penalty = TRUE), please also cite:

@Article{Cribari_Costa_Fonseca2025,
  title   = {Numerical stability enhancements in beta autoregressive moving
             average model estimation},
  author  = {Francisco Cribari-Neto and Everton Costa and Rodney V. Fonseca},
  journal = {Brazilian Journal of Probability and Statistics},
  year    = {2025},
  volume  = {39},
  number  = {4},
  pages   = {410--437},
  doi     = {10.1214/25-BJPS645}
}

If your application involves hydro-environmental modeling, please consider citing:

@Article{Costa_Cribari_Scher2024,
  title     = {Test inferences and link function selection in dynamic beta 
               modeling of seasonal hydro-environmental time series with 
               temporary abnormal regimes},
  author    = {Costa, E. and Cribari-Neto, F. and Scher, V. T.},
  journal   = {Journal of Hydrology},
  year      = {2024},
  note      = {forthcoming}
}

Contributing

Contributions are welcome! If you find any issues or have suggestions for improvements, please open an issue or submit a pull request.

License

This project is licensed under the MIT License. See the LICENSE file for details.

Contact

For questions, suggestions, or issues related to the code, please contact:

Everton da Costa 📧 [email protected]

Code of Conduct

Please note that the betaARMA project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.