Generalized Kumaraswamy Regression Models for Bounded Data • gkwreg

Overview

The gkwreg package provides a comprehensive and computationally efficient framework for regression modeling of data restricted to the standard unit interval $(0, 1)$ , including proportions, rates, fractions, percentages, and bounded indices.

While Beta regression is the traditional approach for such data, gkwreg focuses on the Generalized Kumaraswamy (GKw) distribution family. This offers exceptional flexibility by encompassing seven important bounded distributions—including Beta and Kumaraswamy—as special or limiting cases.

The package enables full distributional regression, where all relevant parameters can be modeled as functions of covariates through flexible link functions. Maximum Likelihood estimation is performed efficiently via the Template Model Builder (TMB) framework, leveraging Automatic Differentiation (AD) for superior computational speed, numerical accuracy, and optimization stability.

Key Features

Flexible Distribution Hierarchy

Model bounded data using the 5-parameter Generalized Kumaraswamy (GKw) distribution and its seven nested subfamilies:

Distribution	Code	Parameters Modeled	Fixed Parameters	# Params
Generalized Kumaraswamy	`gkw`	$\alpha, \beta, \gamma, \delta, \lambda$	None	5
Beta-Kumaraswamy	`bkw`	$\alpha, \beta, \gamma, \delta$	$\lambda = 1$	4
Kumaraswamy-Kumaraswamy	`kkw`	$\alpha, \beta, \delta, \lambda$	$\gamma = 1$	4
Exponentiated Kumaraswamy	`ekw`	$\alpha, \beta, \lambda$	$\gamma = 1, \delta = 0$	3
McDonald (Beta Power)	`mc`	$\gamma, \delta, \lambda$	$\alpha = 1, \beta = 1$	3
Kumaraswamy	`kw`	$\alpha, \beta$	$\gamma = 1, \delta = 0, \lambda = 1$	2
Beta	`beta`	$\gamma, \delta$	$\alpha = 1, \beta = 1, \lambda = 1$	2

Each family offers distinct flexibility-parsimony tradeoffs. Start simple (kw or beta) and compare nested models using likelihood ratio tests or information criteria.

Advanced Regression Modeling

Extended formula syntax for parameter-specific linear predictors:
```
y ~ alpha_predictors | beta_predictors | gamma_predictors | delta_predictors | lambda_predictors
```
Example: yield ~ batch + temp | temp | 1 | temp | batch
Multiple link functions with optional scaling:
- Positive parameters ( $\alpha, \beta, \gamma, \lambda$ ): log (default), sqrt, inverse, identity
- Probability parameters ( $\delta \in (0,1)$ ): logit (default), probit, cloglog, cauchy
- Link scaling: Control transformation intensity via link_scale (useful for numerical stability)
Flexible control via gkw_control():
- Multiple optimizers: nlminb (default), BFGS, Nelder-Mead, CG, SANN, L-BFGS-B
- Custom starting values, convergence tolerances, iteration limits
- Fast fitting mode (disable Hessian computation for point estimates only)

Computational Efficiency

TMB-powered estimation: Compiled C++ templates with automatic differentiation.
- Exact gradients and Hessians (machine precision).
- 10-100× faster than numerical differentiation.
- Superior convergence stability.
Performance optimizations:
- Intelligent caching of intermediate calculations.
- Vectorized operations via Eigen/Armadillo.
- Memory-efficient for large datasets ( $n > 100,000$ ).

Comprehensive Inference Tools

Standard R Methods (familiar workflow):

summary(), print(), coef(), vcov(), confint()
logLik(), AIC(), BIC(), nobs()
fitted(), residuals(), predict()
anova() for nested model comparisons

Advanced Prediction (predict.gkwreg):

Multiple prediction types: "response", "parameter", "link", "variance", "density", "probability", "quantile".
Predictions under alternative distributional assumptions.

Sophisticated Diagnostics

6 Diagnostic Plot Types (plot.gkwreg):

Residuals vs Indices: Detect autocorrelation.
Cook’s Distance: Identify influential observations.
Leverage vs Fitted: Flag high-leverage points.
Residuals vs Linear Predictor: Check linearity/heteroscedasticity.
Half-Normal Plot with Envelope: Assess distributional adequacy.
Predicted vs Observed: Overall goodness-of-fit.

Installation

You can install the stable version from CRAN (once accepted) or the development version from GitHub.

# Install from CRAN (stable release):
install.packages("gkwreg")

# Install companion distribution package:
install.packages("gkwdist")

# Or install development versions from GitHub:
# install.packages("remotes")
remotes::install_github("evandeilton/gkwdist")
remotes::install_github("evandeilton/gkwreg")

Quick Start

Basic Regression

library(gkwreg)
library(gkwdist)

# Simulate data
set.seed(123)
n <- 500
x1 <- runif(n, -2, 2)
x2 <- rnorm(n)

# True parameters (log link)
alpha_true <- exp(0.8 + 0.3 * x1)
beta_true <- exp(1.2 - 0.2 * x2)

# Generate response from Kumaraswamy distribution
y <- rkw(n, alpha = alpha_true, beta = beta_true)
y <- pmax(pmin(y, 1 - 1e-7), 1e-7) # Ensure strict bounds
df <- data.frame(y = y, x1 = x1, x2 = x2)

# Fit Kumaraswamy regression
# Formula: alpha ~ x1, beta ~ x2 (intercept-only models also supported)
fit_kw <- gkwreg(y ~ x1 | x2, data = df, family = "kw")

# View results
summary(fit_kw)

Advanced Prediction

# Create prediction grid
newdata <- data.frame(
  x1 = seq(-2, 2, length.out = 100),
  x2 = 0
)

# Predict different quantities
pred_mean <- predict(fit_kw, newdata, type = "response") # E(Y|X)
pred_var <- predict(fit_kw, newdata, type = "variance") # Var(Y|X)
pred_alpha <- predict(fit_kw, newdata, type = "alpha") # α parameter
pred_params <- predict(fit_kw, newdata, type = "parameter") # All parameters

# Evaluate density at y = 0.5 for each observation
dens_values <- predict(fit_kw, newdata, type = "density", at = 0.5)

# Compute quantiles (10th, 50th, 90th percentiles)
quantiles <- predict(fit_kw, newdata,
  type = "quantile",
  at = c(0.1, 0.5, 0.9), elementwise = FALSE
)

Model Comparison

# Fit nested models
fit0 <- gkwreg(y ~ 1, data = df, family = "kw") # Null model
fit1 <- gkwreg(y ~ x1, data = df, family = "kw") # + x1
fit2 <- gkwreg(y ~ x1 | x2, data = df, family = "kw") # + x2 on beta

# Information criteria comparison
AIC(fit0, fit1, fit2)

# Likelihood ratio tests
anova(fit0, fit1, fit2, test = "Chisq")

Diagnostic Plots

# All diagnostic plots (base R graphics)
par(mfrow = c(3, 2))
plot(fit_kw, ask = FALSE)

# Select specific plots with customization
plot(fit_kw,
  which = c(2, 5, 6), # Cook's distance, Half-normal, Pred vs Obs
  type = "quantile", # Quantile residuals (recommended)
  caption = list(
    "2" = "Influential Points",
    "5" = "Distributional Check"
  ),
  nsim = 200, # More accurate envelope
  level = 0.95
) # 95% confidence

# Modern ggplot2 version with grid arrangement
plot(fit_kw,
  use_ggplot = TRUE,
  arrange_plots = TRUE,
  theme_fn = ggplot2::theme_bw
)

# Extract diagnostic data for custom analysis
diag <- plot(fit_kw, save_diagnostics = TRUE)
head(diag$data) # Access Cook's distance, leverage, residuals, etc.

Real Data Example

# Food Expenditure Data (proportion spent on food)
data("FoodExpenditure")
food <- FoodExpenditure
food$prop <- food$food / food$income

# Fit different distributional families
fit_beta <- gkwreg(prop ~ income + persons, data = food, family = "beta")
fit_kw <- gkwreg(prop ~ income + persons, data = food, family = "kw")
fit_ekw <- gkwreg(prop ~ income + persons, data = food, family = "ekw")

# Compare families
comparison <- data.frame(
  Family = c("Beta", "Kumaraswamy", "Exp. Kumaraswamy"),
  LogLik = c(logLik(fit_beta), logLik(fit_kw), logLik(fit_ekw)),
  AIC = c(AIC(fit_beta), AIC(fit_kw), AIC(fit_ekw)),
  BIC = c(BIC(fit_beta), BIC(fit_kw), BIC(fit_ekw))
)
print(comparison)

# Visualize best fit
best_fit <- fit_kw
plot(food$income, food$prop,
  xlab = "Income", ylab = "Food Proportion",
  main = "Food Expenditure Pattern", pch = 16, col = "gray40"
)
income_seq <- seq(min(food$income), max(food$income), length = 100)
pred_df <- data.frame(income = income_seq, persons = median(food$persons))
lines(income_seq, predict(best_fit, pred_df), col = "red", lwd = 2)

Advanced Features

Custom Optimization Control

library(gkwreg)
library(gkwdist)

# Simulate data
set.seed(123)
n <- 500
x <- runif(n, 1, 5)
x1 <- runif(n, -2, 2)
x2 <- rnorm(n)
x3 <- rnorm(n, 1, 4)

# True parameters (log link)
alpha_true <- exp(0.8 + 0.3 * x1)
beta_true <- exp(1.2 - 0.2 * x2)

# Generate response from Kumaraswamy distribution
y <- rkw(n, alpha = alpha_true, beta = beta_true)
y <- pmax(pmin(y, 1 - 1e-7), 1e-7) # Ensure strict bounds
df <- data.frame(y = y, x = x, x1 = x1, x2 = x2, x3 = x3)

# Default control (used automatically)
fit <- gkwreg(y ~ x1, data = df, family = "kw")

# Increase iterations for difficult problems
fit_robust <- gkwreg(y ~ x1,
  data = df, family = "kw",
  control = gkw_control(maxit = 1000, trace = 1)
)

# Try alternative optimizer
fit_bfgs <- gkwreg(y ~ x1,
  data = df, family = "kw",
  control = gkw_control(method = "BFGS")
)

# Fast fitting without standard errors (exploratory analysis)
fit_fast <- gkwreg(y ~ x1,
  data = df, family = "kw",
  control = gkw_control(hessian = FALSE)
)

# Custom starting values
fit_custom <- gkwreg(y ~ x1 + x2 | x3,
  data = df, family = "kw",
  control = gkw_control(
    start = list(
      alpha = c(0.5, 0.2, -0.1), # Intercept + 2 slopes
      beta  = c(1.0, 0.3) # Intercept + 1 slope
    )
  )
)

Link Functions and Scaling

# Default: log link for all parameters
fit_default <- gkwreg(y ~ x | x, data = df, family = "kw")

# Custom link functions per parameter
fit_links <- gkwreg(y ~ x | x,
  data = df, family = "kw",
  link = list(alpha = "sqrt", beta = "log")
)

# Link scaling (control transformation intensity)
# Larger scale = gentler transformation, smaller = steeper
fit_scaled <- gkwreg(y ~ x | x,
  data = df, family = "kw",
  link_scale = list(alpha = 5, beta = 15)
)

Working with Large Datasets

# Large dataset example
set.seed(456)
n_large <- 100000
x_large <- rnorm(n_large)
y_large <- rkw(n_large, alpha = exp(0.5 + 0.2 * x_large), beta = exp(1.0))
df_large <- data.frame(y = y_large, x = x_large)

# Fast fitting
fit_large <- gkwreg(y ~ x,
  data = df_large, family = "kw",
  control = gkw_control(hessian = FALSE)
)

# Diagnostic plots with sampling (much faster)
plot(fit_large,
  which = c(1, 2, 4, 6), # Skip computationally intensive plot 5
  sample_size = 5000
) # Use random sample of 5000 obs

Mathematical Background

The Generalized Kumaraswamy Distribution

The GKw distribution is a five-parameter family for variables on $(0, 1)$ with cumulative distribution function:

$F(x; \alpha, \beta, \gamma, \delta, \lambda) = I_{[1-(1-x^{\alpha})^{\beta}]^{\lambda}}(\gamma, \delta)$

where $I_z(a,b)$ is the regularized incomplete beta function. The probability density function is:

$$f(x; \alpha, \beta, \gamma, \delta, \lambda) = \frac{\lambda \alpha \beta x^{\alpha-1}}{B(\gamma, \delta)} (1-x^{\alpha})^{\beta-1} \left[1-(1-x^{\alpha})^{\beta}\right]^{\gamma\lambda-1} \\{1-\left[1-(1-x^{\alpha})^{\beta}\right]^{\lambda}\\}^{\delta-1}$$

Parameter Roles:

$\alpha, \beta$ : Control basic shape (inherited from Kumaraswamy)
$\gamma, \delta$ : Govern tail behavior and concentration
$\lambda$ : Additional flexibility for skewness and peaks

Regression Framework

For response $y_i \in (0,1)$ following a GKw family distribution, each parameter $\theta_{ip} \in \{\alpha_i, \beta_i, \gamma_i, \delta_i, \lambda_i\}$ depends on covariates via link functions:

$g_p(\theta_{ip}) = \eta_{ip} = \mathbf{x}_{ip}^\top \boldsymbol{\beta}_p$

Maximum likelihood estimation maximizes:

$\ell(\Theta; \mathbf{y}, \mathbf{X}) = \sum_{i=1}^{n} \log f(y_i; \theta_i(\Theta))$

TMB computes exact gradients $\nabla \ell$ and Hessian $\mathbf{H}$ via automatic differentiation, enabling fast and stable optimization.

Computational Engine: TMB

Template Model Builder (TMB) translates statistical models into optimized C++ code with automatic differentiation:

Under the Hood:

R Formula -> TMB C++ Template -> Automatic Differentiation -> 
Compiled Object -> Fast Optimization (nlminb/optim) -> 
Standard Errors (Hessian inversion)

Comparison with Other Packages

Feature	gkwreg	betareg	gamlss	brms
Distribution Family	GKw hierarchy (7)	Beta	100+	50+
Estimation	MLE (TMB/AD)	MLE	GAMLSS	Bayesian MCMC
Parameter Modeling	All parameters	Mean, precision	All parameters	All parameters
Speed (n=10k)	Fast (~1s)	Fast (~1s)	Moderate (~5s)	Slow (~5min)
Link Functions	9 options + scaling	Fixed	Many	Many
Optimization	`gkw_control()` (detailed)	Basic	Moderate	Extensive
Diagnostic Plots	6 types, dual graphics	4 types	Extensive	Via bayesplot
Dependencies	gkwdist, TMB, Formula	Minimal	Many	Stan, many

When to use gkwreg:

Need flexible bounded distributions beyond Beta.
Large datasets requiring fast computation.
All parameters depend on covariates.
Frequentist inference preferred.

Documentation and Support

Reference Manual: help(package = "gkwreg")
Vignettes: browseVignettes("gkwreg")
Function Help: ?gkwreg, ?predict.gkwreg, ?plot.gkwreg, ?gkw_control
GitHub Issues: Report bugs or request features

Contributing

Contributions are welcome! Please see our Contributing Guidelines and Code of Conduct.

How to Contribute

Report bugs or request features via GitHub Issues
Submit pull requests for bug fixes or new features
Improve documentation or add examples
Share your use cases and feedback

Community Guidelines

This project follows a Code of Conduct. Please read it before contributing.

Citation

If you use gkwreg in your research, please cite:

citation("gkwreg")

Or use the BibTeX entry:

@Manual{,
  title = {gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data},
  author = {José Evandeilton Lopes},
  year = {2025},
  note = {R package version 2.1.4},
  url = {https://github.com/evandeilton/gkwreg},
}

License

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

## Author and Maintainer

José Evandeilton Lopes (Lopes, J. E.) | evandeilton@gmail.com | GitHub | ORCID

LEG - Laboratório de Estatística e Geoinformação | UFPR - Universidade Federal do Paraná, Brazil

gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data