grbkw: Gradient of the Negative Log-Likelihood for the BKw...
In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data
grbkw R Documentation
Gradient of the Negative Log-Likelihood for the BKw Distribution
Description
Computes the gradient vector (vector of first partial derivatives) of the
negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution
with parameters alpha (\alpha), beta (\beta),
gamma (\gamma), and delta (\delta). This distribution
is the special case of the Generalized Kumaraswamy (GKw) distribution where
\lambda = 1. The gradient is typically used in optimization algorithms
for maximum likelihood estimation.
Computes the gradient vector (vector of first partial derivatives) of the
negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution
with parameters alpha (\alpha), beta (\beta),
gamma (\gamma), and delta (\delta). This distribution
is the special case of the Generalized Kumaraswamy (GKw) distribution where
\lambda = 1. The gradient is typically used in optimization algorithms
for maximum likelihood estimation.
Usage
grbkw(par, data)
Arguments
par
A numeric vector of length 4 containing the distribution parameters
in the order: alpha (\alpha > 0), beta (\beta > 0),
gamma (\gamma > 0), delta (\delta \ge 0).
data
A numeric vector of observations. All values must be strictly
between 0 and 1 (exclusive).
Details
The components of the gradient vector of the negative log-likelihood
(-\nabla \ell(\theta | \mathbf{x})) for the BKw (\lambda=1) model are:
-\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i)
+ \sum_{i=1}^{n}\left[x_i^{\alpha} \ln(x_i) \left(\frac{\beta(\delta+1)-1}{v_i} -
\frac{(\gamma-1) \beta v_i^{\beta-1}}{w_i}\right)\right]
-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - (\delta+1)\sum_{i=1}^{n}\ln(v_i)
+ \sum_{i=1}^{n}\left[\frac{(\gamma-1) v_i^{\beta} \ln(v_i)}{w_i}\right]
-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(w_i)
-\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] -
\beta\sum_{i=1}^{n}\ln(v_i)
where:
-
v_i = 1 - x_i^{\alpha}
-
w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}
-
\psi(\cdot) is the digamma function (digamma).
These formulas represent the derivatives of -\ell(\theta), consistent with
minimizing the negative log-likelihood. They correspond to the general GKw
gradient (grgkw) components for \alpha, \beta, \gamma, \delta
evaluated at \lambda=1. Note that the component for \lambda is omitted.
Numerical stability is maintained through careful implementation.
The components of the gradient vector of the negative log-likelihood
(-\nabla \ell(\theta | \mathbf{x})) for the BKw (\lambda=1) model are:
-\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i)
+ \sum_{i=1}^{n}\left[x_i^{\alpha} \ln(x_i) \left(\frac{\beta(\delta+1)-1}{v_i} -
\frac{(\gamma-1) \beta v_i^{\beta-1}}{w_i}\right)\right]
-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - (\delta+1)\sum_{i=1}^{n}\ln(v_i)
+ \sum_{i=1}^{n}\left[\frac{(\gamma-1) v_i^{\beta} \ln(v_i)}{w_i}\right]
-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(w_i)
-\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] -
\beta\sum_{i=1}^{n}\ln(v_i)
where:
-
v_i = 1 - x_i^{\alpha}
-
w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}
-
\psi(\cdot) is the digamma function (digamma).
These formulas represent the derivatives of -\ell(\theta), consistent with
minimizing the negative log-likelihood. They correspond to the general GKw
gradient (grgkw) components for \alpha, \beta, \gamma, \delta
evaluated at \lambda=1. Note that the component for \lambda is omitted.
Numerical stability is maintained through careful implementation.
Value
Returns a numeric vector of length 4 containing the partial derivatives
of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with
respect to each parameter:
(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta).
Returns a vector of NaN if any parameter values are invalid according
to their constraints, or if any value in data is not in the
interval (0, 1).
Returns a numeric vector of length 4 containing the partial derivatives
of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with
respect to each parameter:
(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta).
Returns a vector of NaN if any parameter values are invalid according
to their constraints, or if any value in data is not in the
interval (0, 1).
Author(s)
Lopes, J. E.
References
Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized
distributions. Journal of Statistical Computation and Simulation,
Kumaraswamy, P. (1980). A generalized probability density function for
double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.
(Note: Specific gradient formulas might be derived or sourced from additional references).
Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized
distributions. Journal of Statistical Computation and Simulation,
Kumaraswamy, P. (1980). A generalized probability density function for
double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.
(Note: Specific gradient formulas might be derived or sourced from additional references).
See Also
grgkw (parent distribution gradient),
llbkw (negative log-likelihood for BKw),
hsbkw (Hessian for BKw, if available),
dbkw (density for BKw),
optim,
grad (for numerical gradient comparison),
digamma.
grgkw (parent distribution gradient),
llbkw (negative log-likelihood for BKw),
hsbkw (Hessian for BKw, if available),
dbkw (density for BKw),
optim,
grad (for numerical gradient comparison),
digamma.
Examples
# Assuming existence of rbkw, llbkw, grbkw, hsbkw functions for BKw
# Generate sample data
set.seed(123)
true_par_bkw <- c(alpha = 2, beta = 3, gamma = 1, delta = 0.5)
if (exists("rbkw")) {
sample_data_bkw <- rbkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4])
} else {
sample_data_bkw <- rgkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4], lambda = 1)
}
hist(sample_data_bkw, breaks = 20, main = "BKw(2, 3, 1, 0.5) Sample")
# --- Find MLE estimates ---
start_par_bkw <- c(1.5, 2.5, 0.8, 0.3)
mle_result_bkw <- stats::optim(par = start_par_bkw,
fn = llbkw,
gr = grbkw, # Use analytical gradient for BKw
method = "BFGS",
hessian = TRUE,
data = sample_data_bkw)
# --- Compare analytical gradient to numerical gradient ---
if (mle_result_bkw$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE)) {
mle_par_bkw <- mle_result_bkw$par
cat("\nComparing Gradients for BKw at MLE estimates:\n")
# Numerical gradient of llbkw
num_grad_bkw <- numDeriv::grad(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)
# Analytical gradient from grbkw
ana_grad_bkw <- grbkw(par = mle_par_bkw, data = sample_data_bkw)
cat("Numerical Gradient (BKw):\n")
print(num_grad_bkw)
cat("Analytical Gradient (BKw):\n")
print(ana_grad_bkw)
# Check differences
cat("Max absolute difference between BKw gradients:\n")
print(max(abs(num_grad_bkw - ana_grad_bkw)))
} else {
cat("\nSkipping BKw gradient comparison.\n")
}
# --- Optional: Compare with relevant components of GKw gradient ---
# Requires grgkw function
if (mle_result_bkw$convergence == 0 && exists("grgkw")) {
# Create 5-param vector for grgkw (insert lambda=1)
mle_par_gkw_equiv <- c(mle_par_bkw[1:4], lambda = 1.0)
ana_grad_gkw <- grgkw(par = mle_par_gkw_equiv, data = sample_data_bkw)
# Extract components corresponding to alpha, beta, gamma, delta
ana_grad_gkw_subset <- ana_grad_gkw[c(1, 2, 3, 4)]
cat("\nComparison with relevant components of GKw gradient:\n")
cat("Max absolute difference:\n")
print(max(abs(ana_grad_bkw - ana_grad_gkw_subset))) # Should be very small
}
# Assuming existence of rbkw, llbkw, grbkw, hsbkw functions for BKw
# Generate sample data
set.seed(123)
true_par_bkw <- c(alpha = 2, beta = 3, gamma = 1, delta = 0.5)
if (exists("rbkw")) {
sample_data_bkw <- rbkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4])
} else {
sample_data_bkw <- rgkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4], lambda = 1)
}
hist(sample_data_bkw, breaks = 20, main = "BKw(2, 3, 1, 0.5) Sample")
# --- Find MLE estimates ---
start_par_bkw <- c(1.5, 2.5, 0.8, 0.3)
mle_result_bkw <- stats::optim(par = start_par_bkw,
fn = llbkw,
gr = grbkw, # Use analytical gradient for BKw
method = "BFGS",
hessian = TRUE,
data = sample_data_bkw)
# --- Compare analytical gradient to numerical gradient ---
if (mle_result_bkw$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE)) {
mle_par_bkw <- mle_result_bkw$par
cat("\nComparing Gradients for BKw at MLE estimates:\n")
# Numerical gradient of llbkw
num_grad_bkw <- numDeriv::grad(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)
# Analytical gradient from grbkw
ana_grad_bkw <- grbkw(par = mle_par_bkw, data = sample_data_bkw)
cat("Numerical Gradient (BKw):\n")
print(num_grad_bkw)
cat("Analytical Gradient (BKw):\n")
print(ana_grad_bkw)
# Check differences
cat("Max absolute difference between BKw gradients:\n")
print(max(abs(num_grad_bkw - ana_grad_bkw)))
} else {
cat("\nSkipping BKw gradient comparison.\n")
}
# --- Optional: Compare with relevant components of GKw gradient ---
# Requires grgkw function
if (mle_result_bkw$convergence == 0 && exists("grgkw")) {
# Create 5-param vector for grgkw (insert lambda=1)
mle_par_gkw_equiv <- c(mle_par_bkw[1:4], lambda = 1.0)
ana_grad_gkw <- grgkw(par = mle_par_gkw_equiv, data = sample_data_bkw)
# Extract components corresponding to alpha, beta, gamma, delta
ana_grad_gkw_subset <- ana_grad_gkw[c(1, 2, 3, 4)]
cat("\nComparison with relevant components of GKw gradient:\n")
cat("Max absolute difference:\n")
print(max(abs(ana_grad_bkw - ana_grad_gkw_subset))) # Should be very small
}
gkwreg documentation built on April 16, 2025, 1:10 a.m.
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| grbkw | R Documentation |
Gradient of the Negative Log-Likelihood for the BKw Distribution
Description
Computes the gradient vector (vector of first partial derivatives) of the
negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution
with parameters alpha (\alpha), beta (\beta),
gamma (\gamma), and delta (\delta). This distribution
is the special case of the Generalized Kumaraswamy (GKw) distribution where
\lambda = 1. The gradient is typically used in optimization algorithms
for maximum likelihood estimation.
Computes the gradient vector (vector of first partial derivatives) of the
negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution
with parameters alpha (\alpha), beta (\beta),
gamma (\gamma), and delta (\delta). This distribution
is the special case of the Generalized Kumaraswamy (GKw) distribution where
\lambda = 1. The gradient is typically used in optimization algorithms
for maximum likelihood estimation.
Usage
grbkw(par, data)
Arguments
par |
A numeric vector of length 4 containing the distribution parameters
in the order: |
data |
A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive). |
Details
The components of the gradient vector of the negative log-likelihood
(-\nabla \ell(\theta | \mathbf{x})) for the BKw (\lambda=1) model are:
-\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i)
+ \sum_{i=1}^{n}\left[x_i^{\alpha} \ln(x_i) \left(\frac{\beta(\delta+1)-1}{v_i} -
\frac{(\gamma-1) \beta v_i^{\beta-1}}{w_i}\right)\right]
-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - (\delta+1)\sum_{i=1}^{n}\ln(v_i)
+ \sum_{i=1}^{n}\left[\frac{(\gamma-1) v_i^{\beta} \ln(v_i)}{w_i}\right]
-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(w_i)
-\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] -
\beta\sum_{i=1}^{n}\ln(v_i)
where:
-
v_i = 1 - x_i^{\alpha} -
w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta} -
\psi(\cdot)is the digamma function (digamma).
These formulas represent the derivatives of -\ell(\theta), consistent with
minimizing the negative log-likelihood. They correspond to the general GKw
gradient (grgkw) components for \alpha, \beta, \gamma, \delta
evaluated at \lambda=1. Note that the component for \lambda is omitted.
Numerical stability is maintained through careful implementation.
The components of the gradient vector of the negative log-likelihood
(-\nabla \ell(\theta | \mathbf{x})) for the BKw (\lambda=1) model are:
-\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i)
+ \sum_{i=1}^{n}\left[x_i^{\alpha} \ln(x_i) \left(\frac{\beta(\delta+1)-1}{v_i} -
\frac{(\gamma-1) \beta v_i^{\beta-1}}{w_i}\right)\right]
-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - (\delta+1)\sum_{i=1}^{n}\ln(v_i)
+ \sum_{i=1}^{n}\left[\frac{(\gamma-1) v_i^{\beta} \ln(v_i)}{w_i}\right]
-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(w_i)
-\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] -
\beta\sum_{i=1}^{n}\ln(v_i)
where:
-
v_i = 1 - x_i^{\alpha} -
w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta} -
\psi(\cdot)is the digamma function (digamma).
These formulas represent the derivatives of -\ell(\theta), consistent with
minimizing the negative log-likelihood. They correspond to the general GKw
gradient (grgkw) components for \alpha, \beta, \gamma, \delta
evaluated at \lambda=1. Note that the component for \lambda is omitted.
Numerical stability is maintained through careful implementation.
Value
Returns a numeric vector of length 4 containing the partial derivatives
of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with
respect to each parameter:
(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta).
Returns a vector of NaN if any parameter values are invalid according
to their constraints, or if any value in data is not in the
interval (0, 1).
Returns a numeric vector of length 4 containing the partial derivatives
of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with
respect to each parameter:
(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta).
Returns a vector of NaN if any parameter values are invalid according
to their constraints, or if any value in data is not in the
interval (0, 1).
Author(s)
Lopes, J. E.
References
Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,
Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.
(Note: Specific gradient formulas might be derived or sourced from additional references).
Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,
Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.
(Note: Specific gradient formulas might be derived or sourced from additional references).
See Also
grgkw (parent distribution gradient),
llbkw (negative log-likelihood for BKw),
hsbkw (Hessian for BKw, if available),
dbkw (density for BKw),
optim,
grad (for numerical gradient comparison),
digamma.
grgkw (parent distribution gradient),
llbkw (negative log-likelihood for BKw),
hsbkw (Hessian for BKw, if available),
dbkw (density for BKw),
optim,
grad (for numerical gradient comparison),
digamma.
Examples
# Assuming existence of rbkw, llbkw, grbkw, hsbkw functions for BKw
# Generate sample data
set.seed(123)
true_par_bkw <- c(alpha = 2, beta = 3, gamma = 1, delta = 0.5)
if (exists("rbkw")) {
sample_data_bkw <- rbkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4])
} else {
sample_data_bkw <- rgkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4], lambda = 1)
}
hist(sample_data_bkw, breaks = 20, main = "BKw(2, 3, 1, 0.5) Sample")
# --- Find MLE estimates ---
start_par_bkw <- c(1.5, 2.5, 0.8, 0.3)
mle_result_bkw <- stats::optim(par = start_par_bkw,
fn = llbkw,
gr = grbkw, # Use analytical gradient for BKw
method = "BFGS",
hessian = TRUE,
data = sample_data_bkw)
# --- Compare analytical gradient to numerical gradient ---
if (mle_result_bkw$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE)) {
mle_par_bkw <- mle_result_bkw$par
cat("\nComparing Gradients for BKw at MLE estimates:\n")
# Numerical gradient of llbkw
num_grad_bkw <- numDeriv::grad(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)
# Analytical gradient from grbkw
ana_grad_bkw <- grbkw(par = mle_par_bkw, data = sample_data_bkw)
cat("Numerical Gradient (BKw):\n")
print(num_grad_bkw)
cat("Analytical Gradient (BKw):\n")
print(ana_grad_bkw)
# Check differences
cat("Max absolute difference between BKw gradients:\n")
print(max(abs(num_grad_bkw - ana_grad_bkw)))
} else {
cat("\nSkipping BKw gradient comparison.\n")
}
# --- Optional: Compare with relevant components of GKw gradient ---
# Requires grgkw function
if (mle_result_bkw$convergence == 0 && exists("grgkw")) {
# Create 5-param vector for grgkw (insert lambda=1)
mle_par_gkw_equiv <- c(mle_par_bkw[1:4], lambda = 1.0)
ana_grad_gkw <- grgkw(par = mle_par_gkw_equiv, data = sample_data_bkw)
# Extract components corresponding to alpha, beta, gamma, delta
ana_grad_gkw_subset <- ana_grad_gkw[c(1, 2, 3, 4)]
cat("\nComparison with relevant components of GKw gradient:\n")
cat("Max absolute difference:\n")
print(max(abs(ana_grad_bkw - ana_grad_gkw_subset))) # Should be very small
}
# Assuming existence of rbkw, llbkw, grbkw, hsbkw functions for BKw
# Generate sample data
set.seed(123)
true_par_bkw <- c(alpha = 2, beta = 3, gamma = 1, delta = 0.5)
if (exists("rbkw")) {
sample_data_bkw <- rbkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4])
} else {
sample_data_bkw <- rgkw(100, alpha = true_par_bkw[1], beta = true_par_bkw[2],
gamma = true_par_bkw[3], delta = true_par_bkw[4], lambda = 1)
}
hist(sample_data_bkw, breaks = 20, main = "BKw(2, 3, 1, 0.5) Sample")
# --- Find MLE estimates ---
start_par_bkw <- c(1.5, 2.5, 0.8, 0.3)
mle_result_bkw <- stats::optim(par = start_par_bkw,
fn = llbkw,
gr = grbkw, # Use analytical gradient for BKw
method = "BFGS",
hessian = TRUE,
data = sample_data_bkw)
# --- Compare analytical gradient to numerical gradient ---
if (mle_result_bkw$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE)) {
mle_par_bkw <- mle_result_bkw$par
cat("\nComparing Gradients for BKw at MLE estimates:\n")
# Numerical gradient of llbkw
num_grad_bkw <- numDeriv::grad(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)
# Analytical gradient from grbkw
ana_grad_bkw <- grbkw(par = mle_par_bkw, data = sample_data_bkw)
cat("Numerical Gradient (BKw):\n")
print(num_grad_bkw)
cat("Analytical Gradient (BKw):\n")
print(ana_grad_bkw)
# Check differences
cat("Max absolute difference between BKw gradients:\n")
print(max(abs(num_grad_bkw - ana_grad_bkw)))
} else {
cat("\nSkipping BKw gradient comparison.\n")
}
# --- Optional: Compare with relevant components of GKw gradient ---
# Requires grgkw function
if (mle_result_bkw$convergence == 0 && exists("grgkw")) {
# Create 5-param vector for grgkw (insert lambda=1)
mle_par_gkw_equiv <- c(mle_par_bkw[1:4], lambda = 1.0)
ana_grad_gkw <- grgkw(par = mle_par_gkw_equiv, data = sample_data_bkw)
# Extract components corresponding to alpha, beta, gamma, delta
ana_grad_gkw_subset <- ana_grad_gkw[c(1, 2, 3, 4)]
cat("\nComparison with relevant components of GKw gradient:\n")
cat("Max absolute difference:\n")
print(max(abs(ana_grad_bkw - ana_grad_gkw_subset))) # Should be very small
}
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