,
Finn Lindgren [aut] | Title: | Hierarchical Bayesian Dirichlet regression models using Integrated Nested Laplace Approximation |
|---|---|
| Description: | The R-package dirinla allows the user to fit models in the compositional data context. In particular, it allows fit Dirichlet regression models using the Integrated Nested Laplace Approximation (INLA) methodology. |
| Authors: | Joaquín Martínez-Minaya [aut, cre]
,
Finn Lindgren [aut] |
| Maintainer: | Joaquín Martínez-Minaya <[email protected]> |
| License: | GPL-2 |
| Version: | 1.0.5.9000 |
| Built: | 2024-11-21 01:24:53 UTC |
| Source: | https://github.com/inlabru-org/dirinla |
The R-package dirinla allows the user to fit models in the compositional data context. In particular, it allows fit Dirichlet regression models using the Integrated Nested Laplace Approximation (INLA) methodology.
See dirinlareg().
Tutorials and more information can be found at https://inlabru-org.github.io/dirinla/
Joaquín Martínez-Minaya [email protected] and Finn Lindgren [email protected]
Useful links:
Fast version of Matrix::.bdiag() – for the case of many (k x k) matrices:
Copyright (C) 2016 Martin Maechler, ETH Zurich
bdiag_m(lmat)bdiag_m(lmat)
lmat |
|
a sparse (Nk x Nk) matrix of class Matrix::dgCMatrix.
data_stack_dirich prepares the data using inla.stack from the package INLA.
data_stack_dirich(y, covariates, share = NULL, data, d, n)data_stack_dirich(y, covariates, share = NULL, data, d, n)
y |
Response variable in a matrix format. |
covariates |
String with the name of covariates. |
share |
Covariates to share in all the cateogries. TODO |
data |
Data.frame which contains all the covariates. |
d |
Number of categories. |
n |
Number of locations. |
Matrix A such as eta = A %*%x
Joaquín Martínez-Minaya [email protected]
n <- 100 d <- 4 V <- matrix(rnorm(4*n, 0, 1), ncol=4) V <- as.data.frame(V) names(V) <- c('v1', 'v2', 'v3', 'v4' ) covariates <- names(V) formula <- y ~ 1 + v1 + v2 | 1 + v1 | 1 + v1 names_cat <- formula_list(formula) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, n*d)), covariates = names_cat, share = NULL, data = V, d = d, n = n )n <- 100 d <- 4 V <- matrix(rnorm(4*n, 0, 1), ncol=4) V <- as.data.frame(V) names(V) <- c('v1', 'v2', 'v3', 'v4' ) covariates <- names(V) formula <- y ~ 1 + v1 + v2 | 1 + v1 | 1 + v1 names_cat <- formula_list(formula) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, n*d)), covariates = names_cat, share = NULL, data = V, d = d, n = n )
data_stack_dirich_formula prepares the data using inla.stack from the package INLA.
data_stack_dirich_formula(y, covariates, share = NULL, data, d, n)data_stack_dirich_formula(y, covariates, share = NULL, data, d, n)
y |
Response variable in a matrix format. |
covariates |
String with the name of covariates. |
share |
Covariates to share in all the cateogries. Not implemented yet. |
data |
Data.frame which contains all the covariates. |
d |
Number of categories. |
n |
Number of locations. |
List with two objects
Object of class inla.stack
Object with class formula
Joaquín Martínez-Minaya [email protected]
n <- 100 d <- 4 V <- matrix(rnorm(4*n, 0, 1), ncol=4) V <- as.data.frame(V) names(V) <- c('v1', 'v2', 'v3', 'v4' ) covariates <- names(V) formula <- y ~ 1 + v1 + v2 | 1 + v1 | 1 + v1 names_cat <- formula_list(formula) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, n*d)), covariates = names_cat, share = NULL, data = V, d = d, n = n )n <- 100 d <- 4 V <- matrix(rnorm(4*n, 0, 1), ncol=4) V <- as.data.frame(V) names(V) <- c('v1', 'v2', 'v3', 'v4' ) covariates <- names(V) formula <- y ~ 1 + v1 + v2 | 1 + v1 | 1 + v1 names_cat <- formula_list(formula) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, n*d)), covariates = names_cat, share = NULL, data = V, d = d, n = n )
digamma_red is the function digamma appropiate for really small values
digamma_red(x, ...)digamma_red(x, ...)
x |
Argument to applied the function digamma. |
... |
Rest of arguments used in the case of digamma functions. |
Result of applying digamma function
Joaquín Martínez-Minaya [email protected]
dirichlet_log_pos_x returns the -log posterior Dirichlet distribution asumming
multivariate normal prior with precision matrix Qx for elements of the latent field.
dirichlet_log_pos_x(A = A, x, Qx = Qx, y)dirichlet_log_pos_x(A = A, x, Qx = Qx, y)
A |
A matrix which links eta with the latent field, i.e., eta = A x. |
x |
Vector with the elements of the latent field, i.e., eta = A x. |
Qx |
Precision matrix for the priors of the latent field. |
y |
Vector with the response variable. |
A real value showing the -log posterior density is returned
Joaquín Martínez-Minaya [email protected]
dirinlareg Main function to do a Dirichlet Regression
dirinlareg( formula, y, data.cov, share = NULL, x0 = NULL, tol0 = 1e-05, tol1 = 0.1, k0 = 20, k1 = 5, a = 0.5, strategy = "ls-quasi-newton", prec = prec, verbose = FALSE, cores = 1, sim = 1000, prediction = FALSE, data.pred.cov = NULL, ... )dirinlareg( formula, y, data.cov, share = NULL, x0 = NULL, tol0 = 1e-05, tol1 = 0.1, k0 = 20, k1 = 5, a = 0.5, strategy = "ls-quasi-newton", prec = prec, verbose = FALSE, cores = 1, sim = 1000, prediction = FALSE, data.pred.cov = NULL, ... )
formula |
object of class formula indicating the response variable and the covariates of the Dirichlet regression |
y |
matrix containing the response variable |
data.cov |
data.frame with the covarites, only the covariates! |
share |
parameters to be fitted jointly. |
x0 |
initial optimization value |
tol0 |
tolerance |
tol1 |
tolerance for the gradient such that |grad| < tol1 * max(1, |f|) |
k0 |
number of iterations |
k1 |
number of iterations including the calling to inla |
a |
step length in the optimization algorithm |
strategy |
strategy to use to optimize |
prec |
precision for the prior of the fixed effects |
verbose |
if TRUE all the computing process is shown. Default is FALSE |
cores |
Number of cores for parallel computation. The package parallel is used. |
sim |
Simulations to call inla.posterior.sample and extract linear predictor, alphas and mus. The bigger it is, better is the approximation, but more computational time. |
prediction |
if TRUE we will predict with the new values of the covariates given in data.pred.cov. |
data.pred.cov |
data.frame with the values for the covariates where we want to predict. |
... |
arguments for the inla command |
model dirinlaregmodel object
Joaquín Martínez-Minaya [email protected]
if (dirinla_safe_inla() && requireNamespace("DirichletReg", quietly = TRUE)) { ### In this example, we show how to fit a model using the dirinla package ### ### --- 1. Loading the libraries --- #### ### --- 2. Simulating from a Dirichlet likelihood --- #### set.seed(1000) N <- 50 #number of data V <- as.data.frame(matrix(runif((4) * N, 0, 1), ncol = 4)) #Covariates names(V) <- paste0('v', 1:4) formula <- y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4 (names_cat <- formula_list(formula)) x <- c(-1.5, 1, -3, 1.5, 2, -3 , -1, 5) mus <- exp(x) / sum(exp(x)) C <- length(names_cat) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, N * C)), covariates = names_cat, data = V, d = C, n = N) A_construct <- data_stack_construct A_construct[1:8, ] eta <- A_construct %*% x alpha <- exp(eta) alpha <- matrix(alpha, ncol = C, byrow = TRUE) y_o <- DirichletReg::rdirichlet(N, alpha) colnames(y_o) <- paste0("y", 1:C) head(y_o) ### --- 3. Fitting the model --- #### y <- y_o model.inla <- dirinlareg( formula = y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4, y = y, data.cov = V, prec = 0.0001, verbose = FALSE) summary(model.inla) }if (dirinla_safe_inla() && requireNamespace("DirichletReg", quietly = TRUE)) { ### In this example, we show how to fit a model using the dirinla package ### ### --- 1. Loading the libraries --- #### ### --- 2. Simulating from a Dirichlet likelihood --- #### set.seed(1000) N <- 50 #number of data V <- as.data.frame(matrix(runif((4) * N, 0, 1), ncol = 4)) #Covariates names(V) <- paste0('v', 1:4) formula <- y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4 (names_cat <- formula_list(formula)) x <- c(-1.5, 1, -3, 1.5, 2, -3 , -1, 5) mus <- exp(x) / sum(exp(x)) C <- length(names_cat) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, N * C)), covariates = names_cat, data = V, d = C, n = N) A_construct <- data_stack_construct A_construct[1:8, ] eta <- A_construct %*% x alpha <- exp(eta) alpha <- matrix(alpha, ncol = C, byrow = TRUE) y_o <- DirichletReg::rdirichlet(N, alpha) colnames(y_o) <- paste0("y", 1:C) head(y_o) ### --- 3. Fitting the model --- #### y <- y_o model.inla <- dirinlareg( formula = y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4, y = y, data.cov = V, prec = 0.0001, verbose = FALSE) summary(model.inla) }
dirinlaregmodel is a new object class
dirinlaregmodel( call = NULL, formula = NULL, summary_fixed = NULL, marginals_fixed = NULL, summary_random = NULL, marginals_random = NULL, summary_hyperpar = NULL, marginals_hyperpar = NULL, summary_linear_predictor = NULL, marginals_linear_predictor = NULL, summary_alphas = NULL, marginals_alphas = NULL, summary_precision = NULL, marginals_precision = NULL, summary_means = NULL, marginals_means = NULL, summary_predictive_alphas = NULL, marginals_predictive_alphas = NULL, summary_predictive_means = NULL, marginals_predictive_means = NULL, summary_predictive_precision = NULL, marginals_predictive_precision = NULL, dic = NULL, waic = NULL, cpo = NULL, nobs = NULL, ncat = NULL, y = NULL, data.cov = NULL )dirinlaregmodel( call = NULL, formula = NULL, summary_fixed = NULL, marginals_fixed = NULL, summary_random = NULL, marginals_random = NULL, summary_hyperpar = NULL, marginals_hyperpar = NULL, summary_linear_predictor = NULL, marginals_linear_predictor = NULL, summary_alphas = NULL, marginals_alphas = NULL, summary_precision = NULL, marginals_precision = NULL, summary_means = NULL, marginals_means = NULL, summary_predictive_alphas = NULL, marginals_predictive_alphas = NULL, summary_predictive_means = NULL, marginals_predictive_means = NULL, summary_predictive_precision = NULL, marginals_predictive_precision = NULL, dic = NULL, waic = NULL, cpo = NULL, nobs = NULL, ncat = NULL, y = NULL, data.cov = NULL )
call |
The call of the function dirinlareg. |
formula |
Formula introduced by the user. |
summary_fixed |
List containing a summary of the marginal posterior distributions of the fixed effects. |
marginals_fixed |
List containing the marginal posterior distributions of the fixed effects. |
summary_random |
List containing a summary of the marginal posterior distributions of the random effects. |
marginals_random |
List containing the marginal posterior distributions of the random effects. |
summary_hyperpar |
List containing a summary of the marginal posterior distributions of the hyperparameters. |
marginals_hyperpar |
List containing the marginal posterior distributions of the hyperparameters. |
summary_linear_predictor |
List containing a summary of the marginal posterior distributions of the linear predictor. |
marginals_linear_predictor |
List containing the marginal posterior distributions of the linear predictor. |
summary_alphas |
List containing a summary of the marginal posterior distributions of the alphas. |
marginals_alphas |
List containing the marginal posterior distributions of the alphas. |
summary_precision |
List containing a summary of the marginal posterior distributions of the precision. |
marginals_precision |
List containing the marginal posterior distributions of the precision. |
summary_means |
List containing a summary of the marginal posterior distributions of the means. |
marginals_means |
List containing the marginal posterior distributions of the means. |
summary_predictive_alphas |
List containing a summary of the marginal posterior predictive distribution of the alphas. |
marginals_predictive_alphas |
List containing the marginal posterior predictive distribution of the alphas. |
summary_predictive_means |
List containing a summary fo the marginal posterior predictive distribution of the means. |
marginals_predictive_means |
List containing the marginal posterior predictive distribution of the means. |
summary_predictive_precision |
List containing a summary of the marginal posterior predictive distribution of the precision. |
marginals_predictive_precision |
List containing the marginal posterior predictive distribution of the precision. |
dic |
List containing the inla output for dic. |
waic |
List containing the inla output for waic. |
cpo |
List containing the inla output for cpo. |
nobs |
Number of observations. |
ncat |
Number of categories. |
y |
matrix containing the response variable |
data.cov |
data.frame with the covarites, only the covariates! |
object of list and dirinlaregmodel class.
extract_fixed is a function to extract summary and marginals distribution corresponding to the fixed effects
extract_fixed(inla_model, names_cat)extract_fixed(inla_model, names_cat)
inla_model |
Object of inla class. |
names_cat |
List generated with extract_formula. |
summary_fixed Summary of fixed effects for each category.
marginals_fixed Marginals for each parameter estimated.
Joaquín Martínez-Minaya [email protected]
extract_linear_predictor extracts the posterior distribution from the linear predictor
extract_linear_predictor( inla_model, n, d, Lk_eta, names_cat = names_cat, sim, verbose, cores )extract_linear_predictor( inla_model, n, d, Lk_eta, names_cat = names_cat, sim, verbose, cores )
inla_model |
An object of class inla. |
n |
Number of observations. |
d |
Number of categories. |
Lk_eta |
Cholesky decomposition of the Hessian matrix. |
names_cat |
List generated with extract_formula. |
sim |
simulations for the function inla.posterior.sample |
verbose |
if TRUE all the computing process is shown. Default is FALSE |
cores |
number of cores to be used in the computations |
summary_linear_predictor List containing a summary of the marginal posterior distributions of the linear predictor.
marginals_linear_predictor List containing simulations of marginal posterior distributions of the linear predictor.
summary_alphas List containing a summary of the marginal posterior distributions of the alphas.
marginals_alphas List containing simulations of the marginal posterior distributions of the alphas.
summary_precision List containing a summary of the marginal posterior distributions of the precision.
marginals_precision List containing simulations of the marginal posterior distributions of the precision.
summary_means List containing a summary of the marginal posterior distributions of the means.
marginals_means List containing the simulations of the marginal posterior distributions of the means.
Joaquín Martínez-Minaya [email protected]
formula_list reads the formula and generates a list with the name of the covariates used in each category
formula_list(form, y = NULL)formula_list(form, y = NULL)
form |
Object of class formula. |
y |
Matrix containing the response variable |
A list with the names of the variables used in each category.
Joaquín Martínez-Minaya [email protected]
formula <- y ~ 1 + v1 + v2 | -1 + v1 | 0 + v2 formula_list(formula)formula <- y ~ 1 + v1 + v2 | -1 + v1 | 0 + v2 formula_list(formula)
g0_vector_eta computes the gradient of -loglikelihood
g0_vector_eta_1(A = A, x, y)g0_vector_eta_1(A = A, x, y)
A |
Matrix which links eta with the latent field, i.e., eta = A x. |
x |
Vector with the elements of the latent field, i.e., eta = A x. |
y |
Vector with the response variable. |
A numeric vector with the gradient in eta.
Joaquín Martínez-Minaya [email protected]
H_matrix_eta_diag computes the expected Hessian in eta of -loglikelihood
H_matrix_eta_diag(eta, d, y)H_matrix_eta_diag(eta, d, y)
eta |
eta vector to compute the expected Hessian. |
d |
Dimension |
y |
Data corresponding to the i-individual |
Elements of the diagonal such as H = H0 + diag
Joaquín Martínez-Minaya [email protected]
H0_matrix_eta_x computes the expected Hessian in eta of -loglikelihood
H0_matrix_eta_x(eta, d, cores)H0_matrix_eta_x(eta, d, cores)
eta |
Linear predictor resulting of the product |
d |
Dimension. |
cores |
Number of cores for parallel computation. The package parallel is used. |
Expected Hessian in eta.
Joaquín Martínez-Minaya [email protected]
H0_matrix_eta_1 computes the expected Hessian in eta of -loglikelihood
H0_matrix_eta1(eta, d)H0_matrix_eta1(eta, d)
eta |
eta vector to compute the expected Hessian. |
d |
Dimension |
Expected Hessian in eta.
Joaquín Martínez-Minaya [email protected]
beta_mult_eta computes the log beta function in eta
log_beta_mult_eta(x)log_beta_mult_eta(x)
x |
Vector of elements. |
Numeric value.
Joaquín Martínez-Minaya [email protected]
look_for_mode_x computes optimization algorithms to find the mode of the posterior
look_for_mode_x( A = A, x0, tol0, tol1, k0, a = 0.5, y, d, n, strategy = "ls-quasi-newton", Qx, verbose, cores )look_for_mode_x( A = A, x0, tol0, tol1, k0, a = 0.5, y, d, n, strategy = "ls-quasi-newton", Qx, verbose, cores )
A |
Matrix which links latent field with linear predictor. |
x0 |
Initial optimization value. |
tol0 |
Tolerance for |x_new - x_old| and |f_new - f_old|. |
tol1 |
Tolerance for the gradient such that |grad| < tol1 * max(1, |f|) |
k0 |
Number of iterations. |
a |
Step length in the algorithm. |
y |
Response variable. Number of columns correspond to the number of categories. |
d |
Number of categories. |
n |
Number of individuals. |
strategy |
Strategy to use to optimize. |
Qx |
Prior precision matrix for the fixed effects. |
verbose |
By default is FALSE. If TRUE, the computation process is shown in the scream. |
cores |
Number of cores for parallel computation. The package parallel is used. |
x_hat Matrix with the x of the iterations.
Hk Hessian in eta. This Hessian is a combination of the real Hessian (when it is positive definite) and the expected Hessian (when the real Hessian is not positive definite).
gk Gradient in eta.
Lk Cholesky decomposition matrix.
eta Linear predictor.
z New pseudo observation conditioned to eta.
Joaquín Martínez-Minaya [email protected]
newton_x computes optimization algorithms to find the mode of the
posterior. Line search strategy with Armijo conditions is implemented
newton_x(A, x_hat, gk, Hk, a, Qx, strategy, y, d = d)newton_x(A, x_hat, gk, Hk, a, Qx, strategy, y, d = d)
A |
Matrix which links eta with the latent field, i.e., eta = A x |
x_hat |
Vector with the elements of the latent field, i.e., eta_hat = A x_hat |
gk |
Gradient in eta. |
Hk |
Hessian in eta. |
a |
Step length. |
Qx |
Precision matrix for the prior of the latent field. |
strategy |
Strategy to use to optimize. Now, line search strategy with quasi-newton algorithm is the only one avaliable. |
y |
Vector with the response variable |
d |
Number of categories. |
g0 : Gradient in x_hat_new. A numeric vector with the gradient in x_hat_new.
x_hat_new: New value of x after apply one iteration.
Joaquín Martínez-Minaya [email protected]
plot.dirinlaregmodel Method which plots a dirinlaregmodel x
## S3 method for class 'dirinlaregmodel' plot(x, ...)## S3 method for class 'dirinlaregmodel' plot(x, ...)
x |
Object of class dirinlaregmodel. |
... |
Other arguments. |
Plotting the posterior of the fixed effects.
Joaquín Martínez-Minaya [email protected]
predict.dirinlaregmodel computes the posterior predictive distribution for some given values of the covariates
## S3 method for class 'dirinlaregmodel' predict(object, data.pred.cov, ...)## S3 method for class 'dirinlaregmodel' predict(object, data.pred.cov, ...)
object |
dirinlaregmodel object. |
data.pred.cov |
Data.frame with the covariate values for the variables to predict. |
... |
Other arguments. |
model dirinlaregmodel object
Joaquín Martínez-Minaya [email protected]
if (dirinla_safe_inla() && requireNamespace("DirichletReg", quietly = TRUE)) { ### In this example, we show how to fit a model using the dirinla package ### ### --- 1. Loading the libraries --- #### library(INLA) library(DirichletReg) ### --- 2. Simulating from a Dirichlet likelihood --- #### set.seed(1000) N <- 50 #number of data V <- as.data.frame(matrix(runif((4) * N, 0, 1), ncol = 4)) #Covariates names(V) <- paste0('v', 1:4) formula <- y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4 (names_cat <- formula_list(formula)) x <- c(-1.5, 1, -3, 1.5, 2, -3 , -1, 5) mus <- exp(x) / sum(exp(x)) C <- length(names_cat) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, N * C)), covariates = names_cat, data = V, d = C, n = N) A_construct <- data_stack_construct A_construct[1:8, ] eta <- A_construct %*% x alpha <- exp(eta) alpha <- matrix(alpha, ncol = C, byrow = TRUE) y_o <- rdirichlet(N, alpha) colnames(y_o) <- paste0("y", 1:C) head(y_o) ### --- 3. Fitting the model --- #### y <- y_o model.inla <- dirinlareg( formula = y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4, y = y, data.cov = V, prec = 0.0001, verbose = FALSE) summary(model.inla) ### --- 4. Predicting for v1 = 0.25, v2 = 0.5, v3 = 0.5, v4 = 0.1 --- #### model.prediction <- predict(model.inla, data.pred.cov= data.frame(v1 = 0.25, v2 = 0.5, v3 = 0.5, v4 = 0.1)) model.prediction$summary_predictive_means }if (dirinla_safe_inla() && requireNamespace("DirichletReg", quietly = TRUE)) { ### In this example, we show how to fit a model using the dirinla package ### ### --- 1. Loading the libraries --- #### library(INLA) library(DirichletReg) ### --- 2. Simulating from a Dirichlet likelihood --- #### set.seed(1000) N <- 50 #number of data V <- as.data.frame(matrix(runif((4) * N, 0, 1), ncol = 4)) #Covariates names(V) <- paste0('v', 1:4) formula <- y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4 (names_cat <- formula_list(formula)) x <- c(-1.5, 1, -3, 1.5, 2, -3 , -1, 5) mus <- exp(x) / sum(exp(x)) C <- length(names_cat) data_stack_construct <- data_stack_dirich(y = as.vector(rep(NA, N * C)), covariates = names_cat, data = V, d = C, n = N) A_construct <- data_stack_construct A_construct[1:8, ] eta <- A_construct %*% x alpha <- exp(eta) alpha <- matrix(alpha, ncol = C, byrow = TRUE) y_o <- rdirichlet(N, alpha) colnames(y_o) <- paste0("y", 1:C) head(y_o) ### --- 3. Fitting the model --- #### y <- y_o model.inla <- dirinlareg( formula = y ~ 1 + v1 | 1 + v2 | 1 + v3 | 1 + v4, y = y, data.cov = V, prec = 0.0001, verbose = FALSE) summary(model.inla) ### --- 4. Predicting for v1 = 0.25, v2 = 0.5, v3 = 0.5, v4 = 0.1 --- #### model.prediction <- predict(model.inla, data.pred.cov= data.frame(v1 = 0.25, v2 = 0.5, v3 = 0.5, v4 = 0.1)) model.prediction$summary_predictive_means }
summary_fast summarise a matrix by rows
summary_fast(A)summary_fast(A)
A |
matrix to be summarised |
A matrix whose columns are "mean", "sd", "0.025quant", "0.5quant", "0.975quant"
Joaquín Martínez-Minaya [email protected]
A <- matrix(rnorm(10000), ncol = 1000) summary_fast(A)A <- matrix(rnorm(10000), ncol = 1000) summary_fast(A)
summary.dirinlaregmodel is a function which gives a summary of a dirinlaregmodel object
## S3 method for class 'dirinlaregmodel' summary(object, ...)## S3 method for class 'dirinlaregmodel' summary(object, ...)
object |
Object of class dirinlaregmodel. |
... |
Other arguments. |
Print summary.
Joaquín Martínez-Minaya [email protected]
trigamma_red is the function trigamma appropiate for really small values
trigamma_red(x, ...)trigamma_red(x, ...)
x |
Argument to applied the function trigamma. |
... |
Rest of arguments used in the case of digamma functions. |
Result of applying trigamma function.
Joaquín Martínez-Minaya [email protected]