Generalized Beta, also need param_estimate and stats_tbl

[spsanderson]

Param Estimate

Function:

#' Estimate Generalized Beta Parameters
#'
#' @family Parameter Estimation
#' @family Generalized Beta
#'
#' @details This function will attempt to estimate the generalized Beta shape1, shape2, shape3, and rate
#' parameters given some vector of values.
#'
#' @description The function will return a list output by default, and if the parameter
#' `.auto_gen_empirical` is set to `TRUE` then the empirical data given to the
#' parameter `.x` will be run through the `tidy_empirical()` function and combined
#' with the estimated generalized Beta data.
#'
#' @param .x The vector of data to be passed to the function.
#' @param .auto_gen_empirical This is a boolean value of TRUE/FALSE with default
#' set to TRUE. This will automatically create the `tidy_empirical()` output
#' for the `.x` parameter and use the `tidy_combine_distributions()`. The user
#' can then plot out the data using `$combined_data_tbl` from the function output.
#'
#' @examples
#' library(dplyr)
#' library(ggplot2)
#'
#' set.seed(123)
#' x <- tidy_generalized_beta(100, .shape1 = 2, .shape2 = 3, 
#' .shape3 = 4, .rate = 5)[["y"]]
#' output <- util_generalized_beta_param_estimate(x)
#'
#' output$parameter_tbl
#'
#' output$combined_data_tbl %>%
#'   tidy_combined_autoplot()
#'
#' @return
#' A tibble/list
#'
#' @author Steven P. Sanderson II, MPH
#'
#' @export
#'

util_generalized_beta_param_estimate <- function(.x, .auto_gen_empirical = TRUE) {
  
  # Tidyeval ----
  x_term <- as.numeric(.x)
  n <- length(x_term)
  
  # Checks ----
  if (!is.vector(x_term, mode = "numeric") || is.factor(x_term)) {
    rlang::abort(
      message = "'.x' must be a numeric vector.",
      use_cli_format = TRUE
    )
  }
  
  if (n < 2) {
    rlang::abort(
      message = "'.x' must contain at least two non-missing distinct values. All values of '.x' must be positive.",
      use_cli_format = TRUE
    )
  }
  
  # Negative log-likelihood function for generalized Beta distribution
  genbeta_lik <- function(params, data) {
    shape1 <- params[1]
    shape2 <- params[2]
    shape3 <- params[3]
    rate <- params[4]
    -sum(actuar::dgenbeta(data, shape1 = shape1, shape2 = shape2, 
                          shape3 = shape3, rate = rate, log = TRUE))
  }
  
  # Initial parameter guesses
  initial_params <- c(shape1 = 1, shape2 = 1, shape3 = 1, rate = 1)
  
  # Optimize to minimize the negative log-likelihood
  opt_result <- stats::optim(
    par = initial_params,
    fn = genbeta_lik,
    data = x_term
  )
  
  shape1 <- opt_result$par[["shape1"]]
  shape2 <- opt_result$par[["shape2"]]
  shape3 <- opt_result$par[["shape3"]]
  rate <- opt_result$par[["rate"]]
  
  # Return Tibble ----
  if (.auto_gen_empirical) {
    te <- tidy_empirical(.x = x_term)
    td <- tidy_generalized_beta(.n = n, .shape1 = round(shape1, 3), .shape2 = round(shape2, 3), .shape3 = round(shape3, 3), .rate = round(rate, 3))
    combined_tbl <- tidy_combine_distributions(te, td)
  }
  
  ret <- dplyr::tibble(
    dist_type = "Generalized Beta",
    samp_size = n,
    min = min(x_term),
    max = max(x_term),
    mean = mean(x_term),
    shape1 = shape1,
    shape2 = shape2,
    shape3 = shape3,
    rate = rate
  )
  
  # Return ----
  attr(ret, "tibble_type") <- "parameter_estimation"
  attr(ret, "family") <- "generalized_beta"
  attr(ret, "x_term") <- .x
  attr(ret, "n") <- n
  
  if (.auto_gen_empirical) {
    output <- list(
      combined_data_tbl = combined_tbl,
      parameter_tbl     = ret
    )
  } else {
    output <- list(
      parameter_tbl = ret
    )
  }
  
  return(output)
}

Example:

> x <- tidy_generalized_beta(100, .shape1 = 2, .shape2 = 3, 
+ .shape3 = 4, .rate = 5)[["y"]]
> output <- util_generalized_beta_param_estimate(x)
> 
> output$parameter_tbl
# A tibble: 1 × 9
  dist_type        samp_size    min   max  mean shape1 shape2 shape3  rate
  <chr>                <int>  <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl> <dbl>
1 Generalized Beta       100 0.0944 0.196 0.153   5.70   5.44   2.05  4.67

Stats Tibble

Function:

#' Distribution Statistics
#'
#' @family Generalized Beta
#' @family Distribution Statistics
#'
#' @details This function will take in a tibble and return the statistics
#' of the given type of `tidy_` distribution. It is required that data be
#' passed from a `tidy_` distribution function.
#'
#' @description Returns distribution statistics in a tibble.
#'
#' @param .data The data being passed from a `tidy_` distribution function.
#'
#' @examples
#' library(dplyr)
#'
#' set.seed(123)
#' tidy_generalized_beta() |>
#'   util_generalized_beta_stats_tbl() |>
#'   glimpse()
#'
#' @return
#' A tibble
#'
#' @author Steven P. Sanderson II, MPH
#'
#' @export
#' @rdname util_generalized_beta_stats_tbl
util_generalized_beta_stats_tbl <- function(.data) {
  
  # Immediate check for tidy_ distribution function
  if (!"tibble_type" %in% names(attributes(.data))) {
    rlang::abort(
      message = "You must pass data from the 'tidy_dist' function.",
      use_cli_format = TRUE
    )
  }
  
  if (attributes(.data)$tibble_type != "tidy_generalized_beta") {
    rlang::abort(
      message = "You must use 'tidy_generalized_beta()'",
      use_cli_format = TRUE
    )
  }
  
  # Data
  data_tbl <- dplyr::as_tibble(.data)
  
  atb <- attributes(data_tbl)
  shape1 <- atb$.shape1
  shape2 <- atb$.shape2
  shape3 <- atb$.shape3
  rate <- atb$.rate
  scale <- 1 / rate
  
  # Generalized Beta statistics calculation
  stat_mean <- ifelse(shape2 > 1, shape1 / (shape2 - 1), "undefined")
  stat_mode <- ifelse((shape1 > 1) & (shape2 > 2), (shape1 - 1) / (shape2 - 2), "undefined")
  stat_var <- ifelse(shape2 > 2, (shape1 * shape2) / ((shape2 - 1)^2 * (shape2 - 2)), "undefined")
  stat_sd <- ifelse(stat_var == "undefined", "undefined", sqrt(stat_var))
  stat_skewness <- ifelse(shape2 > 3, (2 * (shape2 - 2 * shape1) * sqrt(shape2 - 2)) / ((shape2 - 3) * sqrt(shape1 * (shape1 + shape2))), "undefined")
  stat_kurtosis <- ifelse(shape2 > 4, 3 + (6 * (shape2^3 - 2 * shape2^2 * (shape1 - 1) + shape1^2 * (shape1 + 1))) / (shape1 * (shape1 + 1) * (shape2 - 3) * (shape2 - 4)), "undefined")
  
  # Data Tibble
  ret <- dplyr::tibble(
    tidy_function = atb$tibble_type,
    function_call = atb$dist_with_params,
    distribution = dist_type_extractor(atb$tibble_type),
    distribution_type = atb$distribution_family_type,
    points = atb$.n,
    simulations = atb$.num_sims,
    mean = stat_mean,
    mode = stat_mode,
    range = paste0("0 to Inf"),
    std_dv = stat_sd,
    coeff_var = ifelse(stat_var == "undefined", "undefined", sqrt(stat_var) / stat_mean),
    skewness = stat_skewness,
    kurtosis = stat_kurtosis,
    computed_std_skew = tidy_skewness_vec(data_tbl$y),
    computed_std_kurt = tidy_kurtosis_vec(data_tbl$y),
    ci_lo = ci_lo(data_tbl$y),
    ci_hi = ci_hi(data_tbl$y)
  )
  
  # Return
  return(ret)
}

Example:

> set.seed(123)
> tidy_generalized_beta() |>
+   util_generalized_beta_stats_tbl() |>
+   glimpse()
Rows: 1
Columns: 17
$ tidy_function     <chr> "tidy_generalized_beta"
$ function_call     <chr> "Generalized Beta c(1, 1, 1, 1, 1)"
$ distribution      <chr> "Generalized Beta"
$ distribution_type <chr> "continuous"
$ points            <dbl> 50
$ simulations       <dbl> 1
$ mean              <chr> "undefined"
$ mode              <chr> "undefined"
$ range             <chr> "0 to Inf"
$ std_dv            <chr> "undefined"
$ coeff_var         <chr> "undefined"
$ skewness          <chr> "undefined"
$ kurtosis          <chr> "undefined"
$ computed_std_skew <dbl> -0.07917738
$ computed_std_kurt <dbl> 1.789811
$ ci_lo             <dbl> 0.03836872
$ ci_hi             <dbl> 0.9413363

AIC

Function:

#' Calculate Akaike Information Criterion (AIC) for Generalized Beta Distribution
#'
#' This function calculates the Akaike Information Criterion (AIC) for a generalized Beta
#' distribution fitted to the provided data.
#'
#' @family Utility
#' 
#' @author Steven P. Sanderson II, MPH
#' 
#' @description
#' This function estimates the shape1, shape2, shape3, and rate parameters of a generalized Beta distribution
#' from the provided data using maximum likelihood estimation,
#' and then calculates the AIC value based on the fitted distribution.
#'
#' @param .x A numeric vector containing the data to be fitted to a generalized Beta distribution.
#'
#' @details
#' This function fits a generalized Beta distribution to the provided data using maximum
#' likelihood estimation. It estimates the shape1, shape2, shape3, and rate parameters
#' of the generalized Beta distribution using maximum likelihood estimation. Then, it
#' calculates the AIC value based on the fitted distribution.
#'
#' Initial parameter estimates: The function uses reasonable initial estimates
#' for the shape1, shape2, shape3, and rate parameters of the generalized Beta distribution.
#'
#' Optimization method: The function uses the optim function for optimization.
#' You might explore different optimization methods within optim for potentially
#' better performance.
#'
#' Goodness-of-fit: While AIC is a useful metric for model comparison, it's
#' recommended to also assess the goodness-of-fit of the chosen model using
#' visualization and other statistical tests.
#'
#' @examples
#' # Example 1: Calculate AIC for a sample dataset
#' set.seed(123)
#' x <- tidy_generalized_beta(100, .shape1 = 2, .shape2 = 3, 
#'                           .shape3 = 4, .rate = 5)[["y"]]
#' util_generalized_beta_aic(x)
#'
#' @return
#' The AIC value calculated based on the fitted generalized Beta distribution to 
#' the provided data.
#'
#' @name util_generalized_beta_aic
NULL

#' @export
#' @rdname util_generalized_beta_aic
util_generalized_beta_aic <- function(.x) {
  # Tidyeval
  x <- as.numeric(.x)
  
  # Negative log-likelihood function for generalized Beta distribution
  neg_log_lik_genbeta <- function(par, data) {
    shape1 <- par[1]
    shape2 <- par[2]
    shape3 <- par[3]
    rate <- par[4]
    -sum(actuar::dgenbeta(data, shape1 = shape1, shape2 = shape2, 
                          shape3 = shape3, rate = rate, log = TRUE))
  }
  
  # Initial parameter estimates
  pe <- TidyDensity::util_generalized_beta_param_estimate(x)$parameter_tbl
  shape1 <- pe$shape1
  shape2 <- pe$shape2
  shape3 <- pe$shape3
  rate <- pe$rate
  initial_params <- c(shape1 = shape1, shape2 = shape2, shape3 = shape3, 
                      rate = rate)
  
  # Fit generalized Beta distribution using optim
  fit_genbeta <- stats::optim(
    par = initial_params,
    fn = neg_log_lik_genbeta,
    data = x
  )
  
  # Extract log-likelihood and number of parameters
  logLik_genbeta <- -fit_genbeta$value
  k_genbeta <- 4 # Number of parameters for generalized Beta distribution (shape1, shape2, shape3, and rate)
  
  # Calculate AIC
  AIC_genbeta <- 2 * k_genbeta - 2 * logLik_genbeta
  
  # Return AIC
  return(AIC_genbeta)
}

Example:

> set.seed(123)
> x <- tidy_generalized_beta(100, .shape1 = 2, .shape2 = 3, 
+                           .shape3 = 4, .rate = 5)[["y"]]
> util_generalized_beta_aic(x)
[1] -498.3238