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Beta Functions

Source: R/beta_functions.R
beta_functions.Rd

Functions to compute the Beta function, normalised incomplete beta function, and their complements, as well as their inverses and derivatives.

Beta Function \(B(a, b)\):

  • beta_boost(a, b)

$$B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$

Incomplete Beta Functions:

  • Normalised (Regularised) Functions:

    • ibeta(a, b, x): Normalised incomplete beta function \(I_x(a, b)\)

$$I_x(a,b) = \frac{1}{B(a, b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$$

  • ibetac(a, b, x): Normalised complement, \(1 - I_x(a, b) = I_{1-x}(b, a)\)

  • Non-normalised Functions:

    • beta_boost(a, b, x): Full incomplete beta function \(B_x(a, b)\)

$$\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$$

  • betac(a, b, x): Full complement , \(1 - B_x(a, b) = B_{1-x}(b, a)\)

Inverse Functions:

  • Primary inverses (solving for x):

    • ibeta_inv(a, b, p): Returns \(x\) such that \(p = I_x(a, b)\)

    • ibetac_inv(a, b, q): Returns \(x\) such that \(q = 1 - I_x(a, b)\)

  • Parameter inverses (solving for a or b):

    • ibeta_inva(b, x, p): Returns a such that \(p = I_x(a, b)\)

    • ibetac_inva(b, x, q): Returns a such that \(q = 1 - I_x(a, b)\)

    • ibeta_invb(a, x, p): Returns b such that \(p = I_x(a, b)\)

    • ibetac_invb(a, x, q): Returns b such that \(q = 1 - I_x(a, b)\)s

Derivatives:

ibeta_derivative(a, b, x): Computes the partial derivative with respect to x of the incomplete beta function

$$\frac{\partial}{\partial x}I_x(a,b) = \frac{(1-x)^{b-1}x^{a-1}}{B(a,b)}$$

Usage

beta_boost(a, b, x = NULL)

ibeta(a, b, x)

ibetac(a, b, x)

betac(a, b, x)

ibeta_inv(a, b, p)

ibetac_inv(a, b, q)

ibeta_inva(b, x, p)

ibetac_inva(b, x, q)

ibeta_invb(a, x, p)

ibetac_invb(a, x, q)

ibeta_derivative(a, b, x)

Arguments

a

First parameter of the beta function (must be positive)

b

Second parameter of the beta function (must be positive)

x

Upper limit of integration (0 <= x <= 1)

p

Probability value (0 <= p <= 1)

q

Probability value (0 <= q <= 1), where q = 1 - p

Value

A single numeric value with the computed beta function, normalised incomplete beta function, or their complements, depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

if (FALSE) { # \dontrun{
# Euler beta function B(2, 3)
beta_boost(2, 3)
# Normalised incomplete beta function I_x(2, 3) for x = 0.5
ibeta(2, 3, 0.5)
# Normalised complement of the incomplete beta function 1 - I_x(2, 3) for x = 0.5
ibetac(2, 3, 0.5)
# Full incomplete beta function B_x(2, 3) for x = 0.5
beta_boost(2, 3, 0.5)
# Full complement of the incomplete beta function 1 - B_x(2, 3) for x = 0.5
betac(2, 3, 0.5)
# Inverse of the normalised incomplete beta function I_x(2, 3) = 0.5
ibeta_inv(2, 3, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(2, 3) = 0.5
ibetac_inv(2, 3, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(a, b)
# with respect to a for x = 0.5 and q = 0.5
ibetac_inva(3, 0.5, 0.5)
# Inverse of the normalised incomplete beta function I_x(a, b)
# with respect to b for x = 0.5 and p = 0.5
ibeta_invb(0.8, 0.5, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(a, b)
# with respect to b for x = 0.5 and q = 0.5
ibetac_invb(2, 0.5, 0.5)
# Derivative of the incomplete beta function with respect to x for a = 2, b = 3, x = 0.5
ibeta_derivative(2, 3, 0.5)
} # }