gaminv

Gamma inverse cumulative distribution function

Syntax

X = gaminv(P,A,B)
[X,XLO,XUP] = gaminv(P,A,B,pcov,alpha)


Description

X = gaminv(P,A,B) computes the inverse of the gamma cdf with shape parameters in A and scale parameters in B for the corresponding probabilities in P. P, A, and B can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs. The parameters in A and B must all be positive, and the values in P must lie on the interval [0 1].

The gamma inverse function in terms of the gamma cdf is

$$x=F^{-1}(p|a,b)=\{x:F(x|a,b)=p\}$$

where

$$p=F(x|a,b)=\frac{1}{b^a\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int }}{t}^{a-1}{e}^{\frac{-t}{b}}dt}$$

[X,XLO,XUP] = gaminv(P,A,B,pcov,alpha) produces confidence bounds for X when the input parameters A and B are estimates. pcov is a 2-by-2 matrix containing the covariance matrix of the estimated parameters. alpha has a default value of 0.05, and specifies 100(1-alpha)% confidence bounds. XLO and XUP are arrays of the same size as X containing the lower and upper confidence bounds.

Examples

This example shows the relationship between the gamma cdf and its inverse function.

a = 1:5;
b = 6:10;
x = gaminv(gamcdf(1:5,a,b),a,b)
x =
  1.0000  2.0000  3.0000  4.0000  5.0000

Algorithms

There is no known analytical solution to the integral equation above. gaminv uses an iterative approach (Newton's method) to converge on the solution.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).