Probability Distribution and Statistical Functions -- Gamma Distribution Module

doc/specs/index.md

@@ -30,6 +30,7 @@ This is and index/directory of the specifications (specs) for each new module/fe
  - [stats_distributions_uniform](./stdlib_stats_distribution_uniform.html) - Uniform Probability Distribution
  - [stats_distributions_normal](./stdlib_stats_distribution_normal.html) - Normal Probability Distribution
  - [stats_distributions_exponential](./stdlib_stats_distribution_exponential.html) - Exponential Probability Distribution
+ - [stats_distributions_gamma](./stdlib_stats_distribution_gamma.html) - Gamma Probability Distribution
  - [string\_type](./stdlib_string_type.html) - Basic string support
  - [stringlist_type](./stdlib_stringlist_type.html) - 1-Dimensional list of strings
  - [strings](./stdlib_strings.html) - String handling and manipulation routines

doc/specs/stdlib_stats_distribution_gamma.md

@@ -0,0 +1,267 @@
+---
+
+title: stats_distribution_gamma
+---
+
+# Statistical Distributions -- Gamma Distribution Module
+
+[TOC]
+
+## `rvs_gamma` - gamma distribution random variates
+
+### Status
+
+Experimental
+
+### Description
+
+With one argument for shape parameter, the function returns a random sample from the standard gamma distribution `Gam(shape)` with `rate = 1.0`. 
+
+With two arguments, the function returns a random sample from gamma distribution `Gam(shape, rate)`.
+
+With three arguments, the function returns a rank one array of gamma distributed random variates.
+
+For complex shape and rate parameters, the real and imaginary parts are sampled independently of each other.
+
+
+### Syntax
+
+`result = [[stdlib_stats_distribution_gamma(module):rvs_gamma(interface)]](shape [, rate] [[, array_size]])`
+
+### Class
+
+Function
+
+### Arguments
+
+`shape` : has `intent(in)` and is a scalar of type `real` or `complex`.
+
+`rate`: optional argument has `intent(in)` and is a scalar of type `real` or `complex`.
+
+`array_size`: optional argument has `intent(in)` and is a scalar of type `integer` with default kind.
+
+### Return value
+
+The result is a scalar or rank one array, with a size of `array_size`, and has the same type of `shape`.
+
+### Example
+
+```fortran
+program demo_gamma_rvs
+    use stdlib_random, only : random_seed
+    use stdlib_stats_distribution_gamma, only: rgamma => rvs_gamma
+
+    implicit none
+    real ::  g(2,3,4)
+    complex :: shape, scale
+    integer :: put, get
+
+    put = 1234567
+    call random_seed(put, get)
+
+    print *, rgamma(2.0)
+    !single standard gamma random variate with shape of 2.0, rate=1.0
+
+! 2.50538206
+
+    print *, rgamma(3.0,2.0)  !gamma random variate with shape=3.0, rate=2.0
+
+! 1.30591583
+
+    g(:,:,:) = 0.5
+    print *, rgamma(g)
+    !a rank 3 array of 60 standard gamma random variates with shape=0.5
+
+!  1.03841162  1.33044529  0.912742674  0.131288037  0.638593793 
+!  1.03565669E-02  0.624804378  1.12179172  4.91380468E-02  6.69969944E-03 
+!  6.67014271E-02  0.132111162  0.101102419  0.648416579  1.14922595 
+!  2.29003578E-02  1.85964716E-04  1.21213868E-02  1.69112933 
+!  7.30440915E-02  0.395139128  0.182758048  0.427981257  0.985665262
+
+    print *, rgamma(0.5,1.0,10)
+    ! an array of 10 random variates with shape=0.5, rate=1.0
+
+!  1.39297554E-04  0.296419382  0.352113068  2.80515051  3.65264394E-04 
+!  0.197743446  5.54569438E-02  9.30598825E-02  1.02596343  1.85311246
+
+    shape = (3.0, 4.0)
+    scale = (2.0, 0.7)
+    print *, rgamma(shape,scale)
+    !single complex gamma random variate with real part of shape = 3.0,
+    !rate=2.0; imagainary part of shape=4.0, rate=0.7
+
+! (0.826188326,3.54749799)
+
+end program demo_gamma_rvs
+```
+
+## `pdf_gamma` - gamma distribution probability density function
+
+### Status
+
+Experimental
+
+### Description
+
+The probability density function (pdf) of the single real variable gamma distribution:
+
+$$ f(x)= \frac{scale^{shape}}{\Gamma (shape)}x^{shape-1}e^{-scale \times x} , for \;\;  x>0, shape, scale>0$$
+
+For a complex variable (x + y i) with independent real x and imaginary y parts, the joint probability density function is the product of the corresponding marginal pdf of real and imaginary pdf (for more details, see
+"Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
+
+$$f(x+\mathit{i}y)=f(x)f(y)$$
+
+### Syntax
+
+`result = [[stdlib_stats_distribution_gamma(module):pdf_gamma(interface)]](x, shape, rate)`
+
+### Class
+
+Elemental function
+
+### Arguments
+
+`x`: has `intent(in)` and is a scalar of type `real` or `complex`.
+
+`shape` has `intent(in)` and is a scalar of type real` or `complex`.
+
+`rate`: has `intent(in)` and is a scalar of type `real` or `complex`.
+
+All arguments must have the same type.
+
+### Return value
+
+The result is a scalar or an array, with a shape conformable to arguments, of type `real`.
+
+### Example
+
+```fortran
+program demo_gamma_pdf
+    use stdlib_random, only : random_seed
+    use stdlib_stats_distribution_gamma, only: rgamma => rvs_gamma,&
+                                             gamma_pdf => pdf_gamma
+
+    implicit none
+    real :: x(2,3,4),g(2,3,4),s(2,3,4)
+    complex :: shape, scale
+    integer :: put, get
+
+    put = 1234567
+    call random_seed(put, get)
+
+    print *, gamma_pdf(1.0, 1.0, 1.0)
+    !a probability density at 1.0 with shape=1.0, rate=1.0
+
+! 0.367879450
+
+    g(:,:,:) = 2.0
+    s(:,:,:) = 1.0
+    x = reshape(rgamma(2.0, 1.0, 24),[2,3,4]) ! gamma random variates array
+    print *, gamma_pdf(x,g,s)     ! a rank 3 gamma probability density array
+
+!  0.204550430  0.320178866  0.274986655  0.348611295  0.101865448 
+!  0.102199331  0.358981341  0.223676488  0.254329354  0.356714427 
+!  0.267390072  0.305148095  0.367848188  7.26194456E-02  1.49471285E-02 
+!  0.246272027  0.360770017  0.339665830  0.101558588  0.358678699 
+!  0.224196941  0.359253854  7.56355673E-02  0.251869917
+
+    shape = (1.0, 1.5)
+    scale = (1.0, 2.)
+    print *, gamma_pdf((1.5,1.0), shape, scale)
+    ! a complex expon probability density function at (1.5,1.0) with real part
+    !of shape=1.0, rate=1.0 and imaginary part of shape=1.5, rate=2.0
+
+! 9.63761061E-02
+
+end program demo_gamma_pdf
+```
+
+## `cdf_gamma` - gamma distribution cumulative distribution function
+
+### Status
+
+Experimental
+
+### Description
+
+Cumulative distribution function (cdf) of the single real variable gamma distribution:
+
+$$ F(x)= \frac{\gamma (shape, scale \times x)}{\Gamma (shape)}, for \;\;  x>0, shape, scale>0} $$
+
+For a complex variable (x + y i) with independent real x and imaginary y parts, the joint cumulative distribution function is the product of corresponding marginal cdf of real and imaginary cdf (for more details, see
+"Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):
+
+$$F(x+\mathit{i}y)=F(x)F(y)$$
+
+### Syntax
+
+`result = [[stdlib_stats_distribution_gamma(module):cdf_gamma(interface)]](x, shape, rate)`
+
+### Class
+
+Elemental function
+
+### Arguments
+
+`x`: has `intent(in)` and is a scalar of type `real` or `complex`.
+
+`shape`: has `intent(in)` and is a scalar of type `real` or `complex`.
+
+`rate`: has `intent(in)` and is a scalar of type `real` or `complex`.
+
+All arguments must have the same type.
+
+### Return value
+
+The result is a scalar of type `real` with the same kind of input arguments.
+
+### Example
+
+```fortran
+program demo_gamma_cdf
+    use stdlib_random, only : random_seed
+    use stdlib_stats_distribution_gamma, only: rgamma => rvs_gamma,&
+                                             gamma_cdf => cdf_gamma
+
+    implicit none
+    real :: x(2,3,4),g(2,3,4),s(2,3,4)
+    complex :: shape, scale
+    integer :: seed_put, seed_get
+
+    seed_put = 1234567
+    call random_seed(seed_put, seed_get)
+
+    print *, gamma_cdf(1.0, 0.5,1.0)
+    ! a standard gamma cumulative at 1.0 with a shape=0.5, rate=1.0
+
+! 0.842700839
+
+    print *, gamma_cdf(2.0, 1.5,2.0)
+    ! a cumulative at 2.0 with a shape=1.5, rate=2.0
+
+! 0.953988254
+
+    g(:,:,:) = 1.0
+    s(:,:,:) = 1.0
+    x = reshape(rgamma(1.0, 1.0, 24),[2,3,4])
+    !gamma random variates array with a shape=1.0, rate=1.0
+    print *, gamma_cdf(x,g,s)        ! a rank 3 standard gamma cumulative array
+
+!  0.710880339  0.472411335  0.578345954  0.383050948  0.870905757 
+!  0.870430350  0.170215249  0.677347481  0.620089889  0.161825046 
+!  4.17549349E-02  0.510665894  0.252201647  0.911497891  0.984424412 
+!  0.635621786  0.177783430  0.414842933  0.871342421  0.338317066 
+!  2.06879266E-02  0.335232288  0.907408893  0.624871135
+
+    shape = (.7, 2.1)
+    scale = (0.5,1.0)
+    print *, gamma_cdf((0.5,0.5),shape,scale)
+    !complex gamma cumulative distribution at (0.5,0.5) with real part of
+    !shape=0.7,rate=0.5 and imaginary part of shape=2.1,rate=1.0
+
+! 2.87349485E-02
+
+end program demo_gamma_cdf
+
+```

src/CMakeLists.txt

@@ -41,6 +41,7 @@ set(fppFiles
     stdlib_stats_distribution_uniform.fypp
     stdlib_stats_distribution_normal.fypp
     stdlib_stats_distribution_exponential.fypp
+    stdlib_stats_distribution_gamma.fypp
     stdlib_stats_var.fypp
     stdlib_quadrature.fypp
     stdlib_quadrature_trapz.fypp