Generate and return the Cumulative Distribution Function (CDF) of a univariate Standard Uniform distribution or a Uniform distribution with the specified support via lower and upper input arguments at the specified input values. More...

Detailed Description

Generate and return the Cumulative Distribution Function (CDF) of a univariate Standard Uniform distribution or a Uniform distribution with the specified support via lower and upper input arguments at the specified input values.

Parameters

[in]	x	: The input scalar or array of the same shape as other input array arguments, of either If `X` is `integer`, the discrete Uniform distribution CDF with support `[lower, upper]` will be returned. If `X` is `integer`, the output argument `CDF` must be of type `real` of kind RK. If `X` is `real`, the continuous Uniform distribution CDF with support `[lower, upper]` will be returned. If `X` is `complex`, the two real and imaginary components of `CDF` will correspond to two independent distributions.
[in]	lower	: The input scalar or array of the same shape as other input array arguments, of the same type and kind as `X`, representing the lower bound of the Uniform distribution. (optional, default = `0`. It must be present if and only if the input argument `upper` is also present.)
[in]	upper	: The input scalar or array of the same shape as other input array arguments, of the same type and kind as `X`, representing the upper bound of the Uniform distribution. (optional, default = `1`. It must be present if and only if the input argument `lower` is also present.)

Returns

cdf : The output scalar or array of the same shape as the input array arguments, of either

type real of default kind RK (if the input value x is integer of kind any supported by the processor (e.g., IK, IK8, IK16, IK32, or IK64)), or
type complex of the same kind as x (if the input value x is of type complex of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128)),
type real of the same kind as x (if the input value x is of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128)),

containing the CDF of the specified discrete or continuous Uniform distribution.

Possible calling interfaces ⛓

: use pm_distUnif, only: getUnifCDF

cdf = getUnifCDF(x)

cdf = getUnifCDF(x, lower, upper)

pm_distUnif::getUnifCDF
Generate and return the Cumulative Distribution Function (CDF) of a univariate Standard Uniform distr...
Definition: pm_distUnif.F90:395

pm_distUnif
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distUnif.F90:274

Warning: The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

Remarks: The procedures under discussion are elemental.

See also: getUnifCDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK ! All kinds are supported.

4 use pm_kind, only: IK ! All kinds are supported.

5 use pm_kind, only: CK ! All kinds are supported.

6 use pm_kind, only: RK ! All kinds are supported.

7 use pm_io, only: display_type

8 use pm_arraySpace, only: getLinSpace

9 use pm_distUnif, only: getUnifCDF

10 use pm_arrayRange, only: getRange

11

12 implicit none

13

14 integer(IK) , parameter :: NP = 1000_IK

15

16 ! 1-dimensional array of x values.

17

18 integer(IK) , allocatable :: point_IK(:)

19 real(RK) , allocatable :: point_RK(:), CDF_RK(:), CDF_IK(:)

20 complex(CK) , allocatable :: point_CK(:), CDF_CK(:)

21

22 integer(IK) :: range_IK, lower_IK = -3_IK , upper_IK = +4_IK

23 real(RK) :: range_RK, lower_RK = -4._RK , upper_RK = +4._RK

24 complex(CK) :: range_CK, lower_CK = (-4._CK,-1._CK) , upper_CK = (+4._CK,+3._CK)

25

26 type(display_type) :: disp

27 disp = display_type(file = "main.out.F90")

28

29 range_IK = upper_IK - lower_IK

30 range_RK = upper_RK - lower_RK

31 range_CK = upper_CK - lower_CK

32

33 point_IK = getRange(lower_IK - range_IK/2, upper_IK + range_IK/2)

34 point_RK = getLinSpace(lower_RK - range_RK/2, upper_RK + range_RK/2, count = NP)

35 point_CK = getLinSpace(lower_CK - range_CK/2, upper_CK + range_CK/2, count = NP)

36 allocate(CDF_IK(size(point_IK)))

37 allocate(CDF_RK(size(point_RK)))

38 allocate(CDF_CK(size(point_CK)))

39

40 call disp%skip()

41 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

42 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

43 call disp%show("! Compute the Cumulative Distribution Function (CDF) of the Uniform distribution at the specified values.")

44 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

45 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

46 call disp%skip()

47

48 call disp%skip()

49 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

50 call disp%show("! Compute the discrete Uniform CDF at an input scalar integer value.")

51 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

52 call disp%skip()

53

54 call disp%skip()

55 call disp%show("CDF_IK(1) = getUnifCDF(x = 0_IK)")

56 CDF_IK(1) = getUnifCDF(x = 0_IK)

57 call disp%show("CDF_IK(1)")

58 call disp%show( CDF_IK(1) )

59 call disp%skip()

60

61 call disp%skip()

62 call disp%show("lower_IK")

63 call disp%show( lower_IK )

64 call disp%show("upper_IK")

65 call disp%show( upper_IK )

66 call disp%show("CDF_IK(1) = getUnifCDF(x = 0_IK, lower = lower_IK, upper = upper_IK)")

67 CDF_IK(1) = getUnifCDF(x = 0_IK, lower = lower_IK, upper = upper_IK)

68 call disp%show("CDF_IK(1)")

69 call disp%show( CDF_IK(1) )

70 call disp%skip()

71

72 call disp%skip()

73 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

74 call disp%show("! Compute the continuous Uniform CDF at an input scalar real value.")

75 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

76 call disp%skip()

77

78 call disp%skip()

79 call disp%show("CDF_RK(1) = getUnifCDF(x = 0._RK)")

80 CDF_RK(1) = getUnifCDF(x = 0._RK)

81 call disp%show("CDF_RK(1)")

82 call disp%show( CDF_RK(1) )

83 call disp%skip()

84

85 call disp%skip()

86 call disp%show("lower_RK")

87 call disp%show( lower_RK )

88 call disp%show("upper_RK")

89 call disp%show( upper_RK )

90 call disp%show("CDF_RK(1) = getUnifCDF(x = 0._RK, lower = lower_RK, upper = upper_RK)")

91 CDF_RK(1) = getUnifCDF(x = 0._RK, lower = lower_RK, upper = upper_RK)

92 call disp%show("CDF_RK(1)")

93 call disp%show( CDF_RK(1) )

94 call disp%skip()

95

96 call disp%skip()

97 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

98 call disp%show("! Compute the continuous Uniform CDF at an input scalar complex value.")

99 call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")

100 call disp%skip()

101

102 call disp%skip()

103 call disp%show("CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK))")

104 CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK))

105 call disp%show("CDF_CK(1)")

106 call disp%show( CDF_CK(1) )

107 call disp%skip()

108

109 call disp%skip()

110 call disp%show("lower_CK")

111 call disp%show( lower_CK )

112 call disp%show("upper_CK")

113 call disp%show( upper_CK )

114 call disp%show("CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK), lower = lower_CK, upper = upper_CK)")

115 CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK), lower = lower_CK, upper = upper_CK)

116 call disp%show("CDF_CK(1)")

117 call disp%show( CDF_CK(1) )

118 call disp%skip()

119

120 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

121 ! Output an example array for visualization.

122 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

123

124 CDF_IK = getUnifCDF(point_IK, lower_IK, upper_IK)

125 CDF_RK = getUnifCDF(point_RK, lower_RK, upper_RK)

126 CDF_CK = getUnifCDF(point_CK, lower_CK, upper_CK)

127

128 block

129

130 integer :: fileUnit, i

131

132 open(newunit = fileUnit, file = "main.unif.cdf.IK.txt")

133 write(fileUnit,"(2(g0,:,' '))") (point_IK(i), CDF_IK(i), i = 1, size(point_IK))

134 close(fileUnit)

135

136 open(newunit = fileUnit, file = "main.unif.cdf.RK.txt")

137 write(fileUnit,"(2(g0,:,' '))") (point_RK(i), CDF_RK(i), i = 1, size(point_RK))

138 close(fileUnit)

139

140 open(newunit = fileUnit, file = "main.unif.cdf.CK.txt")

141 write(fileUnit,"(4(g0,:,' '))") (point_CK(i), CDF_CK(i), i = 1, size(point_CK))

142 close(fileUnit)

143

144 end block

145

146end program example

pm_arrayRange::getRange
Generate minimally-spaced character, integer, real sequences or sequences at fixed intervals of size ...
Definition: pm_arrayRange.F90:135

pm_arraySpace::getLinSpace
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Definition: pm_arraySpace.F90:108

pm_io::show
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726

pm_io::skip
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508

pm_arrayRange
This module contains procedures and generic interfaces for generating ranges of discrete character,...
Definition: pm_arrayRange.F90:36

pm_arraySpace
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
Definition: pm_arraySpace.F90:33

pm_io
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252

pm_io::disp
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393

pm_kind
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268

pm_kind::RK
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
Definition: pm_kind.F90:543

pm_kind::CK
integer, parameter CK
The default complex kind in the ParaMonte library: real64 in Fortran, c_double_complex in C-Fortran I...
Definition: pm_kind.F90:542

pm_kind::IK
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540

pm_kind::SK
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539

pm_io::display_type
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler ⛓
1#!/usr/bin/env sh

2rm main.exe

3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe

4./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓
1del main.exe

2set PATH=..\..\..\lib;%PATH%

3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe

4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓
1#!/usr/bin/env sh

2rm main.exe

3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe

4./main.exe

Example output ⛓
1

2!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4! Compute the Cumulative Distribution Function (CDF) of the Uniform distribution at the specified values.

5!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

6!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

7

8

9!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

10! Compute the discrete Uniform CDF at an input scalar integer value.

11!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

12

13

14CDF_IK(1) = getUnifCDF(x = 0_IK)

15CDF_IK(1)

16+0.50000000000000000

17

18

19lower_IK

20-3

21upper_IK

22+4

23CDF_IK(1) = getUnifCDF(x = 0_IK, lower = lower_IK, upper = upper_IK)

24CDF_IK(1)

25+0.50000000000000000

26

27

28!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

29! Compute the continuous Uniform CDF at an input scalar real value.

30!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

31

32

33CDF_RK(1) = getUnifCDF(x = 0._RK)

34CDF_RK(1)

35+0.0000000000000000

36

37

38lower_RK

39-4.0000000000000000

40upper_RK

41+4.0000000000000000

42CDF_RK(1) = getUnifCDF(x = 0._RK, lower = lower_RK, upper = upper_RK)

43CDF_RK(1)

44+0.50000000000000000

45

46

47!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

48! Compute the continuous Uniform CDF at an input scalar complex value.

49!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

50

51

52CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK))

53CDF_CK(1)

54(+0.0000000000000000, +1.0000000000000000)

55

56

57lower_CK

58(-4.0000000000000000, -1.0000000000000000)

59upper_CK

60(+4.0000000000000000, +3.0000000000000000)

61CDF_CK(1) = getUnifCDF(x = (0._CK,0._CK), lower = lower_CK, upper = upper_CK)

62CDF_CK(1)

63(+0.50000000000000000, +0.25000000000000000)

64

65

Postprocessing of the example output ⛓
1#!/usr/bin/env python

2

3import matplotlib.pyplot as plt

4import pandas as pd

5import numpy as np

6

7fontsize = 17

8

9marker ={ "IK" : "."

10 , "RK" : "-"

11 , "CK" : "-"

12 }

13ylab = { "IK" : "Discrete Integer Uniform CDF"

14 , "RK" : "Continuous Real Uniform CDF"

15 , "CK" : "Continuous Complex Uniform CDF"

16 }

17

18for kind in ["IK", "RK", "CK"]:

19

20 df = pd.read_csv("main.unif.cdf."+kind+".txt", delimiter = " ")

21

22 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)

23 ax = plt.subplot()

24

25 if kind == "CK":

26 plt.plot( df.values[:, 0]

27 , df.values[:,2]

28 , marker[kind]

29 , color = "r"

30 )

31 plt.plot( df.values[:, 1]

32 , df.values[:,3]

33 , marker[kind]

34 , color = "blue"

35 )

36 ax.legend ( ["real", "imaginary"]

37 , fontsize = fontsize

38 )

39 else:

40 plt.plot( df.values[:, 0]

41 , df.values[:, 1]

42 , marker[kind]

43 , color = "r"

44 )

45

46 ax.set_xlabel("X", fontsize = 17)

47 ax.set_ylabel(ylab[kind], fontsize = 17)

48

49 plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")

50 ax.tick_params(axis = "y", which = "minor")

51 ax.tick_params(axis = "x", which = "minor")

52

53 plt.savefig("getUnifCDF."+kind+".png")

Visualization of the example output ⛓

Test:: test_pm_distUnif

Todo:: Normal Priority: This generic interface can be extended to input string arguments to make it compatible with setUnifRand.

Final Remarks ⛓

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright: Computational Data Science Lab

Author:: Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 395 of file pm_distUnif.F90.

The documentation for this interface was generated from the following file:

src/fortran/main/pm_distUnif.F90