Generate and return the Cumulative Distribution Function (CDF) of the univariate Normal distribution. More...

Detailed Description

Generate and return the Cumulative Distribution Function (CDF) of the univariate Normal distribution.

Parameters

[out]	cdf	: The output scalar or array of the same shape as the input array-like arguments, of, type `real` of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the CDF of the distribution at the specified input `x`.
[in]	x	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as the output `cdf`, representing the point(s) at which the CDF must be computed.
[in]	mu	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as the output `cdf` representing the location parameter of the distribution. (optional, default = `0`)
[in]	invSigma	: The input scalar of the same type and kind as the output `quantile`, representing the inverse of the scale parameter of the distribution. (optional, default = `1.`)

Possible calling interfaces ⛓

: use pm_distNorm, only: setNormCDF

call setNormCDF(cdf, x)

call setNormCDF(cdf, x, mu)

call setNormCDF(cdf, x, mu, invSigma)

pm_distNorm::setNormCDF
Generate and return the Cumulative Distribution Function (CDF) of the univariate Normal distribution.
Definition: pm_distNorm.F90:886

pm_distNorm
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distNorm.F90:179

Warning: The condition invSigma > 0. must hold for the corresponding procedure argument.
This condition is verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.; 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: getNormCDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK

4 use pm_kind, only: IK, RK ! all real kinds are supported.

5 use pm_distNorm, only: setNormCDF

6 use pm_arraySpace, only: setLinSpace

7 use pm_arraySpace, only: setLogSpace

8 use pm_io, only: display_type

9

10 implicit none

11

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

13 real(RK), dimension(NP) :: Point, mu, invSigma, CDF

14

15 type(display_type) :: disp

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

17

18 call setLinSpace(mu, x1 = -5._RK, x2 = +5._RK)

19 call setLinSpace(Point, x1 = -10._RK, x2 = +10._RK)

20 call setLogSpace(invSigma, logx1 = log(0.1_RK), logx2 = log(10._RK))

21

22 call disp%skip()

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

24 call disp%show("! Compute the Cumulative Distribution Function (CDF) of the (Standard) Normal distribution.")

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

26 call disp%skip()

27

28 call disp%skip()

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

30 call disp%show("! Standard CDF.")

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

32 call disp%skip()

33

34 call disp%skip()

35 call disp%show("Point(NP/2)")

36 call disp%show( Point(NP/2) )

37 call disp%show("call setNormCDF(CDF(1), Point(NP/2))")

38 call setNormCDF(CDF(1), Point(NP/2))

39 call disp%show("CDF(1)")

40 call disp%show( CDF(1) )

41 call disp%skip()

42

43 call disp%skip()

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

45 call disp%show("! CDF with a mean.")

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

47 call disp%skip()

48

49 call disp%skip()

50 call disp%show("mu(1)")

51 call disp%show( mu(1) )

52 call disp%show("Point(1)")

53 call disp%show( Point(1) )

54 call disp%show("call setNormCDF(CDF(1), Point(1), mu(1))")

55 call setNormCDF(CDF(1), Point(1), mu(1))

56 call disp%show("CDF(1)")

57 call disp%show( CDF(1) )

58 call disp%skip()

59

60 call disp%skip()

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

62 call disp%show("! CDF with a standard deviation.")

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

64 call disp%skip()

65

66 call disp%skip()

67 call disp%show("invSigma(1)")

68 call disp%show( invSigma(1) )

69 call disp%show("Point(1)")

70 call disp%show( Point(1) )

71 call disp%show("call setNormCDF(CDF(1), Point(1), invSigma(1))")

72 call setNormCDF(CDF(1), Point(1), invSigma(1))

73 call disp%show("CDF(1)")

74 call disp%show( CDF(1) )

75 call disp%skip()

76

77 call disp%skip()

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

79 call disp%show("! CDF with a mean and a standard deviation.")

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

81 call disp%skip()

82

83 call disp%skip()

84 call disp%show("mu(1)")

85 call disp%show( mu(1) )

86 call disp%show("invSigma(1)")

87 call disp%show( invSigma(1) )

88 call disp%show("Point(1)")

89 call disp%show( Point(1) )

90 call disp%show("call setNormCDF(CDF(1), Point(1), mu(1), invSigma(1))")

91 call setNormCDF(CDF(1), Point(1), mu(1), invSigma(1))

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

93 call disp%show( CDF(1) )

94 call disp%skip()

95

96 call disp%skip()

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

98 call disp%show("! A vector of CDF at different points with the same mean and standard deviation.")

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

100 call disp%skip()

101

102 call disp%skip()

103 call disp%show("mu(1)")

104 call disp%show( mu(1) )

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

106 call disp%show( invSigma(1) )

107 call disp%show("Point(1:NP:NP/5)")

108 call disp%show( Point(1:NP:NP/5) )

109 call disp%show("call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1), invSigma(1))")

110 call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1), invSigma(1))

111 call disp%show("CDF(1:NP:NP/5)")

112 call disp%show( CDF(1:NP:NP/5) )

113 call disp%skip()

114

115 call disp%skip()

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

117 call disp%show("! A vector of CDF at the same point but with different means and standard deviations.")

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

119 call disp%skip()

120

121 call disp%skip()

122 call disp%show("mu(1:NP:NP/5)")

123 call disp%show( mu(1:NP:NP/5) )

124 call disp%show("invSigma(1:NP:NP/5)")

125 call disp%show( invSigma(1:NP:NP/5) )

126 call disp%show("Point(1)")

127 call disp%show( Point(1) )

128 call disp%show("call setNormCDF(CDF(1:NP:NP/5), Point(1), mu(1:NP:NP/5), invSigma(1:NP:NP/5))")

129 call setNormCDF(CDF(1:NP:NP/5), Point(1), mu(1:NP:NP/5), invSigma(1:NP:NP/5))

130 call disp%show("CDF(1:NP:NP/5)")

131 call disp%show( CDF(1:NP:NP/5) )

132 call disp%skip()

133

134 call disp%skip()

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

136 call disp%show("! A vector of CDF at different points with different means and a standard deviations.")

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

138 call disp%skip()

139

140 call disp%skip()

141 call disp%show("mu(1:NP:NP/5)")

142 call disp%show( mu(1:NP:NP/5) )

143 call disp%show("invSigma(1:NP:NP/5)")

144 call disp%show( invSigma(1:NP:NP/5) )

145 call disp%show("Point(1:NP:NP/5)")

146 call disp%show( Point(1:NP:NP/5) )

147 call disp%show("call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1:NP:NP/5), invSigma(1:NP:NP/5))")

148 call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1:NP:NP/5), invSigma(1:NP:NP/5))

149 call disp%show("CDF(1:NP:NP/5)")

150 call disp%show( CDF(1:NP:NP/5) )

151 call disp%skip()

152

153 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

154 ! Output an example array for visualization.

155 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

156

157 block

158 integer(IK) :: fileUnit, i

159 real(RK) , parameter :: mu(*) = [0._RK, 0._RK, 0._RK, -2._RK]

160 real(RK) , parameter :: invSigma(*) = 1._RK / [3.0_RK, 1.0_RK, 0.3_RK, 1.0_RK]

161 open(newunit = fileUnit, file = "setNormCDF.RK.txt")

162 do i = 1, NP

163 call setNormCDF(CDF(1:4), Point(i), mu, invSigma)

164 write(fileUnit,"(5(g0,:,' '))") Point(i), CDF(1:4)

165 end do

166 close(fileUnit)

167 end block

168

169end program example

pm_arraySpace::setLinSpace
Return the linSpace output argument with size(linSpace) elements of evenly-spaced values over the int...
Definition: pm_arraySpace.F90:324

pm_arraySpace::setLogSpace
Return the logSpace output argument with size(logSpace) elements of logarithmically-evenly-spaced val...
Definition: pm_arraySpace.F90:765

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_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::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! Compute the Cumulative Distribution Function (CDF) of the (Standard) Normal distribution.

4!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

5

6

7!%%%%%%%%%%%%%%

8! Standard CDF.

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

10

11

12Point(NP/2)

13-0.10010010010010006E-1

14call setNormCDF(CDF(1), Point(NP/2))

15CDF(1)

16+0.49600665046901393

17

18

19!%%%%%%%%%%%%%%%%%

20! CDF with a mean.

21!%%%%%%%%%%%%%%%%%

22

23

24mu(1)

25-5.0000000000000000

26Point(1)

27-10.000000000000000

28call setNormCDF(CDF(1), Point(1), mu(1))

29CDF(1)

30+0.28665157186802404E-6

31

32

33!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

34! CDF with a standard deviation.

35!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

36

37

38invSigma(1)

39+0.10000000000000003

40Point(1)

41-10.000000000000000

42call setNormCDF(CDF(1), Point(1), invSigma(1))

43CDF(1)

44+0.0000000000000000

45

46

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

48! CDF with a mean and a standard deviation.

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

50

51

52mu(1)

53-5.0000000000000000

54invSigma(1)

55+0.10000000000000003

56Point(1)

57-10.000000000000000

58call setNormCDF(CDF(1), Point(1), mu(1), invSigma(1))

59CDF(1)

60+0.30853753872598683

61

62

63!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

64! A vector of CDF at different points with the same mean and standard deviation.

65!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

66

67

68mu(1)

69-5.0000000000000000

70invSigma(1)

71+0.10000000000000003

72Point(1:NP:NP/5)

73-10.000000000000000, -5.9959959959959956, -1.9919919919919913, +2.0120120120120113, +6.0160160160160174

74call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1), invSigma(1))

75CDF(1:NP:NP/5)

76+0.30853753872598683, +0.46033110585969739, +0.61821680114072308, +0.75841126983960572, +0.86468254413044943

77

78

79!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

80! A vector of CDF at the same point but with different means and standard deviations.

81!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

82

83

84mu(1:NP:NP/5)

85-5.0000000000000000, -2.9979979979979978, -0.99599599599599564, +1.0060060060060056, +3.0080080080080087

86invSigma(1:NP:NP/5)

87+0.10000000000000003, +0.25142033481427983, +0.63212184758124557, +1.5892828656229783, +3.9957803018952722

88Point(1)

89-10.000000000000000

90call setNormCDF(CDF(1:NP:NP/5), Point(1), mu(1:NP:NP/5), invSigma(1:NP:NP/5))

91CDF(1:NP:NP/5)

92+0.30853753872598683, +0.39166134321215629E-1, +0.62917015486441130E-8, +0.0000000000000000, +0.0000000000000000

93

94

95!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

96! A vector of CDF at different points with different means and a standard deviations.

97!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

98

99

100mu(1:NP:NP/5)

101-5.0000000000000000, -2.9979979979979978, -0.99599599599599564, +1.0060060060060056, +3.0080080080080087

102invSigma(1:NP:NP/5)

103+0.10000000000000003, +0.25142033481427983, +0.63212184758124557, +1.5892828656229783, +3.9957803018952722

104Point(1:NP:NP/5)

105-10.000000000000000, -5.9959959959959956, -1.9919919919919913, +2.0120120120120113, +6.0160160160160174

106call setNormCDF(CDF(1:NP:NP/5), Point(1:NP:NP/5), mu(1:NP:NP/5), invSigma(1:NP:NP/5))

107CDF(1:NP:NP/5)

108+0.30853753872598683, +0.22549737584883295, +0.26448116263651789, +0.94507059915700919, +1.0000000000000000

109

110

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

2

3import matplotlib.pyplot as plt

4import pandas as pd

5import numpy as np

6import glob

7import sys

8

9linewidth = 2

10fontsize = 17

11

12marker ={ "CK" : "-"

13 , "IK" : "."

14 , "RK" : "-"

15 }

16xlab = { "CK" : "X ( real/imaginary components )"

17 , "IK" : "X ( integer-valued )"

18 , "RK" : "X ( real-valued )"

19 }

20legends = [ r"$\mu = 0.0,~\sigma = 3.0$"

21 , r"$\mu = 0.0,~\sigma = 1.0$"

22 , r"$\mu = 0.0,~\sigma = 0.3$"

23 , r"$\mu = -2.,~\sigma = 1.0$"

24 ]

25

26for kind in ["IK", "CK", "RK"]:

27

28 pattern = "*." + kind + ".txt"

29 fileList = glob.glob(pattern)

30 if len(fileList) == 1:

31

32 df = pd.read_csv(fileList[0], delimiter = " ")

33

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

35 ax = plt.subplot()

36

37 if kind == "CK":

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

39 , df.values[:,1:5]

40 , marker[kind]

41 , linewidth = linewidth

42 #, color = "r"

43 )

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

45 , df.values[:,1:5]

46 , marker[kind]

47 , linewidth = linewidth

48 #, color = "blue"

49 )

50 else:

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

52 , df.values[:,1:5]

53 , marker[kind]

54 , linewidth = linewidth

55 #, color = "r"

56 )

57 ax.legend ( legends

58 , fontsize = fontsize

59 )

60

61 plt.xticks(fontsize = fontsize - 2)

62 plt.yticks(fontsize = fontsize - 2)

63 ax.set_xlabel(xlab[kind], fontsize = 17)

64 ax.set_ylabel("Cumulative Distribution Function (CDF)", fontsize = 17)

65

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

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

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

69

70 plt.savefig(fileList[0].replace(".txt",".png"))

71

72 elif len(fileList) > 1:

73

74 sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:: test_pm_distNorm

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 886 of file pm_distNorm.F90.

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

src/fortran/main/pm_distNorm.F90