ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
pm_distNorm::getNormCDF Interface Reference

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
[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]sigma: The input scalar of the same type and kind as the output cdf representing the scale parameter of the distribution.
(optional, default = 1.)
Returns
cdf : The output scalar or array of the same shape as the input array-like arguments, of,
  1. 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.


Possible calling interfaces ⛓

use pm_distNorm, only: getNormCDF
cdf = getNormCDF(x, mu = mu, sigma = sigma)
pm_distNorm::getNormCDF
Generate and return the Cumulative Distribution Function (CDF) of the univariate Normal distribution.
Definition: pm_distNorm.F90:754
pm_distNorm
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distNorm.F90:179
Warning
The condition 0. < sigma 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
setNormCDF


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: getNormCDF
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, Sigma, 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(Sigma, 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("CDF(1) = getNormCDF(Point(NP/2))")
38 CDF(1) = getNormCDF(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("CDF(1) = getNormCDF(Point(1), mu(1))")
55 CDF(1) = getNormCDF(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("Sigma(1)")
68 call disp%show( Sigma(1) )
69 call disp%show("Point(1)")
70 call disp%show( Point(1) )
71 call disp%show("CDF(1) = getNormCDF(Point(1), Sigma(1))")
72 CDF(1) = getNormCDF(Point(1), Sigma(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("Sigma(1)")
87 call disp%show( Sigma(1) )
88 call disp%show("Point(1)")
89 call disp%show( Point(1) )
90 call disp%show("CDF(1) = getNormCDF(Point(1), mu(1), Sigma(1))")
91 CDF(1) = getNormCDF(Point(1), mu(1), Sigma(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("Sigma(1)")
106 call disp%show( Sigma(1) )
107 call disp%show("Point(1:NP:NP/5)")
108 call disp%show( Point(1:NP:NP/5) )
109 call disp%show("CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1), Sigma(1))")
110 CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1), Sigma(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("Sigma(1:NP:NP/5)")
125 call disp%show( Sigma(1:NP:NP/5) )
126 call disp%show("Point(1)")
127 call disp%show( Point(1) )
128 call disp%show("CDF(1:NP:NP/5) = getNormCDF(Point(1), mu(1:NP:NP/5), Sigma(1:NP:NP/5))")
129 CDF(1:NP:NP/5) = getNormCDF(Point(1), mu(1:NP:NP/5), Sigma(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("Sigma(1:NP:NP/5)")
144 call disp%show( Sigma(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("CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(1:NP:NP/5))")
148 CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(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 open(newunit = fileUnit, file = "getNormCDF.RK.txt")
160 write(fileUnit,"(5(g0,:,' '))") (Point(i), getNormCDF(Point(i), [0._RK, 0._RK, 0._RK, -2._RK], sigma = [3.0_RK, 1.0_RK, 0.3_RK, 1.0_RK]), i = 1, NP)
161 close(fileUnit)
162 end block
163
164end 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
14CDF(1) = getNormCDF(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
28CDF(1) = getNormCDF(Point(1), mu(1))
29CDF(1)
30+0.28665157186802404E-6
31
32
33!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
34! CDF with a standard deviation.
35!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
36
37
38Sigma(1)
39+0.10000000000000003
40Point(1)
41-10.000000000000000
42CDF(1) = getNormCDF(Point(1), Sigma(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
54Sigma(1)
55+0.10000000000000003
56Point(1)
57-10.000000000000000
58CDF(1) = getNormCDF(Point(1), mu(1), Sigma(1))
59CDF(1)
60+0.0000000000000000
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
70Sigma(1)
71+0.10000000000000003
72Point(1:NP:NP/5)
73-10.000000000000000, -5.9959959959959956, -1.9919919919919913, +2.0120120120120113, +6.0160160160160174
74CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1), Sigma(1))
75CDF(1:NP:NP/5)
76+0.0000000000000000, +0.0000000000000000, +1.0000000000000000, +1.0000000000000000, +1.0000000000000000
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
86Sigma(1:NP:NP/5)
87+0.10000000000000003, +0.25142033481427983, +0.63212184758124557, +1.5892828656229783, +3.9957803018952722
88Point(1)
89-10.000000000000000
90CDF(1:NP:NP/5) = getNormCDF(Point(1), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
91CDF(1:NP:NP/5)
92+0.0000000000000000, +0.0000000000000000, +0.0000000000000000, +0.21777024628022446E-11, +0.56609157067133431E-3
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
102Sigma(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
106CDF(1:NP:NP/5) = getNormCDF(Point(1:NP:NP/5), mu(1:NP:NP/5), Sigma(1:NP:NP/5))
107CDF(1:NP:NP/5)
108+0.0000000000000000, +0.0000000000000000, +0.57554482194544898E-1, +0.73663111418286575, +0.77421378819753417
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[:, 1]
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.

  1. 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.
  2. 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 754 of file pm_distNorm.F90.

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