Generate and return the Probability Density Function (PDF) of the Raised Cosine distribution for an input x within the support of the distribution $x \in [\mu - \sigma, \mu + \sigma]$. More...

Detailed Description

Generate and return the Probability Density Function (PDF) of the Raised Cosine distribution for an input x within the support of the distribution $x \in [\mu - \sigma, \mu + \sigma]$.

See the documentation of pm_distCosRaised for more information on the Raised Cosine distribution.

Parameters

[in]	x	: The input scalar or array of the same shape as other array like arguments, of type `real` of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the value at which the PDF 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 the input argument `x`, containing the location parameter of the distribution. (optional, default = $0.$)
[in]	sigma	: The input scalar or array of the same shape as other array-like arguments, of the same type and kind as `x`, containing the scale parameter of the distribution. (optional, default = $1.$)

Returns: pdf : The output scalar or array of the same shape as any input array-like argument, of the same type and kind the input argument x, containing the PDF of the distribution.

Possible calling interfaces ⛓

: use pm_distCosRaised, only: getCosRaisedPDF

pdf = getCosRaisedPDF(x, mu = mu, sigma = sigma)

pm_distCosRaised::getCosRaisedPDF
Generate and return the Probability Density Function (PDF) of the Raised Cosine distribution for an i...
Definition: pm_distCosRaised.F90:196

pm_distCosRaised
This module contains classes and procedures for computing various statistical quantities related to t...
Definition: pm_distCosRaised.F90:83

Warning: The condition 0. < sigma must hold for the corresponding input arguments.
The condition x >= mu - sigma must hold for the corresponding input arguments.
The condition x <= mu + sigma must hold for the corresponding input arguments.
These conditions are 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: setCosRaisedPDF

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK

4 use pm_kind, only: RKG => RK ! all real kinds are supported.

5 use pm_arrayMembership, only: operator(.inrange.)

6 use pm_mathSubAdd, only: operator(.subadd.)

7 use pm_distCosRaised, only: getCosRaisedPDF

8 use pm_arraySpace, only: getLinSpace

9 use pm_arraySpace, only: setLinSpace

10 use pm_arraySpace, only: setLogSpace

11 use pm_io, only: display_type

12

13 implicit none

14

15 integer(IK), parameter :: NP = 5_IK

16 real(RKG), dimension(NP) :: mu, sigma, PDF

17

18 type(display_type) :: disp

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

20

21 call disp%skip()

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

23 call disp%show("! The standard PDF.")

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

25 call disp%skip()

26

27 call setLinSpace(mu, x1 = -5._RKG, x2 = +5._RKG)

28 call setLogSpace(sigma, logx1 = log(0.1_RKG), logx2 = log(10._RKG))

29

30 call disp%skip()

31 call disp%show("PDF(1) = getCosRaisedPDF(0._RKG)")

32 PDF(1) = getCosRaisedPDF(0._RKG)

33 call disp%show("PDF(1)")

34 call disp%show( PDF(1) )

35 call disp%skip()

36

37 call disp%skip()

38 call disp%show("PDF(1:5) = getCosRaisedPDF([real(RKG) :: -1., -.5, 0., .5, 1.])")

39 PDF(1:5) = getCosRaisedPDF([real(RKG) :: -1., -.5, 0., .5, 1.])

40 call disp%show("PDF(1:5)")

41 call disp%show( PDF(1:5) )

42 call disp%skip()

43

44 call disp%skip()

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

46 call disp%show("! PDF with a mean.")

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

48 call disp%skip()

49

50 call disp%skip()

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

52 call disp%show( mu(1) )

53 call disp%show("PDF(1) = getCosRaisedPDF(mu(1), mu(1))")

54 PDF(1) = getCosRaisedPDF(mu(1), mu(1))

55 call disp%show("PDF(1)")

56 call disp%show( PDF(1) )

57 call disp%skip()

58

59 call disp%skip()

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

61 call disp%show("! PDF with a scale parameter.")

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

63 call disp%skip()

64

65 call disp%skip()

66 call disp%show("sigma(1)")

67 call disp%show( sigma(1) )

68 call disp%show("PDF(1) = getCosRaisedPDF(0._RKG, sigma = sigma(1))")

69 PDF(1) = getCosRaisedPDF(0._RKG, sigma = sigma(1))

70 call disp%show("PDF(1)")

71 call disp%show( PDF(1) )

72 call disp%skip()

73

74 call disp%skip()

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

76 call disp%show("! PDF with a mean and a standard deviation.")

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

78 call disp%skip()

79

80 call disp%skip()

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

82 call disp%show( mu(1) )

83 call disp%show("sigma(1)")

84 call disp%show( sigma(1) )

85 call disp%show("PDF(1:2) = getCosRaisedPDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))")

86 PDF(1:2) = getCosRaisedPDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))

87 call disp%show("PDF(1:2)")

88 call disp%show( PDF(1:2) )

89 call disp%skip()

90

91 call disp%skip()

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

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

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

95 call disp%skip()

96

97 call disp%skip()

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

99 call disp%show( mu(1) )

100 call disp%show("sigma(1)")

101 call disp%show( sigma(1) )

102 call disp%show("PDF(1:NP) = getCosRaisedPDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))")

103 PDF(1:NP) = getCosRaisedPDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))

104 call disp%show("PDF(1:NP)")

105 call disp%show( PDF(1:NP) )

106 call disp%skip()

107

108 call disp%skip()

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

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

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

112 call disp%skip()

113

114 call disp%skip()

115 call disp%show("mu(1:NP)")

116 call disp%show( mu(1:NP) )

117 call disp%show("sigma(1:NP)")

118 call disp%show( sigma(1:NP) )

119 call disp%show("PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))")

120 PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))

121 call disp%show("PDF(1:NP)")

122 call disp%show( PDF(1:NP) )

123 call disp%skip()

124

125 call disp%skip()

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

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

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

129 call disp%skip()

130

131 call disp%skip()

132 call disp%show("mu(1:NP)")

133 call disp%show( mu(1:NP) )

134 call disp%show("sigma(1:NP)")

135 call disp%show( sigma(1:NP) )

136 call disp%show("PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))")

137 PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))

138 call disp%show("PDF(1:NP)")

139 call disp%show( PDF(1:NP) )

140 call disp%skip()

141

142 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

143 ! Output an example array for visualization.

144 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

145

146 block

147 integer(IK) :: fileUnit, i, j

148 real(RKG) :: point(1000), PDF(4), mu(4), sigma(4)

149 open(newunit = fileUnit, file = "getCosRaisedPDF.RK.txt")

150 call setLinSpace(point, x1 = -4._RKG, x2 = +4._RKG)

151 sigma = [+3._RKG, +1._RKG, +.3_RKG, 1._RKG]

152 mu = [+0._RKG, +0._RKG, +0._RKG, -2._RKG]

153 do i = 1, size(point)

154 do j = 1, size(PDF)

155 if(point(i) .inrange. (mu(j) .subadd. sigma(j))) then

156 PDF(j) = getCosRaisedPDF(point(i), mu(j), sigma(j))

157 else

158 PDF(j) = 0._RKG

159 end if

160 end do

161 write(fileUnit,"(5(g0,:,' '))") point(i), PDF

162 end do

163 close(fileUnit)

164 end block

165

166end program example

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

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_arrayMembership
This module contains procedures and generic interfaces for assessing whether particular value(s) or a...
Definition: pm_arrayMembership.F90:75

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_mathSubAdd
This module contains procedures and generic interfaces for evaluating the mathematical operator acti...
Definition: pm_mathSubAdd.F90:67

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! The standard PDF.

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

5

6

7PDF(1) = getCosRaisedPDF(0._RKG)

8PDF(1)

9+1.0000000000000000

10

11

12PDF(1:5) = getCosRaisedPDF([real(RKG) :: -1., -.5, 0., .5, 1.])

13PDF(1:5)

14+0.0000000000000000, +0.50000000000000000, +1.0000000000000000, +0.50000000000000000, +0.0000000000000000

15

16

17!%%%%%%%%%%%%%%%%%

18! PDF with a mean.

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

20

21

22mu(1)

23-5.0000000000000000

24PDF(1) = getCosRaisedPDF(mu(1), mu(1))

25PDF(1)

26+1.0000000000000000

27

28

29!%%%%%%%%%%%%%%%%%%%%%%%%%%%%

30! PDF with a scale parameter.

31!%%%%%%%%%%%%%%%%%%%%%%%%%%%%

32

33

34sigma(1)

35+0.10000000000000003

36PDF(1) = getCosRaisedPDF(0._RKG, sigma = sigma(1))

37PDF(1)

38+9.9999999999999964

39

40

41!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

42! PDF with a mean and a standard deviation.

43!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

44

45

46mu(1)

47-5.0000000000000000

48sigma(1)

49+0.10000000000000003

50PDF(1:2) = getCosRaisedPDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))

51PDF(1:2)

52+5.0000000000000293, +5.0000000000000293

53

54

55!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

56! A vector of PDF at different points with the same mean and standard deviation.

57!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

58

59

60mu(1)

61-5.0000000000000000

62sigma(1)

63+0.10000000000000003

64PDF(1:NP) = getCosRaisedPDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))

65PDF(1:NP)

66+0.0000000000000000, +5.0000000000000293, +9.9999999999999964, +5.0000000000000293, +0.0000000000000000

67

68

69!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

71!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

72

73

74mu(1:NP)

75-5.0000000000000000, -2.5000000000000000, +0.0000000000000000, +2.5000000000000000, +5.0000000000000000

76sigma(1:NP)

77+0.10000000000000003, +0.31622776601683800, +1.0000000000000004, +3.1622776601683813, +10.000000000000002

78PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))

79PDF(1:NP)

80+9.9999999999999964, +3.1622776601683786, +0.99999999999999956, +0.31622776601683772, +0.99999999999999978E-1

81

82

83!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

84! A vector of PDF at different points with different means and a standard deviations.

85!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

86

87

88mu(1:NP)

89-5.0000000000000000, -2.5000000000000000, +0.0000000000000000, +2.5000000000000000, +5.0000000000000000

90sigma(1:NP)

91+0.10000000000000003, +0.31622776601683800, +1.0000000000000004, +3.1622776601683813, +10.000000000000002

92PDF(1:NP) = getCosRaisedPDF(mu(1:NP), mu(1:NP), sigma(1:NP))

93PDF(1:NP)

94+9.9999999999999964, +3.1622776601683786, +0.99999999999999956, +0.31622776601683772, +0.99999999999999978E-1

95

96

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("Probability Density Function (PDF)", 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_distCosRaised

Todo:: Very Low Priority: This generic interface can be extended to complex arguments.

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 196 of file pm_distCosRaised.F90.

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

src/fortran/main/pm_distCosRaised.F90