pm_distCosRaised::getCosRaisedCDF Interface Reference

Generate and return the Cumulative Distribution Function (CDF) 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 Cumulative Distribution Function (CDF) 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 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 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

cdf : 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 CDF of the distribution.

Possible calling interfaces ⛓

use pm_distCosRaised, only: getCosRaisedCDF
 
cdf = getCosRaisedCDF(x, mu = mu, sigma = sigma)

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

setCosRaisedCDF

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK
    use pm_kind, only: RKG => RK ! all real kinds are supported.
    use pm_arrayMembership, only: operator(.inrange.)
    use pm_mathSubAdd, only: operator(.subadd.)
    use pm_distCosRaised, only: getCosRaisedCDF
    use pm_arraySpace, only: getLinSpace
    use pm_arraySpace, only: setLinSpace
    use pm_arraySpace, only: setLogSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 5_IK
    real(RKG), dimension(NP) :: mu, sigma, CDF
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%")
    call disp%show("! The standard CDF.")
    call disp%show("!%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call setLinSpace(mu, x1 = -5._RKG, x2 = +5._RKG)
    call setLogSpace(sigma, logx1 = log(0.1_RKG), logx2 = log(10._RKG))
 
    call disp%skip()
    call disp%show("CDF(1) = getCosRaisedCDF(0._RKG)")
                    CDF(1) = getCosRaisedCDF(0._RKG)
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("CDF(1:5) = getCosRaisedCDF([real(RKG) :: -1., -.5, 0., .5, 1.])")
                    CDF(1:5) = getCosRaisedCDF([real(RKG) :: -1., -.5, 0., .5, 1.])
    call disp%show("CDF(1:5)")
    call disp%show( CDF(1:5) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a mean.")
    call disp%show("!%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("CDF(1) = getCosRaisedCDF(mu(1), mu(1))")
                    CDF(1) = getCosRaisedCDF(mu(1), mu(1))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a scale parameter.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("sigma(1)")
    call disp%show( sigma(1) )
    call disp%show("CDF(1) = getCosRaisedCDF(0._RKG, sigma = sigma(1))")
                    CDF(1) = getCosRaisedCDF(0._RKG, sigma = sigma(1))
    call disp%show("CDF(1)")
    call disp%show( CDF(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! CDF with a mean and a standard deviation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("sigma(1)")
    call disp%show( sigma(1) )
    call disp%show("CDF(1:2) = getCosRaisedCDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))")
                    CDF(1:2) = getCosRaisedCDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))
    call disp%show("CDF(1:2)")
    call disp%show( CDF(1:2) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with the same mean and standard deviation.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1)")
    call disp%show( mu(1) )
    call disp%show("sigma(1)")
    call disp%show( sigma(1) )
    call disp%show("CDF(1:NP) = getCosRaisedCDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))")
                    CDF(1:NP) = getCosRaisedCDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))
    call disp%show("CDF(1:NP)")
    call disp%show( CDF(1:NP) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at the same point but with different means and standard deviations.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1:NP)")
    call disp%show( mu(1:NP) )
    call disp%show("sigma(1:NP)")
    call disp%show( sigma(1:NP) )
    call disp%show("CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))")
                    CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))
    call disp%show("CDF(1:NP)")
    call disp%show( CDF(1:NP) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! A vector of CDF at different points with different means and a standard deviations.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    call disp%skip()
    call disp%show("mu(1:NP)")
    call disp%show( mu(1:NP) )
    call disp%show("sigma(1:NP)")
    call disp%show( sigma(1:NP) )
    call disp%show("CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))")
                    CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))
    call disp%show("CDF(1:NP)")
    call disp%show( CDF(1:NP) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer(IK) :: fileUnit, i, j
        real(RKG) :: point(1000), CDF(4), mu(4), sigma(4)
        open(newunit = fileUnit, file = "getCosRaisedCDF.RK.txt")
        call setLinSpace(point, x1 = -4._RKG, x2 = +4._RKG)
        sigma = [+3._RKG, +1._RKG, +.3_RKG, 1._RKG]
        mu = [+0._RKG, +0._RKG, +0._RKG, -2._RKG]
        do i = 1, size(point)
            do j = 1, size(CDF)
                if(point(i) .inrange. (mu(j) .subadd. sigma(j))) then
                    CDF(j) = getCosRaisedCDF(point(i), mu(j), sigma(j))
                elseif (point(i) > mu(j) + sigma(j)) then
                    CDF(j) = 1._RKG
                else
                    CDF(j) = 0._RKG
                end if
            end do
            write(fileUnit,"(5(g0,:,' '))") point(i), CDF
        end do
        close(fileUnit)
    end block
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%
! The standard CDF.
!%%%%%%%%%%%%%%%%%%
 
 
CDF(1) = getCosRaisedCDF(0._RKG)
CDF(1)
+0.50000000000000000
 
 
CDF(1:5) = getCosRaisedCDF([real(RKG) :: -1., -.5, 0., .5, 1.])
CDF(1:5)
-0.19490859162596877E-16, +0.90845056908104654E-1, +0.50000000000000000, +0.90915494309189537, +1.0000000000000000
 
 
!%%%%%%%%%%%%%%%%%
! CDF with a mean.
!%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
CDF(1) = getCosRaisedCDF(mu(1), mu(1))
CDF(1)
+0.50000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! CDF with a scale parameter.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
sigma(1)
+0.10000000000000003
CDF(1) = getCosRaisedCDF(0._RKG, sigma = sigma(1))
CDF(1)
+0.50000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! CDF with a mean and a standard deviation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
sigma(1)
+0.10000000000000003
CDF(1:2) = getCosRaisedCDF(mu(1) .subadd. sigma(1) / 2, mu(1), sigma(1))
CDF(1:2)
+0.90845056908105654E-1, +0.90915494309189437
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with the same mean and standard deviation.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1)
-5.0000000000000000
sigma(1)
+0.10000000000000003
CDF(1:NP) = getCosRaisedCDF(getLinSpace(mu(1) - sigma(1), mu(1) + sigma(1), count = NP), mu(1), sigma(1))
CDF(1:NP)
+0.15066625053317021E-16, +0.90845056908105654E-1, +0.50000000000000000, +0.90915494309189437, +0.99999999999999989
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at the same point but with different means and standard deviations.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1:NP)
-5.0000000000000000, -2.5000000000000000, +0.0000000000000000, +2.5000000000000000, +5.0000000000000000
sigma(1:NP)
+0.10000000000000003, +0.31622776601683800, +1.0000000000000004, +3.1622776601683813, +10.000000000000002
CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))
CDF(1:NP)
+0.50000000000000000, +0.50000000000000000, +0.50000000000000000, +0.50000000000000000, +0.50000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! A vector of CDF at different points with different means and a standard deviations.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
mu(1:NP)
-5.0000000000000000, -2.5000000000000000, +0.0000000000000000, +2.5000000000000000, +5.0000000000000000
sigma(1:NP)
+0.10000000000000003, +0.31622776601683800, +1.0000000000000004, +3.1622776601683813, +10.000000000000002
CDF(1:NP) = getCosRaisedCDF(mu(1:NP), mu(1:NP), sigma(1:NP))
CDF(1:NP)
+0.50000000000000000, +0.50000000000000000, +0.50000000000000000, +0.50000000000000000, +0.50000000000000000
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import glob
import sys
 
linewidth = 2
fontsize = 17
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : "X ( real/imaginary components )"
        , "IK" : "X ( integer-valued )"
        , "RK" : "X ( real-valued )"
        }
legends =   [ r"$\mu = 0.0,~\sigma = 3.0$"
            , r"$\mu = 0.0,~\sigma = 1.0$"
            , r"$\mu = 0.0,~\sigma = 0.3$"
            , r"$\mu = -2.,~\sigma = 1.0$"
            ]
 
for kind in ["IK", "CK", "RK"]:
 
    pattern = "*." + kind + ".txt"
    fileList = glob.glob(pattern)
    if len(fileList) == 1:
 
        df = pd.read_csv(fileList[0], delimiter = " ")
 
        fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
        ax = plt.subplot()
 
        if kind == "CK":
            plt.plot( df.values[:, 0]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            plt.plot( df.values[:, 1]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1:5]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            ax.legend   ( legends
                        , fontsize = fontsize
                        )
 
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        ax.set_xlabel(xlab[kind], fontsize = 17)
        ax.set_ylabel("Cumulative Distribution Function (CDF)", fontsize = 17)
 
        plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
        ax.tick_params(axis = "y", which = "minor")
        ax.tick_params(axis = "x", which = "minor")
 
        plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        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.

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 576 of file pm_distCosRaised.F90.

---

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

• src/fortran/main/pm_distCosRaised.F90