pm_sampleVar::getVarCorrection Interface Reference

Generate and return the bias correction factor for the computation of the variance of a (weighted) sample.
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Detailed Description

Generate and return the bias correction factor for the computation of the variance of a (weighted) sample.

When the population mean is replaced with sample mean, multiplying the output of this generic interface with the output (co)variance from setVar or setCov yields an unbiased estimate of the sample variance.
See the documentation of the parent module pm_sampleVar for algorithmic details and variance definitions.

Parameters

[in]	weisum	: The input scalar of, type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing either, the size of an unweighted sample, only if the input argument correction is missing or is not rweight_type, the quantity sum(weight) of a weighted sample, only if the input argument correction is missing or is not rweight_type, based upon which the bias correction is computed. The vector weight(1:nsam) refers to the weights of a sample of size nsam.
[in]	weisqs	: The input scalar of the same type and kind as weisum, containing the sum of squared sample weight: sum(weight**2). (optional. If missing, the output corresponds to the Bessel correction (i.e., frequency weight is assumed). If present, the correction corresponding to reliability weights is returned.)

Returns

normfac : The output scalar of the same type and kind as weisum containing the bias correction factor of variance equation.
Multiplying normfac with the sum of the sample weight (or its size, if unweighted) yields in the Bias Correction factor.

Possible calling interfaces ⛓

use pm_sampleVar, only: getVarCorrection
 
normfac = getVarCorrection(weisum) ! unweighted or frequency-weighted sample variance correction
normfac = getVarCorrection(weisum, weisqs) ! reliability-weighted sample variance correction
!

Warning

The condition 0 < weisum must hold for the corresponding input arguments.
The condition 0 < weisqs must hold for the corresponding input arguments.
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.

Note

Note the effects of bias-correction in computing the variance become noticeable only for very sample sample sizes (i.e., when nsam is small).

See also

getVar
setVar
getCov
setCov

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, LK
    use pm_io, only: display_type
    use pm_distUnif, only: getUnifRand
    use pm_sampleVar, only: getVarCorrection
 
    implicit none
 
    integer(IK) :: itry, ntry = 10
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    block
        use pm_kind, only: RKG => RKS ! All other real types are also supported.
        integer(RKG), allocatable :: weight(:)
        real(RKG) :: correction
        do itry = 1, ntry
            call disp%skip()
            call disp%show("weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))")
                            weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
            call disp%show("weight")
            call disp%show( weight )
            call disp%show("correction = getVarCorrection(real(sum(weight), RKG))")
                            correction = getVarCorrection(real(sum(weight), RKG))
            call disp%show("correction")
            call disp%show( correction )
            call disp%show("correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))")
                            correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
            call disp%show("correction")
            call disp%show( correction )
            call disp%skip()
        end do
    end block
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example rand array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        use pm_kind, only: RKG => RK
        use pm_arraySpace, only: getLogSpace
        use pm_io, only: getErrTableWrite, trans
        real(RKG), allocatable :: sampleSize(:)
        sampleSize = getLogSpace(log(1.1_RKG), log(1001._RKG), 1000_IK)
        if (0 /= getErrTableWrite("getVarCorrection.RK.txt", reshape([sampleSize, getVarCorrection(sampleSize)], [size(sampleSize), 2]), header = SK_"Sample Size,Bessel Correction")) error stop 'table write failed.'
    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 ⛓

 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+7, +3, +8, +6, +1, +8
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.03125000
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.25750577
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+1, +6, +3, +4
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.07692313
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.46268654
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+2, +9
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.10000002
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+3.36111116
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+5, +3, +6, +1, +7, +9, +3, +2, +3, +7, +5, +6, +1, +9, +9, +6, +7
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.01136363
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.07915533
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+4, +6, +5, +4, +8, +9, +1, +4, +4
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.02272725
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.15450394
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+2, +4, +5, +3, +7, +8, +8, +2, +7, +2, +5, +6, +8
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.01515150
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.10132480
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+8, +3
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.10000002
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+2.52083325
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+8, +2, +7
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.06250000
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.68023252
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+3, +2, +3, +4, +4, +8, +3, +4, +2, +5, +7, +1, +2, +4, +4, +8, +2
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.01538467
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.08089328
 
 
weight = getUnifRand(1, 9, getUnifRand(2_IK, 20_IK))
weight
+9, +2, +5, +3, +5
correction = getVarCorrection(real(sum(weight), RKG))
correction
+1.04347825
correction = getVarCorrection(real(sum(weight), RKG), real(sum(weight**2), RKG))
correction
+1.33333337
 
 

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
 
pattern = "*.RK.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()
 
    plt.plot( df.values[:, 0]
            , df.values[:,1:]
            , linewidth = linewidth
           #, color = "r"
            )
    ax.legend   ( list(df.columns.values[1:])
                , fontsize = fontsize
                )
 
    plt.xticks(fontsize = fontsize - 2)
    plt.yticks(fontsize = fontsize - 2)
    ax.set_xscale("log")
    ax.set_yscale("log")
    ax.set_xlabel(df.columns.values[0], fontsize = 17)
    ax.set_ylabel("Correction", 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_sampleVar

Final Remarks ⛓

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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:

Fatemeh Bagheri, Monday 02:15 AM, September 27, 2021, Dallas, TX

Definition at line 481 of file pm_sampleVar.F90.

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

• src/fortran/main/pm_sampleVar.F90