ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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pm_distGeomCyclic::isFailedGeomCyclicFit Interface Reference

Generate and return .true. if the parameters of a least-squares fit to the histogram representing a Cyclic-Geometric-distributed sample can be successfully inferred, otherwise, return .false..
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Detailed Description

Generate and return .true. if the parameters of a least-squares fit to the histogram representing a Cyclic-Geometric-distributed sample can be successfully inferred, otherwise, return .false..

See the documentation of pm_distGeomCyclic for more details on the Cyclic Geometric distribution.
Given a set of Cyclic Bernoulli trial success steps \(\ms{stepSuccess}\) and the corresponding number of successes \(\ms{freqSuccess}\) at these steps, with the supplied Cyclic Bernoulli trial \(\ms{period}\), the procedures of this generic interface return the best-fit parameters \((\ms{probSuccess}, \ms{normFac})\) of the following curve,

\begin{eqnarray} \large \ms{freqSuccess} &=& \ms{normFac} \times \pi_{\mathcal{CG}} (X = \ms{stepSuccess} ~|~ \ms{probSuccess}, \ms{period}) ~, \nonumber \\ &=& \ms{normFac} \times \frac{\ms{probSuccess} (1 - \ms{probSuccess})^{\ms{stepSuccess} - 1}}{1 - (1 - \ms{probSuccess})^{\ms{period}}} ~, \end{eqnarray}

where the probability of success on each trial is \(\ms{probSuccess}\), and \(\pi_{\mathcal{CG}}\) is the probability mass function of the \(\ms{stepSuccess}\)th trial being the first success in a cyclic trial set with the cycle \(\ms{period}\).

Note
This fitting appears frequently in Fork-Join parallel MCMC simulations of the ParaMonte library.
See Amir Shahmoradi, Fatemeh Bagheri (2020). ParaDRAM: A Cross-Language Toolbox for Parallel High-Performance Delayed-Rejection Adaptive Metropolis Markov Chain Monte Carlo Simulations. for details and discussion.
Parameters
[in]stepSuccess: The input contiguous positive vector of
  1. type real of the same kind as that of probSuccess,
  2. type integer of default kind IK,
containing the steps ( \(x\)) at which contribution frequencies represented by input vector freqSuccess have occurred.
[in]freqSuccess: The input contiguous positive vector of the same type, kind, and size as the input stepSuccess, containing the contribution frequencies at the corresponding stepSuccess in the histogram.
[in]period: The input positive scalar of type integer of default kind IK, containing the period of the Cyclic Geometric distribution.
The period represents the maximum number of Bernoulli trials in the experiment.
Note that the condition maxval(stepSuccess) <= period must hold and the sizes of stepSuccess and freqSuccess vectors must be smaller than the specified period.
[out]probSuccess: The output positive scalar of
  1. type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
representing the best-fit probability-of-success parameter of the Cyclic Geometric distribution inferred from the fitting.
[in]normFac: The output positive scalar of the same type and kind as the output probSuccess, containing the best-fit normalization constant of the histogram represented by the input stepSuccess and freqSuccess`.
Returns
failed : The output scalar of type logical of default kind LK that is .true. if and only if the best-fit output parameters are successfully inferred.
The algorithm can fail only if the optimizer fails to converge to set of best-fit parameters.


Possible calling interfaces

failed = isFailedGeomCyclicFit(stepSuccess(1:period), freqSuccess(1:period), period, probSuccess, normFac)
Generate and return .true. if the parameters of a least-squares fit to the histogram representing a C...
This module contains classes and procedures for computing various statistical quantities related to t...
Warning
The condition all(0 < stepSuccess) must hold for the corresponding input arguments.
The condition 1 < size(freqSuccess)) must hold for the corresponding input arguments.
The condition size(stepSuccess) == size(freqSuccess) must hold for the corresponding input arguments.
The condition all(stepSuccess <= size(stepSuccess)) must hold for the corresponding input arguments.
The condition all(stepSuccess <= period) must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.
Remarks
The procedures under discussion are impure.
See also
setGeomCyclicRand
setGeomCyclicLogPMF
setGeomCyclicLogCDF


Example usage

1program example
2
3 use pm_kind, only: SK, IK
9 use pm_io, only: getErrTableWrite
10 use pm_io, only: display_type
11
12 implicit none
13
14 integer(IK) :: period, i
15 type(display_type) :: disp
16 integer(IK), allocatable :: rand(:)
17 integer(IK), allocatable :: freqSuccess(:)
18 integer(IK), allocatable :: stepSuccess(:)
19 disp = display_type(file = "main.out.F90")
20
21 block
22 use pm_kind, only: RKG => RKH
23 real(RKG) :: probSuccess, probSuccessFit, normFacFit
24 real(RKG), allocatable :: x(:), y(:)
25 call disp%skip()
26 call disp%show("probSuccess = .1; period = 20")
27 probSuccess = .1; period = 20
28 call disp%show("rand = getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)])")
29 rand = getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)])
30 call disp%show("if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.rnd.txt', table = rand)) error stop 'table output failed.'")
31 if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.rnd.txt', table = rand)) error stop 'table output failed.'
32 call disp%show("call setUnique(getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)]), unique = stepSuccess, count = freqSuccess, order = -1_IK)")
33 call setUnique(getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)]), unique = stepSuccess, count = freqSuccess, order = -1_IK)
34 call disp%show("stepSuccess")
35 call disp%show( stepSuccess , format = SK_"(sp,20(g0,:,', '))")
36 call disp%show("freqSuccess")
37 call disp%show( freqSuccess , format = SK_"(sp,20(g0,:,', '))")
38 call disp%show("if (isFailedGeomCyclicFit(stepSuccess, freqSuccess, period, probSuccessFit, normFacFit)) error stop 'Fitting failed.'")
39 if (isFailedGeomCyclicFit(stepSuccess, freqSuccess, period, probSuccessFit, normFacFit)) error stop 'Fitting failed.'
40 call disp%show("[probSuccessFit, normFacFit]")
41 call disp%show( [probSuccessFit, normFacFit] )
42 call disp%show("x = getLinSpace(1._RKG, real(period, RKG), 500_IK) ! for visualization.")
43 x = getLinSpace(1._RKG, real(period, RKG), 500_IK) ! for visualization.
44 call disp%show("y = normFacFit * probSuccessFit * (1 - probSuccessFit)**(x - 1) / (1 - (1 - probSuccessFit)**period) ! for visualization.")
45 y = normFacFit * probSuccessFit * (1 - probSuccessFit)**(x - 1) / (1 - (1 - probSuccessFit)**period) ! for visualization.
46 call disp%show("y = normFacFit * exp(getGeomCyclicLogPMF(x, probSuccessFit, period)) ! for visualization.")
47 y = normFacFit * exp(getGeomCyclicLogPMF(x, probSuccessFit, period)) ! for visualization.
48 call disp%show("if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.fit.txt', table = reshape([x, y], [size(x), 2]))) error stop 'table output failed.'")
49 if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.fit.txt', table = reshape([x, y], [size(x), 2]))) error stop 'table output failed.'
50 call disp%skip()
51 end block
52
53end program example
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Return a vector of unique values in the input array in place of the array itself.
Generate and return the natural logarithm of the Probability Mass Function (PMF) of the Cyclic Geomet...
Generate and return a scalar (or array of arbitrary rank of) random value(s) from the Cyclic Geometri...
Generate and return the iostat code resulting from writing the input table of rank 1 or 2 to the spec...
Definition: pm_io.F90:5940
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains procedures and generic interfaces for finding unique values of an input array of...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
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
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
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Definition: pm_kind.F90:858
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
2probSuccess = .1; period = 20
3rand = getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)])
4if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.rnd.txt', table = rand)) error stop 'table output failed.'
5call setUnique(getGeomCyclicRand(probSuccess, [(period, i = 1, 1000)]), unique = stepSuccess, count = freqSuccess, order = -1_IK)
6stepSuccess
7+1, +2, +4, +3, +5, +6, +8, +9, +7, +11, +12, +10, +13, +14, +15, +17, +16, +20, +18, +19
8freqSuccess
9+114, +97, +91, +90, +84, +76, +55, +55, +52, +51, +36, +33, +31, +26, +25, +24, +23, +15, +12, +10
10if (isFailedGeomCyclicFit(stepSuccess, freqSuccess, period, probSuccessFit, normFacFit)) error stop 'Fitting failed.'
11[probSuccessFit, normFacFit]
12+0.105651955925097061021975721637957888, +976.572977671411732916580527403386634
13x = getLinSpace(1._RKG, real(period, RKG), 500_IK) ! for visualization.
14y = normFacFit * probSuccessFit * (1 - probSuccessFit)**(x - 1) / (1 - (1 - probSuccessFit)**period) ! for visualization.
15y = normFacFit * exp(getGeomCyclicLogPMF(x, probSuccessFit, period)) ! for visualization.
16if (0 /= getErrTableWrite(file = 'isFailedGeomCyclicFit.fit.txt', table = reshape([x, y], [size(x), 2]))) error stop 'table output failed.'
17
18

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 }
16label = [ r"probSuccess = .2, period = 20"
17 , r"Histogram Fit"
18 ]
19
20fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
21ax = plt.subplot()
22
23dfrnd = pd.read_csv("isFailedGeomCyclicFit.rnd.txt", delimiter = ",", header = None)
24for j in range(len(dfrnd.values[0,:])):
25 plt.hist( dfrnd.values[:,j]
26 , histtype = "stepfilled"
27 , density = False
28 , alpha = 0.5
29 , bins = 75
30 )
31dffit = pd.read_csv("isFailedGeomCyclicFit.fit.txt", delimiter = ",", header = None)
32for j in range(1, len(dffit.values[0,:])):
33 plt.plot( dffit.values[:, 0]
34 , dffit.values[:,j]
35 , linewidth = 2
36 , c = "red"
37 )
38 ax.legend ( label
39 , fontsize = fontsize
40 )
41plt.xticks(fontsize = fontsize - 2)
42plt.yticks(fontsize = fontsize - 2)
43#ax.set_xlim(0, 10)
44ax.set_xlabel("Cyclic Geometric Random Value ( integer-valued )", fontsize = 17)
45ax.set_ylabel("Count", fontsize = 17)
46ax.set_title("{} Cyclic Geometric random values and histogram fit".format(len(dfrnd.values[:, 0])), fontsize = 17)
47
48plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
49ax.tick_params(axis = "y", which = "minor")
50ax.tick_params(axis = "x", which = "minor")
51
52plt.savefig("isFailedGeomCyclicFit.png")

Visualization of the example output
Test:
test_pm_distGeomCyclic


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.

Author:
Amir Shahmoradi, Monday March 6, 2017, 3:22 pm, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin.

Definition at line 2555 of file pm_distGeomCyclic.F90.


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